L2 Hodge Theory and Vanishing Theorems Jean-Pierre Demailly Université de Grenoble I, Institut Fourier, BP 74, 38402 Saint-Martin d'Hères, France J.-P. DEMAILLY, L2 HODGE THEORY AND VANISHING THEOREMS 2 0. Introduction The aim of these notes is to describe two fundamental applications of L2 Hilbert space techniques to analytic or algebraic geometry: Hodge theory, and the theory of L2 estimates for the d operator. The point of view adopted here is essentially analytic. The first part is focussed on Hodge theory and it is intended to be rather introductory. Thus the reader will find here only the most elementary topics, mostly those due to W.V.D. Hodge himself [Hod41] or to A. Weil [Wei57]. Hodge theory, as first conceived by its creator, consists of the study of the cohomology of Riemannian or Kahlerian manifolds, by means of a description of harmonic forms and their properties. We refer to the treatment of J. Bertin-Ch. Peters [BePe95] and L. Illusie [11195] for a presentation of more advanced topics and applications (variation of Hodge structure, application of periods, Hodge theory in characteristic p > 0 ...). We consider a Riemannian manifold X and a Euclidean or Hermitian bundle E over X. We assume that E is equipped with a connection D compatible with the metric: A connection is by definition a differential operator analogous to exterior differentiation, acting on forms of arbitrary degree with values in E, and satisfies Leibniz rule for the exterior product. The Laplace-Beltrami operator is the self-adjoint differential operator of second order A# = DeD*e + DeDe, where DE is the Hilbert space adjoint of De- One easily shows that A# is an elliptic operator. The fmiteness theorem for elliptic operators shows then that the space l-Lq(X,E) of harmonic g-forms with values in E is finite dimensional if X is compact (we say that a form u is harmonic if Aeu = 0). If we assume in addition that the connection satisfies DE = 0, the operator De acting on forms of all degrees defines a complex called the de Rham complex with values in the local system of coefficients defined by E. The corresponding cohomology groups will be denoted by ifpR(I', E). The fundamental observation of Hodge theory is that any cohomology class contains a unique harmonic representative, since X is compact. It leads then to an isomorphism, called the Hodge isomorphism (0.1) HlR(X,E)~nlR(X,E). When the manifold X and the bundle E are holomorphic, there exists a unique connection De called the Chern connection, compatible with the Hermitian metric on E and has the following properties: De splits into a sum De = D'E + DE of a connection D'E of type (1,0) and a connection DE of type (0,1), such that D'2 = D"2 = 0 and D'ED'E + D"ED'E = &{E) (Chern curvature tensor of the bundle). The operator D'E acting on the forms of bidegree (p, q) defines then for fixed p, a complex called the Dolbeault complex. When X is compact, the Dolbeault cohomology groups Hp'q(X,E) satisfy a Hodge isomorphism analogous to (0.1), namely (0.2) HP'9(X, E) ~ np'q(X, E), where "HP'9(X, E) denotes the space of harmonic (p, q)-iorms with values in E, relative to the anti-holomorphic Laplacian A^ = D'ED'E + DE*DE. By utilizing this latter result, one easily proves the Serre duality theorem (0.3) HP'9(X,E)* ~ Hn-p'n~q(X,E"), n = dimcX, 3 0. INTRODUCTION which is the complex version of the Poincare duality theorem. The central theorem of Hodge theory concerns compact Kahler manifolds: A Hermitian manifold {X, uj) is called Kdhlerian if the Hermitian (l,l)-form ui = i£\ kujjkdzj A dzk satisfies du = 0. A fundamental example of a compact Kahlerian manifold is given by the projective algebraic manifolds. If X is compact Kahlerian and if E is a local system of coefficients on X, the Hodge decomposition theorem asserts that (0.4) H^R(X,E) = 0 Hp'q(X,E) (Hodge decomposition) p+q=k (0.5) Hpq(X,C) satisfy the symmetry property hp'q = hq>p = /j«-p<«-« = /j"-«,»-p; and that they are connected to the Betti numbers bu = dime HfjR(X, C) by the relation bu = X^p+o=fc hp,q. A certain number of other remarkable cohomological properties of compact Kahler manifolds are obtained by means of the primitive decomposition and the hard Lefschetz theorems (which in turn is a result of the existence of an sl(2, C) action on harmonic forms). These results allow us to describe in a precise way the structure of the Picard group Pic(X) = H1(X,0*) in the Kahlerian case. In a more general setting, we discuss the Hodge-Frolicher spectral sequence (the spectral sequence connecting Dolbeault to de Rham cohomology), and we show how one can utilize this spectral sequence to obtain some general results on the Hodge numbers hp'q of compact complex manifolds. Finally, we establish the semi-continuity of the dimension of the cohomology groups Hq(Xt,Et) of bundles arising from a proper and smooth holomorphic fibration X -> S (result due to Kodaira-Spencer), and we deduce from it that the Hodge numbers hp,q{Xt) are constant if the fibers Xt are Kahlerian (invariance of the hp'q under deformations); the holomorphic nature of the Hodge filtration FpHk(Xt,C) = (Br>pHr'k-r(Xt,C) relative to the Gauss-Manin connection is proven by means of the theorem on the coherence of direct images, applied to the relative de Rham complex il^/s of X ->- S. In the second part, after recalling some of the relevant concepts of positivity and pseudoconvexity, we establish the Bochner-Kodaira-Nakano identity connecting the Laplacians A^ and A^. The identity in question furnishes an explicit expression of the difference A^ - A^ in terms of the curvature ®(E) of the bundle. Under adequate hypothesis (weak pseudoconvexity of X, positivity of the curvature of E), one arrives a priori at the estimate ||Z^«||2 + ||£>£«||2> / X(z)\u\2dV(z) where A is a positive function depending on the eigenvalues of curvature. The inequality is valid here for any form u of bidegree (n,q), n = dimX, q > 1, with values in E, u belonging to the Hilbert space domains of D'^ and D'^ . By an argument of Hilbert space duality one deduces from this the following fundamental theorem, essentially due to Hormander [H6r65] and Andreotti-Vesentini [AV65]: 0.6. Theorem. Let (X,ui) be a Kahler manifold, dimX = n. Assume that X is weakly pseudoconvex. Let E be a Hermitian line bundle and suppose that the J.-P. DEMAILLY, L2 HODGE THEORY AND VANISHING THEOREMS 4 eigenvalues of the curvature form i@(E) with respect to the metric ui at each point x £ X, satisfy 71 (x) 0 everywhere. Then for any form g £ L2(X, An>qTx ® E) such that D'^g = 0 and / (71 + - - - + 7g)~1|), made up of the germs of holomorphic functions / £ Ox,x such that Jv |/|2e-2^ converges in a sufficiently small neighbourhood V of x. A recent result of A. Nadel [Nad89] guarantees that 3( eui in the sense of currents. Then Hq(X,0(Kx®E)®J(h))=Q for all q> 1. In spite of the relative simplicity of the techniques involved, it is an extremely powerful theorem, which by itself contains many of the most fundamental results of analytic or algebraic geometry. Theorem (0.7) also contains the solution of the Levi problem (equivalence of holomorphic convexity and pseudoconvexity), the vanishing theorems of Kodaira-Serre, Kodaira-Akizuki-Nakano and Kawamata-Viehweg for projective algebraic manifolds, as well as the Kodaira embedding theorem characterizing these manifolds among the compact complex manifolds. By its intrinsic character, the "analytic" statement of Nadel's theorem appears useful even for purely algebraic applications. (The algebraic version of the theorem, known as the Kawamata-Viehweg vanishing theorem, utilizes the resolution of singularities and does not give such a clear description of the multiplier sheaf J(h).) In a recent work [Siu96], Y.T. Siu has shown the following remarkable result, by utilizing only the Riemann-Roch formula and an inductive Noetherian argument for the multiplier sheaves. The technique is described in §16 (with some improvements developed in [Dem96]). 5 0. INTRODUCTION 0.8. Theorem [Siu96], [Dem96]). Let X be a projective manifold and L an ample line bundle (i.e. has positive curvature) on X. Then the bundle K® L®m is very ample for m > rrio(n) = 2 + ( ?^~ ), where n = dimX. The importance of having an effective bound for the integer mo(n) is that one can also obtain embeddings of manifolds X in projective space, with a precise control of the degree of the embedding. As a consequence of this, one has a rather simple proof of a significant fmiteness theorem, namely "Matsusaka's big theorem" (cf. [Mat72], [KoM83], [Siu93], [Dem96]): 0.9. Matsusaka's Big Theorem. Let X be a projective manifold and L an ample line bundle over X. There exists an explicit boundmi = mi(n, Ln,Kx-Ln~1) depending only on the dimension n = dim X and on the first two coefficients of the Hilbert polynomial of L, such that mL is very ample for m > mi. From this theorem, one easily deduces numerous fmiteness results, in particular the fact that there exist only a finite number of families of deformations of polarized projective manifolds (X, L), where L is an ample line bundle with given intersection numbers Ln and Kx - L"-1. Part I: L2 Hodge Theory 1. Vector bundles, connections and curvature The goal of this section is to recall some basic definitions of Hermitian differential geometry with regard to the concepts of connection, curvature and the first Chern class of line bundles. l.A. Dolbeault cohomology and the cohomology of sheaves. Assume given X a C-analytic manifold of dimension n. We denote by Ap'qTx the bundle of differential forms of bidegree (p, q) on X, i.e. differential forms which can be written u = 2_, ui,jdz A~zj, dzj := dz^ A - - - A dzip, dzj := dzj1 A - - - A dzjq, \I\=P, \J\ = Q where (z±,... ,zn) are local holomorphic coordinates, and where / = (ii,... ,ip) and J = (ji,... ,jq) are multi-indices (increasing sequences of integers in the interval [1,... , n], with lengths |/| = p, \J\ = q). Let Ap,q be the sheaf of germs of differential forms of bidegree (p,q) with complex valued C°° coefficients. We recall that the exterior derivative d decomposes into d = d' + d" where d'u = } , ' dzu A dzj A dzj, OZh |/|=p,|J| = g,l 0 are locally d"-exact (this is the analogue for d" of the usual Poincare Lemma for d, see for example [Hor66]). In other words, the complex of sheaves (Ap'',d") is exact in degree q > 0: and in degree q = 0, Kerd" is the sheaf Vlvx of germs of holomorphic forms of degree p on X. More generally, if E is a holomorphic vector bundle of rank r over X, there exists a natural operator d" acting on the space C°°(X, Ap'qTx ® E) of C°° (p,q)-forms with values in E. Indeed, if s = X^ie\; by first observing that the transition matrices corresponding to a change of holomorphic frame are holomorphic, and which commute with the operation of d". It then follows that the Dolbeault-Grothendieck Lemma still holds for forms with values in E. For every integer p = 0, 1,..., n, the Dolbeault cohomology groups Hp'q(X, E) are defined as being the cohomology of the complex of global forms of type (p, q) (indexed by q): (1.1) Hp'q(X,E) =Hq{C°°{X,Kp''T*x®E)). There is the following fundamental result of sheaf theory (de Rham-Weil Isomorphism Theorem): Let (C',6) be a resolution of a sheaf T by acyclic sheaves, i.e. a complex (£-, 6) given by an exact sequence of sheaves 0 -> T -4 C° A £} -> >? £« A Cq+1 -?---, 6 7 1. VECTOR BUNDLES, CONNECTIONS AND CURVATURE where HS(X, Cq) = 0 for all q > 0 and s > 1. (To arrive at this latter condition of acyclicity, it is enough for example that the Cq are flasque or soft, for example a sheaf of modules over the sheaf of rings C°°.) Then there is a functorial isomorphism (1.2) Hq(Y(X, £")) -»? ff9(X, .F). We apply this in the following situation. Let Ap,q{E) be the sheaf of germs of C°° sections of Ap'qTx ® E. Then (^--(JE), d") is a resolution of the locally free Ox-module Qpx0(E) (Dolbeault-Grothendieck Lemma), and the sheaves Ap,q{E) are acyclic as C°°-modules. According to (1.2), we obtain 1.3. Dolbeault Isomorphism Theorem (1953). For all holomorphic vector bundles E on X, there exists a canonical isomorphism Hp'q(X,E) ~Hp(X,npx0(Ej). If X is projective algebraic and if E is an algebraic vector bundle, the theorem of Serre (GAGA) [Ser56] shows that the algebraic cohomology groups Hq(X, flx ® O(Ej) computed via the corresponding algebraic sheaf in the Zariski topology are isomorphic to the corresponding analytic cohomology groups. Since our point of view here is exclusively analytic, we will no longer need to refer to this comparison theorem. l.B. Connections on differentiable manifolds. Assume given a real or complex C°° vector bundle E of rank r on a differentiable manifold M of class C°°. A connection D on E is a linear differential operator of order 1 D : C°°{M,AqT*M®E) -»? C°°(M, Aq+1T*M ® E) such that D satisfies Leibnitz rule: (1.4) D(f Au)=df Au+(-l)deg// ADu for all forms / ¤ C°°(M, APT^), u £ C°°{X, AqT^ ® E). On an open set Q C M where E admits a trivialization r : E\n ^» fl x C, a connection D can be written Du ~r du + T A u where Y £ CO0(fl,A1T^I ® Hom(C,Cr)) is a given matrix of 1-forms and where d acts componentwise on u ~r (uA)i C°°(M, Kv+qT*M ® C) (u,v) i->- {u,v} given by Am The connection D is called Hermitian if it satisfies the additional property d{u, v} = {Du, v} + (-l)deg"{u, Dv}. By assuming that (e\) is orthonormal, one easily verifies that D is Hermitian if and only if T* = -r. In this case ©(£>)* = -0(D), therefore i0(D) e C°°(M,A2T^(g,ReTm(E,E)). 1.7. A particular case. For a complex line bundle L (a complex vector bundle of rank 1), the connection form T of a Hermitian connection D can be taken to be a 1-form with purely imaginary coefficients T = \A (A real). We then have Q(D) = dT = id A. In particular i@(L) is a closed 2-form. The first Chern class of L is defined to be the cohomology class ci(i)R={^©(JD)}eJffDR(M,M). This cohomology class is independent of the choice of connection, since any other connection D\ differs by a global 1-form, D\u = Du + B A u, so that 0(-Di) = Q(D) + dB. It is well-known that ci(L)r is the image in H'2(M, K) of an integral class ci(L) e H2(M,Z). Indeed if A = C°° is the sheaf of C°° functions on M, then via the exponential exact sequence 0 -> Z ->- A ^-+ A* ->- 0, ci(L) can be defined in Cech cohomology as the image of the cocycle {gjk} G H1 (M, ^4*) defining L by the coedge map HX(M,A*) ^H2(M,Z). See for example [GH78] for more details. l.C. Connections on complex manifolds. We now study those properties of connections governed by the existence of a complex structure on the base manifold. If M = X is a complex manifold, any connection D on a complex C°° vector bundle E can be split in a unique manner as a sum of a (1, 0)-connection and a (0, l)-connection, D = D' + D". In a local trivialization r given by a C°° frame, one can write (1.8') D'u ~r d'u + T'Au, (1.8") D"u ~r d"M + r"A«, with r = r" + r". The connection is Hermitian if and only if r" = - (r")* relative to any orthonormal frame. As a consequence, there exists a unique Hermitian connection D associated to a (0, l)-connection prescribed by D". 9 2. DIFFERENTIAL OPERATORS ON VECTOR BUNDLES Now suppose that the bundle E is endowed with a holomorphic structure. The unique Hermitian connection whose component D" is the operator d" defined in §1.A is called the Chern connection of E. With respect to a local holomorphic frame (ex) of E\n, the metric is given by the Hermitian matrix H = (hx^) where h\u = (eA,e"). We have {u,v} = ^hxnUxAVu = v) AHv, where v) is the transpose matrix of u, and an easy calculation gives d{u, v} = (du)] AHv + (-l)des V A (dH Av + Hlfo) = (du + H 1d'Hf\uy AHv + (-l)d^uu] A (dv + H 1(l'HAv), by using the fact that dH = d'H + d'H and H = H. Consequently the Chern connection D coincides with the Hermitian connection defined by J Du ~r du+~H~ld'~H Au, \ D' ~r d' + H^d'H A. = H~1dl (#-), D"=d". These relations show that D'2 = D"'2 = 0. Consequently D2 = D'D" + D"D', and the curvature tensor Q(D) is of type (1,1). Since d'd" + d"d' = 0, we obtain (D'D" + D"D')u ~r H~ld'H A d"u + d" (H~l d'H Au) = d" (H~l d'H) A u. 1.10. Proposition. The Chern curvature tensor ®(E) := @(D) satisfies i@(E) G C^p^A1'1^ Rerm(E,E)). If t : E^q -> fi x C is a holomorphic trivialization and if H is the Hermitian matrix representative of the metric along the fibers of E^q, then i&(E) ~r id"(H'1 d'H) on Q. D If (z\,... , zn) are holomorphic coordinates on X and if (eA)i- C^iM, F), u H> Pu of the form Pu(x) = >^ aQ(x)-Dau(a;), M<<5 where -Efn ~ !lxKr, F\q ~ f2 x Kr are local trivializations on an open chart fi C M with local coordinates (xi,... ,xm), and the coefficients aa(x) are r' x r matrices (aa\n(x))i<\- Hom(P, F) defined by (2.2) T*Mx B £ ->- £) £ Hom^.F,), £) = ^ aa(x)Za. \a\=5 Then ctp(x,£) is a C00 function of the variables (x,£) ¤ P^-, and this function is independent of the choice of coordinates or trivialization used for E, F. ap is called the principal symbol of P. The principal symbol of a composition Q o P of differential operators is simply the product. (2.3) 0 is a C°° density. If E is a Euclidean or Hermitian vector bundle, we can define a Hilbert space L'2(M,E) of global sections with values in E, being the space of forms u with measurable coefficients which are square summable sections with respect to the scalar product (2.5) IMI2 = / \u(x)\2dV(x), Jm (2.5') ((u,v))= [ (u(x),v(x))dV(x), u,v £ L2(M,E). Jm 2.6. Definition. If P : C°°(M,£;)-^ C°°(M,F) is a differential operator and if the bundles E, F are Euclidean or Hermitian, there exists a unique differential operator P* : C°°(M,F) ->- C°°{M,E), called the formal adjoint of P, such that for all sections u £ C°°(M,E) and v £ C°°(M, F) one has an identity ((Pu,v)) = ((u, P*v)), whenever Supp u fl Supp v CC M. 11 3. FUNDAMENTAL RESULTS ON ELLIPTIC OPERATORS Proof. The uniqueness is easy to verify, being a consequence of the density of C°° forms with compact support in L2 (M, E). By a partition of unity argument, we reduce the verification of the existence of P* to the proof of its local existence. Now let Pu(x) = Yl\a\? M be a C°° Hermitian vector bundle of rank ronM. 3.A. Sobolev spaces. For any real number s, we define the Sobolev space Ws(M.m) to be the Hilbert space of tempered distributions u £ S'(M.m) such that the Fourier transform u is a L2oc function satisfying the estimate (3.i) \\u\\i= f (i + \erwo\2d\(o < +oo. If s ¤ N, we have Hi ~ / E \Dau{x)\2d\{x), |a| k + ^j, we have WS(M,E) C Ck(M,E) and the inclusion is continuous. ? It follows immediately from the Sololev lemma that P| Ws(M,E) =C°°(M,E), s>0 |J WS(M,E) =V'(M,E). s<0 3.3. Rellich lemma. For all t > s, the inclusion W\M,E) ^WS(M,E) is a compact linear operator. ? 3.B. Pseudodifferential operators. If P = X^iaK* aa(x)Da is a differential operator on Mm, the Fourier inversion formula gives Pu{x)= f J2 aa(x)(2m0au(0e27Tix'id\(0, V«eD(Rm), Jum\a\<6 where u(£) = JRm u(x)e~27rlx'^d\(x) is the Fourier transform of u. We call Ws~s (M, F). D In particular if a £ S_00(M; £, F) := f| 55(M; £, F), then Op^ is a continuous operator sending an arbitrary distributional section of V (M, E) into C°° (M, F). Such an operator is called a regular operator. It is a standard result in the theory of distributions that the class 1Z of regular operators coincides with the class of operators defined by means of a C°° kernel K(x,y) £ Rom(Ey, Fx). That is, the operators of the form R:V'(M,E)^C°°(M,F), u ^ Ru, Ru(x) = f K(x,y) ? u(y)dV(y). Jm Conversely, if dV(y) = ry(y)dyi - ? - dym on ftj and if we write Ru = ~^2 R{6ju) , where (6j) is a partition of unity, the operator R(6j») is the pseudodifferential operator associated to the symbol a defined by the partial Fourier transform °M = (l(y)9j(y)K(x,y))$(x,£), a £ S~°° {M; E,F). When one works with pseudodifferential operators, it is customary to work modulo the regular operators and to allow operators more generally of the form Op,,- + R where R £ 1Z is an arbitrary regular operator. 3.7. Composition. If a G S6{M;E,F) and a' £ S5'{M;F,G), 8, 5' £ K, there exists a symbol a'(}a £ Ss+S (M;E,G) such that Opa> o Op,,- = Opo-'^o- mod 1Z. Moreover a'<)a -a' ? a £ S^^'^M; E, G). 3.8. Definition. A pseudodifferential operator Op,,- of degree S is called el liptic if it can be defined by a symbol a £ SS(M, E, F) such that k(a,£)-w|>c|£r>|, vfefler;, yU£Ex for |£| large enough, the estimate being uniform for x £ M. If E and F have the same rank, the ellipticity condition implies that a(x,£) is invertible for large £. By taking a suitable truncating function #(£) equal to 1 for large £, one sees that the function a'(x,t;) = 0(0a(xy0~1 defines a symbol in the space S~5(M;F,E), and according to (3.8) we have Op,,-' o Op,j = Id + Opp, p £ 5_1 (M; E, E). Choose a symbol r asymptomatically equivalent (at infinity) to the J.-P. DEMAILLY, PART I: L2 HODGE THEORY 14 expansion Id - p + p^2 H h (-l)Jp^J + - - -. It is clear then that one obtains an inverse OpT$a> of Op,,- modulo 1Z. An easy consequence of this observation is the following: 3.9. Garding inequality. Assume given P : C°°(M,£;) ->- C^iM^F) an elliptic differential operator of degree 8, where rank E = rank F = r, and let P be an extension of P with distributional coefficient sections. For all u ¤ W°(M, E) such that Pu ¤ WS(M, F), one then has u £ WS+S(M, E) and IH|g+<5- C°°(M,F) an elliptic differential operator of degree 5. Then: i) Ker P is finite dimensional. ii) P(C°°(M,E)) is closed and of finite codimension in Cco(M,F); moreover, if P* is the formal adjoint of P, there exists a decomposition. C°°(M,F) =P(C°°(M,E)) ©KerP* as an orthogonal direct sum in W°(M, F) = L2(M,F). Proof, (i) The Garding inequality shows that ||it||g+(j < Cg||u||o for all u ¤ KerP. By the Sobolev Lemma, this implies that KerP is closed in W°(M,E). Moreover, the 11 | |o-closed unit ball of Ker P is contained in the 11 | |,$-ball of radius Co, therefore it is compact according to the Rellich Lemma. Riesz Theorem implies that dim KerP < +oo. (ii) We first show that the extension P : WS+S(M,E) -> WS(M,F) has closed image for all s. For any e > 0, there exists a finite number of elements V!,... ,vN £ WS+S(M,F), N = N(e), such that N (3.11) ||U||0«i»o|- Indeed the set: K(vj} = Lg Ws+s(M,F) ; e\\u\\a+s + £ \((u,Vj))0\ < l}, 15 3. FUNDAMENTAL RESULTS ON ELLIPTIC OPERATORS is relatively compact in W°(M, F) and f],v.-. K(Vj) = {0}. It follows that there are elements (vj) such that K(Vj) are contained in the unit ball of W°(M,E), as required. Substituting the main term ||w||o given by (3.11) in the Garding inequality; we obtain / N (1 - Cse)\\u\\s+S < Cs (\\Pu\\s + J2 l««,«i»o ^ 3 = 1 Define T = {u ¤ WS+S(M,E) ; u±Vj, 1 < j < n} and put e = 1/2C8. It follows that \\u\\s+5<2Cs\\Pu\\s, VmGT. This implies that P(T) is closed. As a consequence P(WS+5(M,E))=P(T) + Yect(P(v1),... ,P(vN)) is closed in WS(M, E). Consider now the case s = 0. Since C°°(M,E) is dense in W\M,E), we see that in W°(M,E) = L'2(M,E), one has P(WS(M,E))\ = f P(C°°(M,£;)) j =KerP*. We have thus proven that (3.12) W°(M,£;) = P(WS(M,E)) ©KerP*. Since P* is also elliptic, it follows that KerP* is finite dimensional and that KerP* = KerP* is contained in C°°(M,F). By applying the Garding inequality, the decomposition formula (3.12) gives (a)WS(M,E) = P(WS+5(M,E)) ©KerP*, (b)C°°(M,E) = P(C°°(M,E)) ©KerP*. We finish this section by the construction of the Green's operator associated to a self-adjoint elliptic operator. 3.15. Theorem. Assume given E a Hermitian vector bundle of rank r on a compact manifold M, and P : Cca(M,E) ->? C°°(M,E) a self-adjoint elliptic differential operator of degree 6. Then if H denotes the orthogonal projection operator H : C°°(M,E) ->- KerP, there exists a unique operator G on C°°{M,E) such that PG + H = GP + H = Id, moreover G is a pseudo-differential operator of degree -6, called the Green's operator associated to P. Proof. According to Theorem 3.10, KerP = KerP* is finite dimensional and Im P = (Ker P)^. It then follows that the restriction of P to (Ker P)1- is a bijective operator. One defines G to be 0 © P_1 relative to the orthogonal decomposition C°°(M, E) = KerP © (KerP)-1. The relations PG + H = GP + H = Id are then obvious, as well as the uniqueness of G. Moreover, G is continuous in the Frechet space topology of C°°(M,E), according to the Banach theorem. One also uses J.-P. DEMAILLY, PART I: L2 HODGE THEORY 16 the fact that there exists a pseudo-differential operator Q of order -5 which is an inverse of P modulo 1Z, i.e. PQ = Id + R, R £ 1Z. It then follows that Q = (GP + H)Q = G(Id + R) + HQ = G + GR + HQ, where GR and HG are regular. (H is a regular operator of finite rank defined by the kernel ^Lp8(x) ip*(y), if (ips) is a basis of eigenfunctions of KerP C C°°{M,E).) Consequently G = Q mod 1Z and G is a pseudodifferential operator of order -6. ? 3.16. Corollary. Under the hypotheses of 3.15, the eigenvalues of P form a real sequence \k such that rimj^+oo |Afc| = +00, the eigenspaces V\k of P are finite dimensional, and one has a Hilbert space direct sum L2(M,E) = ®VXk. For any integer m £ N, an element u = J^fc uk £ L2(M, E) is in Wm5(X, E) if and 0n^/£|Afc|2m|K||2<+oo. Proof. The Green's operator extends to a self-adjoint operator G :L2(M,E) -> L2(M,E) which factors through W5(M, E), and is therefore compact. This operator defines an inverse to P : WS(M,E) ->- L2(M,E) on (KerP)-1. The spectral theory of compact self-adjoint operators shows that the eigenvalues fik of G form a real sequence tending to 0 and that L2(M,E) is a direct sum of Hilbert eigenspaces. The corresponding eigenvalues of P are Xk = /zjjT if /ik 7^ 0 and according to the ellipticity of P - A^Id, the eigenspaces V\k = Ker(P - A^Id) are finite dimensional and contained in Cco(M,E). Finally, if u = J^k uk £ L2(M,E), the Carding inequality shows that u ¤ WmS (M, E) if and only if Pmu ¤ L2 (M,E)= W° (M,E), which easily gives the condition J2 l^fc|2m|lMfc||2 < +oo. 4. Hodge theory of compact Riemannian manifolds The establishment of Hodge theory as a well developed subject, was carried out by W.V.D Hodge during the decade 1930-1940 (see [Hod41], [DR55]). The principal goal of the theory is to describe the de Rham cohomology algebra of a Riemannian manifold in terms of its harmonic forms. The principal result is that any cohomology class has a unique harmonic representative. 4.A. Euclidean structure of the exterior algebra. Let (M,g) be an oriented Riemannian C°° manifold of dimension m, and let E -> M be a Hermit-ian vector bundle of rank r on M. We denote respectively by (£1,... ,£m) and (ei,... , er) orthonormal frames of Tm and of £ on a coordinate chart ft C M, and let (£i;--- >£m)> (eiT ? ? ie*) be the corresponding dual coframes of T^, E* respectively. Further, let dV be the Riemannian volume element on M. The exterior algebra A'T^ is endowed with a natural inner product (-, -), given by (4.1) (m A --- Aup,V! A --- Avp) = det((uj,vk))i E. 17 4. HODGE THEORY OF COMPACT RlEMANNIAN MANIFOLDS 4.2. Hodge star operator. The Hodge-Poincare-de Rham - operator is the endomorphism of A'TM defined by a collection of linear maps such that - : kpT*M -»? Km~pT*M, uA*v= (u, v)dV, Vu, v £ APTM. The existence and uniqueness of this operator follows easily from the duality pairing APTM x Km~pT*M ->- K (4.3) (u, i>) h-» u A i>/c2F = ^ e(J, Gl)uivCl, where it = J2\i\=pui£i' v = H\j\=m-pvJ^*j^ and where e(I,Zl) is the sign of the permutation (1,2,... , m) i->- (I, C/) defined by I followed by the complementary (ordered) multi-indices Ci. From this, we deduce (4.4) *v= £e(/,C/)^cY |/|=P More generally, the sesquilinear pairing {-,-} defined by (1.6) induces an operator - on the vector-valued forms, such that (c)- : KVT*M E* defined by s A #t = (s,t)dV, where the exterior product A is combined with the canonical pairing E x E* -> C. We have (4.8) #*= E e(/,C/)t/,A£c/®eA- l-f|=P,A 4.9. Contraction by a vector field. Assume given a tangent vector 0 £ 7m and a form it e APTM. The contraction 6>jw £ AP_1TM is defined by 0_iit(r7i,... ,%_i) = u{9,riu... ,r]p-i), rjj ¤ TM. In terms of the basis (£,-), »j» is the bilinear operator characterized by 0 if I g- {«i,... ,ip}, (_l)*-i£A... £...£? in = ifc. This same formula is also valid when (£j) is not orthonormal. An easy calculation shows that #j» is a derivation of the exterior algebra, i.e. that 6Li(u A v) = {9au) A v + (-l)des«u A (0j«). J.-P. DEMAILLY, PART I: L2 HODGE THEORY 18 Moreover, if 9 = (-,8) ¤ TM, the operator #j» is the adjoint of 6 A -, i.e., (4.10) {Oju,v) = (u,6Av), Vu,v ¤ A'TjJf. Indeed, this property is immediate when 9 = £j, it = £|, v = £}? 4.B. Laplace-Beltrami operator. Let £ be a Hermitian vector bundle on M, and let DE be a Hermitian connection on E. We consider the Hilbert space L2(M,APT^[ £) and t e C°°(M, A^T^ ®£T) have compact support, we have {{DEs,t))= f (DEs,t)dV= f {DEs,*t} Jm Jm = [ d{s,*t} - (-l)p{s,DE*t} = (-l)p+1 f {s,DE*t} Jm Jm by an application of Stokes theorem. As a consequence, (4.5) and (4.7) imply ((DES,t)) = (-ir+i(-ir^m-^ f |S,**JD£*o = (-i)mp+1((^*^**))- Jm The desired formula follows. ? 4.12. Remark. In the case of the trivial connection d on E = M x C, the formula becomes d* = (-l)m+1 *d*. If m is even, these formulas reduce to d* = - * d*, -DJ; = - * DE * . 4.13. Definition. The Laplace-Beltrami operator is the second order differ ential operator acting on the bundle, APTM ® E, such that AE = DEDE + D*EDE. In particular, the Laplace-Beltrami operator acting on APTM is A = dd* + d*d. This latter operator does not depend on the Riemannian structure (M,g). It is clear that the Laplacian A is formally self-adjoint i.e. ((AEs,t)) = ((s,AEt)) whenever the forms s,t are C°° and that one of them has compact support. 19 4. HODGE THEORY OF COMPACT RlEMANNIAN MANIFOLDS 4.14. Calculation of the symbol. For every C°° function /, Leibnitz rule gives e~tf DE{etf s) = tdf A s + Des. By definition of the symbol, we therefore find From formula (2.8), we obtain od* = -((JcE)*, therefore VDiix,® ? s = -|js where £ ¤ Tm is the adjoint tangent vector of £. The equality sidxi, ds = } --dxjAdxj, *-^ *-^ OXj \I\=P \I\=P,3 ,* sr^dsi d d s = - > -- -- jctej, *-^ OXj OXj 1,3 }2 sfelf)^- Consequently A has the same expression as the elementary Laplacian operator, up to a minus sign. 4.C. Harmonic forms and the Hodge isomorphism. Let E be a Hermit-ian vector bundle on a compact Riemannian manifold (M,g). We assume that E is given a Hermitian connection De such that Q{DE) = DE = 0. Such a connection is said to be integrable or flat. It is known that this is equivalent to such an E given by a representation tti(M) ->- U(r). Such a bundle is called a flat bundle or a local system of coefficients. A standard example is the trivial bundle E = M x C with its obvious connection De = d. Our assumption implies that De defines a generalized de Rham complex C°°(M,E) -^C°°(M,A1TZI®E) -> >C°°(M,ApT^®E) -^ - - - . The cohomology groups of this complex are denoted by HER(M, E). The space of harmonic forms of degree p relative to the Laplace-Beltrami operator AE = DEDE + DEDE is defined by np(M,E) = {seC°°(M,ApT^®£;) ; AEs = 0}. Since ((AEs,s)) = \\Des\\2 + \\D*Es\\2, we see that s ¤ W{M,E) if and only if DEs = D*Es = 0. J.-P. DEMAILLY, PART I: I? HODGE THEORY 20 4.16. Theorem. For all p, there exists an orthogonal decomposition C°°(M,APT^ ®E)= np(M,E) © lmDE © ImD*E, where ImDE=DE(C°°(M,Ap-1T;i®E)), lmDE = DE{C°°{M,Kp+1T*M®E)). Proof. It is immediate that HP(M,E) is orthogonal to the two subspaces ImDE and lmDE. The orthogonality of these two subspaces is also obvious, as a result of the hypothesis DE = 0, namely: ((DES,D*Et)) = ((D%s,t)) = 0. We now apply th. 3.10 to the elliptic operator AE = AE acting on the p-forms, i.e. the operator A^ : C°°(M, F) ->- C°°(M, F) acting on the bundle F = APT^ E. We obtain C°°(M, ApT*M ®E)= np(M,E) © AE(C°°(M, APT*M ® Ej), ImAB = Im(DED*E + D*EDE) C ImDE + ImD*E. Further, since ImD^ and ImD^ are orthogonal to "HP(M, E), these spaces are contained in ImA^. D 4.17. Hodge Isomorphism Theorem. The de Rham cohomology groups H^R(M,E) are finite dimensional; moreover H^K{M,E) ~ "HP(M,E). Proof. From the decomposition in (4.16), we obtain BlK(M,E)=DE(C°°(M,S?-1TtI®E)), Z&R(M,E) = KerDB = (ImZT^ = W{M,E) ®lmDE. This shows that any de Rham cohomology class contains a unique harmonic repre sentative. ? 4.18. Poincare duality. The pairing HgR(M, E) x H?~P(M, E*) -> C, (a, t) ^ f s At is a non-degenerate bilinear form, and thus defines a duality between H^R(M,E) and H?~P(M,E*). Proof. First observe that there is a naturally defined flat connection DE* such that for all s ¤ C°°(M, A'T^ E), t ¤ C°°(M,A'T^®E*), one has (4.19) d(s At) = (DEs) A t + (-l)desss A DE,t. It then follows from Stokes theorem that the bilinear map (s, t) i->- JM s At factors through the cohomology groups. For s ¤ C°°(M, APT^ ® E), the reader can easily verify the following formulas (use (4.19) in a similar way to that which was done for the proof of th. 4.11): (4.20) DE*(#s) = (-l)p#D*Es, (DE.)*(#s) = (-l)p+1#DEs, A£,(#s) = #Afj. 21 5. Hermitian and kahler manifolds Consequently #s dzki l 0. This simple observation already implies that a compact Kahler manifold must satisfy certain restrictive topological conditions: 5.3. Consequence. (h)If (X,ui) is compact Kahler and if {ui} denotes the cohomology class of cu in H2(X,R), then {w}n ^ 0. (i)If X is compact Kahler, then H'2k(X,R) ^ 0 for 0 < k < n. Indeed, {oj}k is a non-zero class of H2k(X,R). J.-P. DEMAILLY, PART I: L2 HODGE THEORY 22 5.4. Example. Complex projective space F? is endowed with a natural Kahler metric ui, called the Fubini-Study metric, defined by p*W = ^'d"log(|Co|2 + |Cl|2 + --- + ICn|2) where Co, Ci> ? - ? > Cn are coordinates of C?