% Complex Analytic and Differential Geometry, Chapter VII % J.-P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France \input analgeom.mac \def\Ll{\langle\!\langle} \def\Gg{\rangle\!\rangle} \def\Herm{{\rm Herm}} \def\Nak{{\rm Nak}} \def\Grif{{\rm Grif}} \titlea{Chapter VII}{\newline Positive Vector Bundles and Vanishing Theorems} \begpet In this chapter, we prove a few vanishing theorems for hermitian vector bundles over {\it compact} complex manifolds. All these theorems are based on an a priori inequality for \hbox{$(p,q)$-forms} with values in a vector bundle, known as the Bochner-Kodaira-Nakano inequality. This inequality naturally leads to several positivity notions for the curvature of a vector bundle (Kodaira 1953, 1954), (Griffiths 1969) and (Nakano 1955, 1973). The corresponding algebraic notion of ampleness introduced by (Grothendieck 196?) and (Hartshorne 1966) is also discussed. The differential geometric techniques yield optimal vanishing results in the case of line bundles (Kodaira-Akizuki-Nakano and Girbau vanishing theorems) and also some partial results in the case of vector bundles (Nakano vanishing theorem). As an illustration, we compute the cohomology groups $H^{p,q}(\bbbp^n,\cO(k))\,$; much finer results will be obtained in chapters 8--11. Finally, the Kodaira vanishing theorem is combined with a blowing-up technique in order to establish the projective embedding theorem for manifolds admitting a Hodge metric. \endpet \titleb{1.}{Bochner-Kodaira-Nakano Identity} Let $(X,\omega)$ be a hermitian manifold, $\dim_\bbbc X=n$, and let $E$ be a hermitian holomorphic vector bundle of rank $r$ over $X$. We denote by $D=D'+D''$ its Chern connection (or $D_E$ if we want to specify the bundle), and by $\delta=\delta'+\delta''$ the formal adjoint operator of $D$. The operators $L,\Lambda$ of chapter 6 are extended to vector valued forms in $\Lambda^{p,q}T^\star X\otimes E$ by taking their tensor product with $\Id_E$. The following result extends the commutation relations of chapter 6 to the case of bundle valued operators. \begstat{(1.1) Theorem} If $\tau$ is the operator of type $(1,0)$ defined by $\tau = [\Lambda, d'\omega]$ on $\ci_{\bu,\bu}(X,E)$, then \medskip\noindent $\cmalign{ \hbox{\rm a)}\hfill\qquad &[\delta''_E,L] &= &\ii(D'_E+\tau),\cr \hbox{\rm b)}\hfill\qquad &[\delta'_E,L] &= -&\ii(D''_E+\ovl \tau),\cr \hbox{\rm c)}\hfill\qquad &[\Lambda,D''_E] &= -&\ii(\delta'_E+\tau^\star),\cr \hbox{\rm d)}\hfill\qquad &[\Lambda,D'_E] &= &\ii(\delta''_E+\ovl \tau^\star). \cr}$ \endstat \begproof{} Fix a point $x_0$ in X and a coordinate system $z=(z_1\ld z_n)$ centered at~$x_0$. Then Prop.~V-12.?? shows the existence of a normal coordinate frame $(e_\lambda)$ at~$x_0$. Given any section $s=\sum_\lambda\sigma_\lambda\otimes e_\lambda\in \ci_{p,q}(X,E)$, it is easy to check that the operators $D_E$, $\delta_E'',\,\ldots~$ have Taylor expansions of the type $$D_Es=\sum_\lambda d\sigma_\lambda\otimes e_\lambda+O(|z|),~~~~ \delta_E''s=\sum_\lambda \delta''\sigma_\lambda\otimes e_\lambda+O(|z|), ~\ldots$$ in terms of the scalar valued operators $d$, $\delta$, $\ldots$. Here the terms $O(|z|)$ depend on the curvature coefficients of $E$. The proof of Th.