+1 and where p : Cn+1 \{0} ->- F? is the projection. Let z = (Ci/Co, ? ? ? , Cn/Co) be the non-homogeneous coordinates of the chart C? C F?. A calculation shows that u, = ^d'd" log(l + \z\2) = ^©(O(l)), J un = 1. Since the only non-zero integral cohomology groups of F? are H2p(Fn,Z) ~ Z for 0 < P < n, we see that h = {uj} £ i72(F?, Z) is a generator of the cohomology ring if(Pn,Z). In other words, ff#(F",Z) ~ Z[/i]/(/in+1) as rings. (j)Example. A complex torus is a quotient AT = Cn/r of C? by a lattice T of rank 2n. This gives a compact complex manifold. Any positive definite Hermitian form ui = i Y^ hjkdzj A dzk with constant coefficients on C? defines a Kahler metric onl. (k)Example. Any complex submanifold AT of a Kahler manifold (Y,oj') is Kahler with the induced metric ui = uj'>x- ^n particular, any projective manifold is Kahler (by definition, a projective manifold is a closed submanifold X C F? of projective space). In this case, if u' denotes the Fubini-Study metric on F?, we have the additional property that the class {uj} := {uj'}\x G H'l,R(X, K) is integral, i.e. is the image of an integral class of H2(X, Z). A Kahler metric w with integral cohomology class is called a Hodge metric. (l)Example. Consider the complex surface x = (c2\{o})/r where T = {A? ; n £ Z}, A ¤ ]0,1[, is viewed as a group of dilations. Since C2\{0} is diffeomorphic to M^ x 53, we have X ~ S1 x S3. As a consequence, H2(X, M) = 0 by an application of the Kiinneth formula, and property 5.3 b) shows that X is not Kahler. More generally, one can take for T an infinite cyclic group generated by the holomorphic contractions of C2, of the form (::MS)- --(sM^f)' where A, Ai,A2 are complex numbers such that 0 < |Ai| < |Aa| < 1, 0 < |A| < 1, and p a positive integer. These non-Kahler surfaces are called Hopf surfaces. ? The following theorem shows that a Hermitian metric w on X is Kahler if and only if the metric ui is tangent to order 2 to a Hermitian metric with constant coefficients at any point of X. 5.8. Theorem. Letcu be a positive definite C°° (l,l)-/orm on X. For to to be Kahler, it is necessary and sufficient to show that at any point xq £ X, there exists a holomorphic coordinate system (z\,... , zn) centered at xq such that (5.9) w = i ^ uJimdziAdzm, ujim = 5im + 0(\z\2). l I - <8> -- j,k,l,m \°Zl' °Zm associated to (Tx,ui) at xq. Such a system (zj) is called a geodesic coordinate system at xq. Proof. It is clear that (5.9) implies dXouj = 0, consequently the condition is sufficient. Assume now that uj is Kahler. Then one can choose local coordinates (Ci,... , Cn) sucn that (d(i,... ,d(n) are a w-orthonormal basis of T* X. As a consequence to = i ^ vimdQ A d~(m, where lkjml- J.-P. DEMAILLY, PART I: L2 HODGE THEORY 24 The Kahler condition dcuim/dQ = duijm/dQ at ( = 0 furnishes the equality a',klm = a'ikjm'i m particular a',klm is invariant under all permutations of j, k,l. If one puts Zm = Cm + g ^2 a'jklmQCkO, 1 < W < n, j,k,l then from (5.13) one finds a'jkimCjCkdO, 1 < m < n, j,k,l uj = i ^2 dzm A dzm + i ^ l Tx)*. i,j at a point a is then given by (6.1) Ka)|2=5>/,.,(e0|2- i,j The Hodge * operator (4.2) can be extended to the complex-valued forms by the formula (6.2) u A*v = (u,v)dV. It follows that * is a C-linear isometry - : Ap>qTx -^Xn-q>n-pTx. 25 6. FUNDAMENTAL IDENTITIES OF KAHLERIAN GEOMETRY The standard Hermitian geometric operators are the operators d, 6 = - * dk, the Laplacian A = dd + 5d already defined, and their complex analogues (6.3) d = d' + d", S = d'* + d"*, d" = (d')* = - - d"*, d"* = (d")* = - - d'*, A' = d'd'* + d"d', A" = d"d"* + d"*d". We say that an operator is of pure degree r if it transforms a form of degree k to a form of degree k + r, and similarly an operator of pure bidegree (s,t) is an operator which transforms the (p,q)-farms to forms of bidegree, (p + s,q + t). (Its total degree is then of course r = s + t.) Thus d', d", d'*, d"*, A', A" are of bidegree (1,0), (0,1), (-1,0), (0,-1), (0,0), (0,0) respectively. Another important operator is the operator L of bidegree (1,1) defined by (6.4) Lu = uj f\u, and its adjoint A = L* = *_1L* of bidegree (-1, -1): (6.5) (u,Av) = (Lu,v). We observe that the unitary group U(Tx) - U(n) has a natural action on the space of (p, g)-forms, given by V(n) x A^TZ B (g,v) » (g-1)^. This action makes Ap'qT^ a unitary representation of U(n). Since the metric uj is invariant, it is clear that L and A commute with the action of U(n). 6.B. Commutivity identities. If A, B are endomorphisms (of pure degree) of the graded module M* = C°°(X, A"'"Ty), their graded commutator (or graded Lie bracket) is defined by (6.6) [A,B] = AB-(-l)abBA where a, b are the degrees of A and B respectively. If C is another endomorphism of degree c, one has the following formal Jacobi identity. (6.7) (-l)co [A, [B, C]] + (-l)ab [B, [C, A]] + (-l)6c [C, [A, B]] = 0. For all a £ Ap'qT^-, we will still denote by a the associated endomorphism of type (p, q), operating on A#'#T^ by the formula ui->aAu. Let 7 £ A1'1^ be a real (l,l)-form. There exists a w-orthogonal basis (Ci, C2, - ? - , Cn) of Tx which diagonalizes the two forms ui and 7 simultaneously: l=(?;)?'-ft!) satisfies the commutivity relations [£,A]=o, [b,£] = 2£, [6,A] = -2A. 6.11. Corollary. There is a natural action of the Lie algebra si(2, C) on the vector space A'^'T^, i.e. a morphism of Lie algebras p : sl(2,C) -> End(A"'"Ty), given by p(£) = L, p(X) = A, p(b) = B. We now mention the other very important commutivity identities. Let us first assume that X = Q C C? is open in C? and that uj is the standard Kahler metric, uj = i 2_, d-Zj A dzj. l'«T£) one has (6.12') d'u (6.12") d"u 27 6. FUNDAMENTAL IDENTITIES OF KAHLERIAN GEOMETRY Since the global L2 scalar product is given by ((u,v)) = / ^ uijvijdV, ,(i'i,j some elementary calculations similar to those of the example in 4.12 show that duij d &zk dzk I,J,k (6.13") d"* = -J2 ^S^Adzi A dzj). fjk dzk dzk We first state a lemma due to Akizuki and Nakano [AN54]. 6.14. Lemma. In Cra, one has [d"*,L] = id'. Proof. Formula (6.13") can more succinctly be written d"*u = -V - J- 4^ dzkJ\dzk We then obtain Since uj has constant coefficients, one has gf-(w Aa) = wA Jj- and consequently k \Wk^)AWk However, it is clear that -sf-jw = idzk, therefore [d"*,L]u = i^2dzkA-^- = id'u. ? We are now ready to establish the basic commutivity relations in the situation of an arbitrary Kahler manifold (X, ui). 6.15. Theorem. If(X,uj) is Kahler, then [d"\L] = id', [d'\L] = -- -id", [A, d"} = -id'*, [A,d'] = -- id"*. J.-P. DEMAILLY, PART I: L2 HODGE THEORY 28 Proof. It suffices to establish the first relation, since the second is the conjugate of the first, and the relations in the second line are the adjoint of the relations in the first line. If (zj) is a geodesic coordinate system at a point xq £ X, then for all (p, (/)-forms u, v with compact support in a neighbourhood of xo, (5.9) implies that / (5ZM/JW/J+ 5Z aVKLUljVKL]dV, Jm ^ IJ I,J,K,L ' with clijkl{z) = 0(|z|2) at xq. An integration by parts analogous to that used to obtain (4.12) and (6.13") gives d"*u = - ^^ a ' T^Adzi A dzj) + ^2 buKLUijdzk A dzL, I,J,k "Zk "Zk I,J,K,L where the coefficients buKL are obtained by differentiation of clijkl ? Consequently we have buxL = 0(|z|). Since duj/dzu = 0{\z\), the proof of lemma 6.14 above implies [d"*,L]u = id'u + 0(\z\). In particular the two terms coincide at the given point xq £ X. ? 6.16. Corollary. If (X,ui) is Kdhler, the complex Laplace-Beltrami opera tors satisfy A' = A" = -A. 2 Proof. We first show that A" = A'. One has A" = [d",d"*] = -i[d",[A,d']]. Since [d',d"] = 0, the Jacobi identity (6.7) implies that -[d",[A,d']] + [d',[d",A]] =0, hence A" = [d!, -i[d", A]] = [d',d'*] = A'. Furthermore, A = [d' + d",d'* + d"*} = A' + A" + [d',d"*] + [d",d"]. It therefore suffices to prove: 6.17. Lemma. [d',d"*]=0, [d",d'*]=0. Proof. We have [d',d"*] = -i[d', [A,d']\ and (6.7) implies that -[d',[A,d'j] + [A,[d',d']\ + [d',[d',A]] =0, hence -2[d', [A, d']] = 0 and [d',d"*] = 0. The second relation [d",d'*] = 0 is the adjoint of the first. ? 6.18. Theorem. If (X,ui) is Kdhler, A commutes with all the operators -, d', d", d1*, d"*, L, A. Proof. The identities [d',A'] = [d'*,A'] = 0, [d",A"] = [d"*,A"] = 0 and [A,*] = 0 are immediate. Moreover, the equality [d',L] = d'uj = 0, combined with the Jacobi identity, implies that [L, A'] = [L, [d1, d"]] = - [d1, [d'*,L]] = i[d', d"] = 0. Taking adjoints, we obtain [A', A] = 0. ? 29 6. FUNDAMENTAL IDENTITIES OF KAHLERIAN GEOMETRY 6.C. Primitive elements and the Lefschetz isomorphism theorem. To establish the Lefschetz Theorem, it is convenient to use the representation of sl(2, C) exhibited in Cor. 6.11. We first recall that if g is a Lie sub-algebra (real or complex) of the Lie algebra sl(r, C) = End(C) of complex matrices and if G = exp(g) C GL(r, C) is the associated Lie group, a representation p : g ->- End(V) of the Lie algebra in a complex vector space V induces by exponentiation a representation p : G ->- GL(F) of the group G. Conversely, a representation p : G ->- GL(F) induces by differentiation a representation p : g ->- End(F) of Lie algebras; there is therefore an identification between these two notions. If G is compact, a classical lemma of H. Weyl shows that all representations of g are broken down into a direct sum of irreducible representations (one says that g is reductive): the Haar measure of G indeed allows the construction of an invariant Hermitian metric on V, and one exploits the fact that the orthogonal complement of a sub-representation is a sub-representation. In particular the Lie algebra su(r) of the compact group SU(r) is reductive. It is the same as for sl(r,C), which is the complexification of su(r). We will need the following well-known lemma from representation theory. 6.19. Lemma. Let p : sl(2,C) ->- End(F) be a representation of the Lie algebra sl(2,C) on a finite dimensional complex vector space V, and let L = p(t), A = p(A), B = p(b) £ End(V) be the endomorphisms ofV associated to the basis elements ofsl(2,C). Then: (m)V = ffi^ezVju is a (finite) direct sum of eigenspaces of B, whose eigenvalues p are integers. An element v £ V^ is said to be an element of pure weight p. (n)L and A are nilpotent, satisfying L(V^) C VM+2, A(VM) C VM-2 for all p £ Z. (o)We denote by P = KerA = {v £ V ; Av = 0}, the set of primitive elements. One then has a direct sum decomposition V = @Lr(P). rGN (p)V is isomorphic to a finite direct sum ffimGN5(m)®"m of irreducible representations, where S(m) ~ 5m(C2) is the representation of si(2, C) induced by the m-th symmetric product of the natural representation o/SL(2,C) on C2, and am = dimPm is the multiplicity of the isotypic component S(m). (q)If P^ = P fl V^, then PM = 0 for p > 0 and P = ©^ez,^ Vm+2r is infective for r < m and zero for r > m. f) vn = ®r&i,r>nLr(Pn-2r), where U : P^r ->- Lr(P^2r) is bijective. g) For any r £ N, the endomorphism Lr : V_r -> Vr is bijective. Proof. We first observe the following fact: If v £ VM, then Lv has pure weight p + 2 and Av has pure weight p - 2. Indeed, one has BLv = LBv + [B, L]v = L(pv) + 2Lv = (p + 2)Lv, BAv = ABv + [B, A]v = A(pv) - 2Av = (p - 2)Aw. Now suppose V ^ 0 and let v £ VM be a non-zero eigenvector. If the vectors (Akv)kem were all non-zero, one would have an infinite number of eigenvectors of B with p - 2k distinct eigenvalues, which is impossible. Therefore there exists an integer r > 0 such that Arv ^ 0 and Akv = 0 for k > r. Consequently Arv is a non-zero primitive element of pure weight p' = p - 2r. Thus we conclude that for J.-P. DEMAILLY, PART I: L2 HODGE THEORY 30 some jjl £ C, there exists w ¤ P, a non-zero element of pure weight jjl. The same reasoning as above applied to the powers Lkw shows that there exists an integer m > 0 such that Lmw ^ 0 and Lm+1w = 0. The vector space W of dimension m + 1 generated by wu = Lkw, 0 < k < m is stable under the action of si(2, C). Indeed one has Bwk = (ft + 2k)wk, Lwk = Wk+i by definition, while Awk = ALkw = LkKw - J2 Lk~j~l [L, A]LJw 0 0. The latter assertion of e) follows from the fact that for 0 ^ w £ P-m, one has Lrw ^ 0 if and only if r < m. f) An immediate consequence of e) and the decomposition V = (BrefqLr(P), if one restricts only to elements of pure weight [i. One can only have Lr(Pli-2r) 7^ 0 if either r < m = -(p - 2r), or r > jjl. g) It suffices to verify the assertion in the case of an irreducible representation V ~ Sm(C2). In this case, the result is clear, since the weights 2k -m, 0 < k < m are distributed symmetrically in the interval [-m,m] and that V is generated by (Lkw)oTx)* is called primitive if Au = 0. The space of primitive forms of total degree k is denoted by PrimfcT^ = 0 Primp'qT^. p+q=k 31 6. FUNDAMENTAL IDENTITIES OF KAHLERIAN GEOMETRY Since the operator A commutes with the action of U(Tx) - U(n) on the exterior algebra, it is clear that Pnmp'qTx C Ap'qTx is a U(n)-invariant subspace. One further sees (prop. 6.24) that Primp'qTx is in fact an irreducible representation of U(n). Properties (6.19 e, f, g) successively imply 6.21. Proposition. We have PrimfcT£ = 0 for k > n. Moreover, if u £ PrimfcT£, k < n, then Lru = 0 for r > n - k. 6.22. Primitive decomposition formula. For any u £ A (C ® Tx)*, there exists a unique decomposition u = ^2 Lruk-2r, uk-2r ¤ Primfc_2rT^. r>(k - n) + Consequently, one obtains a decomposition into a direct sum of representations of U(n) Afc(C ®Tx)* = 0 LrPrimfc-2rT^, r>(k - n) + Ap'«(C®Tx)* = 0 LrPnmp-r'q-rTx. r>(p+q-n) + 6.23. Lefschetz Isomorphism Theorem. The linear operators Ln-k . Afc(C (8) Tx)* -> A2n"fc(C (g> Tx)*, Ln-p-q . AP,qT^ _^ A»i-g,n-pT^ are isomorphisms for all integers k < n and (p, q) satisfying p + q < n. 6.24. Proposition. For any (p,q) £ N2 satisfying p + q < n, PrimpqT*x = 0 LrPrimp-r>q-rTx 0E). Using D'E = 0, one arrives at the following result, in the same way as obtained in §4.C. 7.1. Theorem. For any bidegree (p,q), there exists an orthogonal decomposi tion C°°(X, k?T*x ®E)= W'q{X, E) ®ImD'E® Im D'E* where U?{X, E) is the space of A'E-harmonic forms in C°°{X, Ap'qT^ ® E). The above decomposition shows that the subspace of g-cocycles of the complex (C°°(X, AP''T% ® E),d") is W'q{X, E) © Im D'E. From here, we deduce the 7.2. Theorem (Hodge isomorphism). The Dolbeault cohomology groups Hp'q(X,E) are finite dimensional, and there is an isomorphism Hp>q(X,E)~np'q(X,E). Another interesting consequence is a proof of the Serre duality theorem for compact complex manifolds. See Serre [Ser55] for a proof in a somewhat more general context. 7.3. Theorem (Serre duality). The bilinear pairing Hp'q(X,E)xHn-p'n-q(X,E*)^C, (s,t)*+ [ s At Jm is a non-degenerate duality. 33 8. COMOHOLOGY OF COMPACT KAHLER MANIFOLDS Proof. Let si e C°°(X, Ap>qT^(g>E), s2 ¤ Cco(X,An-p'n-q-1T^(g>E). Since si A s2 is of bidegree (n, n - 1), we have (7.4) d(s! A s2) = d"(Sl A s2) = d"Sl A s2 + (-l)p+9si A d"s2. Stokes theorem implies that the bilinear pairing above can be factored through the Dolbeault cohomology groups. The operator # defined is §4.A satisfies # : C°°(X,AMT^ ®E) -»? C°°(X,An-p>n-qT^®E*). Moreover, (4.20) implies d'e.{*s) = (-i)degs#(^)*s, (£>£.)*(#«) = (-i)dess+1#^s, A£.(#a) = #A£S) where 27e* is the Chern connection of E*. Consequently, s £ Hp'q(X,E) if and only if #s ¤ 'Hn~p'n~q(X, E*). Theorem 7.3 is then a consequence of the fact that the integral ||s||2 = Jx s A #s is non-vanishing if s ^ 0. 8. Comohology of compact Kahler manifolds 8.A. Bott-Chern cohomology groups. Let X be a complex manifold, for the moment not necessarily compact. The following "cohomology groups" are useful for describing certain aspects of the Hodge theory of compact complex manifolds, which are not necessarily Kahler. 8.1. Definition. The Bott-Chern cohomology groups of X are given by Hg£(X,C) = (C°°(X, Ap'qT^) nKeid)/d'd"C0O(X,Ap-1'q-1T^). The cohomology H^''(X,C) has a bigraded algebra structure, which we call the Bott-Chern cohomology algebra of X. Since the group d'd"CO0(X,Ap~1'q~1Tx) is also contained in the group of coboundaries d"C°°(X, Ap'q~1Tx) of the Dolbeault complex as well as that in coboundaries of the de Rham complex dCco(X,Ap+q-1(C(g>Tx)*), there are canonical morphisms (r)H?(X,C) -^Hp>q(X,C), (s)Hgq(X,C) ^Hp+q(X,C), of the Bott-Chern cohomology to the Dolbeault or de Rham cohomology. These morphisms are C-algebra homomorphisms. It is also clear from the definition that we have the symmetry property H^'P(X,C) = H^'q(X,C). One can show from the Hodge-Frolicher spectral sequence (see §10) that H^(X,C) is always finite dimensional if X is compact. 8.B. Hodge decomposition theorem. We assume from now on that {X, uj) is a compact Kahler manifold. The equality A = 2A" shows that A is homogeneous with respect to bidegree and that there is an orthogonal decomposition (8.4) Hk(X,C) = 0 np'9(X,C). p+q=k J.-P. DEMAILLY, PART I: L2 HODGE THEORY 34 Since A" = A' = A", one has the equality W'P(X,C) = W>«(X,C). By applying the Hodge isomorphism theorem for de Rham cohomology and for Dolbeault cohomology, one obtains: 8.5. Theorem (Hodge Decomposition). On a compact Kahler manifold, there are canonical isomorphisms H^K(X,<£) ^ 0 Hp-q(X,C) (Hodge decomposition), p+q=k Hq-p(X,C) ~HP>9(X,C) (Hodge symmetry). The only point that is not a priori obvious is that isomorphisms are independent of the choice of Kahler metric. To show that this is indeed the case, one can use the following lemma, which will allow us to compare the three types of cohomology groups considered in §8.A. 8.6. Lemma. Let u be a d-closed (p,q)-form. The following properties are equivalent: a) u is d- exact; b') u is d'-exact; b") u is d"-exact; (t)u is d'd"-exact, i.e. u can be written u = d'd"v. (u)u is orthogonal to "Hp'q(X,C). Proof. It is evident that c) implies a), b'), b"), and that a) or b') or b") implies d). It suffices therefore to prove that d) implies c). Since du = 0, we have d'u = d"u = 0, and since u is assumed orthogonal to Hp'q(X, C), th. 7.1 implies that u = d"s, s £ C°°(X, AP>q-lT^). The analogous theorem to th. 7.1 for d! (which can be deduced by complex conjugation) shows that one can write s = h + d'v + d'*w, where h ¤ W'q-x(X,C), v £ C°°(X, A^-1'*-1^-) and w £ C00(X, Ap+1^~1T^). Consequently u = d"d'v + d"d'*w = -d'd"v - d'*d"w by an application of Lemma 6.16. Since d'u = 0, the component d'*d"w orthogonal to Kerd' must be zero. ? From Lemma 8.6 we deduce the following corollary, which in turn implies that the Hodge decomposition does not depend on the choice of Kahler metric. 8.7. Corollary. Let X be a compact Kahler manifold. Then the natural morphisms H?(X, C) -> H*«{X,C), 0 H^(X,C) -> H*R(X, C) p+q=k are isomorphisms. Proof. The surjectivity of Hp>q(X,C) ->- H?(X,C) follows from the fact that any class in Hp'q(X,C) can be represented by a harmonic (p, q)-iorm, therefore by a (i-closed (p, g)-form; the injectivity property is nothing more than the equivalence (8.5b") <=> (8.5c). Therefore H?(X,C) ~ H?(X,<£) - W>q(X,C), and the isomorphism 0 h^(x,o ^h^x,q p+q=k 35 8. COMOHOLOGY OF COMPACT KAHLER MANIFOLDS is a consequence of (8.4). ? We now mention two simple consequences of Hodge theory. The first concerns the calculation of the Dolbeault cohomology of Pra. Since Hp'p(Fn, C) contains the non-zero class {ojp} and since i7D^(Fn,C) = C, the Hodge decomposition formula implies: 8.8. Consequence. The Dolbeault cohomology groups of Pra are Hp'p(Tn,C) =C for 0p(X,E*), J.-P. DEMAILLY, PART I: L2 HODGE THEORY 36 via the antilinear operator # considered in §4 and §7. These observations are useful for the study of variations of Hodge structure. 8.C. Primitive decomposition and hard Lefschetz theorems. We first introduce some standard notation. The Betti numbers and the Hodge numbers of X are by definition (8.12) bk = dimcHk(X,C), hp>q = dimcHp>q(X,C)- According to the Hodge decomposition, the numbers satisfy the relations (8.13) bk = J2 hP'"> h9'P = hP'9- p+q+k Consequently, the Betti numbers &2fc+i of a compact Kahler manifold are even. Note that the Serre duality theorem gives the additional relation hp'q = hn~p'n~q, provided that X is compact. As we will see, the existence of the primitive decomposition implies many other interesting characteristic properties of the cohomology algebra of a compact Kahler manifold. 8.14. Lemma. If u = X]r>(fc-n) Lrur is the primitive decomposition of a harmonic k-form u, then all the components ur are harmonic. PROOF. Since [A, L] = 0, one obtains 0 = \u = Y,r LrA.ur, therefore Aur = 0 according to the uniqueness of the decomposition. ? Denote by 'Hprim(X, C) = 0p+g=fc ^2,(1, C) the space of primitive harmonic fc-forms and let h££m be the dimension of the component of bidegree (p, q). Lemma (8.14) gives (8.15) np>q(x,c)= 0 Lrn^-r(x,o, r>(p+q-n) + (8.16) hp'q= y, K Formula (8.16) can be written as lip + qq = hppr\m + hp;iLmq-1 .16') If p + q>n, hp>q = hnp^n~P + /V7m~ 1'""P_1 + 8.17. Corollary. The Hodge and Betti numbers satisfy the following in equalities. (v)Ifk=p + qq>hp-1>q-1, bk>bk-2, (w)// k = p + q > n, then hp'q > hp+1>q+1, bk>bk+2. D Another important result of Hodge theory (that is in fact a direct consequence of Cor. 6.23) is the 8.18. Hard Lefschetz Theorem. The cup product morphisms Ln~k :Hk(X,C) -^H'2n-k(X,C), kq(X, C) -> Hn-q>n-p(X, C), p + q < n, 37 8. COMOHOLOGY OF COMPACT KAHLER MANIFOLDS are isomorphisms. ? Another way of stating the hard Lefschetz Theorem is to introduce the Hodge-Riemann bilinear form on H^R(X,C), defined by (8.19) Q(u,v) = (-l)^"1)/2 f uAvAujn-k. Jx The hard Lefschetz Theorem combined with Poincare duality says that Q is non-degenerate. Moreover Q is of parity (-l)k (symmetric if k is even, alternating if k is odd). When w is a Hodge metric, that is a Kahler metric such that {u>} ¤ H2(X, Z), it is clear that Q takes integer values when restricted to Hk(X, Z)/(torsion). The Hodge-Riemann bilinear form satisfies the following additional properties: For p + q = k, (8.20') Q(F'«,^) = 0 if (pV) # (9.P), (8.20") If 0 # u G H$m(X,C), then iP-«Q(«,u) = ||u||2 > 0. In fact (8.20') is clear and (8.20") will be shown if we can check that any (p,q)-primitive form u satisfies (-l)k(k-1)/'2ip-qujn-k Au = *u. Since Primp'qT^ is an irreducible representation of U(n), it suffices to verify the formula for a conveniently chosen (p, g)-form u. One can take for example u = dz\ A - - - A dzp A dzp+i A - - - A dzp+q from an orthonormal basis for ui. The necessary verification is easy for the reader to work out as an exercise. 8.D. A description of the Picard group. Another important application of Hodge theory is a description of the Picard group H1 (X, O*) of a compact Kahler manifold. We assume here that X is connected. The exponential exact sequence O^Z^0^e>*->l gives (8.21) 0->-H"1(X,Z)->.H"1(X,e>) -> Hx{X,0*) A H2(X, Z) ->- H2(X, O), taking into account the fact that the map exp(27ri») : H°(X, O) = C -> H°(X, O*) = C* is surjective. One has i71(X, O) ~ H°'1(X, C) by the Dolbeault isomorphism theorem. The dimension of this group is called the irregularity of X and it is usually denoted by (8.22) q = q(X) = h0'1 = h1'0. Consequently we have b\ = 2q and (8.23) H1(X,0)~Cl, H°(X,n1x)=H1>°(X,C) ~Cq. 8.24. Lemma. The image of H1^, Z) in H1(X,0) is a lattice. Proof. Consider the morphism H1(X,Z)^-H1(X,R) -^(A^C) -^^{X.O) induced by the inclusions Z C M C C C O. Since the Cech cohomology groups with values in Z or K can be calculated by a finite covering of open sets for which each is diffeomorphic to an open convex set, and the same for all their mutual intersections, it is clear that H1(X, Z) is a Z-module of finite type and that the J.-P. DEMAILLY, PART I: L2 HODGE THEORY 38 image of the H1(X, Z) in i71(X, M) is a lattice. It suffices therefore to show that the map HX(X, M) ->- Hx(X,0) is an isomorphism. However, the commutative diagram 0 -> C -> ,4° A ^ A .A2 ->--- y y 4- 4- 0 -> O -> .4°'° A ^O'1 A .4°>2 ->--- shows that the map H1(X, K) -? i71(X, O) corresponds, for de Rham and Dol-beault cohomology, to the composite map HhR(X,R) C HhR(X,C) -? H°^(X,C)- Since H1,0(X,C) and i70,1(X, C) are complex conjugate subspaces in the com- plexification H^R(X, C) of H^R(X, K), we can easily deduce that i^R(X, M) ->? H°'1(X, C) is an isomorphism. ? As a consequence of this lemma, H1(X, Z) is of rank 2g, i.e. H1(X, Z) ~ Z2q. The complex torus of dimension q (8.25) Jac(X) = H1(X,0)/H1(X,Z) is called the Jacobian variety of X. It is isomorphic to the subgroup of HX(X, O*) corresponding to the line bundles with zero first Chern class. In other words, the kernel of the arrow H2(X,Z)^H2(X,0) = H°'2(X,C), which defines the integral cohomology classes of type (1,1), is equal to the image of the morphism ci(») in H2(X, Z). This subgroup is called the Neron-Severi group of X, and is denoted by NS(X). Its rank p(X) is called the Picard number of X. The exact sequence (8.21) then gives (8.26) 0 -> Jac(AT) -> H1 (X, O*) A NS(X) ->- 0. The Picard group HX(X, 0*) is therefore an extension of the complex torus Jac(AT) by the Z-module of finite type NS(X). 8.27. Corollary. The Picard group o/P" is ^(F?,^*) ~ Z with 0(1) as generator, i.e. any line bundle over Pra is isomorphic to one of the line bundles 0(k), kGZ. PROOF. We have Hk(¥n, O) = H°>k(Fn,C) = 0 for k > 1 by applying conseq. 8.8, therefore Jac(P") = 0 and NS(Tn) = H2(P",Z) ~ Z. Moreover, ci(C(l)) is a generator of H2(Pn,Z). ? 9. The Hodge-Frolicher spectral sequence Assume given X a complex manifold (i.e. not necessarily compact) of dimension n. We consider the double complex Kp'q = C°°(X,Ap'qT^-) with its total differential d = d! + d". The Hodge-Frolicher spectral sequence (or Hodge to de Rham spectral sequence) is by definition the spectral sequence associated to this double complex. 39 9. THE HODGE-FROLICHER SPECTRAL SEQUENCE We first recall the algebraic machinery of spectral sequences, which applies to an arbitrary double complex (Kp'q, d! + d") of modules over a ring. We assume here for simplicity that Kp'q = 0 if p < 0 or q < 0. One first associates to K''' the total complex (K',d) such that Kl = (Bp+q=iKp'q, equipped with the total differential d = d' + d". Then K' admits a decreasing filtration formed from the subcomplexes FPK' where (9.1) FpKl = 0 Kj>'-j. P 1, equipped with differentials dr : Efq -> Ep+r'q-r+l of bidegree (r, -r + 1) such that Er+1 = H'(Er) is calculated recursively as the cohomology of the complex (E''',dr), and where the limit E? = limr_>.+00 Ep'q is identified with the graded module G'H'(K'), more precisely E? = GpHp+q(K'). The E\ terms are defined as the cohomology groups of the partial complex d" : Kp'q -? Kp'q+1 by passing to the second differential, that is (9.4) Ev{q = Hq((Kp'',d")), and the differential d\ : Ev,q -? Ev+ ,q is induced by the first differential d'': (9.5) d' : H« ((Kp<% d")) -»? IT' ((Kp+1'#, d")) ? In fact, one has Epq = 0 unless p, q > 0, and the limit i^ = limSr is stationary, more precisely Ep'q = Ev;qx = ---=Epf when r > max(p + 1,g + 2), as one sees by considering the indices in which dr can be non-zero. One says that the spectral sequence converges to the graded filtered module H'(K'), and it is customary to represent this situation by the notation Ef'9 ^GpHp+q{K'). A careful examination of the terms of small degree leads to the exact sequence (9.6) 0 -»? E1/ -> H^K') -> E0/ A E'22ft -»? H2{K'). One says that the spectral sequence degenerates at Ero if dr = 0 for all r > ro and for all bidegree (p, q). In this case one has E':' = E*'*+1 = - - - = _E^#. In the case of the Hodge-Frolicher spectral sequence, the E\ terms are the Dolbeault cohomology groups Ep'q = Hp'q(X, C), and the cohomology of the total complex is precisely the de Rham cohomology i7JR(X, C). One therefore obtains a spectral sequence (9.7) Ep'q = Hp>q{X,<£) => GpH^q{X,q(X,C), Hq>p(X,C) =Hp>q(X,C). P+q=l In terms of this decomposition, the filtration FPH!DR (X, C) is given by FpHlR(X,C)=($Hi>l-i(X,Q. j>p In particular, the conjugate filtration F'H^ is opposed to the filtration F'H^, i.e. HU(X,C) = FpHlw{X,C) (BF'-P+iH^X,®. 9.11. Definition. If X is a compact complex manifold, we say that X admits a Hodge decomposition if the Hodge-Frolicher spectral sequence degenerates at E\ and if the conjugate filtration F'H^R is opposed to F'H^, i.e. H^R = FPH^)R © F!-p+1H1T)R for all p. If X admits a Hodge decomposition in the sense of def. 9.11 and if p + q = I, then it is immediate from the equality -ffDR = ^p+1-fffjR © FqH\yR that ?DR = F'+^r © (FPH^R n FqH^R). Therefore one obtains a canonical isomorphism (9.12) Hp>q(X,C) ~ FpHlvKlFp+lHlvK ~ FPH^R n FqH^R C i£R. 41 9. THE HODGE-FROLICHER SPECTRAL SEQUENCE We deduce from this that there are canonical isomorphisms H^R(X,C) = 0 Hp>q(X,C), Hq>p(X,C) =Hp<9(X,C), p+q=l as expected. Note that (9.12) furnishes another proof of the fact that the Hodge decomposition of a compact Kahler manifold does not depend on the choice of Kahler metric (all the groups and morphisms concerned in (9.12) are intrinsic). In fact, we have shown that a compact Kahler manifold satisfies a still stronger property, that will be convenient to call a strong Hodge decomposition, since this one trivially implies the existence of a Hodge decomposition in the sense of Definition 9.11. 9.13. Definition. If X is a compact complex manifold, we say that X admits a strong Hodge decomposition if the morphisms H£«(X, C) -> H?(X, C), 0 H?(X,C) -»? HU(X,C) P+q=l are isomorphisms. 9.14. Remark. Deligne [Del68, 72] has given an algebraic criterion for the degeneration of the Hodge spectral sequence, including the case of the relative situa tion. More recently, Deligne and Illusie [DeI87] have given a proof of the degenera tion of the Hodge spectral sequence which does not use analytic methods (their idea is to work in characteristic p and to relate the result in characteristic 0). It is neces sary to observe that the degeneration of the Hodge-Frolicher spectral sequence does not automatically imply the Hodge symmetry property Hq,p(X,C) = H.P'Q(X,C) nor the existence of a canonical decomposition of de Rham groups. In fact, it is not difficult to show that the Hodge-Frolicher spectral sequence of a compact com plex surface always degenerates at E\; however if X is not Kahler, then b\ is odd, and one can show using the index theorem of Hirzebruch that ft0'1 = ft.1'0 + 1 and b\ = 2ft,1'0 + 1 (see [BPV84]). One can show that the existence of a Hodge decom position (resp. strong Hodge) is preserved by contraction morphisms (replacement of X by X', if n : X -> X' is a modification); this is an easy consequence of the ex istence of a direct image functor //* acting on all the cohomology groups concerned, such that /it*//* = Id. In the analytic context, //* is easily constructed by calculating cohomology with the aid of currents, since one has on those a natural direct image functor. As any Moishezon manifold admits a projective algebraic modification, we deduce that Moishezon manifolds also admit a strong Hodge decomposition. It would be interesting to know if there exists examples of compact complex manifolds possessing a Hodge decomposition without having a strong Hodge decomposition (there are indeed immediate examples of abstract double complexes having this property). ? In general, when X is not Kahler, a certain amount of interesting information can be deduced from the spectral sequence. For example, (9.6) implies (9.15) h > dim£2'° + (dim^0,1 - dim^'V- In addition, E2' is the cohomology group defined by the sequence d^d'-.E^^E^^E2,'0, J.-P. DEMAILLY, PART I: I? HODGE THEORY 42 and since _E1' is the space of global holomorphic functions on X, the first arrow d\ is zero (by the maximum principal, the holomorphic functions are constant on each connected component of X). Therefore dimE2' > h1,0 - ft2'0. Similarly, E.2' is the kernel of the map E®'1 -> S1'1, therefore dim^0,1 > ft0'1 - h1'1. From (9.15) we deduce (9.16) h > (ft1'0 - ft2>°)+ + (ft0'1 - ft1'1 - ft2'°)+. Another interesting relation concerns the topological Euler-Poincare characteristic XtopW = b0 - h H b2n-i + hn- We utilize the following simple lemma. 9.17. Lemma. Let (C',d) be a bounded complex of finite dimensional vector spaces over a field. Then the Euler characteristic x(C-) = £(-l)9dimC"> is equal to the Euler characteristic x(H'(C')) of the cohomology module. Proof. Set cq= dimC, zq= dim Zq(C), bq = dim Bq(C), hq = dim Hq(C). Then cq = zq + bq+1, hq = zq-bq. Consequently we find E(-x)% = E(-!)% - E(-1)9^ = Et-^v D In particular, if the term E* of the spectral sequence of a filtered complex K* is a bounded complex of finite dimension, one has X(E'r) = X(E'r+1) = ??? = X(^) = x(H'(K-)) because E'+1 = H'(E') and dimE^ = dimHl{K'). In the Hodge-Frolicher spectral sequence one additionally has dim_E{ = ^2v+q=l hp'q, therefore: 9.18. Theorem. For any compact complex manifold X, the topological Euler characteristic can be written xtoP(*)= E (-!)'6'= E (-i)p+9^'9- 0<«<2n 0- C (a morphism 43 10. DEFORMATIONS AND THE SEMI-CONTINUITY THEOREM of complexes of sheaves inducing an isomorphism 7ik(A*) -? 7ik(jC') on the coho-mology of sheaves). It is easy to see that hypercohomology does not depend up to isomorphism on the complex of acyclic sheaves C chosen. Hypercohomology is a functor from the category of complexes of sheaves of abelian groups to the category of graded groups. By definition, if A' ->- B' is a quasi-isomorphism, then mk(X,A') ->- Wk(X,B') is an isomorphism; moreover hypercohomology reduces to the usual cohomology Hk(X,£) of the sheaf £ for a complex A* reduced to a single term A0 = £. Suppose that one has for each term Ap of the complex A* a resolution Ap -> jCp'* by acyclic sheaves Cp'q, giving rise to a double complex of sheaves (£.p'q,d' + d"). Then the associated total complex (C*,d) is an acyclic complex quasi-isomorphic to A', and one therefore has mk(x,A') = Hk(r(x,r)). Further, the double complex Kp'q = T(X, Cp,q) defines a spectral sequence such that Ep'q = Hq(Kp'',d") =Hq(X,Ap), converges to the associated graded cohomology of the total complex Hk(K') = Mk(X, A'). One therefore obtains a spectral sequence called the hypercohomology spectral sequence (9.19) E? = Hq(X,Ap) => GpW+q{X,A'). The filtration Fp of hypercohomology groups is by definition obtained by taking the image of the morphism Uk(X,FpA') ->Uk(X,A'), where FPA' denotes the complex truncated to the left >? 0 -»? 0 -> Ap -> Ap+1 -»? > AN ??? . Consider now the case where X is any given complex manifold and where A' = £l*x is the holomorphic de Rham complex (with the usual exterior differential). The holomorphic Poincare Lemma shows that flx is a resolution of the constant sheaf Cx, i.e., one has a quasi-isomorphism of complexes of sheaves Cx -> Qx> where Cx denotes the complex reduced to a single term in degree 0. By definition of hypercohomology, one therefore has (9.20) Hk(x,cx) = mk(x,n'x), and the exact sequence of hypercohomology of the complex flx furnishes a spectral sequence (9.21) Ep'q = Hq(X,Qpx) => GpHp+q(X,Cx)- Because the groups Mk (X, Q'x) can be calculated by using the resolution of Q'x by the Dolbeault complex Cp,q = C°°(Ap'qTx) (these sheaves are certainly acyclic!), one then sees that the hypercohomology spectral sequence (9.21) is precisely the Hodge-Frolicher spectral sequence previously defined. J.-P. DEMAILLY, PART I: L2 HODGE THEORY 44 10. Deformations and the semi-continuity theorem The purpose of this section is to study the dependence of the groups Hp'q(Xt,C) or more generally the cohomology groups Hq(Xt,Et), when the pair (Xt,Et) depends holomorphically on a parameter t in a certain complex space S. Our approach is to adopt the point of view of Kodaira-Spencer, such as is developed in their original work on the theory of deformations (see for example the complete works of Kodaira [Kod75]). The method of Kodaira-Spencer exploits the continuity properties or semi-continuity of proper spaces of Laplacians as a function of the parameter t. Another approach furnishing more precise results consists of utilizing the theorem of direct images of Grauert [Gra60]. 10.1 Definition. A deformation of compact complex manifolds is given by a proper analytic morphism a : X -> S of connected complex spaces, for which all the fibers Xt = tr_1(t) are smooth manifolds of the same dimension n, and satisfy the following local condition: (H) Any point (el admits a neighbourhood U such that there exists a biholo-morphism ip : U x V ->- U where U is open in C? and V is a neighbourhood of t = cr((), satisfying a o ip = pr2 : U x V ^ V (second projection). We say that (Xt)tes is a holomorphic family of deformations of any given fiber Xto, and that S is the base of the deformation. A holomorphic family of vector bundles (resp. sheaves) Et -? Xt is given by a family of bundles (resp. sheaves) obtained from a global bundle (resp. global sheaf) E ->- X, by restriction to the fibers Xt. If S is smooth, the hypothesis (H) is equivalent to assuming that a is a holomorphic submersion, as a consequence of the theorem of constant rank. There are nevertheless situations where one must necessarily consider also the case of a singular base S (for example when one seeks to construct the "universal deformation" of a manifold). In a topological setting (differentiable or smooth), we have the following lemma, known as Ehresmann's Lemma. 10.2. Ehresmann's Lemma. Let a : X ->- S be a smooth and proper differentiable submersion. a) If S is contractible, then for any to £ S, there exists a commutative diagram X A Xt0 x S pri \ ?/ a S where $ is a diffeomorphism. b) For any given base S, X -> S is a locally trivial bundle (differentiable). In particular, if S is connected, the fibers are all diffeomorphic. Proof, a) Let H : S x [0,1] -> S be a differentiable homotopy between H(»,0) = Ids and H(», 1) = constant map S ->- {to}- The fiber product X = {{x, s, t) e X x S x [0,1] ; a{x) = H{s, t)} with projection a = pr2 x pr3 : X ->- S x [0,1] is still a differentiable submersion, as one can easily verify. One deduces that there exists a vector field £ on X which lifts the vector field |onSx [0,1], i.e. tr*£ = Jj. (There exists a local lifting by the 45 10. DEFORMATIONS AND THE SEMI-CONTINUITY THEOREM submersive property, and one glues together these liftings by means of a partition of unity.) Let ipt be a flow of this lifting: Then, if (x, s, 0) ¤ £fSx{o} - £, one has by construction tpt(x,s,0) = (?,s,t), therefore $ = tp\ defines a diffeomorphism of 3£fSX{o} - X on Xf5X{i} ~ Xto x S, commuting with the projection on S. b) is deduced immediately from a). ? It follows from b) that the bundle t t-> Hk(Xt, C) is a locally trivial bundle of C-vector spaces of finite dimension. Furthermore, in each fiber we have a free abelian subgroup ImHk(Xt, Z) C Hk(Xt, C) of rank bk which generates Hk(Xt, C) as a C-vector space. The transition matrices of this locally constant system are in SLj,fc (Z). Since the transition matrices are locally constant, the bundle t *-? Hk(Xt,C) is equipped with a connection D such that D2 = 0: This connection is called the Gauss-Manin connection. The following lemma is useful. 10.3. Lemma. Let a : X ->- S be a smooth and proper differentiable submersion and £ a C°° vector bundle over X. Consider a family of elliptic operators Pt : C^iXuEt) ^ C°°(Xt,Et) of degree 6. We assume that Pt is self-adjoint semipositive relative to a metric ht on Et and a volume form dVt on Xt, and that the coefficients of Pt, ht and dVt are C°° on X. Then the eigenvalues of Pt, computed with multiplicity, can be arranged in a sequence A0(t)< Ai(t) <-"+oo, where the k-th eigenvalue Xk(t) is a continuous function oft. Moreover, ifX is not in the spectrum {Afc(to)}fceN of Pt0, the direct sum W\j C Cco{Xt,Et) of eigenspaces of Pt with eigenvalues Xk(t) < A defines a C°° vector bundle, t i->- Wt,\, in a neighbourhood of to. PROOF. Since the results are local over S, one can assume that X = Xto x S and £ = prlEto, that is, their fibers Xt and Et are independent oft (but the forms dVt on Xt and the metrics ht on Et are in general dependent on t). Let 11^,4 be the orthogonal projection operator on W\j in L2(Xt,Et) ~ L2(Xt0,Eto). If T(0, A) denotes the circle with center 0 and with radius A in the complex plane, Cauchy's formula gives n\,t = -^ f {zid-Pt^dz, 27Tl Jr(o,\) where the integral is viewed as an integral with vector values in the space of bounded operators on L2 (Mto, Eto). (It suffices to verify the formula on the eigenvectors of Pt, which is elementary.) The arguments made in §3 show that there exists a family of pseudodifferential operators Qt of order -6, for which the symbol depends in a C°° manner with t (and with uniform estimates by differentiation in t), such that PtQt =ld + Rt for regular operators Rt, for which the kernel also depends in a C°° manner in t. Since Qt is a family of compact operators on L2(Xt, Et) which depend in a C°° manner in t, the eigenvalues of Qt depend continuously in t. Up to changing Qt0 on a subspace of finite dimension, one can assume that Qt0 is an isomorphism of L2 (Xto, Eto) onto W5 (Xto, Eto). It will be the same for Qt in a neighbourhood of to, and consequently zld - Pt is invertible if and only if (zld - Pt)Qt = ld + Rt + zQt is invertible. If A is not in the spectrum of Pto, it follows that for all z £ T(0, A), the inverse (zld - Pj)_1 = <5t(Id + Rt + zQt)~x depends in a C°° way in t. This J.-P. DEMAILLY, PART I: L2 HODGE THEORY 46 implies that t >-> Wt,\ is a locally trivial C°° fibration in a neighbourhood of to- The continuity of the eigenvalue Xk(t) of Pt follows from the constant rank of W\j in a neighbourhood of to, for A = Afc(to) ± e. D 10.4. Semi-continuity Theorem (Kodaira-Spencer). IfX^-Sisa smooth, proper C-analytic morphism and if £ is a locally free sheaf on X, the dimensions hq(t) = hq(Xt,£t) are upper semi-continuous functions. More precisely, the alter nating sums hq(t) - h9-1^) + ??? + (-l)qh°(t), 0 0 which does not belong in the spectrum of the operators A"o9,0 < q < n = dim Xt. Then Wq = Wq t = direct sum of eigenspaces of A"9 with eigenvalues < A defines a C°° bundle Wq in a neighbourhood of to- Moreover the differential d" commutes with A" and thus sends the eigenspaces of A"9 into the eigenspaces of A"9+ associated to the same eigenvalues. This shows that (W*,d") is a sub-complex of finite dimension of the Dolbeault complex (C°°(Xt, A0,9T^t (g) Et),d"). The cohomology of this subcomplex coincides with Hq(Xt,Et) since the relation d"d"* + d'l*d'l = A" shows that j-d"* is a homotopy operator on the subcomplex formed from the eigenspaces with eigenvalue A^ when \k ^ 0. If Zf denotes the kernel of the morphism d"9 : Wt9 ->- Wt9+1, then zq{t) := dimZq is an upper semi-continuous function in the Zariski topology, as one can easily see by considering the rank of the minors of the matrix defining the morphism d"q : Wq -> Wq+1. From the truncated complex 0 ->- VFt° ->- Wj -^ > Wf1 -* Zq ->- 0 having for the cohomology the groups HJ(ATt,i?t) with indices 0 < j < q, one obtains hq(t) - h^it) + ??? + (-l)qh°(t) = zq(t) - w9"1 + wq~2 + ??? + (-l)qw°, where wq denotes the rank of Wq. The upper semi-continuity of the term on the left follows, and that of hq(t) is then immediate by induction on q. ? 10.5. Invariance of the Hodge numbers. Let X ->? S be a smooth and proper C-analytic morphism. We assume that the fibers Xt are Kahler manifolds. Then the Hodge numbers hp'q(Xt) are constant. Moreover, in the decomposition Hk(Xt,C) = 0 HP'q(Xt,C), p+q=k the bundles t *-? Hp'q(Xt,C) define C°° subbundles (in general, not holomorphic subbundles) of the bundle t >-)- Hk(Xt,C). 47 10. DEFORMATIONS AND THE SEMI-CONTINUITY THEOREM PROOF. Lemma 10.2 implies that the Betti numbers bk = dim Hk (Xt, C) are constant. Since, according to th. 10.4, hp'q(Xt,C) = hq{Xt,^lpx ) is upper semi-continuous, and h?{Xt) =bk- J2 hr'"(Xt), r-\-s=k,(r,s)^(p,q) these functions are likewise lower semi-continuous. Consequently they are contin uous and therefore constant. A theorem of Kodaira [Kod75] shows that if a fiber Xto is Kahler, then the neighbouring fibers Xt are Kahler and the Kahler metrics ujt can be chosen so that they depend in a C°° way with t. The spaces of harmonic (p, (/)-forms therefore depend in a C°° way with t according to th. 10.4, and one deduces that t h-> Hp'q(Xt, C) is a C°° subbundle of Hk(Xt,£). ? It is possible to obtain more precise and general results by means of the theorem of direct images of Grauert [Gra60]. Recall that if we are given a continuous map / : X -> Y between topological spaces and a sheaf £ of abelian groups on X, then one can define the direct image sheaf Rkf*£ on Y, as being the sheaf associated to the presheaf U t-> Hk(f~1(U),£), for all open U in Y. More generally, being given a complex of sheaves A', we have the direct image sheaves Rq ft, A', obtained from the hypercohomology presheaves U ^Mk{f-1{U),A'). The proof of the theorem of direct images as given by [FoK71] and [KiV71] (also see [DoV72]) furnishes the following fundamental result. 10.6. Theorem of direct images. Let a : 3L ->- S be a proper morphism of complex analytic spaces and A* a bounded complex of coherent sheaves of Ox-modules. Then (aa)The direct image sheaves W.ka*A* are coherent sheaves on S. (bb)Any point of S admits a neighbourhood U C S on which there exists a bounded complex W of sheaves of locally free Os-modules in which the cohomology sheaves Hk(W°) are isomorphic to the sheafM^a^A'. (cc)If the fibers of a are equidimensional ("geometrically flat morphism"), the hypercohomology of the fiber Xt = a-1^) with values in A' = A'®ox ®xt (where Oxt = £>x/o-*ms,t) is given by Hk(Xt,A't) = Hk(Wt-), where (W°) is the complex of finite dimensional spaces Wk = Wk- Wk+1 are of constant rank at each point t £ S. From (10.6b) one then deduces the following result due to [Fle81] with an identical argument to that in th. 10.4. 10.7 Semi-Continuity Theorem. If 3£ ^ S is a proper analytic morphism with equidimensional fibers and if £ is a coherent sheaf on X, then the alternating J.-P. DEMAILLY, PART I: L2 HODGE THEORY 48 hq(t)-hq-1(t) + --- + (-l)qh0(t), with dimensions hk(t) = hk(Xt,£t), are upper semi-continuous functions oft in the analytic Zariski topology (topology of whose closed are the analytic sets). Let a : X -> S be a C-analytic proper and smooth submersion. One assumes that the Hodge spectral sequence of the fibers Xt degenerates at E\ for all t ¤ S (according to (10.7) this is in fact an open property for the analytic Zariski topology on S). If U C S is open and contractible, then tr_1(?7) ~l(x(7 for any fiber over t ¤ U. If Z%,Cx, denotes the locally constant sheaves with base X and with fibers Z, C, one obtains T(U,Rko-*Zx) = Hk(a~1(U),I1) = fffc(Xt,Z), T(U,Rk Hk(Xt,C), equipped with the flat connection D (Gauss-Manin connection), possesses a canonical holomorphic structure induced by the component D0'1 of the Gauss-Manin connection. The flat bundle (BkHk(Xt,C) is called the Hodge bundle of the fibration X -> S. Now consider the relative de Rham complex (ft^,s,dx/s) of the fibration X ->- S. This complex furnishes a resolution of the sheaf cr~1Os ("purely sheafified" inverse image of Os), consequently (10.8) RkaM'x/s = Rko-*{o--1Os) = {RkaXx) ®c Os. The latter equality is obtained immediately by an argument using Os(U) linearity for the cohomology calculated on the open set cr~1(Lr) (the complex structure of tr_1(Lr) does not intervene here). In other words, Mfccr*f)^,s is the locally free Os-module associated to the flat bundle t h->- Hq(Xt,C). One has a relative hyper-cohomology spectral sequence Ef'9 = ROaMx/s => GpW+qo-Mx/s = GpRp+qaXx (the relative spectral sequence is obtained simply by a "sheafification" of the absolute hypercohomology spectral sequence (9.19) of the complex Sl'x/s over *^e open set c-1 ([/)). Since the cohomology of ^lpx/s on the fiber Xt is precisely the space Hq(Xt, Qpx ) of constant rank, th. 10.6d) shows that the direct image sheaves RPaMx/s are locally free. In addition, the filtration FPHk(Xt,C) C Hk{Xt,<£) is obtained on the level of locally free Og-modules associated with taking the image of the O^-linear morphism M 0~*F l£^-/e -y M <7*l2^>/e, which is therefore a coherent subsheaf (and likewise a locally free subsheaf, according to the property of constant rank on the fibers Xt). From (10.8) one deduces the 10.9. Theorem (holomorphic Hodge filtration). The Hodge filtration FpHk(Xt,C) C Hk(Xt,C) defines a holomorphic subbundle relative to the holomorphic structure defined by the Gauss-Manin connection. One sees that in general there is no reason for Hp'q(Xt,C) = FpHk(Xt,C) fl FqHk(Xt,C) to be a holomorphic subbundle of Hk(Xt,C) for any p + q = k, although Hp'q(Xt,C) possesses a natural holomorphic bundle structure (obtained from the coherent sheaf i?9tr*f2^ ,s, or as a quotient of FpHk(Xt,Cj). In other words, this is the Hodge decomposition which is not holomorphic. 10.10 Example. Let S = {t e C; Im r > 0} be the upper half plane and X ->- S the "universal" family of elliptic curves over S, defined by XT = C/(Z + TLt). The two basis elements of the Hodge fiber H1(XT,C), dual to the basis (1,t) of the lattice of periods, are a = dx - Re r/Im rdy and /3 = (Im r)~1dy (z = x + \y £ C denotes the coordinates on XT). These elements therefore satisfy Da = D(] = 0 and define the holomorphic structure of the Hodge bundle; the subbundle H1'°(XT,C) generated by the 1-form dz = a + r/3 is clearly holomorphic (as it should be!), however one sees that the components /31'0 = - ^(Im r)~1dz and /30'1 = - i(Im r)~ldz are not holomorphic in r. 49 J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 50 Part II: L2 Estimations and Vanishing Theorems 11. Concepts of pseudoconvexity and of positivity The statements and proofs of the vanishing theorems brings into play many concepts of pseudoconvexity and positivity. We first present a summary, by bringing together the concepts that we deem necessary. 11.A. Plurisubharmonic functions. The plurisubharmonic functions were introduced independently by Lelong and Oka in 1942 in the study of holomorphic convexity. We refer to [Lel67, 69] for more details. 11.1. Definition. A function u : fl ->- [-oo,+oo[ defined on an open set fl C C? is called plurisubharmonic (abbreviated psh) if (dd)u is upper semi-continuous; (ee)for any complex line L C C?, itfnnL is subharmonic on ft n L, that is, for any a £ fl and £ £ C? satisfying |£| < d(a,Zfl), the function u satisfies the mean inequality 1 f27r u(a) < - / u(a + eie£)d9. 2?r Jo The set of psh functions on fl is denoted by Psh(fi). We give below a list of some fundamental properties satisfied by the psh functions. All these properties come about easily from the definition. 11.2. Fundamental properties. a) Any function u £ Psh(fi) is subharmonic in the 2n real variables, i.e. satisfies the mean value inequality on the Euclidean ball (or sphere): u(a) < ~-r~. / u(z)d\(z) IT r jn\ JB(a,r) for all a £ fl and all r < d(a, Cfl). In this case, one has either u = -oo or u £ L\oc on every connected component of ft. (ff)For any decreasing sequence of psh functions Uk ¤ Psh(fJ), the limit u = limits is psh on ft. (gg)Assume given u £ Psh(fi) such that u ^ - oo on all connected components of ft. If (pt) is a family of regular kernels, then u* p¤ is C°° and psh on ft, = {xGfl; d(x,Cfl) >e}, the family (u*pe) is increasing in e, and lime_>.ow*pe = u. d) Assume given u\,... ,up £ Psh(fJ) and x : Kp ->? M a convex function such that x(tii ? ? ? >tp) is increasing in each variable tj. Then x(ui> ? ? ? tup) IS Psn on ft. In particular u\-\ \-up, maxjui,... ,up}, log(eUl H he"p) are psh on ft. D 11.3. Lemma. A function u £ C2(fl, ffi) is psh on fl if and only if the Hermit- ian form Hu(a)(£) = Xa<7- k Y is a holomorphic map and if v £ C2(Y, ffi), we have d'd"(v o $) = $*d'd"v, therefore H(v o $)(a,0 = Hv(Ma)^'(a).0- In particular Hu, viewed as a Hermitian form on Tx, is independent of the choice coordinates (z\,... , zn). Consequently, the notion of a psh function makes sense on any complex manifold. More generally, we have (hh)Proposition. // $ : X ->- Y is a holomorphic map and v £ Psh(F), thenvo® e Psh(X). ? (ii)Example. It is well known that log|z| is psh (i.e. subharmonic) on C Therefore log|/| £ Psh(X) for any holomorphic function / £ H°(X,Ox)- More generally iog(i/ir + --- + i/gr)£Psh(x) for any choice of functions fj £ H°(X, Ox) and real ctj > 0 (apply property 11.2d with Uj = ctj log \fj\). We will be interested more particularly with singularities of this function along the variety of zeros fi = - ? - = fq = 0, when the olj are rational numbers. ? 11.7. Definition. One says that a psh function u £ Psh(X) has analytic singularities (resp. algebraic) if u can be written locally in the form ^ = fl0g(|/l|2 + --- + |/Af)+«, with holomorphic functions (resp. algebraic) /j, a £ ffi+, (resp. a £ Q+), and where v is a bounded function. J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 52 We introduce then the ideal 3 = 3(u/a) of germs of holomorphic functions h such that there exists a constant C > 0 for which \h\ < Ce"/a, i.e. W where P = \d'd"\z\2 = \ J2 dzi A <**!' P' l? (zii:... ,zip) define a finite ramified covering over the intersection A n A/ of A with a small polydisk A/ = A^ x Ay of C" = ¤P x Cn~p, over the polydisk Ay of Cp. Let m be the number layers of each of these coverings. Then, if A = flAj, the p-dimensional area of A n A is bounded above by the sum of the areas of its projections computed with multiplicities, i.e. Surface Area(A nA)<^ n7Vol(Ay). The fact that [A] is positive is easy. In fact, in terms of local coordinates (w\,... , wv) on Aleg, one has i«i A ai A - - - A \ap A ap = \ det(ajfc)|2iwi A wJi A - - - vwv A wp J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 54 if olj = ^ctjkdwk- This shows that such a product of forms is > 0 by comparison to the canonical orientation defined by iw\ A w\ A - - - A \wp A wp. A deeper result, also proven by P. Lelong [Lel57], is that [A] is a (i-closed current on X, in other words, the set Asing (which is of real dimension < 2p - 2) does not contribute to the boundary current d[A\. Finally, in connection with example 11.12, we have the important 11.14. Lelong-Poincare equation. Let / £ H°(X,Ox) be a nonzero holo-morphic function, Zf = J2rrijZj, rrij £ N, the divisor of zeros of /, and [Zf] = J2rrij[Zj] the associated current of integration. Then -ddiog\f\ = [zf]. PROOF (outline) . It is clear that id'd" log |/| = 0 in a neighbourhood of each point x £" Supp(Zf) = UZj, consequently it suffices to verify the equation in a neighbourhood of any point of Supp(Zf). Let A be the set of singular points of Supp(Zf), i.e. the union of the intersections ZjHZf. and of their singularities Zj^-nlg; we then have dim A < n - 2. In a neighbourhood of any point x £ Supp(Zf)\A there exists local coordinates (zi,... ,zn) such that f(z) = z"l\ where rrij is the multiplicity of / along the component Zj which contains x, and where z\ = 0 is a local equation of Zj near x. Since -d'd" \og\z\ = Dirac measure 8$ in C, we find j^d'd" log \zi\ = [hyperplane z\ = 0], therefore -d'd" log |/| = rrij-d'd" log \Zl\ = m^ZA IT IT in a neighbourhood of x. This shows that the equation is valid on X\A. Con sequently, the difference ^d'd"log|/| - [Zf] is a closed current of degree 2 with measurable coefficients for which the support is contained in A. This current is necessarily zero because A is of too small a dimension for to be able to carry its support. (A is stratified into submanifolds of real codimension > 4, whereas the current itself is of real codimension 2.) ? To conclude this section we now revisit the de Rham and Dolbeault cohomology in the context of the theory of currents. A basic observation is that the Poincare and Dolbeault-Grothendieck Lemmas are still valid for currents. More precisely, if (Vq,d) and (V(F)p'q,d") denotes the complexes of sheaves of currents of degree q (resp. currents of bidegree (p, q) with values in a holomorphic vector bundle F), one still has resolutions of de Rham and of Dolbeault sheaves o->w^v", o^npx(g>o(F) -^v'(f)p'\ As a result, there are canonical isomorphisms (11.15) H&R(M,R) = H«((T(M,V),d)), Hp>q(X,F) = H"((r(X,V'(F)^'),d")). In other words, one can attach a cohomology class {©} £ HpR(M, R) to any closed current 0 of degree q, resp. a cohomology class {0} £ Hp'q(X, F) for any - H9l+-+9m (M,K), ({91},...,{01})^{01}A---A{em}. In particular, if M is a compact oriented manifold and if q\ + - ? - + qm = dimM, one obtains a well defined intersection number {©i} - {©2} {0m} = / {©i} A - - - A {0m}. J M We note however that the specific product 0i A - - - A ®m does not exist in general. 11.C. Positive vector bundles. Let (E, h) be a Hermitian holomorphic vector bundle on a complex manifold X. Its Chern curvature tensor @(E) = ^2 Cjkxudzj A dzk ® e^ ® eM l E, viz. (11.16) @(E)(£®v)= ^ Cjk\u,Zjtkv>J>fi, CjkXy, =CkJu,\- l 0 for any non-zero tensor r = J^ Tj\d/dzj e\ £ Tx <8> E. b) positive in the sense of Griffiths if: ®(E)(£ ® v) > 0 for any non-zero decomposable tensor £ ® v £ Tx E. The corresponding concepts of semi-positivity are defined by replacing the strict inequalities by the broader inequalities. 11.18. The particular case of rank 1 bundles. Suppose that E is a line bundle. The Hermitian matrix H = (/in) associated to a trivialization r : E\q ~ fl x C is then simply a positive function, and it will be convenient to denote it by e~2ip, f ¤ CO0(f),ffi). In this case, the curvature form ®(E) can be identified with the (1, l)-form 2d'd"- f2' x C is another trivialization, tp' the associated weight and g £ 0*(Vl n fi') the transition function, then r'(£) = <7(x)r(£) for all £ £ -Ex, and therefore 0. The corresponding line bundle can be given a singular metric defined by ||w|| = \u\ (modulus of the meromorphic function u). If gj is a generator of the ideal of Dj on an open set fl C X, then t(u) = u FJ g"1 defines a trivialization of O(D) on fl, thus our singular metric is associated to the weight tp = J2atj log |<7j|. The Lelong-Poincare equation implies that -Q(0(D)) = ddctp = [D], IT where [D] = ^2ot.j[Dj] denotes the current of integration on D. ? 11.22. Example. Suppose that tri,... , 0, which is very ample. We adopt here the additive notation for Pic(X) = HX(X, O*), the symbol mE representing the line bundle E®m. By refering to example 11.22, it follows that any ample line bundle E has a C°° Hermitian metric, having a positive definite curvature form. Indeed, if the linear system \mE\ gives an embedding in projective space, then one obtains a C°° Hermitian metric on E®m, and the m-th root gives a metric on E such that ^Q(E) = m^mEl^s- Conversely, Kodaira's embedding theorem [Kod54] says that any positive line bundle E is ample (see exercise 15.11 for a direct analytic proof of this fundamental theorem). 12. Hodge theory of complete Kahler manifolds The goal of this section is primarily to extend to the case of complete Kahler manifolds the results of Hodge theory already proven in the compact case. 12.A. Complete Riemannian manifolds. Before treating the complex situation, we will need to discuss some general results on the Hodge theory of complete Riemannian manifolds. Recall that a Riemannian manifold {M,g) is said to be complete if the geodesic distance Sg is complete, or what amounts to the same thing (Hopf-Rinow Lemma below), if the closed geodesic balls are all compact. We will need the following more precise characterization. 12.1. Lemma (Hopf-Rinow). The following properties are equivalent: (ll)(M,g) is complete; (mm)the closed geodesic balls Bg(a,r) are compact; (nn)there exists an exhaustive function ip £ C°°(M, ffi) such that \dip\g < 1; (oo)there exists in M an exhaustive sequence (Kv)vg$ of compact sets and functions dv eC°°(M,M) such that on a neighbourhood of KVl Supp 9V C K°_^^, 0 <0V < 1 and \d9v\g < Tv. Proof, a) => b). The point x being fixed, one denotes by ro = ro(x), the supremum of the real numbers r > 0 such that Bg(a,r) is compact. Suppose ro < +00. Being given a sequence of points (x") in Bg(a,ro) and e > 0, one chooses a sequence of points xVtC ¤ B(a,ro - e) such that Sg(xv,xVte) < 2e. By compactness of Bg(a, ro - e), one can extract from {xv^) a convergent subsequence for each e > 0. By applying a diagonal process, one easily sees that one can extract from (x") a Cauchy subsequence. Consequently this sequence converges and Bg(a,ro) is compact. The local compactness of M implies that Bg(a,ro + n) is still compact for n > 0 small enough, which is a contradiction if ro < +00. b) =^ c). Suppose M is connected. Choose a point xq £ M and set ipo(x) = ^6(xo,x). Then ipo is exhaustive, and this is a Lipschitz function of order |, therefore ipo is differentiable almost everywhere on M. One obtains the sought for function ip by regularization. J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 58 c) =>- d). Let ip be as in a) and let p £ C°°(M, M) be a function such that p = 1 on ] - oo, 1.1], p = 0 on [1.99, +oo[ and 0 < p' < 2 on [1, 2]. Then K" = {x£ M; ip(x) < T+l), 6"{x) = p(2-v-1^(x)) satisfies the desired properties. d) =>. c). Set il) = J22v-l{l-9v). c) =>- b). The inequality \dip\g < 1 implies \ip(x) - ip(y)\ < 6g(x,y) for any x,y £ M, therefore the geodesic ball Bg(a,r) C {x £ M; <5fl(x, a) < V'(a) + r} is relatively compact. b) => a). This is obvious! ? Let (M, 5) be a Riemannian manifold, not necessarily complete for the moment, E a Hermitian vector bundle on M, with a given Hermitian connection D. One considers the unbounded operator between Hilbert spaces, still denoted by D D : L2(M,APT*M®E) ->- L2\M, Ap+1T*M ® E), for which the domain Dom D is defined as follows: A section u £ L2 is said to be in Dom D if Du calculated in the sense of distributions is still in L2. The domain thus defined is always dense in L2, because Dom D contains the space V(M,APT^E) of C°° sections with compact support, which is itself dense in L2. Moreover, the operator D thus defined, albeit not bounded, is closed, that is to say its graph is closed; this follows at once from the fact that the differential operators are continuous in the weak distribution topology. In the same way, the formal adjoint D* admits an extension to a closed operator D* :L2(M,Ap+1T^(g>E) -> L2(M, kp,T*M ® E). Some well-known elementary results of spectral theory due to Von Neumann guarantees, in addition, the existence of a closed operator D^ with dense domain, called the Hilbert space adjoint of D, defined as follows: An element v £ L2(M,AP+1T^® E) is in Dom D^ if the linear form L2 -> C, u *-? ((Du,v)) is continuous. It is thus written u *-? ((u,w)) for a unique element w £ L2(M,AP,T^I E). One sets D^v = w, so that D^ is defined by the usual adjoint relation ((Du, v)) = ((u, D*nv)) Vw £ Dom D. (Note that the formal adjoint D*, itself, is defined by requiring only the validity of their relation for u £ V(M,AP,T^ ® E).) It is clear that one always has Dom D^ C Dom D* and that D^ = D* on Dom D^. In general, however, the domains are distinct (this is the case for example if M =]0,1[, g = dx2, D = d/dx !). A fundamental observation is that this phenomenon cannot occur if the Riemannian metric is complete. 12.2. Proposition. If the manifold {M,g) is complete, then: a) The space V(M, A'T^E) is dense in Dom D, Dom D* and Dom D n Dom D* respectively, for the norms of the graphs u ^ \\u\\ + \\Du\\, u^\\u\\ + \\D*u\\, u^\\u\\ + \\Du\\ + \\D*u\\. b) D^ = D* (i.e. the two domains coincide), and D^ = D** = D. 59 12. HODGE THEORY OF COMPLETE KAHLER MANIFOLDS c) Let A = DD* + D*D be the Laplacian calculated in the sense of distributions. Foranyu £ Dom A C L2(M,hmT*M®E), one has (u,Au) = ||L>u||2 + ||D*w||2. In particular Dom A C Dom D n Dom D*, KerA = KerD n KerL>*, and A is self adjoint. d) If D2 = 0, there is an orthogonal decomposition L2(M, A'T; ®E)= H'L2 (M, E) © Im~D © ImD«, Ker £> = H'L2 (M, £) © Im~D, w/iere H'L2(M,E) = {u £ L2(M7h*T*M E); Au = 0} is i/ie space o/ L2 harmonic forms on M. Proof, a) It is necessary to show for example that any element u £ Dom 2} can be approximated in the norm of the graph of D by C°° forms with compact support. By assumption, u and Du are in L2. Let (0V) be a sequence of truncating functions as in Lemma 12.1 d). Then 9vu -^ it in L2(M, A'T^ ® E) and D(9vu) = #j,Z)u + d^v A u where \ Du. By replacing u by 9"u, one can assume that u has compact support, and with the aid of a partition of unity, one is reduced to the case where Supp u is contained in a coordinate chart of M on which E is trivial. Let (pf) be a family of regular kernels. A classical lemma in the theory of PDE (Friedrich's Lemma), shows that for any differential operator P of order 1 with C1 coefficients, one has ||-P(pe * it) - pfPu\\L2 ->- 0, as e tends to 0 (u being an L2 section with compact support in the coordinate chart considered). By applying this lemma to P = D, P = D* respectively, one arrives at the desired properties of density. b) is equivalent to the fact that ((Du,v)) = ((u,D*v)), Vu e Dom D, Vi> ¤ Dom D*. However, according to a), one can find uv, vv £ V(M, A'T^ E) such that uv -> u, vv -> v, Duv -> Du and D*vv -> D*v in L2{M,K'T*M®E). The desired equality is then the limit of the equality {{Duv,vv)) = {{uv,D*vv)). c) Let u £ Dom A. Since Am £ L2 and that A is an elliptic operator of order 2, one obtains u £ W2oc by applying the local version of the Garding inequality. In particular Du, D*u £ W^oc C L20C, and we can apply integration by parts as needed, after multiplying the respective forms by C°° functions 0V with compact support. Some simple calculations then give \\evDu\\2 + \\evD*u\\2 = = ((0lDu,Du)) + ((u,D(0lD*u))) = {{D{62v,u),Du)) + ((u, e2"DD*u)) - 2{{6vd6v A u, Du)) + 2((u, Bvddv A D*u)) = ((6lu, Au)) - 2{{d6" A u, 9vDu)) + 2((u, dBv A {9"D*U))) ((0lu,Au)) + 2-"(2||0"£>u||||u|| + 2||6»v£>*u|||H|) ((6lu,Au)) + 2-»{\\evDu\\2 + ||^£>*«||2 + 2|M|2). J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 60 Consequently \\evDu\\2 + \\ovd*u\\2 < T_L_(«02U) Au>> + 21-"|H|2). By letting v tend to +00, one obtains ||-Dw||2 + ||_D*m||2 < ((u,Au)), in particular Du, D*u are in L2. This implies ((u, Av)) = ((Du, Dv)) + ((D*u, D*v)), \/u, v ¤ Dom A, because the equality holds for 9"u and v, and that 9vu -> u, D(9vu) -> Du and D*(9"u) -> D*u in L2. It follows from this that A is self-adjoint. d) If P is a closed operator with dense domain on a Hilbert space %, then KerP is closed and Kerf* = (Im P)±. Consequently (KerP*)^ = (Im P)-11- = Im P. Since KerP* itself is also closed, we have % = KerP* © (KerP*)-1 = KerP* ®InTP. This result applied to P = A gives n'L2 (M, E) = Ker A © Im~A, and it is clear according to (12.2 c) that Im A C Im D (B Im D*. Furthermore, one easily sees that Ker A, Im D and Im D* are pairwise orthogonal by using (12.2 a,c). Property d) follows as in the case where M is compact. ? 12.3. Definition. Assume given a Riemannian manifold (M,g) and a Her- mitian bundle E with a flat Hermitian connection D. We denote by H^R L2 (M, E), the L2 de Rham cohomology groups, namely the cohomology groups of the complex (K',D) defined by Kp = {«eI2(M,A%»£); Du£L2}. In other words, one has Pt^R Li (M, E) = Ker D/Im D, where D is the L2 extension of the connection calculated in the sense of distributions. Since "H^2 (M, E) = KerP>/Im D according to (12.2 d), it follows that: 12.4. Proposition. There is a canonical isomorphism Ul2{M,E)~HlKL2{M,E)sep between T~LPL2 (M, E) and the separated space associated to the L2 de Rham cohomology- In general the space H^R L2 (M, E) is not always separated, but it is in the important case where the L2 cohomology is finite dimensional: 12.5. Corollary. If(M,g) is complete and if H^RL2(M,E) is finite dimen sional, then this space is separated and there is a canonical isomorphism npL2(M,E)~H^RL2(M,E). 61 12. HODGE THEORY OF COMPLETE KAHLER MANIFOLDS Proof. The space Kp can be considered as the Hilbert space with norm u i->- (||w||L2 + ||_Dm||L2)1/2. It is a question of seeing that ImO = D(KP~1) is closed in KerD, KerD being itself closed in K'p. Now D : Kp~1 -t KerD is continuous and its image is of finite codimension by hypothesis. The fact that the image is closed is then a direct consequence of the Banach Theorem. ? 