~1.1 is then reduced to the case of scalar valued operators, which is granted by Th.~VI-10.1.\qed \endproof The Bochner-Kodaira-Nakano identity expresses the antiholomorphic La\-pla\-ce operator $\Delta''=D''\delta''+\delta''D''$ acting on $\ci_{\bu,\bu}(X,E)$ in terms of its conjugate operator $\Delta'=D'\delta'+\delta'D'$, plus some extra terms involving the curvature of $E$ and the torsion of the metric $\omega$ (in case $\omega$ is not K\"ahler). Such identities appear frequently in riemannian geometry (Weitzenb\"ock formula). \begstat{(1.2) Theorem} $\Delta''=\Delta'+[\ii\Theta(E),\Lambda]+[D',\tau^\star]- [D'',\ovl\tau^\star]$. \endstat \begproof{} Equality 1.1~d) yields $\delta''=-\ii[\Lambda,D']- \ovl\tau^\star$, hence $$\Delta''=[D'',\delta'']=-\ii[D'',\big[\Lambda,D']\big]-[D'',\ovl\tau^\star].$$ The Jacobi identity~VI-10.2 and relation 1.1~c) imply $$\big[D'',[\Lambda,D']\big]=\big[\Lambda,[D',D'']]+\big[D',[D'',\Lambda]\big] =[\Lambda,\Theta(E)]+\ii[D',\delta'+\tau^\star],$$ taking into account that $[D',D'']=D^2=\Theta(E)$. Theorem~1.2 follows.\qed \endproof \begstat{(1.3) Corollary (Akizuki-Nakano 1955)} If $\omega$ is K\"ahler, then $$\Delta''=\Delta'+[\ii\Theta(E),\Lambda].$$ \endstat In the latter case, $\Delta''-\Delta'$ is therefore an operator of order $0$ closely related to the curvature of $E$. When $\omega$ is not K\"ahler, Formula~1.2 is not really satisfactory, because it involves the first order operators $[D',\tau^\star]$ and $[D'',\ovl\tau^\star]$. In fact, these operators can be combined with $\Delta'$ in order to yield a new positive self-adjoint operator $\Delta'_\tau$. \begstat{(1.4) Theorem (Demailly 1985)} The operator $\Delta'_\tau=[D'+\tau,\delta'+ \tau^\star]$ is a positive and formally self-adjoint operator with the same principal part as the Laplace operator~$\Delta'$. Moreover $$\Delta''=\Delta'_\tau+[\ii\Theta(E),\Lambda]+T_\omega,$$ where $T_\omega$ is an operator of order $0$ depending only on the torsion of the hermitian metric $\omega\,:$ $$T_\omega=\Big[\Lambda,\big[\Lambda,{\ii\over 2}d'd''\omega\big]\Big]- \big[d'\omega,(d'\omega)^\star\big].$$ \endstat \begproof{} The first assertion is clear, because the equality $(D'+\tau)^\star=\delta'+\tau^\star$ implies the self-adjointness of $\Delta'_\tau$ and $$\Ll\Delta'_\tau u,u\Gg=\|D'u+\tau u\|^2+\|\delta'u+\tau^\star u\|^2\ge 0$$ for any compactly supported form $u\in\ci_{p,q}(X,E)$. In order to prove the formula, we need two lemmas. \endproof \begstat{(1.5) Lemma} {\rm a)}$\qquad[L,\tau]=3d'\omega,$ $\qquad~~$ {\rm b)}$\qquad[\Lambda,\tau]=-2\ii\ovl\tau^\star.