12.6. Remark. For L2 de Rham cohomology, observe that one obtains the identical cohomology groups when working with the subcomplex of global L2 C°°- forms, that is Kp = {u£ C°°{M,kpT*M ®E); uG L2 and Du £ L2} C Kp. For that, it suffices to construct an operator K' ->- K' which is a homotopic inverse to the inclusion. This can be done by using a regularization process by flows of vector fields tending to 0 sufficiently quickly, near infinity. 12.B. Case of Hermitian and complete Kahler manifolds. The preceding results admit of course complex analogs, with almost identical proofs (the details will be therefore left to the reader). One says that a Hermitian or Kahler manifold {X, uj) is complete if the underlying Riemannian manifold is complete. 12.7. Proposition. Let (X, ui) be a complete Hermitian manifold and E a Hermitian holomorphic vector bundle over X. There is a canonical isomorphism HpL2q(M,E)~HpLq(M,E)sep between the space of L2 harmonic forms and the separated L2 Dolbeault cohomology group, this latter space being itself equal to H? (M, E) if the Dolbeault cohomology is finite dimensional. 12.8. Corollary. Let (X,ui) be a Kahler manifold and E a flat Hermitian bundle over X. a) Without further assumptions, there is, for any k, an orthogonal decomposition hUm,e)= 0 npJ(M,E), npLUM,E) = nqL!(M,E*). p+q=k b) If moreover (X, ui) is complete, there are canonical isomorphisms HkL2(M,E)sep~ 0 H?(M,E)sep, H?(M,E)sep ~ HqLf(M,E*)sep. p+q=k c) If (X, ui) is complete, and if the L2 de Rham and Dolbeault cohomology groups are finite dimensional, there are canonical isomorphisms HkL2(M,E)~ 0 Hp'2q(M,E), Hp<2q(M,E)~HqLf(M,E*). p+q=k 12.C. Hodge theory of weakly pseudoconvex Kahler manifolds. The weakly pseudoconvex Kahler manifolds furnish an important example of complete Kahler manifolds. J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 62 12.9. Definition. A complex manifold X is said to be weakly pseudoconvex if there exists a C°° psh exhaustion function ip on X. (Recall that a function ip is said to be exhaustive if for any c > 0 the level set Xc = V'~1(c) 1S relatively compact, i.e. ip(z) tends to +00 when z tends towards infinity, according to the stratification of the complements of compact parts of X.) In particular, the compact complete manifolds X are weakly pseudoconvex (take ip = 0), as well as the Stein manifolds. For example the affine algebraic subvarieties of C^ (take ip(z) = \z\2), the open balls X = B(zo,r) (take ip{z) = l/(r - \z - £o|2)), the open convex sets, and so on. A basic observation is the following: 12.10. Proposition. Any weakly pseudoconvex Kdhler manifold (X,ui) has a complete Kdhler metric ill. Proof. For any increasing convex function \ £ CO0(ffi,ffi), we will consider the closed (1,1)-form ujx = uj + i d'd"(x ° VO = w + x'(V0i d'd"ip + x"(i>)i d'ip A d"ip. Since the three terms are positive or zero, this is a Kahler metric. The presence of the third term implies that the norm of x"^)1 dip by comparison to ujx is less than or equal to 1, therefore if p is a choice of (x")1^2 we have \d(p o ip)\Wx < 1. According to (12.1 c), ujx will be complete as long as p o ip is exhaustive, that is, as long as lim+00 p(t) = +00. We therefore obtain the sufficient condition /?+00 / x"W1/2* = +oo, which is realized, for example, for the choice x(t) = t2 or x(t) = t ~ log*) t>l.D We have now established a Hodge decomposition theorem for weakly pseudoconvex Kahler manifolds having "sufficiently many strictly pseudoconvex directions". Following Andreotti-Grauert [AG62], we introduce the: (pp)Definition. A complex manifold X is said to be ^-convex (resp. absolutely ^-convex) if X has an exhaustion function (resp. a psh exhaustion function) ip, which is strongly ^-convex on the complement X\K of a compact part, i.e. such that i d'd"ip has at least n - £+1 positive eigenvalues at any point of X\K, where n = dime AT. (qq)Example. Let X be a smooth projective variety such that there exists a surjective morphism F : X ->- Y onto another smooth projective variety Y. Let D be a divisor of Y and let X = X\F~1(D), Y = Y\D. We assume that F induces a submersion X\F~1(D) ->- Y\D and that O(D) \r, is ample. Then X is absolutely £-convexfor I = dim X - dim Y +1. Indeed, the hypothesis of ampleness oiO{D)\D implies that there exists a Hermitian metric on O(D) for which the curvature is positive definite in a neighbourhood of D, that is on an open set of the form Y\K' where K' is a compact part of Y\D. Let a £ H0(Y,O(Dj) be the canonical section of the divisor D. Then - log|c|2 is strongly psh on Y\K', consequently ip = - log I(7 o F\2 is psh and strongly ^-convex on X\K, where K = F_1(i4''). In addition, ip clearly defines an exhaustion on X. Nothing is known of ip on K, but 63 12. HODGE THEORY OF COMPLETE KAHLER MANIFOLDS it is enough to truncate ip by taking a maximal regularized ipc = max£(^, C) with a constant C > sup^ ip to obtain an everywhere psh function ipc on X. We now can state the Hodge decomposition theorem for absolutely ^-convex manifolds. This result is due to T. Ohsawa [Ohs81, 87]; we present here a simplified description of a proof of it in [Dem90a]. A purely algebraic approach of these results was obtained by Bauer-Kosarew [BaKo89,91] and [Kos91]. 12.13. Theorem (Ohsawa [Ohs81,87], [OT88]). Let (X,u) be a Kahler manifold and n = dime X, and assume that X is absolutely l-convex. Then, in suitable degrees, there is a Hodge decomposition and symmetry: ff£R(X,Q ~ 0 Hp>q(X,C), Hp<9(X,C) ~Hq>p(X,C), k>n + £, p+q=k 4R,C(I, C) ~ 0 Hp'q(X, C), HF(XX) ~ H?(X, C), k < n - £, p+q=k all these groups being finite dimensional fff^ C(X, C) and Hp'q(X, C) denotes here the cohomology groups with compact support). Moreover, there is a Lefschetz isomorphism un-P-q A,:Hp Hn-q n + £ are located in K and their number is finite. This implies that the groups H^R(X, C) of degree k >n + £ are finite dimensional. The finiteness of the Dolbeault cohomology groups Hp'q(X,C) = Hq(X,Qpx) is a result of the theorem of Andreotti-Grauert [AG62] (all the cohomology groups of higher degree than £ with values in a given coherent sheaf are separated and finite dimensional if the manifold is £-convex). It is noted however, that the £-convexity, although sufficient to ensure the finiteness of the various groups involved, is not sufficient to guarantee the existence of a Hodge decomposition, nor even the Hodge symmetry. The reader will find a simple counterexample in Grauert-Riemenschneider [GR70]. Now let w be a Kahler metric on X and ip a strongly ^-convex psh exhaustion function on X\K. As one can see, the existence of a Hodge decomposition follows directly from the fact that one has such a decomposition for the L'2 harmonic forms. The key point resides in the observation that any L20C form of degree k >n + £ becomes globally L2 for a suitable choice of metric ux = uj + i d'd'^x0^)-The groups HJ^R(X,C) and Hp'q(X,C) could then be considered as the inductive limit of L2 cohomology groups. In the sequel, we will use notation such as J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 64 L^j (X,Ap'qTx), l-L? (X,C), to denote the spaces of L2-forms (resp. harmonic forms) relative to uix. Since uix is Kahler, one has (12.14) nkL2^(M,C)= 0 ft£t,x(M,Q, K'^x(^C)=^fWx(M,C), p+q=k with an isomorphism "H*2 (M, C) ~ if*2 (M, C)sep as long as ujx is complete. In the sequel, we always assume that cux is complete. It is enough, for example, to impose x"(t) > 1 on [0, +00[. 12.15. Lemma. Let u be a form of bidegree (p,q) with L20C coefficients on X. If V + Q > n + £, then u £ L2^ (X,Ap,qT^) as long as x grows sufficiently quickly near infinity. Proof. At a fixed point x £ X, there exists an orthogonal basis (d/dzi,... , d/dzn) of Tx,x for which uj(x) = i y, dzj A dzj, ojx(x) = i 2, ^j(x)dzj A dlj, l1, ll + cx'(ip), i 0 is the f-th eigenvalue of i d'd"ip{x) and c(x) > 0 on X\K. If we assume 'x! >1, then we can easily deduce \u\ixdvWx < {i + clX'W)n-p-10- + Cix,W + c2x"W) MldVu - (i + cx'W)q~t+1 n + t, this is smaller or equal to c3{x'M-1 + x"Mx'(1>)-2), 65 12. HODGE THEORY OF COMPLETE KAHLER MANIFOLDS and it is easy to show that this quantity can be made arbitrarily small towards infinity on X as x grows sufficiently quickly to infinity on M. ? Proof of the theorem (12.13), conclusion. A well-known result of the Andreotti-Grauert [AG62] guarantees that the natural topology of the cohomology groups Hq (X, J7) of any given coherent sheaf T on a ^-convex manifold is separated for q > £. If T = O(E) is the sheaf of sections of a holomorphic vector bundle, the groups Hq(X, O(E)) are algebraically and topologically isomorphic to the cohomology groups of the Dolbeault complex of forms of type (0, q) with L20C coefficients for which the ^''-differential has I? coefficients in terms of the Frechet topology defined by the semi-norms u i->- ||tt||z,2(ii-) + IK'MHz,2(if)- To see this, one can begin again word for word the proof of Theorem (1.3), by observing that the L2oc complex still furnishes a resolution of O(E) by the (acyclic) sheaves of C°°-modules. It follows from what proceeds this that the morphism Llx{X,k^T*x) D KerD'>x -»? H?(X,C) = Hq{X,Wx) is continuous and with closed kernel. Consequently this kernel contains the image Im D1' , and we obtain a factorization %? (X, C) ~ Ker D'^ /Irn-^ -+ H? (X, C). The proof of proposition (12.2) further shows that Im £>" coincides with the image of D"(V(X, XP^TX)) in Llx(X,SP>iTx). Consider the limit morphism (12.16) WmWjqx{X, C) -> Hp>q(X, C), x where the inductive limit is extended to the set of increasing C°° convex functions X, such that x"(t) > 1 on [0, +oo[, with the order relation Xi 1 X2 ^ Xi < X2 and L2^ (X, \?T*X) C L^ (X, A?TX) for k=p + q. It is easy to see that this order is filtered by again taking the arguments used for Lemma (12.15). Furthermore, it is well-known that the de Rham cohomology groups are always separated in the induced topology from the Frechet topology on the space of forms, consequently one has a limit morphism (12.16Dr) \imntx(X,C) ^ H^r(X,C) x analogous to (12.16). The decomposition formula of Theorem (12.13) follows now from (12.14), and from the following elementary lemma. 12.17. Lemma. The limit morphisms (12.16), (12.16)dr are bijective for k = p + q > n + £. Proof. Let us treat for example the case of the morphism (12.16), and let u be a L[20C n + £. Then there exists a choice of x f°r which u ¤ L^j , therefore u £ KerD" and (12.16) is surjective. If a class {u} £ "H^'9 (X, C) is sent to zero in Hp'q(X, C), one can write u = d"v for a certain form v with L20C coefficients and of bidegree (p, q - 1). In the case p + q > n + I, we will have v £ L'^ for x h Xo large enough, therefore the class of u = D" v in HP'q(X, C) is zero and (12.16) is injective. When p + q = n + t, the form v does J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 66 not necessarily belong anymore to one of the spaces L^ , but it suffices to show that u = d"v is in the image of Im £>" for \ large enough. Let 6 ¤ CO0(ffi, ffi) be a truncating function such that 6(t) = 1 for t < 1/2, 6(t) = 0 for t > 1 and \6'\ < 3. Then d"(0(eV>)u) = 0(e^)d"v + eO'(e^)d"i> A v. According to the proof of lemma (12.15), there exists a continuous function C(x) > 0 such that \v\lxdVUx < C(l +x"(V0/x'(V0)M*dK;, whereas \d"$\lx < l/x"W according to the same definition of ujx. We see therefore that the integral f le'ie^d'^AvlljV^ < f C(l/X"W + l/x'W)\v\2dV Jx J x is finite for \ large enough, and by dominated convergence d"(0(eip)v) converges to d"v = uin Llx(X,AP'«T%). D Poincare and Serre duality show that the spaces HfjRc(X, C) and HP'q(X,C) with compact support are dual to the spaces H^^k(X,C) and Hn~p'n~q(X, C) since the latter are separated and of finite dimension, which is very much the case if k=p + q'""« (X, C) ->- #""*>'"-« (X, C), which is surjective for x large enough according to Lemma (12.17) and the fmiteness of the group Hn~p'n~q(X,C). Therefore (12.20) is injective for \ large, and after composition with the Lefschetz isomorphism (12.19), we obtain an injection un-P-q A. = un-P-g A.. Hp,i(X,c) -»? h229jx~p(x,c)sep ~nz~q'n~p(x,c). (The equality ujn~p~q A - = w"~p~9 A - follows from the fact that ux has the same cohomology class as ui.) By taking the inductive limit on x and in combination with the limit isomorphism (12.16), we obtain an injective map (12.21) w"-p-«A. : HP'q(X,C) ^Hn-q-n-p(X,C), p + q«(X, C) or through H^'n~P(X, C), we deduce from this that the natural morphisms H?(X,C) ^Hp'q(X,C) are injective for p+ q < n - £ and surjective for p + q > n + i. Of course, there are entirely analogous properties for the de Rham cohomology groups. 13. Bochner techniques and vanishing theorems Let X be a complex manifold with a given Kahler metric ui = Y^ Wjkdzj A dzk-Let (E, h) be a Hermitian holomorphic vector bundle over X. We denote by D = D' + D" the Chern connection and @(E) the associated curvature tensor. 13.1. Basic commutivity relations. Let L be the operator Lu = ui A u acting on the vector valued forms, and let A = L* be its adjoint. Then [D"*,L]=id!, [D'*,L] = -id", [A,D"] = -id'*, [A,D']=id"*. Proof (outline). This is a simple consequence of the commutivity relation (6.14) already shown for the trivial connection d = dl + d" on E = X x C. Indeed, for any point xq £ X, there exists a local holomorphic frame (eA)i e\ with s\ £ C°°(X, AP'«TJ), we obtain D"s = J2 d"s\ ® eA + O(M), D"*s = Y, d"*sx ® eA + 0(\z\). The stated relations follow easily. ? 13.2. The Bochner-Kodaira-Nakano identity. If (X, ui) is a Kahler man ifold, the complex Laplacians A' and A" acting on the forms with values in E satisfy the identity A" = A' + [i0(£O,A]. Proof. The latter equality (13.1) gives D"* = -i[A,D'], therefore A" = [£>",£>"*] = -i[£>", [A, £>']]. The Jacobi identity implies [D",[A,D']] = [A, [£>',£>"]] + [D',[D",A]] = [A, ©(£?)] + i[D', £>'*], which is based on the fact that [D',D"] = D2 = @(E). The stated identity fol lows. ? Assume that X is compact and let u £ C°°(X, Ap'qT^ ® E) be an arbitrary (p, (/)-form. Integration by parts gives (A'u,u) = \\D'u\\2 + \\D'*u\\2 >0, J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 68 and one has an analogous equality for A". From the Bochner-Kodaira-Nakano identity, one deduces a priori the inequality (13.3) \\D"u\\2 + \\D"*u\\2> [ ([i®(E),A]u,u)dVL0. Jx This inequality is the well-known Bochner-Kodaira-Nakano inequality (see [Boc48], [Kod53], [Nak55]). When u is A"-harmonic, we obtain f ([i ®(E),A]u,u)dV < 0. Jx If the Hermitian operator [i Q(E),A] is positive on each fiber of Ap'qT^- q(X,E) =np'9(X,E) = 0 according to Hodge theory. In this approach, the essential point is to know how to calculate the curvature form ®(E) and to find sufficient conditions for which the operator [i ®(E),A] is positive definite. Some elementary (albeit somewhat agonizing) calculations yields the following formula: If the curvature of E is written in the form (11.16) and if u = 22 uJ,K,\dzi A dzj 1. ? Another approachable case is the case where E is a line bundle (r = 1). Indeed, at each point x £ X, we can then choose a coordinate system, which simultaneously diagonalizes the Hermitian forms ui(x) and Q(E)(x), in such a way that u>(x) = i 2_, dzj A dzj, Q(E)(x) = i \_] Ijdzj A dlj l= £ (£7i + £7i_ J2 7iWl2 |J|=p,|if|=« S'GJ j£if l (71 H h 7g - 7n-p+l 7n)l"|2- Assume that i 0(S) is positive. It is then natural to provide X with the particular Kahler metric ui = i @(E). Then 7^ = 1 for j = 1,2,... , n and we obtain <[i Q(E), A}u,u) = (p + q-n)\u\2. As a consequence: 13.7. Kodaira-Akizuki-Nakano Vanishing Theorem ([AN54]). If E is a positive line bundle over a compact complex manifold X, then Hp>q(X,E)=Hq(X,npx(g>E)=0 ioip + q>n + l. D More generally, if E is a positive vector bundle in the sense of Griffiths (or ample), of rank r > 1, Le Potier [LP75] has proven that Hp'q(X,E) = 0 for p + q > n + r. The proof is not a direct consequence of the Bochner technique. A simple enough proof has been obtained by M. Schneider [Sch74], by utilizing the Leray spectral sequence associated to the projection on X of the projective bundle T(E) -> X. 13.8. Exercise. It is significant for various applications to formulate vanish ing theorems which are also valid in the case of semi-positive line bundles. There is, for example, the following result due to J. Girbau [Gir76] : Let (X,uj) be a compact Kahler manifold, assume that E is a line bundle and that i @(E) > 0 has at least n - k positive eigenvalues at each point, for a certain integer k > 0. Then HP'«(X, E) = 0 for p + q > n + k + 1. Indication. Use the Kahler metric cue = i Q(E) + ecu with small e > 0. A more natural and powerful version of this result has been obtained by A. Sommese [Som78, ShSo85] : Following these authors, we say that E is fc-ample if a certain multiple mE is such that the canonical map $|m£| : X\B]mE] -> P- J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 70 has all its fibers of dimension < k and dimi?|mB| < k. If X is projective and if E is fc-ample, then Hp'q(X, E) = 0 for p + q > n + k + 1. Indication. Prove the dual result, that Hp>q{X,E~r) = 0 torp+q nx <2) e-1 ® o(-Y) -^npx® e-1 ->- (npx ® e-1) ry ->- o, 0 -> f)y_1 ® E~A ->- (f^ <2) S"1) ry -> rjy ® e~a -> o. ? 14. L2 estimations and existence theorems The starting point is the following L2 existence theorem, which is essentially due to Hormander [H6r65, 66], and Andreotti-Vesentini [AV65]. We only sketch the principal ideas, while referring for example to [Dem82] for a detailed exposition of the techniques considered in the situation here. 14.1. Theorem. Let (X,ui) be a complete Kdhler manifold, and let E be a Hermitian vector bundle of rank r on X, such that the curvature operator A = Ae9^ = [i Q(E),AU] is semi-positive on all the fibers of Kp,qTx ® E, q > 1. Let g £ L2(X, Ap'qTx (g> E) be a form satisfying D"g = 0 and / (A-1g,g)dVul < +oo. (At the points where A is not positive definite, we assume as a precondition that A~lg exists almost everywhere. We then choose the preconditional term A~xg of minimal norm, orthogonal to Ker A.) Then there exists f £ L2(X,Xp'q-1Tx ® E) such that D"f = g and f |/|2dVw < f (A-1g,g)dVbJ. Jx Jx Proof. Let u e L'2(X,K^qTx®E) be a form such that D"u £ L2 and D"*u ¤ L2 in the sense of distributions. Lemma (12.2 a) shows (under the indispensable hypothesis that uj is complete) that u is the limit of a sequence of C°° forms uv with compact support in such a way that uv -> u, D"uv -> D"u and D"*uv -> D"*u in L2. It follows that a priori the inequality (13.3) extends to arbitrary forms u such that u, D"u, D"*u £ L2. Now, since KerD" is weakly (and therefore strongly) closed, we obtain an orthogonal decomposition of the Hilbert space L2(X, Ap'qTx ® E), namely L2(X,Ap'qTx(g>E) =KerD"©(KerD")±. Let v = v\ + V2 be the corresponding decomposition of a C°° form v £ Vp'q(X,E) with compact support (in general, v±, «2 do not have compact support!). Since (KerD")-1- = Im D"* C Ker D"* by duality and g, v\ £ Ker_D" by hypothesis, we obtain D"*V2 = 0 and \(g,v)\2 = |(ff,Vl)|2 < f (A-1g,g)dVL0 f (Avuv1)dVL0 Jx Jx 71 14. L2 ESTIMATIONS AND EXISTENCE THEOREMS by applying the Cauchy-Schwartz inequality. The inequality (13.3) a priori, applied to u = v\ gives / (Av^v^dV^ < ||£>"ui||2 + \\D"*Vl\\2 = ||^"*i-i||2 = \\D"*v\\2. Jx Combining these two inequalities we find that \(g,v)\2<^Jx(A-1g,g)dV0?j\\D"*v\\2 for any C°° (p, g)-form v with compact support. This shows that there is a well-defined linear form w = D"*v ^ {v,g), L^X^k^-^^E) D D"*(Vp>q{E)) -»? C on the image of D"*. This linear form is continuous in the L2 norm, and its norm is < C with C= U(A-1g,g)dV0J According to the Hahn-Banach Theorem, there exists an element /£ L2(X,Ap'q-1TZ®E) such that ll/H < C and (v,g) = (D"*v,f) for any v, consequently D"f = g in the sense of distributions. The inequality ||/|| < C is equivalent to the latter estimation in the theorem. ? The preceding L2 existence theorem can be applied in the general context of weakly pseudoconvex Kahler manifolds (see definition (12.9)), and the same if the Kahler metric considered ui is not complete. Indeed, according to Proposition (12.10), we arrive at complete Kahler metrics by setting uje = uj + ei d'd'V2 = uj + 2e(2i ij;d'd"ip + i d'tp A d"ip) with a C°° psh exhaustion function ip > 0. As a consequence, the L2 existence theorem (14.1) applies to each Kahler metric u¤. Indeed one can show (the calculations being left to the reader!) that the quantities Iffl^dK, and ((AgquJ)~1g,g)wdVw are decreasing functions of ui when p = n = dime X. For a D"-closed form g of bidegree (n, q), we therefore obtains solutions ft of the equation D"ft = g satisfying / \MledVUe < [ ((A^lX'^dhJV^ < f ((AtfJ-ig^^dVu. Jx Jx Jx These solutions ff can be uniformly bounded in the L2 norm on any compact set. Thus we can extract a weakly convergent subsequence in L2. The limit / is a solution of D"f = g and satisfies the required L2 estimation relative to the metric ui initially given (which, to repeat, is not necessarily complete). A particularly important case is the following: J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 72 14.2. Theorem. Let (X,ui) be a Kdhler manifold, dimX = n. Assume that X is weakly pseudoconvex. Let E be a Hermitian line bundle and let 7i(x) < --- < 7"(x) be the eigenvalues of curvature (i.e. the eigenvalues of i Q(E) with respect to the metric ui) at any point x. Assume that the curvature is semi-positive, i.e. 71 > 0 everywhere. Then for any form g £ L2(X,Xn'qT^[ ® E) satisfying D"g = 0 and / (71 + - - - + 7g)-1|5f dK, <+00, Jx (one assumes therefore g(x) = 0 almost everywhere at all points where 71 (x) + - - - + lq(x) = 0), there exists f E) such that D"f = g and 2 / |/|2dVw < / (7l + - - - + ^y1 \g\2dV^. Jx Jx Proof. Indeed, for p = n, formula (13.6) shows that (Au,u) > (7i + --- + 7g)M2, therefore (A~1u,u) > (71 + - - - An important observation is that the above theorem still applies when the Hermitian metric of E is a singular metric with positive curvature in the sense of currents. Indeed, by a process of regularization (convolution of psh functions by regular kernels), the metric can made C°° and the solutions obtained by means of Theorems (14.1) or (14.2), since the regular metrics have limits satisfying the desired estimates. In particular, we obtain the following corollary. 14.3. Corollary. Let (X,ui) be a Kdhler manifold, dim AT = n. Assume that X is weakly pseudoconvex. Let E be a holomorphic bundle provided with a singular metric for which the local weight is denoted by tp £ L\oc. Assume that i@(E)= 2i d'd"ip > euj for a certain e > 0. Then for any form g £ L2(X, An,qT^ ® E) satisfying D"g = 0, there exists f ¤ L2(X, AP'""1^ E) such that D" f = g and [ |/|2e-2^Ww <- f \g\2e-2^dVw. ? Jx Qe Jx We denoted here somewhat incorrectly the metric in the form |/|2 e~2v, as if the weight ip were globally defined on X (certainly, this is not possible if E is globally trivial). By abuse of notation, we will nevertheless use this same notation because it clearly underlines the dependence of the L2 norm on the psh function associated to the weight. 15. Vanishing theorems of Nadel and Kawamata-Viehweg We begin by introducing the concept of multiplier ideal sheaves, following A. Nadel [Nad89]. The principal idea in fact goes back to the fundamental work of E. Bombieri [Bom70] and H. Skoda [Sko72]. 73 15. VANISHING THEOREMS OF NADEL AND KAWAMATA-VIEHWEG 15.1. Definition. Let

n + s for a certain integer s > 0, then e~2v > C\z - x\~2n~2s in a neighbourhood of x and J((p)x C tUq^1, where mn^x denotes the maximal ideal ofOQ,x. Proof. The proof rests on some classical estimations of complex potential theory, see H. Skoda [Sko72]. ? 15.4. Proposition ([Nad89]). For any psh function ip on SI C X, the sheaf J(ip) is a coherent sheaf of ideals on ft. Proof. Since the result is local we can assume that fl is the unit ball in C?. Let 7iv(fl) be the set of the holomorphic functions / on fl such that fn \f\'2e~2ipdX < +00. According to the strong Noetherian property of coherent sheaves, the set 7iv(fl) generates a coherent sheaf of ideals 0 C On- It is clear that 3 C J{- X be a modification of non-singular com plex varieties (i.e. a proper holomorphic map that is generically 1 : 1), and let tp be a psh function on X. Then H. {0(KX>) ® J(

X\S is a biholomorphism. By definition of multiplier ideal sheaves, 0{Kx) ® J(^p) is identified with the sheaf of holomorphic n-forms / on some open set U C X, satisfying i? /A/e-2*5 ¤ L\0C{U). Since ip is locally bounded above, we can likewise consider the forms / which a priori are defined only on U\S, because / is in L20C(U), and thus automatically extends through S. The change of variables formula gives / i"V A /e"2^ = / i"> / A IFfe-2^, Ju J^-Hu) therefore / ) is reduced to a purely algebraic problem. The first observation is that J((p) is easily calculated if ip = J2 aj l°g \dj\ where Dj = 571(0) are smooth irreducible divisors with normal crossings. Then J{dv<+?. j u Since the gj can be taken as coordinate functions in suitable local coordinate systems (zi,... , zn), the integrability condition is that h is divisible by Yi 9j ', where m,j - aj > -1 for each j, i.e. m,j > [ctj\ (where |_ J denotes the integral part). Consequently j{- X of X such that /x*-3 is an invertible sheaf 0(-D) associated to a divisor with normal crossings D = J2 ^jDj, where (Dj) are the components of the exceptional divisor of X. (Consider the blow-up X' of X along the ideal 0, so that the inverse image of 3 on X' becomes an invertible sheaf O(-D'), then blow-up X' again so as to render X' smooth and D' with normal crossings, by invoking Hironaka [Hi64].) We then have Kx = 75 15. VANISHING THEOREMS OF NADEL AND KAWAMATA-VIEHWEG fj,*Kx + R where R = "^ZpjDj is the divisor of zeros of the jacobian JM of the blow-up map. From the direct image formula 15.5, we deduce J( 0. Indication. By using Parseval's formula and polar coordinates Zj = rje1®', show that the problem is equivalent to determining for which p-tuples (/3i,... , fip) ¤ W the integral f r21^---r2/*r1dr1---rpdrp = f t[Pl+1)/ai - - .t{f*+1)/a* dh dtp J[o,i]v r2ai +??? + r2pap J[o,i]p h+--- + tp h tp is convergent. Deduce from this that J( 1- (This exercise shows that the analytic definition of J((f) is also sometimes very convenient for calculations). ? Let E be a line bundle over X with a given singular metric h with curvature current Qh(E). If

ecu. Then Hq [X, 0{KX +E)® J{h)) = 0 for all q > 1. Proof. Let Cq be the sheaf of germs of (n,g)-forms u with values in E and with measurable coefficients, for which |u|2e-2^ and \d"u\2e~'2,p are simultaneously locally integrable. The operator d" defines a complex of sheaves (£',d") which is a resolution of the sheaf 0(Kx + E) J(f): Indeed, the kernel of d" in degree 0 consists of the germs of holomorphic n-forms with values in E which satisfy the integrability condition. Therefore the coefficient function belongs to J(f), and the exactness at degree q > 1 arises from Corollary 14.3 applied to arbitrary small balls. Since each sheaf Cq is a C^-module, C is a resolution by acyclic sheaves. Let ip J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 76 be a C°° psh exhaustion function on X. We apply Corollary 14.3 globally on X, with the initial metric of E multiplied by the factor e~x°^, where \ is an increasing convex function of arbitrary growth at infinity. This factor can be used to ensure convergence of integrals at infinity. From Corollary 14.3, we then deduce that Hq(T(X, £*)) = 0 for q > 1. The theorem follows by virtue of the de Rham-Weil Isomorphism Theorem (1.2). ? 15.9. Corollary. Let (X,ui), E and

- J(ip) ->- Ox ->- Ox/J(f) ->- 0, twisted by 0(KX + E), and apply Theorem 15.8 to obtain the vanishing of the first group H1. The stated surjective property follows. ? 15.10. Corollary. Let (X,ui), E and

n + s at a given point x £ X for which v( 0 depending only on L and N such that for any s £ N, the group H°(X, G(mL)) gen erates the jets of order s at any point Xj, for m > as + b. Indication. Apply Corollary 15.9 to E = -Kx +mL, with a singular metric on L of the form h = hoe~e^, where ho is C°° with positive curvature, e > 0 small, and ip{z) ~ log 12; - Xj\ in a neighbourhood of Xj. Deduce from this the Kodaira embedding theorem: 15.12. Kodaira Embedding Theorem. If L is a line bundle on a compact complex manifold, then L is ample if and only if L is positive. ? An equivalent way to state the Kodaira embedding theorem is the following: 15.13. Kodaira criterion for projectivity. A compact complex manifold X is projective algebraic if and only if X contains a Hodge metric. That is, a Kahler metric with integral cohomology class. 77 15. VANISHING THEOREMS OF NADEL AND KAWAMATA-VIEHWEG Proof. If X C F^ is projective algebraic, then the restriction of the Fubini-Study metric to X is a Hodge metric. Conversely, if X has a Hodge metric uj, the cohomology class representative {uj} in H2(X, Z) defines a complex topological (i.e. C°°) line bundle, say L. Since uj is of type (1,1), the exponential exact sequence (8.20) H^X^Ox) -> ff2(AT,Z)-> H2(X,0) = H°>2(X,C) shows that the line bundle L can be represented by a cocycle in i71(X, O^). In other words, L is endowed with a complex structure. Moreover, there exists a Hermitian metric h on L such that ^-©^(L) = w. Consequently, L is ample and X is projective algebraic. 15.14. Exercise (Riemann conditions characterizing Abelian varieties). A complex torus X = Cn /T is called an Abelian variety if X is projective algebraic. Show by using (15.13) that a torus X is an Abelian variety if and only if there exists a positive definite Hermitian form H on C? such that Im #(71,72) G Z for all 71,72 in the lattice T. Indication. Use a process of averaging to reduce the proof to the case of Kahler metric invariant by translations. Observe that the real torus Z71 + Z72 defines a system of generators of the homology group H2(X, Z) and that Jz +z w = ^(71,72). 15.15. Exercise (solution of the Levi problem). Show that the following two properties are equivalent. (rr)X is strongly pseudoconvex, i.e. X admits a strongly psh exhaustion function. (ss)AT is a Stein, i.e. the global holomorphic functions separate points, furnishing a system of local coordinates at every point, and X is holomorphically convex. (By definition, this means that for any discrete sequence (zv) in X, there exists a function / £ H°(X, Ox) such that \f{zv)\ ->- 00.) ? 15.16. Remark. As long as one is interested only in the case of forms of bidegree (n, q),n = dimX, the L2 estimates extend to the complex spaces acquiring arbitrary singularities. Indeed, if X is a complex space and tp a psh weight function on X, one can still define a sheaf Kx(ip) on X, such that the sections of Kx((p) on an open set U are the holomorphic n-forms / on the regular part U fl Xleg, satisfying the integrability condition i? / A fe~2v £ L\oc(U). In this context, the functorial property 15.5 can be written (or is written) H*(Kx<((pon)) = Kx(f), and it is valid for arbitrary complex spaces X, X', jjl : X' -> X being a modification. If X is non-singular, one has Kx(f) = 0{KX) ® J{y>), however, if X is singular, the symbols Kx and J(f) do not have to be dissociated. The statement of the Nadel vanishing theorem becomes Hq(X, O(E) ® Kx(ip)) = 0 for q > 1, under the same hypothesis (X Kahler and weakly pseudoconvex, curvature of E > eui). The proof is obtained by restricting all the situations to Xleg. Although in general Xreg is not weakly pseudoconvex (a necessary condition being codimXs;ng = 1), Xreg is always Kahlerian complete (the complement of an analytic subset in a weakly pseudoconvex Kahler space is Kahlerian complete, see for example [Dem82]). As a consequence, the Nadel vanishing theorem is essentially insensitive to the presence of singularities. ? J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 78 We now deduce an algebraic version of the Nadel vanishing theorem obtained independently by Kawamata [Kaw82] and Viehweg [Vie82]. (The original proof relies on a different method using cyclic coverings to reduce to the case situation of the ordinary Kodaira Theorem.) Before stating the theorem, we need a definition. 15.17. Definition. A line bundle L on a compact complex manifold is called large if its Kodaira dimension is equal to n = dim AT, that is, if there exists a constant c > 0 such that dimH0(X,O(kL)) >ckn, k > k0. 15.18. Definition. A line bundle L on a projective algebraic manifold is called numerically effective (nef for short) if L satisfies one of the following three equivalent properties: (tt)For any irreducible algebraic curve C C X, one has L ? C = fc c\(L) > 0. (uu)If A is an ample line bundle, then kL + A is ample for all k > 0. (vv)For any e > 0, there exists a C°° Hermitian metric ht on L such that 0^s(L) > -euj, where w is a fixed Hermitian metric on X. The equivalence of properties 15.18 a) and b) is well-known and we will omit it here (see for example Hartshorne [Har70] for the proof). It is clear in addition that 15.18 c) implies 15.18 a), while 15.18 b) implies 15.18 c). Indeed if oj = ^Q{A) is the curvature of a metric of A with positive curvature, and if hk is a metric on L inducing a metric with positive curvature on kL + A, it becomes k-^-Q(L) + ^Q(A) > 0, where ^Q(L) > -\u. Now, if D = J2aJDJ > 0 is an effective Q-divisor, we define the multiplier ideal sheaf J{D) to be the sheaf J((f) associated to the psh function tp = ^Q!jlog|<7j| defined by the generators gj of O(-Dj). According to remark 15.6, the calculation of J{D) can be done algebraically by making use of desingularizations [i : X -> X such that fj,*D becomes a divisor with normal crossings on X. 15.19. Kawamata-Viehweg Vanishing Theorem. Let X be a projective algebraic manifold, and let F be a line bundle on X such that a multiple mF of F can be written in the form mF = L + D, where L is a nef and large line bundle, and D an effective divisor. Then Hq(X,O(Kx + F)(g>J(m-1D))=0 for q > 1. 