$ \endstat \begproof{} a) Since $[L,d'\omega]=0$, the Jacobi identity implies $$[L,\tau]=\big[L,[\Lambda,d'\omega]\big]=-\big[d'\omega,[L,\Lambda]\big]= 3d'\omega,$$ taking into account Cor.~VI-10.4 and the fact that $d'\omega$ is of degree $3$. \medskip \noindent b) By 1.1~a) we have $\tau=-\ii[\delta'',L]-D'$, hence $$[\Lambda,\tau]=-i\big[\Lambda,[\delta'',L]\big]-[\Lambda,D']= -i\big(\big[\Lambda,[\delta'',L]\big]+\delta''+\ovl\tau^\star\big).$$ Using again VI-10.4 and the Jacobi identity, we get $$\eqalign{ \big[\Lambda,[\delta'',L]\big]&=-\big[L,[\Lambda,\delta'']\big]-\big[\delta'', [L,\Lambda]\big]\cr &=-\big[[d'',L],\Lambda\big]^\star-\delta''=-[d''\omega,\Lambda]^\star-\delta'' =\ovl\tau^\star-\delta''.\cr}$$ A substitution in the previous equality gives $[\Lambda,\tau]= -2\ii\ovl\tau^\star$.\qed \endproof \begstat{(1.6) Lemma} The following identities hold: \smallskip \item{\rm a)} $[D',\ovl\tau^\star]=-[D',\delta'']=[\tau,\delta''],$ \smallskip \item{\rm b)} $-[D'',\ovl\tau^\star]=[\tau,\delta'+\tau^\star]+T_\omega.$ \endstat \begproof{} a) The Jacobi identity implies $$-\big[D',[\Lambda,D']\big]+\big[D',[D',\Lambda]\big]+\big[\Lambda,[D',D'] \big]=0,$$ hence $-2\big[D',[\Lambda,D']\big]=0$ and likewise $\big[\delta'',[\delta'',L] \big]=0$. Assertion a) is now a consequence of 1.1~a) and d). \medskip \noindent b) In order to verify b), we start from the equality $\ovl\tau^\star={\ii\over 2}[\Lambda,\tau]$ provided by Lemma~1.5 b). It follows that $$[D'',\ovl\tau^\star]={\ii\over 2}\big[D'',[\Lambda,\tau]\big].\leqno(1.7)$$ The Jacobi identity will now be used several times. One obtains $$\leqalignno{ \qquad\big[D'',[\Lambda&,\tau]\big]=\big[\Lambda,[\tau,D'']\big]+ \big[\tau,[D'',\Lambda]\big]~;&(1.8)\cr [\tau,D'']&=[D'',\tau]=\big[D'',[\Lambda,d'\omega]\big] =\big[\Lambda,[d'\omega,D'']\big]+\big[d'\omega,[D'',\Lambda]\big]&(1.9)\cr &=[\Lambda,d''d'\omega]+[d'\omega,A]\cr}$$ with $A=[D'',\Lambda]=\ii(\delta'+\tau^\star)$. From (1.9) we deduce $$\big[\Lambda,[\tau,D'']\big]=\big[\Lambda,[\Lambda,d''d'\omega]\big]+ \big[\Lambda,[d'\omega,A]\big].\leqno(1.10)$$ Let us compute now the second Lie bracket in the right hand side of (1.10: $$\leqalignno{ \qquad\big[\Lambda,[d'\omega,A]\big]&=\big[A,[\Lambda,d'\omega]\big]- \big[d'\omega,[A,\Lambda]\big]=[\tau,A]+\big[d'\omega,[\Lambda,A]\big]~; &(1.11)\cr [\Lambda,A]&=\ii[\Lambda,\delta'+\tau^\star]=\ii[D'+\tau,L]^\star.&(1.12)\cr}$$ Lemma 1.5 b) provides $[\tau,L]=-3d'\omega$, and it is clear that $[D',L]=d'\omega$. Equalities (1.12) and (1.11) yield therefore $$\leqalignno{ [\Lambda,A]&=-2\ii(d'\omega)^\star,\cr \big[\Lambda,[d'\omega,A]\big]&=\big[\tau,[D'',\Lambda]\big]-2\ii [d'\omega,(d'\omega)^\star].&(1.13)\cr}$$ Substituting (1.10) and (1.13) in (1.8) we get $$\leqalignno{ \big[D'',[\Lambda,\tau]\big]&=\big[\Lambda,[\Lambda,d''d'\omega]\big]+ 2\big[\tau,[D'',\Lambda]\big]-2\ii\big[d'\omega,(d'\omega)^\star\big]&(1.14)\cr &=2\ii\big(T_\omega+[\tau,\delta'+\tau^\star]\big).