15.20. Corollary. If F is nef and large, then H«(X,0(Kx + F)) = 0 for q>l. Proof. Let A be a non-singular very ample divisor. There is an exact sequence 0 -»? H°(X, 0(kL - A)) -»? H°(X, O(kL)) -> H°(A, O(kL) ]A), and dim H0{A,O(kL)\A) < Cfcn_1 for a certain constant C > 0. Since L is large, there exists an integer fc0 ^> 0 such that 0{k§L - A) has a non-trivial section. If E is the divisor of this section, we have 0(koL - A) ~ O(E), therefore O(koL) ~ 0(A + E). Now, for k > k0, we arrive at O(kL) = O((k-k0)L + A + E). According to 15.18 b), the line bundle 0((k - ko)L + A) is ample, therefore it comes with a C°° Hermitian metric hk = e~'fik, and with positive definite curvature form ujk = j-@((k - ko)L + A). Let 1. ? 16. On the conjecture of Fujita Given an ample line bundle L, a fundamental question is of determining an effective integer mo such that mL is very ample for m > mo. The example where X is a hyperelliptic curve of genus g and where L = G(p) is associated to one of the 2g+2 Weierstrass points, shows that mo must be at least equal to 2g+l (additionally it is checked rather easily that mo = 2g+1 always answers the question for a curve). It follows from this that mo must necessarily depend on the geometry of X, and cannot depend only on the dimension of X. However, when mL is replaced by the "adjoint" line bundle Kx + mL, a simple universal answer seems likely to emerge. 16.1. Fujita's conjecture ([Fuj87]). If L is an ample line bundle on a projective manifold of dimension n, then i) Kx + (n + 1)L is generated by its global sections; ii) Kx + (n + 2)L is very ample. The bounds predicted by the conjecture are optimal for (X,L) = (F",C(1)), since in this case Kx = 0(-n - 1). The conjecture is easy to verify in the case of curves (exercise!), and I. Reider [Rei88] has solved the conjecture in the affirmative in the case n = 2. Ein-Lazarsfeld [EL93] and Fujita [Fuj93] arrived at establishing part i) in dimension 3, and a very thorough refinement of their technique allowed Kawamata [Kaw95] to also arrive at the case of dimension 41. The other cases of the conjecture, namely i) for n > 5 and ii) for n > 3, remain for the time being unsolved. The first step in the direction of this conjecture for arbitrary dimension n has been realized in 1991 (work published 2 years later in [Dem93]), by means of an analytic method relying on a resolution of a Monge-Ampere equation. Similar results were obtained by Kollar [Kol92] employing entirely algebraic 1The technique of Fujita [Fuj93] and Kawamata [Kaw95] has just been simplified considerably and clarified by S. Helmke [Hel96]. J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 80 methods. We refer to [Laz93] for an excellent article devoted to the synthesis of these developments, as well as [Dem94] for the analytic version of the theory. This section is devoted to the proof of some results dependent on Kujita's conjecture in arbitrary dimension. The principal ideas of interest here are inspired by some recent work of Y.T. Siu [Siu96]. Siu's method, which is naturally algebraic and relatively elementary, consists of combining the Riemann-Roch formula with the Kawamata-Viehmeg vanishing theorem (however, it will be much more convenient to use this Nadel's formulation of the theorem, using the multiplier ideal sheaves). Subsequently, X will denote a projective algebraic manifold of dimension n. The first useful observation is the following classical consequence of the Riemann-Roch formula: 16.2. Particular case of the Riemann-Roch formula. Let 3 C Ox be a coherent sheaf of ideals on X such that the variety of zeros V(3) is of dimension d (with possibly some components of lower dimension). Let Y = J2 ^j^j be the effective algebraic cycle of dimension d associated to the components of dimension d of y(0) (the multiplicities Xj taking into account the multiplicity of the length of the ideal 3 along each component). Then, for any line bundle E, the Euler characteristic x(X, 0(E + niL) Ox/0{Z)) is a polynomial P(m) of degree d and with leading coefficient Ld ? Y/dl ? The second useful fact is an elementary lemma concerning the numerical polynomials (polynomials with rational coefficients, defining a map of Z into Z). 16.3. Lemma. Let P(m) be a numerical polynomial of degree d > 0 and with leading coefficient aa/d\, ad £ Z, ad > 0. We assume that P(m) > 0 for all m > rriQ. Then (ww)For all N > 0, there exists m £ [mo, mo + Nd] such that P(m) > N. (xx)For all k £ N, there exists m £ [mo, mo + kd] such that P(m) > a(ikd/2d~1. (yy)For all N > 2d2, there exists m £ [mo, mo + N] such that P(m) > N. PROOF, a) Each one of the N equations P(m) = 0, P(m) = 1,... , P(m) = N - l has at most d roots, therefore there is necessarily an integer m ¤ [mo, nio+dN] which is not a root of these equations. b) By virtue of Newton's formula for the iterated differences AP(m) = P(m + 1) - P(m), we obtain AdP(m) = J2 (-l)j(d.)p(m + d-j) = ad, Vm G Z. lad, whereby we obtain the existence of an integer m £ [mo, mo + d] with P(m) > ac{/2d~1. The result is therefore proven for k = 1. In the general case, we apply this particular result to the polynomial Q(m) = P(km - (k - l)mo), for which the leading coefficient is a,dkd/d\ 81 16. ON THE CONJECTURE OF FUJITA c) If d = 1, part a) already gives the result. If d = 2, a glance at the parabola shows that f a2N2/8 if N is even, max P(m) > { , " m¤[m0,m0+N] ~ { a2(N2 - l)/8 if N is odd; therefore maxmG[m0imo+Ar] P(m) > N whenever N > 8. If 3, we apply b) with k equal to the smallest integer satisfying kd/2d~1 > N, i.e. k = |~2(7V/2)1/d], where \x\ £ Z denotes the greater integer. Then kd< (2(N/2)1/d + l)d 2d2, as one sees after a short calculation. ? We now apply the Nadel vanishing theorem in an analogous way to that of Siu [Siu96], with some simplifications in the technique and some improvements for the bounds. Their method simultaneously gives a simple proof of a fundamental classical result due to Fujita. 16.4. Theorem (Fujita). If L is an ample line bundle on a projective manifold X of dimension n, then Kx + (n + 1)L is nef. Using the theory of Mori and the "base point free theorem" ([Mor82], [Kaw84]), one can show in fact that Kx + (n + 1)L is semi-ample, and that there exists a positive integer m such that m(Kx + (n + 1)L) is generated by its sections (see [Kaw85] and [Fuj87]). The proof is based on the observation that n + 1 is the maximum length of the extremal rays of smooth projective varieties of dimension n. Their proof of (16.4) is different and was obtained at the same time as the proof of th. (16.5) below. 16.5. Theorem. Let L be an ample line bundle and let G be a nef line bundle over a projective manifold X of dimension n. Then the following properties hold. a) 2Kx + mL + G simultaneously generates the jets of order s±,... , sp £ N at arbitrary points x±,... ,xp £ X, i.e., there exists a surjective map H0(X,O(2Kx+mL + G))^ 0 0(2KX + mL + G) ® Ox,Xj/m^x], i2 + E1 2 + ( "^ ). b) 2Kx + (n + 1)L + G simultaneously generates the jets of order s\,... , sp at arbitrary points x±,... ,xp £ X so long as the intersection numbers Ld ? Y of L on all the algebraic subsets Y of X of dimension d are such that rd Y^ ^ V f3n + 2Si-l\ Ln/dJ'^A » / Proof. The proofs of (16.4) and (16.5a, b) are completely parallel, that is why we will present them simultaneously (in the case of (16.4), it is simply agreed that {xi,... ,xp} = 0). The idea is to find an integer (or a rational number) mo and a singular Hermitian metric ho on Kx + m^L for which the curvature current is strictly positive, 0/jo > euj, such that V(J(ho)) is of dimension 0 and such that the weight ipo of ho satisfies v( n + Sj for all j. Since L and G are J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 82 nefs, 15.18 c) implies that (m - rrio)L + G has for all m > mo a metric b! for which the curvature 0/j< has an arbitrarily small negative part, say 0/^ > - |w. Then O/j0 + 0^- > |w is positive definite. An application of Cor. 15.9 to F = Kx + mL + G = (Kx + m^L) + ((m - mo)L + G) with metric ho <8> h' guarantees the existence of sections of Kx + F = 2Kx + mL + G producing the desired jets for m > mo. Fix an embedding $|ml| : X -> FN, /i^> 0, given by the sections Ao,... , Xn G H0(X,/j,L), and let h^ be the associated metric on L, with positive definite curvature form ui = &(L). To obtain the desired metric ho on Kx + moL, one fixes an integer a £ W and one uses a process of double induction to construct singular metrics {hk,v)v>\ on aKx + bkL, for a decreasing sequence of positive integers b\ > &2 > - - - > bk > - ? - ? Such a sequence is necessarily stationary and mo will be precisely the stationary limit mo = limbk/a. The metrics hk,v are chosen to be the type that satisfy the following properties: o) hu,v is an "algebraic" metric of the form \\^\\hk,v | (a+l)/i/ aii _ x(a+l)6*-amis|2u/(a+i)M' \2^1- 7fc(£) is an arbitrary local trivialization of aKx + bkL. Observe that (a + l)(n + Sj) for all i,j; l) J(hk,v+i) 3 J(hk,v) and J(hk,v+i) ^ J(hk,v) as long as the variety of zeros V{J{hk,v)) is positive dimensional. The weighty = j^logZ\Tia+1>( 1, therefore the difference ipk,v - 2(a+i)u 1°§S lr(Aj')|2 1S &lso plurisubharmonic. Consequently ~i^®hk^{a-Kx + bkL) = ±d'd"ipk,v > T^hrw. Moreover, condition /?) clearly implies that v{tpk,v,Xj) > a(n + Sj). Finally, condition 7) combined with the strong Noetherian property of coherent sheaves guarantees that the sequence (hk,i>)i>>i will eventually produce a subscheme V(J(hk,v)) of dimension 0. One can check that the sequence (hk,v)v>\ terminates at this point, and we set hk = hk,v to be the final metric thus reached, such that dimV(J(hk)) = 0. For k = 1, it is clear that the desired metrics (/ii,")">i exist if b\ is chosen large enough. (For example, such that (a + \)Kx + (&i - 1)£ generates the jets of order (a + l)(n + max Sj) at every point. Then the sections <7i,... ,ov can be chosen such that mi = ? ? ? = mv = b\ - 1.) We assume that the metrics {hk,v)v>i and hk are already constructed, and proceed with the construction of (hk+i,v)v>i-We use again induction on 1/, and assume that hk+\,v is already constructed and that dimV(J(hk+i,v)) > 0. We begin our induction with v = 0, and let us declare in this case that J(hk+i,o) = 0 (this corresponds to an infinite metric of weight identically equal to -00). By virtue of the Nadel vanishing theorem applied to Fm = aKx +mL= (aKx + bkL) + (m - bk)L for the metric hk ® {hL)®m-bk, we 83 16. ON THE CONJECTURE OF FUJITA obtain Hq(X,0((a + 1)KX + mL) ® J(hk)) =0 for q > l,m>bk. Since V(J(hkj) is of dimension 0, the sheaf Ox/J{hk) is a skyscraper sheaf and the exact sequence 0 -> J(hk) ->- Ox ->- Ox I' J{hk) -> 0 twisted by the invertible sheaf 0((a + l)Kx + mL) shows that Hq(X, 0((a + \)KX + mL)) = 0 for q > 1, m > bk. Analogously, we find H"(X, 0((a + \)KX + mL) ® J(hk+hv)) =0 for q > 1, m > bk+1 (it is therefore true for v = 0, since J{hk+i^) = 0), and when m > max(&fc,&fc+i) = &<., the exact sequence 0 -> J(hk+i^) ->- Ox ->- Ox/J{hk+i,v) ->- 0 implies ff«(X, C((a + 1)2^ + mL) Ox/J(hk+1,v)) =0 for g > 1, m > 6*. In particular, since the group H1 above is zero, any section u' of (a + l)Kx + mL on the sub-scheme V(J(hk+i^j) has an extension u to X. Fix a basis u'ly... ,u'N of sections of this sheaf on V{J{hk+-\_,")) and take arbitrary extensions u\,... , un to X. Consider the linear map allotting to each section aonl the collection of jets of order (a + l)(n + Sj) - 1 at the points Xj, i.e. «= e oi«ii->©4rl)(B+'i)_l(u)- l 0, m > bk. The polynomial P is of degree d = dimV(J(hk+i^j) > 0. We therefore obtain the existence of an integer m £ [bk, bk + rj\ such that N = P(m) > S + 1, for some explicit integer tjER (For example, rj = n(5 + 1) is always appropriate according to (16.3 a), but it will be equally important to use the other possibilities to optimize the choices.) We then find a section av+i G H°(X, (a + l)Kx + mL) having a non-trivial restriction a'v+1 to V(J(hk+\^)), vanishing to order > (a + l)(n + Sj) at each point Xj. Now set mv+i = m, and the condition m"+i < s^bkjr\ is realized if bk + j] < ^^Hfc-i-i. This shows that one can choose recursively fc+i a ,i \ -r(h+ri) 1. J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 84 By definition, hk+i,v < hk+i,v, therefore J{hk+i,v+i) 3 J{hk+i,v)- It is the case that J(hk+i,v+i) ¥" J{hk+i,v), because J(hk+i,v+i) contains the sheaf of ideals associated to the divisor of zeros of cr"+i, whereas cr"+i is not identically zero on V(J(hk+i,v))- Now, an easy calculation shows that the iterated sequence bk+i = L^+r(^fc + *7)J + 1 stabilizes to the limit value bk = a(rj + 1) + 1, for any initial value b\ greater than this limit. In this way, we obtain a metric /loo with positive definite curvature on aKx + (a(rj + 1) + 1)L, such that dim V(J(hoo)) = 0 and v(tfioo,Xj) > a(n + sj) at each point Xj. Proof of (16.4). In this case, the set {xj} is taken to be the empty set, therefore 6 = 0. By virtue of (16.3 a), the condition P(m) > 1 is realized for at least one integer m ¤ [bk, bk + n], therefore one can take rj = n. Since jjlL is very ample, [iL has a metric having an isolated logarithmic pole of Lelong number 1 at each given point (for example, the algebraic metric defined by the sections of jjlL vanishing at xq). Therefore F'a = aKx + (a(n + 1) + l)L + n/iL has a metric h'a such that V(J(h'a)) is of dimension zero and contains {xo}. By virtue of Cor. (15.9), we conclude that Kx + F'a = (a + 1)KX + (a(n + 1) + 1 + n/j,)L is generated by its sections, in particular Kx + +av^ +"ML is nef. By letting a tend to +oo, we deduce that Kx + (n + 1)L is nef. D PROOF of (16.5 a). It suffices here to choose a = 1. Then _ y^ (3n + 2sj - 1 ~ ^ \ n i (3nn_1) + 1 > 2n2 for n > 2. Lemma (16.3 c) shows that P(m) > 6 + 1 for at least one m £ [bk, bk + rj\ with rj = 6 + 1. We begin the induction procedure k i->- k + 1 with bi=rj + l = 5 + 2, because the only necessary property for the induction step is the vanishing property Hq(X,2Kx+mL) = 0 for q > 1, m > h, which is realized according to Kodaira's vanishing theorem and the ampleness prop erty of Kx + b\L. (We use here the result of Fujita (16.4), by observing that bi > n+1.) The recursive formula bk+i = [t; (bk+7])\+l then gives bk = rj+1 = (5+2 for all k, and (16.5 a) follows. ? Proof of (16.5 b). Completely similar to (16.5 a), except that we choose rj = n, a = 1 and bk = n + 1 for all k. By applying Lemma (16.3 b), we have P(m) > a,dkd/2d~1 for at least one integer m £ [mo,mo + kd\, where ad > 0 is the leading degree coefficient of P. By virtue of Lemma (16.2), we have aa > infdimy=did - Y. Take k = [n/d\. The condition P(m) > S + 1 can then be realized for an integer m £ [mo, mo + kd\ C [mo, mo + n], provided that inf Ld-Y[n/d\d/2d-1 > 5, dimY=d that which is equivalent to the condition in (16.5 b). 85 17. AN EFFECTIVE VERSION OF MATSUSAKA'S BIG THEOREM The big disadvantage of the described technique is that one must necessarily utilize multiples of L to avoid the zeros of the Hilbert polynomial, in particular it is not possible to directly obtain a criterion of large ampleness for 2Kx + L in the statement of (16.5 b). Such a criterion can nevertheless be obtained with the aid of the following elementary lemma. 16.6. Lemma. Suppose that there exists an integer n £ N* such that jjlF simultaneously generates all the jets of order fi(n + Sj) + 1 at every point Xj of a subset {xi,... ,xp} C X. Then Kx + F simultaneously generates all the jets of order Sj at the point Xj. Proof. Choose the algebraic metric on F defined by a basis oi,... , unchanged everywhere else (this is possible by taking C > 0 suf ficiently large). Then ip'(z) = \z\'2 - C + (n + Sj) log \z - Xj\ in a neighbourhood of Xj, in particular cp1 is strictly plurisubharmonic near Xj. In this way, we obtain a metric h' on F with semi-positive curvature everywhere on X, and has positive definite curvature in a neighbourhood of {x\,... ,xp}. The resulting conclusion then is a direct application of the L2 estimates (14.2). ? 16.7. Theorem. Let X be a projective manifold of dimension n and L an ample line bundle on X. Then 2Kx + L simultaneously generates the jets of order si,... , sp at arbitrary points xi,... ,xp £ X so long as the intersection numbers Ld -Y of L on all the algebraic subsets Y C X of dimension d satisfy ^->i^ie(("+i,<4"+;"+i)-2). ><-«-»? Proof. Lemma (16.6) applied with F = Kx + L and [/, = n + 1 shows that the desired property for the jets of 2Kx + L occurs if (n + l)(Kx + L) generates the jets of order (n + l)(n + Sj) + 1 at the points Xj. Lemma (16.6) applied again with F = pKx + (n + 1)L and n = 1 shows by descending induction on p that it suffices that F generates all the jets of order (n + l)(n + Sj) + 1 + (n + 1 - p)(n + 1) at the points Xj. In particular, for 2Kx + (n + 1)L it suffices to obtain all the jets of order (n + l)(2n + Sj - 1) + 1. Th. (16.5 b) then gives the desired condition. ? We conclude by mentioning some immediate consequences of th. 16.5, obtained by taking L = ±Kx- (zz)Corollary. Let X be a projective manifold of general type, with Kx ample and dimX = n. Then mKx is very ample for m > mo = ( ?+ ) + 4. (aaa)Corollary. Let X be a Fano variety (that is, a projective manifold such that -Kx is ample), of dimension n. Then -mKx is very ample for m > J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 86 17. An effective version of Matsusaka's big theorem We encounter here the problem of finding an explicit integer mo such that mL is very ample for m > mo. The existence of such a bound mo, depending only on the dimension and the coefficients of the Hilbert polynomial of L, was first established by Matsusaka [Mat72]. Further Kollar and Matsusaka [KoM83] have shown that one could indeed find a bound mo = mo(n,Ln,Kx ? Ln~l) dependent only on n = dimX and on the first two coefficients. Recently, Siu [Siu93] has obtained an effective version of the same result furnishing an explicit "reasonable" bound mo (although this bound is unfortunately still far from being optimal). We explain here the method of Siu, starting from some simplifications and improvements suggested in [Dem96]. The starting point is the following lemma. 17.1. Lemma. Let F andG be nef line bundles on X. If Fn > nFn~x -G, then any positive multiple k(F - G) admits a non-trivial section for k > ko sufficiently large. Proof. The lemma can be proven as a special case of the holomorphic Morse inequalities (see [Dem85], [Tra91], [Siu93], [Ang95]). We give here a simple proof, following a suggestion of F. Catanese. We can assume that F and G are very ample (if not, it suffices to replace F and G by F' = pF + A and G' = pG + A with A very ample and sufficiently positive to ensure large ampleness of any sum with an nef bundle, then to choose p > 0 large enough for which F' and G' satisfy the same numerical hypothesis as F andG). Then 0{k{F-G)) ~ 0{kF-Gi Gk) for arbitrary elements G\,... ,Gk of the linear system \G\. If we choose such elements Gj in general position, the lemma follows from the Riemann-Roch formula applied to the restriction morphism H°(X, O(kF)) -> 0 H0{GJ7O{kF^Gj). ? 17.2. Corollary. Let F andG be nef line bundles over X. If F is big and if m > nFn~l ? GIFn, then 0(mF - G) can be given a (possibly singular) Hermitian metric h, having a positive definite curvature form, i.e. such that 0(,(mF - G) > eui, e > 0, for a Kdhler metric ui. Proof. In fact, if A is ample and e ¤ Q+ is small enough, Lemma (17.1) implies that a certain multiple k(mF - G - eA) admits a section. Let E be the divisor of this section and let u = Q(A) £ c\{A) be a Kahler metric representing the curvature form of A. Then mF - G = eA + jE can be given a singular metric h with curvature form 0/,(mF - G) = e@(A) + j[E] > ew. D We now consider the problem of obtaining a non-trivial section of mL. The idea of [Siu93] is to obtain a more general criterion for the ampleness of mL - B when B is nef. In this way, we will be able to subtract from mL any undesired multiple of Kx that would be added to L, by application of the Nadel Vanishing Theorem (for this, we simply replace B, by B plus a multiple of Kx + (n + 1)£). 17.3. Proposition. Let L be an ample line bundle on a projective manifold X of dimension n, and let B be an nef line bundle on X. Then Kx + mL - B admits a non-zero section for an integer m satisfying L"-1?B m < n hn+1 87 17. AN EFFECTIVE VERSION OF MATSUSAKA'S BIG THEOREM Proof. Let mo be the smaller integer > nL L"'B? Then m^L - B can be given a singular Hermitian metric h with positive definite curvature. By virtue of the Nadel vanishing theorem, we obtain Hq{X,O{Kx+mL-B)®J{h))=0 for q > 1, therefore P(m) = h°(X, 0(Kx + mL - B) J{h)) is a polynomial for m > mo-Since P is a polynomial of degree n which is not identically zero, there exists an integer m £ [mo,mo + n] which is not a root. Therefore there exists a non-trivial section of H°(X, 0{KX +mL- B)) D H°(X, 0{KX + mL - B) ® J{h)) for some m ¤ [mo,mo + n], as stated. ? 17.4. Corollary. If L is ample and B is nef then mL - B has a non-zero section for at least one integer , L?-1 - B + Ln~l ? Kx m < n\ hn + 1 Ln Proof. According to the result of Fujita (16.4), Kx + (n + l)L is nef. We can therefore replace B by B + Kx + (n + 1)L in Prop. (17.3). Corollary (17.4) follows. ? 17.5. Remark. We do not know if the bound obtained in the above corollary is optimal, but it is certainly not very far from being it. Indeed, even for B = 0, the multiplicative factor n cannot be replaced by a number smaller than n/2. To see this, take for example for X a product C\ x - - - x Cn of curves Cj of large enough genus gj, and L = 0(ai[pi]) ® - - - ® 0(an[pn]), B = 0. Our sufficient condition so that \mL\ ^ 0 becomes in this case m < J2(^9j ~ ^)/aj + n(n + 1)> while for a generic choice of pj the bundle mL admits sections only if maj > gj for all j. The inaccuracy of our inequality thus plays more on one multiplicative factor 2 when ai = - - - = an = 1 and g\ ^> g^ S> - - - S> gn ->- +00. In addition, the additive constant n + 1 is already the best possible when B = 0 and X = F?. ? Up to this point, the method was not really sensitive to the presence of singularities (Lemma (17.1) is still true in the singular case as is easily seen by passing to a desingularization of X). In the same way, as we observed with remark (15.16), the Nadel vanishing theorem still remains essentially valid. Prop. (17.3) can then be generalized as follows: 17.6. Proposition. Let L be an ample line bundle on a projective manifold X of dimension n, and let B be an nef line bundle on X. For any (reduced) algebraic subvariety Y of X of dimension p, there exists an integer LP'1 BY ? < P-LP Y h p + 1 such that the sheaf toy <8> Oy{mL - B) has a non-zero section. ? By applying a suitable induction procedure relying on the results above, we can now improve the effective bound obtained by Siu [Siu93] for Matsusaka's big theorem. Our statement will depend on the choice of a constant A" such that J.-P. DEMAILLY, PART II: L2 ESTIMATIONS AND VANISHING THEOREMS 88 m(Kx + (n + 2)L) + G is very ample for m > A" and all nef line bundles G. Theorem (0.2 c) shows that A" < (3n+1) - 2n (a more elaborate argument concerning the recent results of Angehrn-Siu [AS94] allows us in fact to see that A" < n3-ri2-n-l for n > 2). Of course, one expects with this that A" = 1 for all n, if one believes that the conjecture of Fujita is true. 17.7. Effective version of Matsusaka's Big Theorem. Let L and B be nef line bundles on a projective manifold X of dimension n. Assume that L is ample and let H = Xn(Kx + (n + 2)L). Then mL - B is very ample for f3n-1_1)/2(L"-1-(^ + H))^""^1)/2^"-1 - #)3"-2(»/2-3/4)-l/4 (Xn)3"-2(n/2-l/4) + l/4 In particular mL is very ample for , Jn-X K x 3"-2(n/2+3/4) + l/4 m>Cn(L"rn-2[n + 2+L LnKx with Cn = (2n)(3""1-1)/2(A")3""2(«/2+3/4)+1/4. Proof. We utilize Th. (3.1) and Prop. (17.6) to construct by induction a sequence of algebraic subvarieties (not necessarily irreducible) X = Yn D Yn-\ 3 - - - D Y2 D Y\ such that Yp = UjYpj is of dimension p, Yp-\ being obtained for each p > 2 as the union of the set of zeros of the sections apjGH0(Ypj,OYpJmpJL-B)) for suitable integers mpj > 1. We proceed by induction on the decreasing values of the dimension p, and we seek to obtain with each step an upper bound mp for the integer mpj. By virtue of Cor. (17.4), we can find an integer mn such that mnL - B admits a non-trivial section an for L"-1 -(B + Kx + (n + 1)L) L"-1 ? (B + H) m" < n < n- Ln ~ Ln Now suppose that the sections an,... , crp+i,j have already been constructed. One then obtains by induction a p-cycle Yp = ^2/J>P,jYpj defined by Yp = sum of the divisors of zeros of the sections crp+i,j on the components Yp+ij, where the multiplicity fipj of Ypj C Yp+itk is obtained by multiplying the corresponding multiplicity nP+i,k by the order of vanishing of crp+i,k along Ypj. We obtain the equality of cohomology classes Yp = ^2(mp+1,kL - B) ? (pp+i^Yp+i^k) < mp+1L ? Yp+1. By induction, we then obtain the numerical inequality Yp < mp+1 ???mnLn~p. Now, for each component Ypj, Prop. (17.6) shows that there exists a section of wyp . ® Oyp j (mpjL - B) for a certain integer U>-~l . B . Y ? mpj < p -- + p + 1 < pmp+1 ? ? ? mnLn~l -B+p+1. 89 17. AN EFFECTIVE VERSION OF MATSUSAKA'S BIG THEOREM We have used here the obvious lower bound Lp~x ? Yp^q > 1 (this bound is besides undoubtly one of weak points of the method...). The degree Yv,q by comparison to H admits the upper bound SPij := Hp ? YpJ < mp+1 ? ? ? mnHp ? Ln~p. The Hovanski-Teissier concavity inequality gives (Ln~p -Hp)p(L")1"p OypJ (mpjL - B) already constructed, we obtain a section of Oypd(mVjL - B + (Spj - p - 2)H) on Ypj. We do not want H appearing at this stage, which is why we will replace B by B + (Sp^q-p - 2)H. We obtain then a section apj of Oypj(mpjL - B) for a certain integer mpj such that mpj < pmp+i ? ? ? mnLn~x ? (B + (Spj - p - 2)H) + p + 1 < pnip+1 ? - ? mnSpjLn~ ? (B + H) < p(mp+1 ? ? ? mnf{ J L"-1 -(B + H). Consequently, by setting m = nLn~x -(B + H), we obtain the descending inductive relation (Ln~l ? H)p mp (mpjL -B)-C>0. In addition, if C = Yij, then (m0L - B) ? C > m0 - B ? Y > m0 - m2 - - - mnLn-x ? B > 0. According to the definition of A" (and the proof where such a constant exists, cf. (0.2c)), H + G is very ample for any nef line bundle G, in particular H + m§L - B is very ample. We again replace B by B + H. This substitution has the effect of replacing M by the new constant m = 77(Ln_1 - (B + 2Hj) and mo by mo = max(m",m"_i,... ,mj,7772 - - - m"L?_1 - (B + Hj). The latter term being the largest estimation of rnp implies mn< ^3"-1-l)/2(^-1-g)'3"-2-1""-^'/2+'"-2'/2£"-1-(B+g)) 1/1-0^ J« ^L"j(3n-2_1)("_1/2)/2 + (n_2)/2 + 1 ^ \L'h) (Ln)3"-2(n/2-l/4) + l/4 Q PROOF of lemma (17.8). Let X C F^ be the embedding given by H, so that H = OxiXj- There exists a projective linear map Pra ->- Fp+1 for which the restriction it : Y ->- Fp+1 to Y is a finite and birational morphism of Y onto an algebraic hypersurface Y' of degree S in Fp+1. Let s £ i7T°(Fp+1, C((5)) be the polynomial of degree 6 defining Y'. We claim that for any small Stein open subset W C Fp+1 and any holomorphic p-form u, L2 on Y'nW, there exists a holomorphic (p + l)-form u, L2 on W, with values in 0(S), such that u\Y'nw = u Ads. In fact, this is precisely the conclusion of the L2 extension theorem of Ohsawa-Takegoshi [OT87], [Ohs88] (also see [Man93] for a more general version of this result). One can equally invoke standard arguments in local algebra (see Hartshorne [Har77], th. III-7.11). Since KVP+i = 0(-p - 2), the form u can be considered as a section of O (S - p - 2) on W, consequently the morphism of sheaves u ^ u Ads extends to a global section of 'Horn (uy, Oy (6 - p - 2)). The inverse image of n* furnishes a section of Horn (n* ujy, Oy{[5 -p- 2)H)). Since 7r is finite and generically 1 : 1, it is easy to see that tt*ujy' = ojy- The lemma follows. ? 17.9. Remark. In the case of surfaces (77 = 2), we can take A" = 1 according to the result of I. Reider [Rei88], and the arguments developed above ensure that 777L is very ample for ^ A(L-(KX+4L))2 m > 4 - . By working through the proof more carefully, it can be shown that the multiplicative factor 4 can be replaced by 2. In fact, Fernandez del Busto has recently shown that 91 17. AN EFFECTIVE VERSION OF MATSUSAKA'S BIG THEOREM mL is very ample for 1 \(L-(Kx+4L) + if " m>2[ V +3 ' and an example of G. Xiao shows that this bound is essentially optimal (see [FdB94]). Matsusaka's big theorem yields a number of other important finiteness results. One of the prototypes of these results is the following statement. 17.10. Corollary. There exists only a finite number of families of deformations of polarized projective manifolds (X,L) of dimension n, where L is an ample line bundle for which the intersection numbers Ln and Kx ? Ln~x are fixed. Proof. Indeed, since Ln and Kx ? Ln~x are fixed, there in fact exists a calculable integer mo such that moL is very ample. We then obtain an embedding $ = $|moL| : X ^ FN such that $*C(1) = ±m0L. The image Y = $(X) is of degree deg(F) = / Cl(0(l))" = f Cl(±moL)n=m$Ln. 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Wells, Differential analysis on complex manifolds, Graduate Texts in Math. 65, 2nd edition, Springer-Verlag, Berlin (1980). Frobenius and Hodge Degeneration Luc Illusie Universite de Paris-Sud, Department de Mathrhatiques, Batiment 425, 91405 Orsay Cedex, France 96 97 0. INTRODUCTION In [D-I], the Hodge degeneration theorem and the Kodaira-Akizuki-Nakano vanishing theorem for smooth projective varieties over a field of characteristic zero are shown by methods of algebraic geometry in characteristic p > 0. These present notes will serve as an introduction to the subject, with the intention of keeping the non-specialist in mind (who will be able to also consult the presentation of Oesterle [O]). Thus we will assume known by the reader only some rudiments of the theory of schemes (EGA I 1-4, [H2] II 2-3). On the other hand, we require of the reader a certain familiarity with homological algebra. The results of [D-I] are expressed simply in the language of derived categories. Although it is possible to avoid there the recourse, see for example [E-V], we prefer to place it in its context, which appears more natural. However, to help the beginner, we recall in n°4 the basic definitions and some essential points. 0. Introduction Let X be a complex analytic manifold. By the Poincare Lemma, the de Rham complex Q x of holomorphic forms on X is a resolution of the constant sheaf C. As a result, the augmentation C -> ftx defines an isomorphism (for all n) (0.1) Hn(X, C) ^ H£R(X) = Hn(X, n-x), where the second term, called the de Rham cohomology of X (in degree n), is the n-th hypercohomology group of X with values in flx. The first spectral sequence of hypercohomology abuts to the de Rham cohomology of X (0.2) E? = Hq(X,ftpx) => #££'(£), which is called the Hodge to de Rham spectral sequence (or Hodge-Frolicher) (cf. [De] n°9). Let us assume X is compact. Then, by the finiteness theorem of Cartan-Serre, the Hq(X, £lx), and therefore all the terms of the spectral sequence (0.2) are finite dimensional C-vector spaces. If we set bn = dimH£R(X) = dimHn(X,C) (n-th Betti number of X) and hp'q = dimHq(X,nx) (Hodge number), we have (0.3) bn< Y, hP9> p+q=n with equality for all n if and only if (0.2) degenerates at E\. Suppose in addition that X is Kahler. Then by Hodge theory, the Hodge spectral sequence of X degenerates at Ei : this is the Hodge degeneration theorem, ([De] 9.9). Denote by 0 = Fn+1 C Fn C - - - C Fp = FpH£R(X) C - - - C F° = H£R(X) the resulting filtration of the Hodge spectral sequence (Hodge filtration). By degeneration, one has a canonical isomorphism (0.4) Ep'q = H" (X, Qpx) ~ E? = Fp/Fp+1. We put Hp'q = Fpf)Fq, where the bar denotes complex conjugation on HQR(X), defined by means of (0.1), and the isomorphism Hn(X, C) ~ Hn(X, K) ® C. It follows that Hp>q = Hq'P. LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 98 Further, Hodge theory furnishes the following results ([De] 9.10): (a) the composite homomorphism H? ^ FpHp+lq(X) -» Fp/Fp+1 is an isomorphism (i.e. Hp,q is a complement of Fp+l in Fp); whence, by composing with (0.4), determines an isomorphism Hq(X,npx); 0 Hp>\ p+q=n (Hodge decomposition). These results apply in particular to the complex analytic manifold X associated to a smooth projective scheme X over C. The difference between (a) and (b), which is of a transcendental nature, utilizes complex conjugation in an essential way. The Hodge degeneration can in this case be formulated in a purely algebraic manner. The de Rham complex of X is indeed the complex of analytic sheaves associated to the algebraic de Rham complex fl'x of X over C (a complex of sheaves in the Zariski topology, for which the components are locally free coherent sheaves). The canonical morphism (of ringed spaces) X -> X induces homomorphisms on the Hodge and de Rham cohomologies (0.7) Hq(x,npx) -> Hg(x,npx), (0.8) H£R(X) -»? ffSR(£), where H£R(X) = Hn(X,Q'x). We make use of the Hodge to algebraic de Rham spectral sequence (0.9) E^=H"(X,npx)^H^\X), and a morphism of (0.9) in (0.2) inducing (0.7) and (0.8) respectively on the initial terms and the abutment. By the comparison theorem of Serre [GAGA], (0.7) is an isomorphism, and therefore the same holds for (0.8). Consequently, the degeneration at Ei of (0.2) is equivalent to that of (0.9). In other words, if one sets hp'"(X) =dimHq(X,npx), hn(X) = dimH£R(X), the Hodge degeneration theorem for X is expressed by the (purely algebraic) relation (0.10) hn(X) = J2 hp'q(X). p+q=n More generally, if X is a smooth and proper scheme over a field k, one can consider the de Rham complex flx/k of X over k, and one still has a Hodge to de Rham spectral sequence (0.11) Epi = H«(X,Slpx/k) => H&?