\cr}$$ Formula b) is a consequence of (1.7) and (1.14).\qed \endproof Theorem 1.4 follows now from Th.~1.2 if Formula~1.6 b) is rewritten $$\Delta'+[D',\tau^\star]-[D'',\ovl\tau^\star]= [D'+\tau,\delta'+\tau^\star]+T_\omega.$$ When $\omega $ is K\"ahler, then $\tau=T_\omega=0$ and Lemma~1.6 a) shows that $[D',\delta'']=0$. Together with the adjoint relation $[D'',\delta']=0$, this equality implies $$\Delta=\Delta'+\Delta''.\leqno(1.15)$$ When $\omega$ is not K\"ahler, Lemma~1.6 a) can be written $[D'+\tau,\delta'']=0$ and we obtain more generally $$[D+\tau,\delta+\tau^\star]=\big[(D'+\tau)+D'',(\delta'+\tau^\star)+\delta'' \big]=\Delta'_\tau+\Delta''.$$ \begstat{(1.16) Proposition} Set $\Delta_\tau=[D+\tau,\delta+\tau^\star]$. Then $\Delta_\tau=\Delta'_\tau+\Delta''$. \endstat \titleb{2.}{Basic a Priori Inequality} Let $(X,\omega)$ be a {\it compact} hermitian manifold, $\dim_\bbbc X=n$, and $E$ a hermitian holomorphic vector bundle over $X$. For any section $u\in\ci_{p,q}(X,E)$ we have $\Ll\Delta''u,u\Gg=\|D''u\|^2+\|\delta''u\|^2$ and the similar formula for $\Delta'_\tau$ gives $\Ll\Delta'_\tau u,u\Gg\ge 0$. Theorem 1.4 implies therefore $$\|D''u\|^2+\|\delta''u\|^2\ge\int_X\big(\langle[\ii\Theta(E),\Lambda]u,u\rangle+ \langle T_\omega u,u\rangle\big)dV.\leqno(2.1)$$ This inequality is known as the {\it Bochner-Kodaira-Nakano} inequality. When $u$ is $\Delta''$-harmonic, we get in particular $$\int_X\big(\langle[\ii\Theta(E),\Lambda]u,u\rangle+ \langle T_\omega u,u\rangle\big)dV\le 0.\leqno(2.2)$$ These basic a priori estimates are the starting point of all vanishing theorems. Observe that $~[\ii\Theta(E),\Lambda]+T_\omega~$ is a hermitian operator acting pointwise on $~\Lambda^{p,q}T^\star X\otimes E$ (the hermitian property can be seen from the fact that this operator coincides with $\Delta''-\Delta'_\tau$ on smooth sections). Using Hodge theory (Cor.~VI-11.2), we get: \begstat{(2.3) Corollary} If the hermitian operator $[\ii\Theta(E),\Lambda]+T_\omega$ is positive definite on $\Lambda^{p,q}T^\star X\otimes E$, then $H^{p,q}(X,E)=0$.\qed \endstat In some circumstances, one can improve Cor.~2.3 thanks to the following ``analytic continuation lemma" due to (Aronszajn 1957): \begstat{(2.4) Lemma} Let $M$ be a connected $\ci$-manifold, $F$ a vector bundle over $M$, and $P$ a second order elliptic differential operator acting on $\ci(M,F)$. Then any section $\alpha\in\ker\,P$ vanishing on a non-empty open subset of $M$ vanishes identically on $M$. \endstat \begstat{(2.5) Corollary} Assume that $X$ is compact and connected. If $$[\ii\Theta(E),\Lambda]+T_\omega\in\Herm\big(\Lambda^{p,q}T^\star X\otimes E\big)$$ is semi-positive on $X$ and positive definite in at least one point $x_0\in X$, then $H^{p,q}(X,E)=0$. \endstat \begproof{} By (2.2) every $\Delta''$-harmonic $(p,q)$-form $u$ must vanish in the neighborhood of $x_0$ where $[\ii\Theta(E),\Lambda]+T_\omega>0$, thus $u\equiv 0$. Hodge theory implies $H^{p,q}(X,E)=0$.