(X/k) (where H^R(X/k) = Hn(X,flx,k)), formed of finite-dimensional fc-vector spaces. If k is of characteristic zero, the Hodge degeneration theorem implies the degeneration of (0.11) at Ei : standard techniques (cf. n°6) indeed make it possible to go back initially to k = C, then with the aid of Chow's Lemma and of the resolution of singularities one reduces the proper case to the projective case ([DO]). There 99 1. SCHEMES: DIFFERENTIALS, THE DE RHAM COMPLEX are those who have long sought for a purely algebraic proof of the degeneration of (0.11) at Ei for k of characteristic zero. Faltings [Fal] was the first to give a proof of it independent of Hodge theory2. A simplification of crystalline techniques due to Ogus [Ogl], Fontaine-Messing [F-M] and Kato [Kal] led, shortly thereafter, to the elementary proof presented in [D-I]. We refer to the introduction of [D-I] and to [O] for a broad overview. We only indicate that the degeneration of (0.11) (for k of characteristic zero) is proven by reduction to the case where k is of characteristic p > 0, where, however, it can happen that the degeneration is automatic! This proof is based however on the help of some additional hypothesis on X (upper bound of the dimension, liftability) which is sufficient for our purposes (see 5.6 for a precise statement). We explain in n°6 the well-known technique which allows us to go from characteristic p > 0 to characteristic zero. The degeneration theorem in characteristic p > 0 to which we have just alluded follows from a decomposition theorem (5.1), relying on some classical properties of differential calculus in characteristic p > 0 (Frobenius endomorphism and Cartier isomorphism), which we recall in n°3, after having summarized, in n°l and 2, the formalism of differentials and smoothness on schemes. The aforementioned decomposition theorem furnishes at the same time an algebraic proof of the Kodaira-Akizuki-Nakano vanishing theorem for the smooth projective varieties over a field of characteristic zero (6.10 and [De] 11.7). The last two sections are of a more technical nature: We outline the evolution of the subject since the publication of [D-I], and, in the appendix, we describe some complementary results due to Mehta-Srinivas [Me-Sr] and Nakkajima [Na]. 1. Schemes: differentials, the de Rham complex We recall here the definition and basic properties of differential calculus over schemes. The reader will find a complete treatment in (EGA IV 16.1-16.6); also see [B-L-R] 2.1 and [H2] II 8 for an introduction. 1.1. We say that a morphism of schemes i : T$ -> T is a thickening of order 1 (or by abuse, that T is a thickening of order 1 of T0) if i is a closed immersion defined by an ideal of Ot of square zero. If T and To are affine, with rings A and A$, such a morphism corresponds to a surjective homomorphism A -> Aq for which the kernel is an ideal of square zero. The schemes T and To have the same underlying space, and the ideal a of i, annihilated by a, is a quasi-coherent Ot0 ( = 0T/a)-module. Let j : X -> Z be an immersion, with ideal / (by definition, j is an isomorphism of X onto a closed subscheme j(X) of a larger open subset U of Z, and / is the quasi-coherent sheaf of ideals of U defining j(X) in U, (EGA I 4.1, 4.2)). Let Z\ be the subscheme3 of Z, with the same underlying space as X, defined by the ideal I2. Then j factors (in a unique way) into X il> Zl -*i> Z 2The purists observe that this proof, which rests on the existence of the Hodge-Tate decomposition for p-adic etale cohomology of a smooth and proper variety over a local field of unequal characteristic, is not entirely "algebraic", in the sense of where it uses the comparison theorem of Artin-Grothendieck between etale cohomology and Betti cohomology for smooth and proper varieties over C. 3At the expense of some abuse of notation, we will allow ourselves the flexibility of interchanging "immersion" (resp. "closed immersion") and "subscheme" (resp. "closed subscheme"); that amounts here to neglecting the isomorphism of X onto j(X). LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 100 where h\ is an immersion, and j\ is a thickening of order 1, with ideal I/I2; one says that (ji,hi), or more simply Z\, is the first infinitesimal neighbourhood of j (or of X in Z). The ideal I /I2 (which is a quasi-coherent Ox-module) is called the conormal sheaf of j (or of X in Z). We denote it by Mx/z ? 1.2. Let / : X -> Y be a morphism of schemes, and let A : X -> Z := X XyX be the diagonal morphism. This is an immersion (closed if and only if X is separated over F) (EGA I 5.3). The conormal sheaf of A is called the sheaf of Kahler 1-differentials of / (or of X over Y) and is denote by flx,Y; we sometimes write &X/A mstead 0I ^x/y ^ ^ 1S amne with ring A. Thus we have a quasi-coherent Ox-module, defined by (1.2.1) nx/Y = i/i2, where / is the ideal of A. Let X -h- Z\ -> Z be the first infinitesimal neighbourhood of A. The two projections of Z = X Xy X on X induce, by composition with Zi -> Z, two F-morphisms pi,p? : Z\ ->- X, which retract Ai. The sheaf of rings of the scheme Z\, which has the same underlying space as X, is called the sheaf of principal parts of order 1 of X over Y, and is denoted by VX,Y- We have, by construction, an exact sequence of abelian sheaves (1-2.2) o -> nx/Y -»? Vx/Y -»? Ox -»? 0, split by each of the ring homomorphisms ji,J2 '? Ox -> T^x/y induced from pi,p2-The difference j'2 - ji is a homomorphism of abelian sheaves of Ox in flx,Y, which is called the differential, and which is denoted by (1.2.3) dx/Y (or d) : Ox -> «x/y If M is an Ox-module, a Y-derivation of Ox in M is any homomorphism of sheaves of /_1(Oy)-modules D : Ox -> M (where /_1 denotes the inverse image functor for abelian sheaves) such that D(ab) =aDb + bDa for all local sections a, b of Ox- We denote by Bevy (Ox, M), the set of Y-derivations of Ox in M, which is in a natural way an abelian group. The differential dx/Y is a F-derivation of Ox in QX,Y. One shows that it is universal, in the sense that for any F-derivation D of Ox in an Ox-module M (not necessarily quasi-coherent), there exists a unique homomorphism of Ox-modules u : Qx/y ~^ ^ such that u o dx/Y = ^> i-e- the homomorphism (1.2.4) Rom(nx/Y,M)->Dery(Ox,M), u^uodx/y is an isomorphism. The sheaf "Hom(Qx,Y, Ox) is called the tangent sheaf of / (or of X over F), and is denoted by (1-2.5) Tx/Y (or sometimes ®x/y)- F°r any open subset U of X, (1.2.4) gives an isomorphism T(U,Tx/y) - Dery(Oc/, 0[/). Recall that one calls a Y-point of X a F-morphism T -> X. By definition, X Xy X "parameterizes" the set of pairs of 101 1. SCHEMES: DIFFERENTIALS, THE DE RHAM COMPLEX F-points of X (i.e. represents the corresponding functor on the category of Y-schemes). The geometric significance of the first infinitesimal neighbourhood Z\ of the diagonal of X over Y is that it parameterizes the pairs of Y-points of X neighbouring of order 1 (i.e. congruent modulo an ideal of square zero): More precisely, if i : T0 ->- T is a thickening of order 1, with ideal a, where T is a F-scheme, and if ti, ti : T ->- X are two F-points of X which coincide modulo a (i.e. such that t\i = t-2,% = to : To ->- X), then there exists a unique F-morphism h : T ->- Z\ such that p\h = t\ and p^h = t^. Moreover, if t\, t\ : Ox -? to*Or 4 are the homomorphisms of sheaves of rings associated to t\ and ti, t\ -1\ is a F-derivation of X with values in to*a, such that (1.2.6) (t?2-tl)(s)=h*(dS) for any local section s of Ox, where h* : ^x/y ~~^ *oa *s tne homomorphism of Ox-modules induced by h (on the corresponding conormal sheaves of X in Z\ and To in T). If / is a morphism of affine schemes, corresponding to a ring homomorphism A -» B, then Z = Spec B ®a B, A corresponds to the ring homomorphism sending &i ® &2 onto &i62, with kernel J = T(Z,T). We have T(X,VX/Y) = (B ®A B)/J2, and we set (1.2.7) r(x,n1x/Y) = n1B/A. The S-module Qg ,A = J/ J2, for which the associated quasi-coherent sheaf is QxiY, is called the module of K'dhler 1-differentials of B over A. The map d = g^/a = r(X, dx/y) '? B -t flg/A is an ^-derivation, satisfying a universal property that we leave to the reader to formulate. The homomorphisms ji, j2 : B ->- (B a B)/J2 of 1.1 are given by jib = class of b 1, J2& = class of 1 ® b. Since J is generated by 1 ® & - b 1, fJ^/^ is generated, as a Ti-module, by the image of d. It follows from this that if / is any given morphism of schemes, Qx/y *s generated, as an Ox-module, by the image of d. 1.3. Any commutative square X' -4 X (1-3-1) f'i if Y' A Y defines in a canonical way, a homomorphism of Ox' -modules (1.3.2) 9*nx/Y^nx,/YI, which sends 1 g~1(dx/Ys) onto dX'/Y'(l ® ff_1(s))- (If -^ is an Ox-module, by definition g*_E = Ox'®fl-i(e>x)5-1(Tv)-) This is an isomorphism if the square (1.3.1) is cartesian, i.e. if the morphism X' -> Y' Xy X is an isomorphism. Moreover, in this case, the canonical homomorphism (1.3.3) f'*nY,/Y(Bg*nx/Y^nx,/Y is an isomorphism. 4 Recall that T and To have the same underlying space. LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 102 1.4. Let i^yAs be morphisms of schemes. Then the canonical sequence of homomorphisms (1.4.1) ftfy/g -> Q^s -»? QX/y -> 0 is exact. 1.5. Let X A Z Y be a commutative triangle, where i is an immersion, with ideal /. The differential dz/y induces a homomorphism d : Mx/z ->- **^z/y> and the sequence (1.5.1) Afx/z ->- **^/y -> fijf/y -> 0 is exact. 1.6. Let X = AY = Y[Ti,... ,Tn] be the affine space of dimension n over Y. The Ox-module Qx/y *s free> wrtn basis dTj (1 < i < n). If Y is affine, with ring A, and if s £ A[Ti,... ,T"], then cfe = J2(ds/dTi)dTi, where the ds/*9^ are the usual partial derivatives. Properties 1.3 to 1.6, for which the verification is completely standard, are fundamental. It is by virtue of these that we can "calculate" the modules of differentials. For more details, see the indicated references above. 1.7. Let / : X ->- Y be a morphism of schemes. For i £ N, we denote by "X/Y - ^ "X/Y the i-th exterior product of the Ox-module ftx/Y- (^ *s agreed that Q°x/y = ®x-) One shows that there exists a unique family of maps d : Qx/y ~~^ ^x/V satisfying the following conditions: (bbb)d is a F-anti-derivation of the exterior algebra 0 Slx/Y-> l-Q- ^ *s /_1(Oy)-linear and d(ab) = da A b + (-l)*a A Y be a morphism of schemes. We say that / is locally of finite type (resp. locally of finite presentation) if, for any point x oi X, there exists an affine open neighbourhood U of x and an affine open neighbourhood V °f V = f(x) sucn that f(U) C V and that the homomorphism of rings A ->- B associated to U -> V makes B an A-algebra of finite type (i.e. a quotient of an algebra of polynomials A[t\,... ,tn]) (resp. of finite presentation (i.e. a quotient of an algebra of polynomials A[t\,... ,tn] by an ideal of finite type)). If Y is locally Noetherian, "locally of finite type" is equivalent to "locally of finite presentation", and if it is, then it follows that X is locally Noetherian. If / : X -> Y is locally of finite presentation, the Ox-module Qx/y 1S °f nmte type for all i, therefore coherent if Y is locally Noetherian. 2.2. Let / : X ->- Y be a morphism of schemes. We say that / is smooth (resp. net (or non-ramified), resp. etale) if / is locally of finite presentation and if the following condition is satisfied: For any commutative diagram X (2.2.1) g0S if t0 At -> f where i is a thickening of order 1 (1.1), there exists, locally in the Zariski topology on T, a (resp. at most one, resp. a unique) F-morphism g : T -t X such that gi = go- It follows immediately from the definition that the composite of two smooth morphisms (resp. net, resp. etale) is smooth (resp. net, resp. etale), and that if / : X -> Y is smooth (resp. net, resp. etale), it is the same with the morphism /' : X' ->- Y' induced by a base change Y' ->- Y. If for i = 1,2, /, : X{ ->- Y is smooth (resp. net, resp. etale), the fiber product / = /i Xy fi2 : X\ Xy X2 -> Y is therefore smooth (resp. net, resp. etale). Additionally it is immediate that the projection of the affine line Ay = Y[t] ->- Y is smooth, and it is therefore the same for the projection of the space AY -> Y. Remarks 2.3. (a) Because of the uniqueness which allows a gluing together, we can omit in the definition of etale, locally in the Zariski topology. On the other hand, we cannot do it in the definition of smooth. There exist a cohomological obstruction that we will later specify, to the existence of a global extension g oi go- LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 104 (b) If n is an integer > 1, we say that a morphism of schemes i : T0 -> T is a thickening of order n if i is a closed immersion defined by an ideal / such that jn+i _ q_ jf rp^ denotes the closed subscheme of T defined by Im+1, % itself factors into a sequence of thickenings of order 1 : To -> Xi ->- - - - ->- Tm -t Tm+i ->----->- Tn. In Definition 2.2, we can therefore replace thickening of order 1 by thickening of order n. The following proposition summarizes the essential properties of differentials associated to smooth morphisms (resp. net, resp. etale). Proposition 2.4. (a) // / : X ->- Y is smooth (resp. net), the Ox-module Qx/y is locally free of finite type (resp. zero). (b) In the situation of 1.4, if f is smooth, the sequence (1.4.1) extended by a zero to the left (2.4.1) o -». rtfyjs -? nx/s -». nx/Y -+ o is ea;aci and locally split. In particular, if f is etale, the canonical homomor-phism f*QY,s ->- £lx/s is an isomorphism. (c) In the situation of 1.5, if f is smooth, the sequence (1.5.1) extended by a zero to the left (2.4.2) 0 -»? Mx/z -> **^z/y -> ^x/y -> 0 is ea;aci anrf locally split. In particular, if f is etale, the canonical homomor-phism Mx/z ~^ i*^z/Y *s an isomorphism. 2.5. The verification of 2.4 is not difficult (EGA IV 17.2.3), but unfortunately somewhat scattered in (EGA Oiv 20). Here is an outline. The key ingredient is the following. If / : X -> Y is a morphism of schemes and / a quasi-coherent Ox-module, we call a Y-extension of X by I, a F-morphism i : X ->- X' which is a thickening of order 1 with ideal /. Two F-extensions i\ : X -> X\ and «2 : X ->- X^ of X by / are said to be equivalent if there exists a F-isomorphism g of X\ onto X^ such that gi\ = «2 and that g induces the identity on /. An analogous construction to this is the "Baer sum" for extensions of modules over a ring associated to the set Exty(AV) of equivalence classes of F-extensions of X by / with a structure of an abelian group, with neutral element the trivial extension defined by the algebra of dual numbers Ox ffi I- Assertion (c) follows immediately from the definition: The smoothness of / indeed implies that the first infinitesimal neighbourhood i\ of i retracts locally onto X, and the choice of a retraction r permits the splitting (2.4.2) (by the derivation associated to Id^ - ii or, cf. (1.2.6)). Assume / is smooth. If / is a quasi-coherent Ox-module and if i : X ->- Z is a y-extension of AT by /, the sequence (2.4.2) is therefore an extension of Ox-modules e(i) of f^x/y by I- One can show that i i-» e(i) gives an isomorphism (2.5.1) Exty(X,J)-> Ext^x(f)x/y,I) 105 SMOOTHNESS AND LIFTINGS (cf. [II] I, chap. II, 1.1.9. We define an inverse of (2.5.1) by associating to an extension M of ^x/y by /, the F-extension Z of X defined in the following way: Identify, via ji, the sheaf of principal parts VX,Y (1.2.2) with the ring of dual numbers Ox © ^x/y anc^ denote by F = Ox © M the ring of dual numbers over M; the extension M makes F an /_1 (Oy )-extension of VX,Y by /. That is, if E = F xvi Ox is the "pull-back" of F by the homomorphism j2 = ji + dx/Y '? Ox -? T-'xiy tnen E is a /_1 (Oy )-extension of Ox by /, which defines the Y-extension Z). Since / is smooth, any F-extension of X by / is locally trivial, and therefore by virtue of (2.5.1), it follows from this that the sheaf Ext^" (flx,Y,I) (associated to the presheaf U t-> ¥jy±l0jJ(Vl]j,Y,I\u)) is zero, and therefore also that Ext^.(n^,y,J) =0 for all open subsets U of X and all quasi-coherent 0[/-modules J. Since Qx/y 1S °^ nmte type (2.1), it follows that Qx/y 1S locally free °f finite type, which proves the part of (a) relative to the smooth case. (The relative part of the net case is immediate: For any F-scheme X, if i : X -> Z is the trivial F-extension of X by a quasi-coherent Ox-module /, the set of Y-retractions of Z on X is identified with ~Kom(£lx,Y,I) by r >-> r - ro, where ro corresponds to the natural injection of Ox in Ox® I, cf. (1.2.6).) In particular, it follows from (a) and (2.5.1) that if X is an affine scheme and is smooth over Y, we have Exty (X, I) = 0 for any quasi-coherent Ox-module /. Finally, we arrive at (b), by using, for X, Y, S affine, and any given /, the natural exact sequence (EGA Orv 20.2.3) (2.5.2) 0 -> Dery(Ox,/) -»? Ders(Ox,/) -»? Ders(0YJJ) -> A Exty(X,I) ->? Exts(X,I) -»? Exts(y,/./), where the arrows other than d are the obvious arrows of functoriality, and d associates to an 5-derivation D : Oy -> f*I the F-extension defined by the ring of dual numbers Ox ffi I and the homomorphism a \-> f*a + Da of Oy in /*(Ox ffi I)- Observe that if / : X -> Y is a morphism locally of finite presentation of affine schemes (i.e. corresponding to a homomorphism of rings A ->? B making B an A-algebra of finite presentation), then, for that / is smooth, it is necessary and sufficient that for any quasi-coherent Ox-module /, we have Exty(X,/) = 0 (the sufficiency rises from the definition, and the necessity was already noted above). Assertions 2.4 (b) and (c) have converses, which furnish a very convenient criteria of smoothness. Their verfication is easy, starting from previous considerations. Proposition 2.6. (a) In the situation of 1.4, assume gf smooth. If the sequence (2.4.1) is exact and locally split, then f is smooth. If the canonical homomorphism f*£lY,s ->- fi-x/s 's an isomorphism, then f is Stale. (b) In the situation of 1.5, assume g smooth. If the sequence (2.4.2) is exact and locally split, then f is smooth. If the canonical homomorphism Afx/z ~^ **^^-/y *s an isomorphism, then f is etale. 2.7. Let / : X -> Y be a smooth morphism, assume given x a point of X, and denote by k(x) the residue field of the local ring Ox,x- Let s\,... , sn be sections of Ox in a neighbourhood of x for which the differentials form a basis of £lx/Y at x, i.e., chosen such that the images (dsi)x of dsi in QX,Yx form a basis of this module over Ox,x, or such that the images (dsi)x of dsi in Qx/y ® ^(x) f°rm a basis of LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 106 this vector space over k(x). Since Qx/y is locally free of finite type, there exists an open neighbourhood U of x such that the Sj are defined over U and that the ds{ form a basis of ^x/y\u- ^ne si then define a F-morphism of U in the affine space of dimension n over Y: a = (si,... ,sn):U -^AY = Y[tu... ,*"]. According to 1.6 and 2.6 (a), s is eto/e. We say that the Sj form a /oca^ coordinate system of X on 7 over U (or, if U is not specified, at x). A smooth morphism is therefore locally composed of an etale morphism and of the projection of a standard affine space. 2.8. Now assume given the situation of 1.5, by assuming g is smooth, and let i be a point of X. According to 2.4 (c) and 2.6 (b), for that / to be smooth in a neighbourhood of x, it is necessary and sufficient that there exists sections si,... , sr of / in a neighbourhood of x, generating Ix and such that the (dsi)(x) are linearly independent in f2^,y(x) = Qlz/y ®k(x) (where k(x) is the residue field of Oz,x, which is also that of Ox,x)- F°r this reason, 2.6 (b) is referred to as the jacobian criterion. Suppose / is smooth in a neighbourhood of x (or at x, like one says sometimes), and let si,... ,sr be sections of / generating /in a neighbourhood of x. Then, for that the Sj defines a minimal system of generators of Ix (i.e. induces a basis of / k(x) = Ix/mxIx, or still forms a basis of I/I2 = Afx/z m a neighbourhood of x), it is necessary and sufficient that the (dsi)(x) are linearly independent in ^?z/y(x)5- Therefore, wherever this is the case, if we supplement the Sj by sections Sj (r+1 < j < r + n) of Oz in a neighbourhood of x such that the (dsi)(x) (1 < i < r + n) form a basis of VLlZjY{x), then the S{ (1 < i < n) define an etale F-morphism s from an open neighbourhood U of x in Z into the affine space AY+r, such that U fl X is the inverse image of the linear subspace with equations t\ = - - - = tr = 0: unx ->- u I if Ay ->- A^ In algebraic geometry, this statement plays the role of the implicit function theorem. 2.9. Let k be a field and let / : X -> Y = Specfc be a morphism. Assuming / smooth, then X is regular (i.e. for any point x of X, the local ring Ox,x is regular, i.e. its maximal ideal mx can be generated by a regular sequence of parameters); moreover, if a; is a closed point, k(x) is a finite separable extension of k, and the dimension of Ox,x is equal to the dimension dim^ X of the irreducible component of X containing x and of the rank of Qx/y a^ x- Conversely, if k is perfect, and if X is regular, then / is smooth. More generally, we have the following criterion, left as an easy verification from 2.7 and 2.8 : 5Or still that the sequence (sj) is O^-regular at x, i.e. that the corresponding Kozul complex is a resolution of Ox in a neighbourhood of x (cf. (SGA 6 VII 1.4) and (EGA IV 17.12.1)). 107 SMOOTHNESS AND LIFTINGS Proposition 2.10. Let f : X -^Y be a morphism locally of finite presentation (2.1). The following conditions are equivalent : (i) / is smooth; (ii) / is flat and the geometric fibers of f are regular schemes. (We say that / is flat if for any point x of X, Ox,x is a flat module over Oyj(x) ? A geometric fiber of / is the reduced scheme of a fiber Xy = X Xy Spec k(y) of / at a point y by an extension of scalars to an algebraic closure of k(y).) If / : X -> Y is smooth, and a; is a point of X, the integer dima:(/) := dimfc(a.) ttlx/Y ® k(x) = ?gox,^x/Y,x is called the relative dimension of / at x. By the classical theory of dimension (EGA IV 17.10.2), this is the dimension of the irreducible component of the fiber Xf(x) containing x. Since Qx/y *s l°caUy free of finite type, it is a locally constant function of x. It is zero if and only if / is etale, in other words, / is etale if and only if / is locally of finite presentation, flat and net (it is this criterion which is taken as the definition of an etale in (SGA 1 I)). If / is smooth and of pure relative dimension r, i.e. of constant relative dimension equal to the integer r, then the de Rham complex ilx/Y (1-7-1) is zero in degree > r, and Qx/Y 1S locally free of rank (^); in particular, flrx ,Y is an invertible Ox-module. Smooth morphisms occupy a central place in the theory of infinitesimal deformations. The following two propositions summarize this. They are however of a more technical nature than the preceeding statements, and as they will be useful only in the proof of 5.1, we will advise the reader to refer to it at that time there. Proposition 2.11. Assume given a diagram (2.2.1), with f smooth. Let L be the ideal of i. (a) There exists an obstruction c(g0) GExt^pSfijf/y,/) for which the vanishing is necessary and sufficient for the existence of a Y-morphism (global) g : T -t X extending go (i.e. such that gi = go). (b) If c(go) = 0, the set of extensions g of go is an affine space under Rom(g^nx/Y,I). Since £lx/Y is locally free of finite type, there is a canonical isomorphism (2.11.1) Ext^fi^/y, J) ~ ^(To.^om^fi^/y,/)) (and 7iom(gQflx,Y,I) - go^x/Y®^ where Tx/y is the tangent sheaf (1.2.5)). Set G = 'Hom(gofl1x,Y,I). According to (1.2.6), if U is an open subscheme of T with corresponding Uq over T0, two extensions of go\u0 to U "differ" by a section of G over Uq (and being given an extension, one can modify it by "adding" a section of G). Since go locally extends by definition of the smoothness of /, we then conclude that the sheaf P over T0 associating to Uq the set of extensions of go\u0 to U, is a torsor under G. Assertions (a) and (b) follow from this: c(go) is the class of this torsor. More explicitly, if (Ui)i¤E is an open covering of T and gi an extension of go over Ui, then, over Ui fl Uj, gi - gj is a F-derivation Dy of Ox with values in LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 108 ffo *(I\Uir\Uj)> i-e- a homomorphism of Qx/Y mt° 9° *(I\UinUj), i-e- finally a section of G over Ui fl Uj, and the (5^-) form a cocycle, for which the class is c{go). Note that if T (or what amounts to the same T0) is affine, then H1(T0,nom(g*nx/Y,I))=0 and consequently go admits a global extension to T. Proposition 2.12. Assume given i : Yq -t Y a thickening of order 1 with ideal I, and /o : Xq -^ Yq 0, smooth morphism. (a) There exists an obstruction W(/o)GExt2(^o/yo,/0*I) for which the vanishing is necessary and sufficient for the existence of a smooth lifting Xq over Y, i.e. by definition, of a smooth Y-scheme X equipped with a Yq-isomorphism Yq Xyl~ Xq6 . (eee)If ui(fo) = 0, the set of isomorphism classes of liftings of Xq over Y is an affine space under Ext (flx ,Y ,fo*I) (where by definition, if X\ and X2 are liftings of Xq, an isomorphism of X\ onto X2 is a Y-isomorphism of X\ on X'2 inducing the identity on Xq). (fff)If X is a lifting of Xq over Y, the group of automorphisms of X (i.e. Y-automorphisms of X inducing the identity on Xq) is naturally identified with nom(nXo/Yoj*i). Since Qx ,Y is locally free of finite type, there is, for all ieZ,a canonical isomorphism (2.12.1) Ext'(n^o/yo)/0*/) ~ H\XQ,nom{VL\olYo,fQ[I)) (and ?{om(QXa ,Y ,f^I) ~ TXo/Yocg>f^I). If X0 is affine, the second term of (2.12.1) is zero for i > 1, and consequently there exists a lifting of X0 over Y, and two such liftings are isomorphic. 2.13. Here is an outline of the proof of 2.12. The data of a lifting X is equivalent to that of a cartesian square Xq -4 X hi if Yq -4 Y, with / smooth, let J be the ideal of thickness j. The flatness of / (2.10) implies that the homomorphism f£I -t J induced from this square is an isomorphism. (It is moreover easy to verify that conversely, if X is a F-extension of Xq by J such that the corresponding homomorphism f£I -> J is an isomorphism, then X is automatically a lifting of X0.) Assertion (c) is therefore a particular case of 2.11 (b). The identification consists of associating with an automorphism u of X the "derivation" u - ldx- Similarly, if X\ and X2 are two liftings of X0, 2.11 (a) implies that Xi and X2 are isomorphisms if X0 is affine, and that the set of isomorphisms 6In this section, when we speak of a lifting of a smooth IVscheme, it will be implicit, unless mentioned to the contrary, that we are thinking of it as a smooth lifting. 109 3. FROBENIUS AND CARTIER ISOMORPHISM of X\ over X2 is then an affine space under Hom(f2^ ,Y , f$I). Assertions (a) and (b) come about formally. The verification of (b) is analogous to that of 2.11: If X\ and X2 are two liftings of X0, the "difference" of their isomorphism classes is the class of the torsor under 7iom(flx ,Y ,/q/) of the local isomorphisms of X\ on X2- (We also observe that the classes of F-extensions X\ and X2 of X0 by JqI differ by a unique Fo-extension of Xq by f$I, and invoke (2.5.1).) Finally, we indicate the construction of the obstruction ui(fo), by assuming for simplicity that X0 is separated. First of all, by the jacobian criterion (2.8), the existence of a global lifting is assured in the case where X0 and Y0 are affine, and /o is associated to a homomorphism of rings Aq ->- B$, where B$ is the quotient of an Ao-algebra of polynomials Ao[t\,... ,tn] by the ideal generated by a sequence of elements (gi,... ,gr) such that the dgi are linearly independent at every point x of X0 (to arbitrarily lift the gi). Since (always according to (2.8)) /o is locally of the preceding form, we can choose an open affine covering U = (([/j)o)jg.e of Xq, and for each i, a lifting Ui of (t/j)o over Y. Since X0 has been assumed separated, each intersection (£/«_/)o = {Ui)o fl (Uj)o is affine, and consequently, we can choose an isomorphism of liftings «y of U^u^^ over Uj\(ui:i)o- On a triple intersection (Uijk)o = (Ui)o n (Uj)0 n ([//t)o, the automorphism uijk = u^UjkUij of t/j|(t/i3.fc)0 differs from the identity by a section Cijk = Ujjfc - Id of the sheaf 7iom(fl1x Y ,/q/)- One verifies that (cy/t) is a 2-cocycle of U with values in Hom(rj^o Y /q I), where the class of this cocycle in ff2(X0,^Om(^0il-0,/*/)) does not depend on the choices, and that it vanishes if and only if on a refinement covering, the U{j can be modified in a way in which they glue on the triple intersection, and also define a global lifting X of X0. This is the stated obstruction. Remark 2.14. The theory of gerbes [Gi] and that of the cotangent complex [II], one or the other, allows us to get rid of the separation assumption made above, and especially gives a more conceptual proof of 2.12. 3. Frobenius and Cartier isomorphism The general references for this section are (SGA 5 XV 1) for the definitions and basic properties of Frobenius morphisms, absolute and relative, and [Kl] 7 for the Cartier isomorphism (cf. also [12] 0 2 and [D-I] 1). In this section, p denotes a fixed prime number. 3.1. We say that a scheme X is of characteristic p if pOx = 0, i.e. if the morphism X ->- Spec Z factors (necessary in a unique way) through Spec Fp. If X is a scheme of characteristic p, we define the absolute Frobenius morphism of X (or, simply Frobenius endomorphism, if there is no fear of confusion) to be the endomorphism of X which is the identity over the underlying space of X, and the raising to the p-th power on Ox ? We denote it by Fx ? If X is affine with ring A, Fx corresponds to the Frobenius endomorphism Fa of A, a \-> av. Let / : X -> Y be LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 110 a morphism of schemes. Then there is a commutative square X 1^ X (3-1-1) fi if Y *> Y. Denote by X^ (or X', if there is no ambiguity) the scheme (Y, Fy) Xy X induced from X by the change of base Fy. The morphism Fx defines a unique y-morphism F = Fx/y '? X -> X', giving rise to a commutative diagram X -^» X' -> X (3-1-2) f\ I if Y ^ Y, where the upper composite is Fx and the square is cartesian. We call F the relative Frobenius of X over Y. The morphisms of the upper line induce homeomorphisms on the underlying spaces (Fy is a "universal homeomorphism", i.e. a homeomor-phism and the remainder after any change of base). If Y is affine with ring A, and X is the affine space Ay = Speci?, where B = A\t\,... ,tn], then X' = AY7, and the morphisms F : X ->- X' and X' -> X correspond respectively to the homomorphisms t{ i->-t\ and at{ i->- aptj (a ¤ A). Proposition 3.2. Let Y be a scheme of characteristic p, and f : X ->- Y a smooth morphism of pure relative dimension n (2.10). Then the relative Frobenius F : X ->- X' is a finite and flat morphism, and the Ox1 -algebra F*Ox is locally free of rank pn. In particular, if f is etale, F is an isomorphism, i.e. the square (3.1.1) is cartesian. We first treat the case where n = 0, which requires some commutative algebra: The point is that F is etale, because according to 2.6 (a), an etale F-morphism between F-schemes is automatically etale, and that a morphism which is both etale and radical8 is an open immersion ((SGA 1 I 5.1) or (EGA IV 17.9.1)). Then the case where X is the affine space Ay is immediate: The monomials n^i?*' wrtn 0 < mi < p - 1 form a basis of F*Ox over Ox< ? The general case is deduced from 2.7. Remarks 3.3. (a) Since, according to 2.10, Qx/y ^s l°cany free over Ox of rank (?), it follows from 3.2 that F*flx,Y is locally free over Ox< of rankp?("). (b) The statement of 3.2 relative to n = 0 admits a converse: If Y is of characteristic p and if AT is a F-scheme such that the relative Frobenius Fx/y is an isomorphism, then X is etale over Y (SGA 5 XV 1 Prop. 2). When Y is the spectrum of a field, this is "Mac Lane's criteria". 7It is not true in general that X and X' are isomorphic as Y-schemes, it is the exceptional case here. 8A morphism g : T -s> S is said to be radical if g is injective and, for any point t of T, with image in S, the residue field extension k(s) -s> k(t) is radical. Ill 3. FROBENIUS AND CARTIER ISOMORPHISM 3.4. Let Y be a scheme of characteristic p and / : X -> Y a morphism. Set d = dx/y(l-2.3). If s is a local section of Ox, one has d(sp) = psp~1ds = 0. Since d(sp) = Fx(ds) = F*(l ® cfe), it follows that (a) the canonical homomorphisms (1.3.2) associated to (Fx,Fy) and F, fx^x/y -^ ^X/Yi F tlx'/Y ~^ ^-x/y are zero; (b) the differential of the complex F*fl'x,Y is Ox'-linear; in particular, the sheaves of cycles Zl, with boundaries Bl and the cohomology W = Z% jB% of the complex F*£Ix/y are Cx'-niodules, and the exterior product acting on the graded Ox'~ module 0 ZlF*Q,x,Y (resp. @/HlF*Slx,Y) is a graded anti-commutative algebra. These facts are at the source of miracles of differential calculus in characteristic p. The principal result is the following theorem, due to Cartier [C] : Theorem 3.5. Let Y be a scheme of characteristic p and f : X ->- Y a morphism. (a) There exists a unique homomorphism of graded Ox-algebras Y' satisfying the following two conditions : (i) for i = 0, 7 is given by the homomorphism F* : Ox1 -> F%Ox', (ii) fori = 1, 7 sends l®ds to the class of sp~1ds in 7^1ir*f2^.,y (where l®ds denotes the image of the section ds offlx,Y in flx,,Y. (b) If f is smooth, 7 is an isomorphism. In case (b), 7 is called the Cartier isomorphism, and is denoted by C_1. Its inverse, or the composite 0Z'F.Jfyy->0n X'/Y of its inverse with the projection of 0 Zl onto 0"H8, where Zl denotes the sheaf of cycles of F*Q,X,Y in degree i, is denoted by C. It is this latter homomorphism which was initially defined by Cartier, and which we sometimes call the Cartier operation. The adopted presentation in 3.5 is due to Grothendieck (handwritten notes), and detailed in [Kl] 7. When Y is a perfect scheme, i.e. such that Fy is an automorphism, for example if Y is the spectrum of a perfect field, one of the most significant cases for applications is this: If / is smooth, C_1 gives by composition with the isomorphism ®Wx/Y^($)(Fy)xM X'/Y (where (Fy)x;X' -> X is the isomorphism induced from Fy by change of base) an isomorphism Cabs : ^Jj^x/Y -^ KjpWFx *^x/Y that we call the absolute Cartier isomorphism. Corollary 3.6. Let Y be a scheme of characteristic p and f : X ->- Y a smooth morphism. Then for any i, the sheaves of Ox1 -modules r*ilxiY, Z r^ilx/Y, H rSfilxiY, rl r*ilxiY LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 112 are locally free of finite type (where Z% resp. Bl denotes the sheaf of cycles resp. boundaries in degree i). Taking into account 3.3 (a) and the exactness of i*1*, it suffices to apply 3.5 (b), while proceeding by descending induction on i. We briefly indicate the proof of 3.5, according to [Kl] 7. For (a), it amounts to the same, taking into account (1.3.2), to construct the composite of 7 with the homomorphism 0f!Ly -> @(-FV)x *^x'/y ^e_ a homomorphism of graded Ox-algebras 7abS : 0<4/y "? 