\qed \endproof \titleb{3.}{Kodaira-Akizuki-Nakano Vanishing Theorem} The main goal of vanishing theorems is to find natural geometric or algebraic conditions on a bundle $E$ that will ensure that some cohomology groups with values in $E$ vanish. In the next three sections, we prove various vanishing theorems for cohomology groups of a hermitian {\it line bundle} $E$ over a {\it compact} complex manifold $X$. \begstat{(3.1) Definition} A hermitian holomorphic line bundle $E$ on $X$ is said to be positive $($resp. negative$)$ if the hermitian matrix $\big(c_{jk}(z)\big)$ of its Chern curvature form $$\ii\Theta(E)=\ii\sum_{1\le j,k\le n}c_{jk}(z)\,dz_j\wedge d\ovl z_k$$ is positive $($resp. negative$)$ definite at every point $z\in X$. \endstat Assume that $X$ has a K\"ahler metric $\omega$. Let $$\gamma_1(x)\le\ldots\le\gamma_n(x)$$ be the eigenvalues of $\ii\Theta(E)_x$ with respect to $\omega_x$ at each point $x\in X$, and let $$\ii\Theta(E)_x=\ii\sum_{1\le j\le n}\gamma_j(x)\,\zeta_j\wedge\ovl\zeta_j, ~~~~\zeta_j\in T^\star_x X$$ be a diagonalization of $\ii\Theta(E)_x$. By Prop.~VI-8.3 we have $$\leqalignno{ \langle [\ii\Theta(E),\Lambda]u,u\rangle&=\sum_{J,K} \Big(\sum_{j\in J}\gamma_j+\sum_{j\in K}\gamma_j -\sum_{1\le j\le n}\gamma_j\Big)|u_{J,K}|^2\cr &\ge(\gamma_1+\ldots+\gamma_q-\gamma_{p+1}-\ldots-\gamma_n)|u|^2 &(3.2)\cr}$$ for any form $u=\sum_{J,K}u_{J,K}\,\zeta_J\wedge\ovl\zeta_K \in\Lambda^{p,q}T^\star X$. \begstat{(3.3) Akizuki-Nakano vanishing theorem (1954)} Let $E$ be a holomorphic line bundle on $X$. \smallskip \item{\rm a)} If $E$ is positive, then~ $H^{p,q}(X,E)=0~~$ for~ $p+q\ge n+1.$ \smallskip \item{\rm b)} If $E$ is negative, then~ $H^{p,q}(X,E)=0~~$ for~ $p+q\le n-1.$ \endstat \begproof{} In case a), choose $\omega=\ii\Theta(E)$ as a K\"ahler metric on $X$. Then we have $\gamma_j(x)=1$ for all $j$ and $x$, so that $$\Ll [\ii\Theta(E),\Lambda]u,u\Gg\ge (p+q-n)||u||^2$$ for any $u\in\Lambda^{p,q}T^\star X\otimes E$. Assertion a) follows now from Corollary~2.3. Property~b) is proved similarly, by taking $\omega=-\ii\Theta(E)$. One can also derive b) from a) by Serre duality (Theorem~VI-11.3).\qed \endproof When $p=0$ or $p=n$, Th.~3.3 can be generalized to the case where $\ii\Theta(E)$ degenerates at some points. We use here the standard notations $$\Omega^p_X=\Lambda^pT^\star X,~~~~K_X=\Lambda^nT^\star X,~~~~ n=\dim_\bbbc X~;\leqno(3.4)$$ $K_X$ is called the {\it canonical line bundle} of $X$. \begstat{(3.5) Theorem (Grauert-Riemenschneider 1970)} Let $(X,\omega)$ be a compact and connected K\"ahler manifold and $E$ a line bundle on $X$. \smallskip \item{\rm a)} If $\ii\Theta(E)\ge 0$ on $X$ and $\ii\Theta(E)>0$ in at least one point $x_0\in X$, then \smallskip \centerline{$H^q(X,K_X\otimes E)=0~~~~\hbox{\rm for}~~q\ge 1.$} \smallskip \item{\rm b)} If $\ii\Theta(E)\le 0$ on $X$ and $\ii\Theta(E)<0$ in at least one point $x_0\in X$, then \smallskip \centerline{$H^q(X,E)=0~~~~\hbox{\rm for}~~q\le n-1.$} \endstat It will be proved in Volume II, by means of holomorphic Morse inequalities, that the K\"ahler assumption is in fact unnecessary. This improvement is a deep result first proved by (Siu 1984) with a different ad hoc method. \begproof{} For $p=n$, formula (3.2) gives $$\Ll [\ii\Theta(E),\Lambda]u,u\Gg\ge (\gamma_1+\ldots+\gamma_q)|u|^2\leqno(3.6)$$ and a) follows from Cor.~2.5. Now b) is a consequence of a) by Serre duality.\qed \endproof \titleb{4.}{Girbau's Vanishing Theorem} Let $E$ be a line bundle over a compact connected K\"ahler manifold $(X,\omega)$. Girbau's theorem deals with the (possibly everywhere) degenerate semi-positive case. We first state the corresponding generalization of Th.~4.5. \begstat{(4.1) Theorem} If $\ii\Theta(E)$ is semi-positive and has at least $n-s+1$ positive eigenvalues at a point $x_0\in X$ for some integer $s\in\{1\ld n\}$, then $$H^q(X,K_X\otimes E)=0~~~~\hbox{\rm for}~~q\ge s.$$ \endstat \begproof{} Apply 2.5 and inequality (3.6), and observe that $\gamma_q(x_0)>0$ for all $q\ge s$.\qed \endproof \begstat{(4.2) Theorem (Girbau 1976)} If $\ii\Theta(E)$ is semi-positive and has at least $n-s+1$ positive eigenvalues at every point $x\in X$, then $$H^{p,q}(X,E)=0~~~~\hbox{\rm for}~~p+q\ge n+s.$$ \endstat \begproof{} Let us consider on $X$ the new K\"ahler metric $$\omega_\varepsilon=\varepsilon\omega +\ii\Theta(E),~~~~\varepsilon>0,$$ and let $\ii\Theta(E)=\ii\sum\gamma_j\,\zeta_j\wedge\ovl\zeta_j$ be a diagonalization of $\ii\Theta(E)$ with respect to $\omega$ and with $\gamma_1\le\ldots\le \gamma_n$. Then $$\omega_\varepsilon=\ii\sum~(\varepsilon+\gamma_j)\,\zeta_j\wedge\ovl\zeta_j.$$ The eigenvalues of $\ii\Theta(E)$ with respect to $\omega_\varepsilon$ are given therefore by $$\gamma_{j,\varepsilon}=\gamma_j/(\varepsilon+\gamma_j)\in[0,1[, ~~~~1\le j\le n.\leqno(4.3)$$ On the other hand, the hypothesis is equivalent to $\gamma_s>0$ on $X$. For $j\ge s$ we have $\gamma_j\ge\gamma_s$, thus $$\gamma_{j,\varepsilon}={1\over 1+\varepsilon/\gamma_j}\ge{1\over 1+ \varepsilon/\gamma_s}\ge1-\varepsilon/\gamma_s,~~~~s\le j\le n.~\leqno(4.4)$$ Let us denote the operators and inner products associated to $\omega_\varepsilon$ with $\varepsilon$ as an index. Then inequality (3.2) combined with (4.4) implies $$\eqalign{ \langle[\ii\Theta(E),\Lambda_\varepsilon]u,u\rangle_\varepsilon &\ge\Big(\big(q-s+1)\big)(1-\varepsilon/\gamma_s)-(n-p)\Big)|u|^2\cr &=\big(p+q-n-s+1-(q-s+1)\varepsilon/\gamma_s\big)|u|^2.\cr}$$ Theorem 4.2 follows now from Cor.~2.3 if we choose $$\varepsilon<{p+q-n-s+1\over q-s+1}~\min_{x\in X}\gamma_s(x).