07^* .n"x/Y satisfying the analogous conditions to (i) and (ii), i.e. given in degree zero by Fx, and in degree 1 sending ds to the class of sp~1ds. However the map of Ox in UXFX Mx/ Y sending a local section s of Ox onto the class of sp 1ds is a F-derivation (this is a result of the identity p~x((X + Y)v - Xp - Yp) = Y.oKiKpP'1 {Pi)XP~iYi in %[X,Y]). By (1.2.4), it defines the desired homomorphism (7abs)1- Since the exterior algebra ($)QX,Y is strictly anti-commutative ("strictly" means to say that the elements of odd degree are of square zero), it is likewise of its sub-quotient 0 WFX *fl'x,Y, and consequently there exists a unique homomorphism of graded algebras 7abs extending the homomorphisms (7abs)° = Fx and (7abs)1- For (b), one can assume, according to 2.7, that / factors into where h is the canonical projection and g is etale. Given the square (3.1.1) relative to g, being cartesian according to 3.2, it is likewise the same of the analogous square with the relative Frobenius to Y X A X' (3.6.1) gl lg> Z A Z', where one sets for abbreviation Ay = Z. According to 2.4 (b), the homomorphism g*fllz,Y -> flx/Y 1S an isomorphism. The square (3.6.1) being cartesian and F finite, thus furnishes an isomorphism of complexes of Ox-modules (3.6.2) g'*FM'z/Y -»? F,STX/Y. Since g' is etale, therefore flat, the homomorphism (3.6.3) g'-H'F.Sl'z/Y ->? WFM'x/y induced from (3.6.2) is an isomorphism. Since on the other hand g'*£llz, ,Y ->- ^x1 /y is an isomorphism (g' being etale), it follows (by functoriality of 7) that it suffices to prove (b) for Z. By analogous arguments (extension of scalars and Kunneth) one can easy reduce to Y = SpecFp and n = 1, i.e. Z = SpecFp[t]. Then Z' = Z, the monomials l,t,... ,tp_1 form a basis of F*Oz over Oz, and since the differential d: F*Oz ->- FMZ = (F*Oz)dt sends t onto if^dt, one concludes that H°F*nz/¥p (resp. 'H1F*Q,ziv ) is free over Oz with basis 1 (resp. tp~1dt), and therefore that 7 is an isomorphism. 113 3. FROBENIUS AND CARTIER ISOMORPHISM 3.7. There is a close link between Cartier isomorphism and Frobenius lifting. This was known by Cartier, and it serves as motivation for its construction. The decomposition and degeneration theorems of [D-I] originates from this, see n°5. It consists of the following. Let i : Tq ->- T be a thickening of order 1 and go : Sq ->- To a flat morphism. By lifting to a To-scheme So over T one extends a flat T-scheme over S equipped with a To-isomorphism To Xy S ~ Sq, i.e. a cartesian square So -4 5 ffo 4- 4-ff T0 -4 T with g flat. If / (resp. J) is the ideal of thickening i (resp. j), the flatness of g implies that the canonical homomorphism g^I -> J is an isomorphism (cf. 2.13). Take for i the thickening SpecFp -? SpecZ/p2Z, of the ideal generated by p. Let F0 be a scheme of characteristic p, and let F be a lifting of Y0 over Z/p'2Z. The ideal of F0 in ^ is therefore pOy, and the flatness of F over TLjp^TL implies that multiplication by p induces an isomorphism (3.7.1) p:OY0^pOY. Now let /o : X0 -> Y0 be a smooth morphism of ¥p-schemes. Denote by To : X0 -> X0 the Frobenius of X0 relative to Y0. Assume given a (smooth) lifting X (resp. X') of Xq (resp. X0) over Y and a F-morphism T : X -? X' lifting T0, i.e. such that the square X0 -> X To 4 IF *o ">- *' commutes. (We have seen that there exists obstructions to the existence of X, X', and T, cf. 2.11 and 2.12, and that these objects, whenever they exist, are not unique. We will return to this later.) Proposition 3.8. Let f0 : X0 ->- F0 and F : X ^ X' be given as in 3.7. Then: (a) multiplication by p induces an isomorphism P-^Xo/Yo ^P^X/Y- (b) the image of the canonical homomorphism r '? '''X'/Y ~* ^*^X/Y is contained in pF*£l^-,Y. (c) Denote by - FoMx0/Yo LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 114 the homomorphism "induced from F* by division by p", i.e. the unique horao-morphism rendering the square commutative 1 ^* 1 X'/Y ^ P-^0 *^X/Y I tp nx^/Y0 ->? F0 *nx0/Y0 ? Then the image of - k is given by (ai,a2) h->- a\. If k = Fp, then W2(k) ~ Z/p2Z, the isomorphism being given by (01,02) ^ t(oi) +pr(a2), where r denotes the multiplicative section of Z/p2Z -> ¥p. (For an overall discussion of the theory of Witt vectors, see [S] II 6, [D-G] V.) In this case, if X0 is a smooth F0-scheme (i.e. a smooth fc-scheme), and since the absolute Frobenius of Fo is an automorphism, lifting X0 over Y = Spec W2(k) is equivalent to lifting X'0, and according to 2.12, the obstruction to the existence 115 4. DERIVED CATEGORIES AND SPECTRAL SEQUENCES of such a lifting is found in Ext2(fix , Ox0) - H'2(X0,TXo) 9- If this obstruction is zero, one can choose a lifting X' of X'0 and a lifting X of X0, and then the obstruction to a lifting F : X -> X' of the relative Frobenius F0 is found in Ext1 (Fo*^,,^) - Ext1^,,^ *0Xo) (2.II)10. In every case, these two obstructions are locally zero, and even as soon as X$ is affine. The choice of a lifting F furnishes then, according to 3.8, a relatively explicit description of the Cartier isomorphism in degree 1 (and therefore in every degree, by multiplicativity). 4. Derived categories and spectral sequences There are many reference sources on this subject at various levels. The reader with pressing obligations can consult [13], which can be used as an introduction and contains a broad bibliography. We will limit ourselves here by recalling some fundamental points which we will use in the following section. 4.1. Let A be an abelian category (in practice, A will be the category of Ox-modules of a scheme X). We denote by C(A) the category of A-complexes, with differential of degree 1, and further denote by L* (or L) for such a complex > V ->- Li+1 ->----. We say that L is with lower bounded degree (resp. upper, resp. with bounded degree) if L% = 0 for i sufficiently small (resp. sufficiently large, resp. outside of a bounded interval of Z). We denote by ZlL = Kerd : Ll ->- Li+1, BlL = Im d : L1'1 -> L%, H%L = Z%L/BlL, respectively the objects of cycles, boundaries and cohomology in degree i. If A is the category of Ox-modules, we write C(X) in place of C(A), and often WL instead of HlL for an object of C(X) (in order to indicate that it acts on the cohomology sheaf in degree i, and not on the global cohomology group H'(X,L)). For n £ Z, the naive truncation L-n (resp. L-n) of a complex L is the quotient (resp. the subcomplex) of L which coincides with L in degree < n (resp. > n) and has zero components elsewhere. The canonical truncation r<"L (resp. t>"L) is the subcomplex (resp. quotient) of L with components L% for i < n, ZlL for i = n and 0 for i > n (resp: U for i > n, Ll/BlL for i = n and 0 for i < n). One sets r? L induces an isomorphism on Hl for i nL induces an isomorphism on Hl for i > n. For n £ Z, the translate L[n] of a complex L is the complex with components L[n]1 = Ln+% and with differential g?l["] = (-l)radt. A complex L is said to be concentrated in degree r (resp. m ifte interval [a, b]) if L* = 0 for i ^ r (resp. i ^ [a, b]). An object i? of A is often considered as a complex concentrated in degree zero. The complex E[-n] is then concentrated in degree n, with component E in this degree. 9We omit here, for abbreviation, /Yg in the notation of differentials. 10One can show ( [Me-Sr] Appendix) that the obstruction to a choice of (X,X',F) such that X' is the inverse image of X by the Frobenius automorphism of W-2(k) is found in Ext1 (0X' > B1F*Q,°X ); more precisely, such a triplet (X, X', F) exists if and only if the extension class 0 -> B1F"n'Xo -> Z1FMx0 "^> ^x' "^ ° (particular case i = 1 of the Cartier isomorphism 3.5) is zero. See [Sr] for an application to another proof of the principal theorem of [D-I]. LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 116 A homomorphism of complexes u : L ->- M is called a quasi-isomorphism if Hlu is an isomorphism for all i. We say that a complex K is acyclic if if* If = 0 for all i. If w : L -> M is a homomorphism of complexes, the cone TV = C(w) of u is the complex denned by N% = Lt+1 ffi Ml, with differential d(x, y) = {-dix, ux + duty)-For that m to be a quasi-isomorphism, it is necessary and sufficient that C(u) is acyclic. 4.2. Denote by K(A) the category of complexes of A up to homotopy, i.e. the category having the same objects as C(A) but for which the set of arrows of L in M is the set of homotopy classes of morphisms of L into M. The derived category of A, denoted by D(A), is the category induced from K(A) by formally reversing the (homotopy classes of) quasi-isomorphisms: The quasi-isomorphisms of K (A) become isomorphisms in D(A) and D{A) is universal for this property. When A is the category of Ox-modules over a ringed space X, we write D(X) instead of D{A). The categories K{A) and D{A) are additive categories, and one has canonical additive functors C(A) -> K{A) -»? D(A). The category D(A) has the same objects as C(A). Its arrows are calculated "by fractions" from those of K(A): An arrow u : L -> M of D{A) is defined by a couple of arrows of C(A) of the type LA L' A M or L^M'^M, where s and t are quasi-isomorphisms. More precisely, one shows that the homotopy classes of quasi-isomorphisms with source M (resp. target L) form a filtered category11 (resp. the opposite of a filtered category) and that one has HomZ5(A)(JL,M) = lim Hom^{A) (L, M') = lim HomJf(i)(L', M) t-.M^M1 s:L'-¥L as i (resp. s) runs over the preceeding category (resp. its opposite). If L, M are complexes, we set, for i £ Z, Ext*(L,M) =HomZ3(j4)(L,M[i]) = HomD(j4)(L[-i], M). The functors H% and the canonical truncation functors tj on C(A) naturally extend to D(A). On the other hand, it is not the same as the naive truncation functors. 4.3. We denote by D+(A) (resp. D~(A), resp. Db(A)) the full subcate gory of D(A) formed from complexes L cohomologically bounded below (resp. above, resp. bounded), i.e. such that HlL = 0 for i small enough (resp. large enough, resp. outside a bounded interval). If A contains sufficiently many injectives (i.e. if any object of A embeds in an injective), for example if A is the category of Ox- modules over a scheme X, then any object of D+{A) is isomorphic to a complex, with bounded below degree, formed from injectives, and the category D+(A) is equivalent to the full subcategory of K(A) formed from such complexes. 11A category I is said to be filtered if it satisfies the following conditions (a) and (b): (ggg)For any two arrows f,g:i-tj, there exists an arrow h : j -S> k such that hf = hg. (hhh)Assume given any objects i and j, there exists an object k and arrows / : i -s> k, g : j -> k. 117 4. DERIVED CATEGORIES AND SPECTRAL SEQUENCES 4.4. The categories K(A) and D{A) are not in general abelian, but possess a triangle category structure, in the sense of Verdier [V]. This structure is defined by the family of distinguished triangles. A triangle is a sequence of arrows T = (L ->- M -> N -> L[l]) of if(A) (resp. D(A)). A morphism of T in T' = (L' ->- M' -> N' -> L'[l]) is a triplet (u : L -> 1/, v.M^M', w: N ->- TV') such that the three squares formed with it, v, w, u[l] commute. A triangle is said to be distinguished if it is isomorphic to a triangle of the form L 4 M 4 C(u) A L[l], where w is the cone of a morphism of complexes u, and « (resp. p) denotes the obvious inclusion (resp. the opposite of the projection). Any short exact sequence of complexes 0 ->? i? A F -» G ->? 0 defines a distinguished triangle D(A), by means of the natural quasi-isomorphism C(u) ->- G, and any distinguished triangle of D{A) is isomorphic to a triangle of this type. Any distinguished triangle T = (L ->- M ->- TV -? £[1]) of -D(A) gives rise to a long exact sequence > HlL -> HlM -> iTAT 4 Hi+1L ->----, >Ext*(£;,L) -^Ext^M) -?Ext^./V) ->. Exti+1 (£, L) ->----, >Ext{(N,E) -^Ext{(M,E) -^Ext{(L,E) -> Exti+1(N, E) -> --- , for E £ ob -D(A). If the triangle T is associated to a short exact sequence given explicitly above, the operator d of the first of these sequences is the usual boundary operator (this is the reason for the convention of sign in the definition of p). 4.5. Let L be a complex of A and i £ Z. The quotient t- T iTL[-i] -> r- r>j_iL -> T>iL ->- H^Ll-i + 2]. Finally, ^_i,t1L := 7>i_iT Z*L -> 0) defines a distinguished triangle W^Ll-i + 1] ->- r^^L -> #*£[-«] ->, which furnishes a canonical element (4.5.1) q e Ext^iTL,^-1!,). The triplet (Ht~1L, HtL, Cj) is an invariant of L in -D(A). It permits its reconstruction up to an isomorphism if L is cohomologically concentrated in degree i - 1 and i. One can show that the Cj universally realizes the differential g?2 of the spectral sequences of derived functors applied to L (cf. Verdier's theorem12, or [D3]). 2Which should be appearing soon in Asterique. LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 118 4.6. Let L be an object of Dh(A). We say that L is decomposable if L is isomorphic, in D(A), to a complex with zero differential. If L is decomposable, and if u : L' -> L is an isomorphism of D(A), with L' having zero differential, then u induces isomorphisms Ln ->- HlL. In particular V has bounded degree and V = <§,Lli[-i] (in C{A)) (4.1), therefore (4.6.1) L~0ff'L[-i] (in F>(A)). Conversely, if F satisfies (4.6.1), F is trivially decomposable. If L is decomposable, one calls a decomposition of F the choice of an isomorphism (4.6.1) inducing the identity on H% for all i. There exists a finite sequence of obstructions to the decomposability of F: The first are the classes Cj (4.5.1); if the Cj are zero, there are secondary obstructions in Ext3(F*F,F8_2F), etc. In addition, if F is decomposable, F admits in general many decompositions. In the following section, we are especially interested in the case when F is concentrated in degree 0 and 1 : L = (F° ->- F1). In this case: (iii)the class c\ £ Ext2(FxF, H°L) is the obstruction to the decomposability of F; (jjj)the giving of a decomposition of F is equivalent to that of a morphism i71L[-1] ->- F inducing the identity on H1; (kkk)The set of decompositions of F is an affine space under Ext^F^F, H°L) ([D-I] 3.1). 4.7. We now return for example to [HI], II for the definition of the derived functors®, RTiom13, FHom, F/*, F/*, RF in the derived category D{X), where X is a variable scheme, and the description of certain remarkable relations between these functors. We need only recall that these functors are, compared to each argument, exact functors, i.e. transform distinguished triangles to distinguished triangles, and are "calculated" in the following way: (lll)For Beob D(X), F £ ob D~(X), E®F ~ E F' if F ~ F' in D(X), with F' having upper bounded degree (4.1) and with flat components. For given F, there exists a quasi-isomorphism F' -> F with F' of the preceding type; moreover the homotopy classes of such quasi-isomorphisms form a coinitial system (in the category of classes of quasi-isomorphisms with target F, cf. 4.2). (mmm)For E £ ob D(X), F £ ob D+(X), if F ~ F', with F' having lower bounded degree and with injective components, then KHom(E,F) ~ /Hom°{E,F') and i?Hom(i?,F) ~ Hom*(_E, F'). For given F, there exists a quasi-isomorphism F -? F' with F' of the preceding type (and the homotopy classes of such quasi-isomorphisms form a cofinal system). (nnn)For / : X -> F and F £ ob £>+(X), if F ~ F', with F' having lower bounded degree and with flasque components (for example, injective), then Rf*E ~ /*F' and RT(X,E) ~ T(X,E'). One simply writes Hl{X,E) instead of HlRT(X,E)-and more generally, one defines in the same way, i?/" : F>+(X, /_1(Oy)) ->- F+(F), where D(X, f~1(Oy)) denotes the derived category of the category of complexes of /_1(0y)-modules (the de Rham complex Q'x/Y is sucn a complex). (ooo)For / : X -> Y and F £ ob F"(F), F/*F ~ /*F' if F ~ F', with F' having upper bounded degrees and with flat components. 13An error of sign slipped into the definition of the complex Horn* (L, M) in [HI] p. 64: For u G Hom(L% M'+n), it necessarily reads du = d o u + ( -l)n+1u o d. 119 5. DECOMPOSITION, DEGENERATION AND VANISHING THEOREMS 4.8. It can be said that spectral sequences are perhaps one of the most avoided objects in mathematics, and yet at the same time, are one of the most useful algebraic tools for cohomology. This is particularly true of derived categories, which sometimes contributes to this, but they remain essential. There are many references, the oldest ([C-E], XV) being one of the best. In these notes, we will be especially interested in the spectral sequence called the Hodge to de Rham, for which we will recall the definition. Let T : A ->- B be an additive functor between abelian categories. Assume that A has sufficiently many injectives. Then T admits a right derived functor RT :D+(A) -+D+(B), which is calculated by RT{K) ~ T(K') if K ->- K' is a quasi-isomorphism with K' with bounded below degree and with injective components. The objects of cohomology Hi o RT : D+(A) ->- B are denoted by RlT. For ifeob D(A), with bounded below degree, there is a spectral sequence (4.8.1) E\j = RiTiK*) => R*T(K), called the first spectral sequence of hypercohomology of T. It is obtained in the following way: Chooses a resolution K ->- L of K by a bicomplex L, such that each column L%% is an injective resolution of K%. If sL denotes the associated simple complex, the resulting homomorphism of complexes K -> sL is a quasi-isomorphism, therefore RT(K) ~ T(sL) = sT(L), RT(K{) ~ T(Li#), and the filtration of sT(L) by the first degree of L given rise to (4.8.1). Let if be a field and X a fc-scheme. The group (cf. (1.7.1) and 4.7 (c)) (4.8.2) HR(X/k) = H'iX,^) = r(Specfe,i?7.(^/*)) (where / : X ->- Spec k is the structure morphism) is called i-th de Rham cohomology group of X/k. This is a k-vector space. The spectral sequence (4.8.1) relative to the functor T(X, -) and the complex Q'x,k is called the Hodge to de Rham spectral sequence of X/k : (4.8.3) E[j = W(X, Wx/k) => H^R(X/k). This is a spectral sequence of fc-vector spaces. The groups H^{X, Qx/k) are caned the Hodge cohomology groups of X over k. If X is proper over k ([H2] II 4) (for example, projective over k, i.e. a closed subscheme of a projective space FJJ), and since the $lxik are coherent sheaves (2.1), the finiteness theorem of Serre-Grothendieck ([H2] III 5.2 in the projective case, (EGA III 3) in the general case) implies that the Hodge cohomology groups of X over k are finite dimensional fc-vector spaces. By the spectral sequence (4.8.3), it follows from this that the de Rham cohomology groups H^K{X/k) are also finite dimensional over k. Moreover, for each n, one has (4.8.4) Y, dim* ffJ'(X' nx/k) ^ dim* HBn(X/k), i+j=n with equality for all n if and only if the Hodge to de Rham spectral sequence of X over k degenerates at Ei, i.e. the differential dr is zero for all r > 1. LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 120 5. Decomposition, degeneration and vanishing theorems in characteristic p > 0 In this section, as in n°3, p will denote a fixed prime number. The main result is the following theorem ([D-I] 2.1, 3.7): Theorem 5.1. Let S be a scheme of characteristic p. Assume given a (flat) lifting T of S over J,/p2J, (3.7). Let X be a smooth S-scheme, and let us denote as in 3.1, F : X ->- X' the relative Frobenius of X/S. Then if X' admits a (smooth) lifting overT, the complex of Ox1 -modules T-modules (4.6). 5.2. Before beginning the proof, note that a decomposition of T- X' admits a global lifting G : Z -» Z', where Z (resp. Z') lifts X (resp. X') over T. Let (5.3.1) tpa-.Qnic./si-il^F.n'x/s be the homomorphism of complexes, with i-th component LplG, defined in the following way: (p°G = F*:Ox^FmOx; ipG:nx,/s^FMx/s is the homomorphism "G*/p" defined in 3.8 (c). For i > 1, tpG is composed with A*y>c? and of the product AtF^fl1x,s -t F*QX,S. Then (pa is a quasi-isomorphism, inducing the Cartier isomorphism C~l on TV for all i. This is immediate. Step B. This is the principal step. We show that the giving of a lifting Z' of X' over T allows us to define a decomposition of T ?? fijfvshl] -> F.n'x/s of D(X') (and not C(X'j) inducing C_1 over %1. With this intention, we need to compare the homomorphisms ipG of (5.3.1) associated to any other lifting of F with target Z'. 121 5. DECOMPOSITION, DEGENERATION AND VANISHING THEOREMS Lemma 5.4. To any pair {G\ : Z\ ->- Z',G2 ? Z2 ->- Z') of liftings of F is associated canonically a homomorphism h(G1,G2):n1x,/s^F,Ox dh{G\,G2). If Gz : Z% ->- Z' is a third lifting of F, one has h(G1,G2) + h(G2,G3) = h(G1,G3). Let us suppose initially that Z\ and Z2 are isomorphic (in the sense of 2.12 (b)). Choose an isomorphism u : Z\ -± Z2. Then G2u and G\ lift F, i.e. extend to Z\ the composite X -> Z <-^ Z1. Therefore according to 2.11 (b), they differ by a homomorphism hu of F*flx, ,s in Ox, or what amounts to the same, of QX'/s in F*Ox? If v is a second isomorphism of Z\ onto Z2, then taking into account 3.4 (a), it follows from 2.11 (b) that u and v differ by a homomorphism "u - v" : ftx/s ~^ ®x' therefore G2u and G2v differ by the composite of "u - v" and the homomorphism F*flx, ,s -t £lx/s, which is zero, a fortiori G2u = G2v. Therefore hu does not depend on the choice of u. Since Z\ and Z2 are locally isomorphic according to 2.11 (a), we deduce from this a homomorphism (5.4.1) characterized by the property that if u is an isomorphism of Z\ onto Z2 over an open subset U of X (recall still that Z\, Z2 and X have the same underlying space), the restriction of h(Gi, G2) to U is the homomorphism hu, the "difference" between G\ and G2u. The formula LpxG - - Z' of F\ui. We then arrange for each i, a homomorphism of complexes /* - fd '? ^x'/s\Uil -1-] "^ ^*^x/s|c/; 14 of (5.3.1), and for each pair (i,j), a homomorphism hij = h{Gi\Uij,Gj\Uij) : ^x'/s\Uij ~^ F*^'x/s\Ui of (5.4.1), where C/y = [/« fl Uj. These datum are connected by fj-fi = dhij (on Uij), hij + hjk = hik (on Uijk =UinUjnUk)- They make it possible to define a homomorphism of complexes of Ox1 -modules where C(U,F*flx,s) is the simple complex associated to the Cech bicomplex of the covering U with values in F*Vlx,s. The components of this complex are given by C(u,F,nx/s)n= 0 C\u,F.nx/s) a+b=n We identify the underlying spaces of X and X' by means of F (3.1). LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 122 with differential d = di+d,2, where d\ is induced by the differential of the de Rham complex and d ?? ^7S[-1] "> F^'x/s to be the arrow of D(X') composed with u ,G-.s and with the inverse of e (4.2). If (U = (Ui)i£i, (Gi)i¤i) and (V = (Vj)jej, (Gj)jgj) are two choices of systems of Frobenius liftings, then by considering the covering W]JV, indexed by I\JJ, formed from the Ui and from Vj, it follows that ipz, does not depend on choices (cf. [D-I] p. 253). Moreover 1, an arrow of D(X') (Vz'fi = ® ? ? ? ® ? (Hx'/shl])®* ">- (FM'x/s)^. Since QX'/s *s l°cauy free of finite type, we have (4.7 (a)) (*) (^x'/si-Mf1 ^ (nx,/s)0i[-il and similarly, since the F*flx,s are locally free of finite type (3.3 (a)), (**) (FMx/sfJi ^ (FMx/sfJi- We then define for i < p, ? tox'/sl-Q ->? F*nx/S as the composite (via (*) and (**)) of the standard antisymmetrization arrow 123 5. DECOMPOSITION, DEGENERATION AND VANISHING THEOREMS l (well defined because of the assumption i < p), of the arrow (ip^,)®1-, and of the product arrow (F*Q'X,S)0J ->- F*Q'x,g. Since the antisymmetrication arrow is a section of the projection of (flx,,s)®t onto flx,,s, the multiplicative property of the Cartier isomorphism results in iplz, inducing C_1 over W, and this completes the proof of the theorem. Taking into account 3.9, we then deduce: Corollary 5.5. Let k be a perfect field of characteristic p, and let X be a smooth scheme over S = Specfc. If X is lifted over T = SpecW^fc), then T- S (where S = Specfc) induces, for all (i,j), an isomorphism FgH^(X,flx,k) ^> H^X', flx, ,fc), and in particular, we have dimkW{X,ilx/K) = dimkW(X',nxl/k). In addition, since F : X ->- X' is a homeomorphism, one has canonically, for all n, Hn(X',FM'x/k) ^ Hn{X,nx/k) = H£R(X/k). Finally, if X is lifted over W^ik), a decomposition

F*Q'X,S of F*flx,s in D(X') induces, for all n, an isomorphism 0 W(X',nxi/k) ^ Hn(X',F.ilx/k). i+j=n It follows from this that one has, for all n, Y, dimfc W{X, Qx/k) = dimfc H£R(X/k), i+j=n and according to 4.8, this results in the degeneration at E\ of the Hodge to de Rham spectral sequence. LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 124 5.7. For the remaining part to follow, the reader can consult [H2] II, III. Let k be a ring and X a projective fc-scheme, i.e. admits a closed fc-immersion i in a standard projective space P = Fk = Yvo]k[to, ? - ? ,tn]- Let L be an invertible sheaf over X. Recall that: (i) L is very ample if one has L ~ i*Op(l) for such a closed immersion i, which means that there exists global sections Sj ¤ T(X,L) (0 < j < r) defining a closed immersion n-> (sq(x), ... , sr(x)) of X in P; (ii) L is ample, if there exists n > 0 such that L®n is very ample. Assume L ample. Then, according to Serre's theorem ([H2] II 5.17, III 5.2): (ppp)For any coherent sheaf E on X, there exists an integer no such that for any n > no, E ® L®n is generated by a finite number of its global sections, i.e. a quotient of Ox for suitable N. (qqq)For any coherent sheaf E on X, there exists an integer no such that for any n > no and all i > 1, one has Hi(X,E®L®n) =0. The theorem which follows is an analog in characteristic p, of the Kodaira-Akizuki-Nakano vanishing theorem [KAN], [AkN]: Theorem 5.8. Let k be a field of characteristic p, and let X be a smooth projective k-scheme. Let L be an ample invertible sheaf on X. Then if X is of pure dimension d < p (cf. 2.10,) and is lifted over W2(k), we have (rrr)Hj(X,L(g>nix/k)=0 for i + j>d, (sss)W{X,L®-1 ®VLix/k)=Q iov i+j ^®-1 ® ^x/fc) are canonically dual. Formulas (5.8.1) and (5.8.2) are therefore equivalent. It will be more convenient to prove (5.8.2). By Serre's vanishing theorem (5.7 (b)), there exists n > 0 such that W{X,L®Pn (gi nx/k) = 0 for all j > 0 and all i. By Serre duality, it follows that H:'{X,L®~pn ® Slx/k) = 0 for all j < d and all i, and in particular for all (i,j) such that i + j < d. Proceeding by descending induction on n, it therefore suffices to prove the following assertion: (*) if M is an invertible sheaf over X satisfying H^{X,M^P (gi £lx/k) = 0 for all (i,j) such that i + j < d, then H^{X,M ® Qxik) = 0 for all (i,j) such that i+j- m®p, and therefore an isomorphism F' *M' ~ M®p, where F : X ->- X' is the relative Frobenius and M' is the inverse image of M over X'. We deduce, for all i, the following isomorphisms of Ox>-modules (**) M' (8) FMx/k - F*(F*M' (8) Qx/k) ~ F*(M0P ® Clx/k). 125 5. DECOMPOSITION, DEGENERATION AND VANISHING THEOREMS Let us consider the spectral sequence (4.8.1) relative to the functor T = Y{X',-) and on the complex K = M' FMx/k) => H*(X',M' ® FM'x/k). The hypothesis and (**) imply that E[J = 0 for i + j < d. Therefore Hn(X',M' ®F*n'x/k) =0 forn- S the structure morphism. Consider then the spectral sequence (4.8.1) relative to the functor /* and the complex £lx/s, Eij = &f.nx/s^iTMnx/s), which is called the relative Hodge to de Rham spectral sequence (of X over S). Then if X is smooth and proper of relative dimension < p, and if X' LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 126 is lifted over T, this spectral sequence degenerates at E\ and the sheaves R^f^x/s are l°cauy free of finite type. ([D-I] 4.1.5). (www)The latter assertion of 5.5 and the conclusions of 5.6 and 5.8 still remain true if one only assumes X of dimension < p ([D-l] 2.3). This is a consequence of Grothendieck duality for the morphism F. (xxx)There exists many examples of smooth and proper surfaces X over an algebraically closed field k of characteristic p for which the Hodge to de Rham spectral sequence does not degenerate at E\ and which does not satisfy the vanishing property of Kodaira-Akizuki-Nakano type of 5.8. (Taking into account (3) if p = 2, or 5.6 and 5.8 if p > 2, these surfaces are therefore not lifted over W^ik).) See ([D-I] 2.6 and 2.10) for a bibliography on this subject. (yyy)Formulas (5.8.1) and (5.8.2) are still useful if d = 2 < p, X is liftable over W2{k) and L is only assumed numerically positive, i.e. satisfies L ? L > 0 and L ? 0(D) > 0 for any effective divisor D, see [D-I] 2. 6. From characteristic p > 0 to characteristic zero 6.0. There exists a standard technique in algebraic geometry, which allows one to prove certain statements of geometric nature15, formulas over a base field of characteristic zero, from analogous statements over a field of characteristic p > 0, even a finite field. Roughly speaking, it consists of a given base field K, which is in characteristic zero, as an inductive limit of its Z-sub-algebras of finite type Af. Data on K, provided that they satisfy certain finiteness conditions, arise by extension of scalars from similar data on one of the Ai, say Aio = B. It is then enough to solve the similar problem on T = Speci?, that which is seemingly more difficult. The advantage however, is that the closed points of T are then the spectrum of a finite field, and that in a sense which one can specify, there are many such points, so that it is enough to check the statement posed on T after sufficient specialization to these points. There is the business dealing with a problem of characteristic p > 0, where one has the range of corresponding methods (Frobenius, Cartier isomorphism, etc.); moreover one can exploit the fact of being able to choose the characteristic large enough. The two ingredients of the method are: (a) results of passing to the limit, presented in great generality in (EGA IV 8), allowing the "spreading out" of certain data and properties on K, to similar data and properties on B; (b) density properties of closed points on schemes such that the schemes are of finite type over a field or over Z (EGA IV 10). 6.1. Let ((Ai)iej, Uij : At -? Aj (i < j)) be a filtered inductive system of rings, with inductive limit A, and denote by Uj : A{ -> A the canonical homomorphism. The two very important examples are: (i) a ring A written as an inductive limit of its sub-Z-algebras of finite type; (ii) the localization Ap of a ring A at a prime ideal p written as an inductive limit of localizations Af (= A[l/f]) for / ^ p. The prototype of problems and results of type (a) above is the following. Let (E{) = ((Ei)i¤i, V{j : Ei -t Ej) be an inductive system of Aj-modules, having for inductive limit the A-module E. Let us agree to say that (Ei) is cartesian if, 15I.e. stable by base extension, as opposed to statements of arithmetic nature, where the base plays an essential role. 127 6. FROM CHARACTERISTIC p > 0 TO CHARACTERISTIC ZERO for any i < j, Vij (which is an Aj-linear homomorphism of Fj into Ej considered as an Aj-module via u,j) induces, by adjunction, an isomorphism (A,-linear) of u\,-Ei = Aj (giAi Ei in Ej. In this case, the canonical homomorphism V{ : Ei -t E induces for all i, an isomorphism u*Et (= A (g)^ Ei) -^» E. Let ((Fj)j ¤ I,Wij) be a second inductive system of Aj-modules. If (Ei) is cartesian, the RomAi(Ei,Fi) form an inductive system of Aj-modules: The transition map for i < j associated to fi : Ei -? Fi is the homomorphism Ej -? F,- composed with the inverse of the isomorphism of Aj ? Aj ® Fi, and from the map of Aj (gi Fi in Fj defined by Wy. If F denotes the inductive limit of the Fi, one has analogous maps of H01114, (Ei,Fi) into Hom^F, F), which defines a homomorphism (6.1.1) indlimHom^OE^Fi) ->? Hoiru(F,F). We can then pose the following two questions : (zzz)Being given an A-module E, does there exist io £ I and an Aj0-module Ei0 such that E results from Eio by an extension of scalars of Aio to A (or, that which amounts to the same, does there exist a cartesian inductive system (Ei), indexed by {i £ I\i > io}, for which the limit is E) ? (aaaa)If there exists io such that (Ei) and (Fj) are cartesian for i > i0, is the map (6.1.1) (where the inductive limit is reached for i > io) an isomorphism ? There is a positive answer to the two questions with the help of hypothesis of finite presentation. (Recall that a module is said to be finitely presented if it is the cokernel of a homomorphism between free modules of the finite type.) More precisely, there is the following statement, which can be verified immediately: Lemma 6.1.2. With the preceding notation: (bbbb)If E is a finitely presented A-module, there exists io £ I and an Ai0 -module of finite presentation Ei0 such that u* Ei0 ~ E. (cccc)Let (Ei), (Fi) be two inductive systems, cartesian for i > io, with respective inductive limits E and F. Then if Ei0 is finitely presented, the map (6.1.1) is an isomorphism. It follows from this that if E is finitely presented, the Eio which arises by extension of scalars is essentially unique, in this sense that if Eit is another choice (Ei0 and F^ both being two finite presentations), there exists V2 with «2 > i\ and «2 > *o such that Fj0 and F^ become isomorphisms by extensions of scalars to Ai2. The Si = Spec At form a projective system of schemes for which S = Spec A is the projective limit. If (Xi, v^ : Xj ->- X{) is a projective system of Sj-schemes, we say that this system is cartesian for i > io if, for io < i < j, the transition arrow v^ gives a cartesian square Xj -> Xj Oj -> Oj. In this case, the 5-scheme induced from Xio by extension of scalars to S is the projective limit of Xi. If (Yi) is a second projective system of 5j-schemes, cartesian for i > i0, the projective limit Y (= S Xj, Yio) of the Homs^Xj,Yi) form a LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 128 projective system, and one has an analogous map to (6.1.1): (6.1.3) projlimHomSj(XuYi) ->- Roms(X,Y). We can then formulate similar questions to (1) and (2) above. They have similar answers, with the condition of replacing the hypothesis of finite presentation for modules by the hypothesis of finite presentation for schemes (a morphism of schemes X -> Y is said to be a finite presentation if it is locally of finite presentation (2.1) and "quasi-compact and quasi-separated", which means that X is a finite union of open affine subsets Ua over an open affine subset Va of Y and that the intersections Ua H Up have the same property; if Y is Noetherian, X is finitely presented over Y if and only if X is of finite type over Y, i.e. locally of finite type over Y (2.1) and Noetherian): Proposition 6.2. (a) If X is an S-scheme of finite presentation, there exists io G I and an Si0 -scheme Xi0 of finite presentation for which X is induced by a base change. (b) If(Xi), (Y{) are two projective systems of Si-schemes, cartesian for i > io, and if Xi0 and Yi0 are finitely presented over Si0, then the map (6.1.3) is bijective. As in the preceding, it follows from this that Xio of 6.2 (a) is essentially unique (two such schemes become ^-isomorphic for i large enough). Moreover, the usual properties of an 5-scheme of finite presentation (or of a morphism between such) are already determined to some extent, over Si for i large enough. Here are some, which are useful statements in themselves (the reader will find a long list in (EGA IV 8, 11.2, 17.7)): Proposition 6.3. Let X be an S-scheme of finite presentation. We assume that X has one of the following properties V: projective, proper, smooth. Then there exists io £ I and an Si0 -scheme Xi0 of finite presentation, having the same property V, for which X is induced by base change. The case where V is "projective" is easy: X is the closed subscheme of a standard projective space P = P£ defined by an ideal locally of finite type. It suffices to lift P, and then the closed immersion (i.e. the corresponding quotient of Op, cf. 6.11). The "proper" case is less immediate, but roughly, it goes back to a classical result, namely Chow's Lemma (cf. EGA IV 8.10.5). The "smooth" case is a little more difficult (which uses criterion 2.10), see (EGA IV 11.2.6 and 17.7.8). With regard to the properties of type (b) evoked in 6.0, we will only have need of the following result: Proposition 6.4. Let S be a scheme of finite type over Z. Then: (dddd)If x is a closed point of S, the residue field k(x) is a finite field, (eeee)All locally closed nonempty components Z of S contain a closed point of S. For the proof, we refer to (EGA IV 10.4.6, 10.4.7), or in the case where S is affine, this goes back to (Bourbaki, Alg. Com. V, by 3, n° 4) (this is a consequence of Hilbert's theorem of zeros). We will need to apply 6.4 (b) to the case where Z is the smooth part of S, S being assumed integral16 : A scheme is said to be integral if it is reduced and irreducible. 129 6. FROM CHARACTERISTIC p > 0 TO CHARACTERISTIC ZERO Proposition 6.5. Let S be an integral scheme of finite type over Z. The set of points x of S for which S is smooth over SpecZ is a nonempty open set of S. In particular, if A is a Z -algebra of finite type, and integral, there exists s £ A, s ^ 0, such that Spec As is smooth over Z. The openness of the set of smooth points of a morphism locally of finite presentation is a general fact, which is a consequence for example of the jacobi criterion 2.6 (a), cf. (EGA IV 12.1.6.). That in the present case this open set is nonempty follows from a local variant of 2.10 and from the fact that the generic fiber of S is smooth over Q at its generic point, Q being perfect. We will finally have to use some standard results of compatability of direct images by a base change (or, as one says sometimes, of cohomological cleanliness). Not wanting to weigh down our exposition, we will state them only in the case where it will be useful for us to have, for the Hodge cohomology and the de Rham cohomology. Proposition 6.6. Let S be an affine scheme17, Noetherian, integral, and f : X ->- S a smooth and proper morphism. (ffff)The sheaves R^ f*flx,s and Rnf*Qx,s are coherent. There exists a nonempty open set U of S such that, for any (i,j) and any n, the restrictions to U of these sheaves are locally free of finite type. (gggg)For any i £ Z and for any morphism g : S1 ->- S, if f : X' ->- S" denotes the induced scheme of X by base change via g, the canonical arrows of D(S') (according to base change) (hhhh)Lg*Rf*nx/s -»? Rf',Wx,/s, (iiii)Lg*Rf,n'x/s -»- Rf'^'x,/S, are isomorphisms. (c) Fix i ¤ Z and assume that for any j, the sheaf Rif*Qx,s is locally free over S, of constant rank h%K Then for any j, the base change arrow (induced from (6.6.1); (6.6.3) g*Rjf.nx/s -»? Rjfi,nxl/s, is an isomorphism. In particular, i?J fl$lx, is, is locally free of rank h%K (d) Suppose that for all n, Rnf*flx,s is locally free of constant rank hn. Then for all n, the change of base arrow (induced from (6.62)J (6.6.4) gmRnf.nx/s -»? Rnfinx,/S, is an isomorphism. In particular, Rn fl$lx, is, is locally free of rank hn. Let us briefly indicate the proof. The fact that the i?Jf*^x/s are coherent is a particular case of the finiteness theorem of Grothendieck (EGA III 3) (or [H2] III 8.8 in the projective case). The coherence of Rnf*fl'x,s follows from this by the relative Hodge to de Rham spectral sequence (5.9(2)). For the second 7The hypothesis "affine" is unnecessary; we use it only to facilitate the proof of (b). LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 130 assertion of (a), denote by A the (integral) ring of S, K its field of fractions, which is therefore the local ring of S at its generic point n. We set for abbreviation Rif*Wx/s = Wj, RnfMx/s = nn. The fiber of Hij (resp. Hn) at n is free of finite type (a if-vector space of finite dimension), and is the inductive limit of ^?\d(s) (resP- ^ld(s))' f°r s transversing A, D(s) denoting "the open complement" of s, i.e. SpecAs = X - V(s). By 6.1.2 it follows from this that there exists s such that T~LVDts\ (resp. 'H1\Dts)) are free of finite type. For (b), we choose a finite covering U of X by open affine sets, denote by W the open covering of X' induced from U by base change. Since S is affine and that X is proper, therefore separated over S, the finite intersections of open sets in U are affine and similarly the finite intersections of open sets in W are (relatively) affine18 over 5". Consequently (cf. [H2] III 8.7), Rf*nx/S (resp. Rf!"nx,/S,) is represented by f*C(U,nx/s) (resp. f[C(U',ilx,/$'))?> where we denote here by C(U,») the alternating complex of cochains. By the compatability of fP by a base change, there is a canonical isomorphism of complexes g*fAu,nx/s) ^> f'Au',nxl/sl). Since the complex f*C(U, ^x/s) 1S bounded and with flat components, this isomorphism realizes the isomorphism (6.6.1). Similarly, Rf*fl'x,s (resp. Rflflx,,s,)) is represented by f*C(U,Q,x,s) (resp. flC(W,Q'X, ,s,j) (where C denotes this time the associated simple complex of the Cech bicomplex), and one has a canonical isomorphism of complexes g'fAWx/s) ^ f'Au',nx,/s,), which realizes the isomorphism (6.6.2). Assertions (c) and (d) follow from (b) and from the following lemma, for which we leave the verification to the reader: Lemma 6.7. Let A be a Noetherian ring and E a complex of A-modules such that H%{E) are projective of finite type for any i and zero for almost all i. Then: (jjjj)E is isomorphic, in D(A), to a bounded complex with projective components of finite type. (kkkk)If E is bounded and with projective components of finite type, for any A-algebra B, and for all i, the canonical homomorphism BtoAWiE) -^HHB^aE) is an isomorphism. Remarks 6.8. (a) A complex of A-modules, isomorphic in D (A), to a bounded complex with projective components of finite type is said to be perfect. One must be aware that if E is perfect, it is not true in general, that the H%(E) are projective of finite type. One can show that under the hypothesis of 6.6, the complexes Rf*flx,s and Rf*Qx,s are perfect over S (and not only over U). The notion of a perfect complex plays an important role in numerous questions in algebraic geometry. (b) In the statements of 6.6 concerning flx/s, one can replace Slx/s ^ an^ l°cany free Ox-module F of finite type (even coherent and relatively flat over S): The 18A morphism of schemes is said to be affine if the inverse image of any affine open set is affine. 131 6. FROM CHARACTERISTIC p > 0 TO CHARACTERISTIC ZERO conclusions of (a), (b) and (c) are still valid on the condition of replacing flx,/s, by the inverse image sheaf F' of F over X'. Similarly, the complex Rf*F is perfect over S. We are now able to state and prove the promised application of 5.6: Theorem 6.9 (Hodge Degeneration Theorem). Let K be a field of characteristic zero, and X a smooth and proper K-scheme. Then the Hodge spectral sequence of X overK (4.8.3) E^ =W{X,nix/K)^H^K{XlK) degenerates at E\. Set AmiKW{X,nx/K) = hij, dim H£R(X/K) = hn. It suffices to prove that for all n, hn = J2i+j=n ^ (c^- (4-8.3)). Write K as an inductive limit of the family (A\)\¤l of its sub-Z-algebras of finite type. According to 6.3, there exists a ¤ L and a smooth and proper 5a-scheme Xa (where Sa = SpecAa) for which X is induced by base change Specif ->- Sa. Even if it means to replace Aa by Aa[i_1] for a suitable nonzero t £ Aa, we can assume, according to 6.5, that Sa is smooth over SpecZ. Abbreviate Aa by A, Sa by S, Xa by X, and denote by / : X ->- S the structure morphism. Again by replacing A by A[t_1], we can according to 6.6 (a), assume that the sheaves R? f*$l\is (resp. Rnf*^x/s) are ^ree °^ constant rank, necessarily equal then to /iy (resp. hn) according to 6.6 (c) and (d). Since the relative dimension of X over S is a locally constant function and that X is quasi-compact, one can in addition choose an integer d which bounds this dimension at any point of X and therefore the dimension of the fibers of X over S at any point of S. Applying 6.4 (b) to Z = SpecA[l/N] for suitable N (say, the product of prime numbers < d), one can choose a closed point s of S, for which the residue field k = k(s) (a finite field) is of characteristic p > d. Since S is smooth over Spec Z, the canonical morphism Spec k -> S (a closed immersion) is extended (by definition of smoothness (2.2)) to a morphism g : SpecVl^fc) ->- S, where W^ik) is the ring of Witt vectors of length 2 over k (3.9). Denote by Y = 3£s the fiber of X over s = Specfc and Y\ the scheme over SpecVl^fc) induced from X by the base change g. We therefore have cartesian squares: Y -> Fi -> X Spec W2 (k) -4 S d, (6.10.2) Hj(X,L®~1 ® nx/K) = 0 iori + j io, with limit X = S Xg, X{0. (llll)If E is a finitely presented Ox -module, there exists i > io and a Oxt -module Ei of finite presentation for which E is induced by extension of scalars. If E is locally free (resp. locally free of rank r), there exists j > i such that Ej = Oxj ®ox- Ei *s locally free (resp. locally free of rank r). If X is projective over S and E is an ample invertible Ox-module (resp. very ample) (5.7), there exists j > i such that Xj is projective over Sj and Ej is ample invertible (resp. very ample). (mmmm)LetEi0, F{0 be finitely presented Ox-modules, and consider the systems (Ei), (Fi) which are induced by extension of scalars over the Xi for i > io, as well as the modules E and F which are induced by extension of scalars over X. Then there is a natural map ind lim Hom0x. (E^, F^) ->- Hom0x (E, F), i>i0 ' which is bijective. The proof of (b), then of the first two assertions of (a), brings us back to 6.1.2. For the latter part of (a), it suffices treat the case where E is very ample, i.e. corresponds to a closed immersion h : X -> P = frs such that h*Op(l) ~ E. For i sufficiently large, one lifts h by an 5j-morphism hi : Xt ->- Pt = Frs_ and E by invertible Ei over Xi. Even if it means to increase i, hi is a closed immersion and the isomorphism h*Op(l) ~ E comes from an isomorphism h^Op{(l) ~ Ei; Et is then very ample. Proving 6.10. Proceeding as in the proof of 6.9, and moreover applying 6.11, one can find a subring A of K of finite type and smooth over Z, a smooth projective morphism / : X -> S = Spec A of pure relative dimension d, for which X ->- Spec K is induced by base change, and an ample invertible Ojf-niodule £ for which L is induced by extension of scalars. By virtue of 6.6 and 6.8 (b), one can assume, even if it means to replace A by A[t_1], that the sheaves B? f*(M®fllx/'S), where M = C (resp. £®-1), are free of finite type, of constant rank, necessarily equal, according to 6.8 (b), to hij(L) = dimKHi(X,L®nx/K) (resp. hij(L®-1) = Hi(X,L®-x Qx,Kj). Let us choose then g : Spec W2(k) ->- S as in the proof of 6.9. The inverse image sheaf C8 of C over Y = Xs is ample. According to 6.6 and 6.8 (b), one has dimfc H'J(Y, Cs n\r/k) = hij(L), and dimfc W(Y, Cf-1® ft^/fc) = hij(L®"1). The conclusion then follows from 5.8. 133 7. RECENT DEVELOPMENTS AND OPEN PROBLEMS Remark 6.12. In a similar manner, the Ramanujam vanishing theorem on surfaces [Ram] follows from the variant of 5.8 relative to the numerically positive sheaves (cf. 5.9 (5)). 7. Recent developments and open problems A. Divisors with normal crossings, semi-stable reduction, and logarithmic structures. 7.1. Let S be a scheme, A a smooth S-scheme, and D a closed subscheme of A. We say that D is a divisor with normal crossings relative to S (or simply, relative) if, "locally for the etale topology on X", the couple (X, D) is "isomorphic" to the couple formed from the standard affine space Ag = S\t\,... ,tn] and from the divisor V{t\ ? ? ? tr) of the equation t\ ? ? ? tr = 0, for 0 < r < n (the case r = 0 corresponds to t\ ? ? -tr = 1 and V(t\ ? ? -tr) = 0). This means that there exists an etale covering (Xi)iei of X (i.e. a family of etale morphisms Xt -> X for which the union of the images is X) such that, if Dt = Aj xx D is the closed subscheme induced by D on Xt, there exists an etale morphism Xt ->- Ag for which there is a cartesian square Di -> X, V(h---tr) -> A? (n and r dependant on i). In other words, that there exists a coordinate system (xi,... ,xn) on Xi in the sense of 2.7 (defining the etale morphism Xj -> Ag) such that Dt is the closed subscheme of the equation x\ ? ? ? xr = 0. This definition is modeled after the analogous definition in complex analytic geometry (cf. [Dl]), where "locally for the etale topology" is replaced by "locally for the classical topology", and "etale morphism" by "local isomorphism". A standard example of a divisor with normal crossings relative to S = Specfc, k a field of characteristic different from 2, is the cubic with double point D = Speck[x,y]j(y2 - x'2(x - 1)) in the affine plane X = Spec k[x,y]. (Observe in this example that there does not exist a system of coordinates (xj) as above on a Zariski open covering of A, an etale extension (extraction of a square root of x - 1) being necessary for to make possible such a system in a neighbourhood of the origin.) The notion of a divisor with the normal crossings D ^ A relative to S is stable by etale localization over A and by base change S" -> S. If D ^ X is a relative divisor with normal crossings, and if j : U = X\D ^ X is the inclusion of the open complement, we define a subcomplex (7.1.1) ) of j*^l\f/s, called the de Rham complex of X/S with logarithmic poles along D, by the condition that a local section uj of j*$l\jis belong to Qx,s(\ogD) if and only if ui and dw have at most a simple pole along D (i.e. are such that if / is a local equation of D, fuj (resp. / duj) is a section of Slx/S (resp. ^^/s) (NB. / is necessarily a nonzero divisor in Ox))- One easily sees that the Ox-modules Qx,s(\ogD) are locally free of finite type, that Qx,s(\ogD) = A1 flx,s(logD), and that if as above, (x\,... ,xn) are coordinates on an A' etale neighbourhood LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 134 over X where D has for equation x\ ? - ? xr = 0, Qx,s(\ogD) is free with basis There is a natural variant in complex analytic geometry of the construction (7.1.1) (cf. [Dl]). If S = SpecC and D C X is a(n algebraic) divisor with normal crossings, the complex of analytic sheaves associated to (7.1.1) on the analytic space Xan associated to X, n^/c(iogr>)an = n5f"/c(iogr>an), calculates the transcendental cohomology of U with values in C: There is a canonical isomorphism (in the derived category D(Xan,C)) (7.1.2) RjX-n'x/s(logDr\ and consequently an isomorphism (7.1.3) H\Uan,C) ~ Hi{Xaa,nx/s{logD)aD) (loc. cit.). Moreover, if X is proper over C, the comparison theorem of Serre [GAGA] allows us to deduce from (7.1.3) the isomorphism (7.1.4) H^U^X) ~iP(X,Jfys(log£>)). Moreover the filtration F of H*(X,flx,s(logD)), being the outcome of the first spectral sequence of hypercohomology of X with values in flx,s(logD), (7.1.5) Ef = H«(X,Slx/s(logD) => H"+"(X,nx/s(logD)) is the Hodge filtration of the natural mixed Hodge structure of i7*(C/an,Z) defined by Deligne, and the spectral sequence (7.1.5) degenerates at E\ ([D2]). Just as in the case where D = 0 (6.9), this degeneration can be shown by reduction to characteristic p > 0. Indeed, we have the following result which generalizes 5.1 and for which the proof is analogous ([D-I] 4.2.3): Theorem 7.2. Let S be a scheme of characteristic p > 0, 5~ a flat lifting of S over Z/p2Z, X a smooth S-scheme and D C X a relative divisor with normal crossings. Denote by F : X ->- X' the relative Frobenius of X/S. If the couple (X',D') admits a lifting (X'~,D'~) over 5~, where X'~ is smooth and Z)'~ C X'~ is a relative divisor with normal crossings, the complex of Ox-modules T xx ? ? ? xn (n > 1). In other words, if S = A^, the scheme AJ, considered as 5-scheme by s, is the sub-5-scheme of Ag = S[x\,... ,xn] = T\x\,... ,xn,t] with equation x\ - ? - xn = t. 135 7. RECENT DEVELOPMENTS AND OPEN PROBLEMS The morphism s is smooth outside 0 and its fiber at 0 is the divisor D with equation (x\ ? - ? xn = 0), a divisor with normal crossings relative to T, but not with S (a "vertical" divisor). More generally, if S is a smooth T-scheme of relative dimension 1 and E C S a relative divisor with normal crossings (if T is the spectrum of an algebraically closed field, E is therefore simply a finite set of rational points of S), we say that the 5-scheme X has semi-stable reduction along E if, locally for the etale topology (over X and over S) the morphism X -> S is of the form soj, with g smooth, s being the morphism considered above. The divisor D = XxsEcXis then a divisor with normal crossings relative to T (but not to 5)19. An elementary example is furnished by the "Legendre family" X = Speck[x,y,t]/(y2 - x(x - l)(x - t)) over S = Specfc[t], (k a field of characteristic 7^ 2), which has semi-stable reduction on {0} U {1}, the fiber at each of these points being isomorphic to the cubic with double point considered above. The interest in the notion of semi-stable reduction comes from the semi-stable reduction conjecture, which roughly asserts that locally, after suitable ramification of the base, a smooth morphism can be extended to a morphism with semi-stable reduction. This conjecture was established by Grothendieck-Deligne-Mumford and Artin-Winters ([G], [A-W], [D-M]) in any characteristic but relative dimension 1, and Mumford ([M]) in characteristic zero and arbitrary relative dimension. If / : X ->- S has semi-stable reduction along E, we define the de Rham complex with relative logarithmic poles (7.3.1) cjx/s = Qx/s(logD/E), with components cux,s = Atuix,s, where ^x/s is the quotient of f)^/T(logD) by the image of f*Q1s,T(\ogE) and the differential is induced from that of flx,T(logD) by passing to the quotient. This complex has locally free components of finite type (in the case of the morphism s above, u!x/s *s isomorphic to ((§)Oxdxi/xi)/Ox(J2dxi/xi) (therefore free with basis dxt/xi, i > 2)). It induces on the smooth open part U of X over S the usual de Rham complex Q\j/S, and one can show that this is the unique extension over X of this complex which has locally free components of finite type. Moreover, if one sets for abbreviation, wx/t = ^x/tQ°&D), <^s/t = ^s7t(1°6^')> there is an exact sequence (7.3.2) 0 -> a4/T (8i ux/s[-i\ -»? uj'x/t -> u)x/s -> 0, where the arrow to the left is given by ab h->- f*aAb. This exact sequence plays an important role in the regularity theorem of the Gauss-Manin connection (cf. [K2] and the article of Bertin-Peters in this volume). There also exists a variant of these constructions in complex analytic geometry. Assume that T = Spec C, that S is a smooth curve over C, E C S the divisor reduced to a point 0, and that / : X -? S is a morphism with semi-stable reduction at {0}, with fiber Y at 0. (Y is therefore a divisor with normal crossings in X relative to C.) We consider the complex (7.3.3) ujy = C{0} <8>os wx/si with components the locally free sheaves of finite type ujy = Oy ox ^x/s- Steen-brink [St] has shown that the complex analogue Wyan over Yan (which is also the 19 One can similarly define a notion of semi-stable reduction along E without the hypothesis on the relative dimension of S over T, cf. [15]. LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 136 complex of sheaves associated to LUy over yan) embodies the complex of neighbouring cycles R^(C) of / at 0, so that if moreover / is proper, H*(Y, ujy) "calculates" ff*(I(a",C) for t "close enough" to 0. Steenbrink also shows (under this extra hypothesis) that the spectral sequences (nnnn)Ef = B?Uu?xls => Rp+qf*tux/s and (oooo)Epq = Hq(Y,ujY) =>? Hp+q(Y,ujY) degenerate at E\ and that the sheaves Rqf*ux,s are locally free of finite type and of formation compatible with any base change. These results form part of the construction of a limiting mixed Hodge structure on H*(X^n,l,) for t tending to 0 (loc. cit.). They can by themselves, be proven by reduction to characteristic p > 0 ([15]). For T of characteristic p > 0, and / : X ->- S with semi-stable reduction along E C S, the complexes ujx/s an(^ (7.3.6) uj'd = Od ®0s wj/s (where D = E x$ X) indeed give rise to Cartier morphisms (of the type of 3.5), and under the hypothesis of a suitable lifting modulo p2, t 0, W = W(k) the ring of Witt vectors over k, X a smooth and proper W-scheme of relative dimension < p. Then for any integer n > 1, the Hodge to de Rham spectral sequence (7.6.1) E{j = Hi(Xn, n^,wJ => Hi+((Xn/Wn) 137 7. RECENT DEVELOPMENTS AND OPEN PROBLEMS degenerates at E\, where Wn = Wn(k) = W/pnW denotes the ring of Witt vectors of length n over k and Xn the scheme over Wn induced from X by reduction modulo pn (i.e. by extension of scalars ofW to Wn). (pppp)For n = 1, we have Wn = Wn(k) = W/pnW and we recover statement 5.6, apart from which in 7.6, we assume given a lifting of X over W (rather than over Wz)'20. Under the hypothesis of 7.6, it is not true in general, for n > 2, that the de Rham complex ftx ,w (which is, a priori, only a complex of sheaves of W^-modules over Xn (or X\, Xn and X\ having the same underlying space)) is decomposable in the corresponding derived category D{X\,Wn). However, the results of Ogus ([Ogl] 8.20) imply that if a denotes the Frobenius automorphism of Wn, a*flx ,w is isomorphic in the derived category D{X\, Wn) of sheaves of Wra-modules over X\, to the complex flx w (p) induced from ftx w by multiplying the differential by p. (NB. For n = 1, we have ^xn/wn W = © ^Jd/fcl-*]-) ^ne conclusion of 7.6 comes about easily, like various additional properties of H£,R(Xn/Wn) (structure called "of Fontaine-Laffaille" - including in particular the fact that the Hodge filtration is formed from direct factors), see [F-M] and [Kal]. (qqqq)The degeneration and decomposition results for which we discussed until now carry over to de Rham complexes of schemes, possibly with logarithmic poles. More generally, we can consider the de Rham complexes with coefficients in modules with integrable connections. Many generalizations of this type have been obtained: For Gauss-Manin coefficients [15], of sheaves of Fontaine-Laffaille [Fa2], of T-crystals [Og2] (besides these last objects providing a common generalization of the previous two). C. Open problems. 7.9. Let k be a perfect field of characteristic p > 0, X a smooth fc-scheme of dimension d, X' the scheme induced from X by base change by the Frobenius automorphism of k, F : X ->- X' the relative Frobenius (3.1). We have seen in 5.9 (1) (a) (with S = Specfc, T = SpecW2(k)) that the following conditions are equivalent : (i) X' - or, that which amounts to the same here, X - is lifted (by a smooth and proper scheme) over W2(k); (ii) T 1 and Z smooth and proper over Wn of dimension < p, then if Z admits a lifting (smooth and proper) over Wn+i, the Hodge to de Rham spectral sequence of Z/Wn degenerates at Ei ([Og2] 8.2.6). This result truly generalizes 5.6. LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 138 (rrrr)If X is proper over k and DR-decomposable, the Hodge to de Rham spectral sequence of X/k degenerates at E\. (ssss)If X is projective over k, of pure dimension d, and DR-decomposable, and if L is an invertible ample sheaf over X, one has the vanishing results of Kodaira-Akizuki-Nakano (5.8.1) and (5.8.2). By virtue of the equivalence between conditions (i) and (ii) above, a necessary condition for that X is DR-decomposable is that X is lifted over W2(k). According to [D-I], it is sufficient if d < p (5.5 and 5.9 (3)). We are unaware if it is always true in general: Problem 7.10. Let X be a smooth fc-scheme of dimension d > p, liftable over W2(k). Is it the case that X is DR-decomposable ? 7.11. Recall (5.5.1) that if X and F lift over W2(k), X is DR-decomposable; this is the case if X is affine, or is a projective space over k. As indicated in [D-I] 2.6 (iv), if X is liftable over W2(k) and if, for any integer n > 1, the product morphism {tiix/k)®n ->- ^x/k admits a section, then X is DR-decomposable (see 8.1 for a proof). This second condition is checked in particular if X is parallelizable, i.e. if Slxik is a free Ox-module (or, that which amounts to the same, the tangent bundle Tx/k, dual of £lx/k> *s trivial), therefore for example if X is an abelian variety. By a theorem of Grothendieck (cf. [Oo] and [17] Appendix 1), any abelian variety over k is lifted over W2(k) (and similarly over W{k)). Therefore any abelian variety over k is DR-decomposable. Another interesting class of liftable fc-schemes (over W(k)) is formed from complete intersections in frk (see the expose of Deligne (SGA 7 XI) for the definitions and basic properties of these objects). But we do not know if those are DR-decomposable. The first unknown case is that of a (smooth) quadric of dimension 3 in characteristic 2. We also don't know if the Grassmannians, and more generally, flag varieties, which are, albeit liftable over W(k), are DR-decomposable (the only known example is projective space!). Problem 7.10, with "liftable over W2(k)" replaced by "liftable over W(k)", is also an open problem. On the other hand, we can replace "liftable over W{k)" by "liftable over AJ\ where A is a totally ramified extension of W(k) ( = ring of complete discrete valuations, finite and flat over W(k), with residue field k, and of degree > 1 over W(k)): Lang [L] has indeed constructed in any characteristicp > 0, a smooth projective fc-surface X liftable over such a ring A of degree 2 over W(k) such that the Hodge to de Rham spectral sequence of X/k does not degenerate at El 7.12. The decomposition statements to which we referred to at the end of 7.3 apply in particular to a smooth curve S over T = Specfc and with a scheme X over S having semi-stable reduction along a divisor with normal crossings E C S (therefore etale over k), for which certain hypothesis of liftability modulo p2 are satisfied. More precisely, if we assume that: (i) There exists a lifting (E~ C T) of (£ C S) over W2 = W2{k) (with S*~ smooth and E~ a relative divisor with normal crossings, i.e. etale over W2), admitting a lifting F~ : 5~ -> 5~ of the Frobenius (absolute) of S such that {F~)-1(E~)=pE~ 21, 21This notation denotes the divisor induced from E~ by the raising to the p-th power of its local equations. 139 8. APPENDIX: PARALLELIZABILITY AND ORDINARY (ii) / is lifted by /~ : X~ ->- 5~ having semi-stable reduction along E~, then t 0, the classical results of Grauert-Riemenschneider or Kawamata-Viehmeg. Neither can we generalize 5.9 (5) in dimension > 2. See [E-V] for a discussion of these questions. 8. Appendix: parallelizability and ordinary In this section, k denotes a perfect field of characteristic p > 0. We denote by Wn = Wn(k) the ring of Witt vectors of length n over k. We begin by giving a proof of the result mentioned in 7.11: Proposition 8.1. Let X be a smooth k-scheme. Assume that X lifts overW2 and that for any n > 1, the product morphism (^x/fc)^? ~~* ^x/k ad'mits a section. Then X is DR-decomposable (7.9). We will have need of the following lemma: LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 140 Lemma 8.2. Let S and T be as in 5.1, X a smooth scheme over S, Z' a (smooth) lifting of X' over T. Let f1 ?? nx>/s[-l] "> Wx/s be the homomorphism Lp\, of D(X') defined in step B of the proof of 5.1, and for n > 1, r : (Slx,/s)®n[-n] -»? FMx/s L L the composite homomorphism ir o (ip1)®11, where ir : (F*flx,s)®n -> F*£lx,s is the product homomorphism. Likewise, we denote by ir : (nx,,s)®n -t £lx,/s ^e product homomorphism. Then for any local section cu of (ilx,/s)®n, one has ?Hn4>n{u))=C-1on{uj), where C_1 : £lX'/s ~~^ 7~LnF*Vlx,s is the Cartier isomorphism. Proof. It suffices to show this for uj of the form u\ ® ? ? ? ® uin, where Wj is a local section of Qx,,s. By functoriality in the Ei £ D(X'), of the product 1 1 L L niE1 ® - - - - Un{Ei® ? ? ? ®En), ai ® ? ? - - - - w") = C_1(wi) A - - - A C-^Wn) in HnFt,nx,s, and therefore that ft"z/>?(wi ® ? ? - ® w") = C_1 (wi A - - - A un) = C"1 o 7r(wi Specfc is the projection and E is the fc-vector space x*(flx,k). Via a, a section of the surjective homomorphism E®n ->- AnE extends to a section of (flx,k)®n ->- ftx/k- Now apply 8.1. Recall (5.5.1) the following definition: Definition 8.4. Let X be a smooth and proper k-scheme. We say that X is ordinary if for any (i,j), one has W{X,B^lxik) = 0, where Bflx,k = d£l%x- F*Ox admits a retraction, i.e. the exact sequence of Ox<-modules (cf. 3.5) (8.5.1) 0 -»? Ox- -> F»Ox 4 F*BQix/k ~> ° is split. We first observe that if X is Frobenius-decomposable, X is liftable over W^ (or, that which amounts to the same (5.9 (1) (a)), T Ox> -»? f%ox 4 F*znx/k A nx,/k -»? o, composed with (8.5.1) and the extension (8.5.2) 0 -»? F.BSlx/k -»? F.ZVLx/k A ilx,/k -> 0, is zero. In general, we are unaware if "Frobenius-decomposable" implies "DR-decomposable". This is the case according to 8.3, if X is parallelizable. But the converse is false. Indeed one has the following result ([Me-Sr] 1.1): If X is a smooth and proper fc-scheme, parallelizable, then the following conditions are equivalent: (tttt)X is Frobenius decomposable; (uuuu)the extension (8.5.2) is split; (vvvv)X is ordinary; (wwww)(for X of pure dimension d) the homomorphism F* : Hd(X', Ox1) -> Hd(X, Ox) induced by the Frobenius is an isomorphism. In particular, if X is ordinary and parallelizable, X lifts over Wi (Nori-Srinivas ([Me-Sr] Appendix) show in fact that for X projective, X lifts to a smooth projective scheme over W). Moreover - this is the principal result of [Me-Sr] - if fc is algebraically closed and X connected, there exists a Galois etale lifting Y ->- X of order of a power of p such that Y is an abelian variety. If X is projective and smooth over fc, ordinary and parallelizable, Nori-Srinivas (loc. cit.) show more precisely that there exists a unique couple (Z,Fz), where Z LUC ILLUSIE, FROBENIUS AND HODGE DEGENERATION 142 is a lifting (projective and smooth) of X over Wi (resp. Wn (n > 2 given), resp. W) and Fz ? Z ->- Z' a lifting of F : X ->? X', where Z' is the inverse image of Z by the Frobenius automorphism of W? (resp. Wn, resp. W). The existence and uniqueness of this lifting, said canonical, was first established by Serre-Tate [Se-Ta] in the case of abelian varieties. As indicated in 5.5.1, this result admits a converse, without the assumption of parallelizability. Proposition 8.6. Let X be a smooth and proper k-scheme. Assume that there exists schemes Z and Z1 lifting respectively X and X' over W^ and a Wz-morphism G : Z -> Z' lifting F : X -> X' 22. Then X is ordinary. This result was obtained independently by Nakkajima [Na]. 8.7. Proof of 8.6. Let G : Z ->- Z' be a lifting of F and

- F*flx the associated homomorphism of complexes, defined in (5.3.1) (one omits jk from the notation of differentials). This homomorphism sends ilx, into F*Zflx (notation of 8.4) and splits the exact sequence (cf. 3.5) (8.7.1) 0 ->- F*Bflx -> F*ZQx A flx, -> 0. We prove, by descending induction on i, that H*(X,BQX) = 0 (i.e. that Hn(X,BQx) = 0 for all n). For i > dimX, BQX = 0. Fix i and assume that we proved H*(X,BQjx) = 0 for j > i. Then we show that H*{X,B^XX) = 0. By the exact cohomology sequence associated to the exact sequence (8.7.2) 0 ->- F^Zfl^1 ->- FM^1 -4 F"Bflx -> 0, the induction hypothesis implies that for any n, one has ^"(jf'.F.zn^1) A ^"(x,^1), and therefore (8.7.3) dimff?(X',F*ZrJ71) = dimi7"(X, f)^1) = dim ir>(X', fl^,1). The sequence (8.7.1) (relative to i - 1) being split, implies that the exact sequence of cohomology gives the short exact sequence 0 -> Hn(X',F»Bil^1) -> Hn{X\F*ZWxx) A iT^X',^,1) -»? 0. 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