\eqno{\square}$$ \endproof \begstat{(4.5) Remark} \rm The following example due to (Ramanujam 1972, 1974) shows that Girbau's result is no longer true for $p0$ (that is, $s=1$) is due to (Kodaira 1953) and (Serre 1956). \begstat{(5.1) Theorem} Let $F$ be a holomorphic vector bundle over a compact complex manifold $X$, $s$ a positive integer and $E$ a hermitian line bundle such that $\ii\Theta(E)$ has at least $n-s+1$ positive eigenvalues at every point $x\in X$. Then there exists an integer $k_0\ge 0$ such that $$H^q(X,E^k\otimes F)=0~~~~\hbox{\rm for}~~q\ge s~~\hbox{\rm and}~~k\ge k_0.$$ \endstat \begproof{} The main idea is to construct a hermitian metric $\omega_\varepsilon$ on $X$ in such a way that all negative eigenvalues of $\ii\Theta(E)$ with respect to $\omega_\varepsilon$ will be of small absolute value. Let $\omega$ denote a fixed hermitian metric on $X$ and let $\gamma_1\le\ldots\le\gamma_n$ be the corresponding eigenvalues of $\ii\Theta(E)$. \begstat{(5.2) Lemma} Let $\psi\in\ci(\bbbr,\bbbr)$. If $A$ is a hermitian $n\times n$ matrix with eigenvalues $\lambda_1\le\ldots\le\lambda_n$ and corresponding eigenvectors $v_1\ld v_n$, we define $\psi[A]$ as the hermitian matrix with eigenvalues $\psi(\lambda_j)$ and eigenvectors $v_j$, $1\le j\le n$. Then the map $A\longmapsto\psi[A]$ is $\ci$ on $\Herm(\bbbc^n)$. \endstat \begproof{} Although the result is very well known, we give here a short proof. Without loss of generality, we may assume that $\psi$ is compactly supported. Then we have $$\psi[A]={1\over 2\pi}\int_{-\infty}^{+\infty}\wh\psi(t)e^{itA}dt$$ where $\wh\psi$ is the rapidly decreasing Fourier transform of $\psi$. The equality\break $\int_0^t(t-u)^pu^q\,du=p!\,q!/(p+q+1)!~$ and obvious power series developments yield $$D_A(e^{itA})\cdot B=i\int_0^t e^{\ii(t-u)A}\,B\,e^{iuA}du.$$ Since $e^{iuA}$ is unitary, we get $\|D_A(e^{itA})\|\le|t|$. A differentiation under the integral sign and Leibniz' formula imply by induction on $k$ the bound \hbox{$\|D_A^k(e^{itA})\|\le|t|^k$.} Hence $A\longmapsto\psi[A]$ is smooth.\qed \endproof Let us consider now the positive numbers $$t_0=\inf_X \gamma_s>0,~~~~M=\sup_X\max_j|\gamma_j|>0.$$ We select a function $\psi_\varepsilon\in\ci(\bbbr,\bbbr)$ such that $$\psi_\varepsilon(t)=t~~\hbox{\rm for}~t\ge t_0, ~~\psi_\varepsilon(t)\ge t~~\hbox{\rm for}~0\le t\le t_0, ~~\psi_\varepsilon(t)=M/\varepsilon~~\hbox{\rm for}~t\le 0.$$ By Lemma~5.2, $\omega_\varepsilon:=\psi_\varepsilon[\ii\Theta(E)]$ is a smooth hermitian metric on $X$. Let us write $$\ii\Theta(E)=\ii\sum_{1\le j\le n}\gamma_j\,\zeta_j\wedge\ovl\zeta_j,~~~~ \omega_\varepsilon=\ii\sum_{1\le j\le n}\psi_\varepsilon(\gamma_j)\, \zeta_j\wedge\ovl\zeta_j$$ in an orthonormal basis $(\zeta_1\ld\zeta_n)$ of $T^\star X$ for $\omega$. The eigenvalues of $\ii\Theta(E)$ with respect to $\omega_\varepsilon$ are given by $\gamma_{j,\varepsilon}=\gamma_j/\psi_\varepsilon(\gamma_j)$ and the construction of $\psi_\varepsilon$ shows that $-\varepsilon\le \gamma_{j,\varepsilon}\le1$, $1\le j\le n$, and $\gamma_{j,\varepsilon}=1$ for $s\le j\le n$. Now, we have $$H^q(X,E^k\otimes F)\simeq H^{n,q}(X,E^k\otimes G)$$ where $G=F\otimes K^\star_X$. Let $e$, $(g_\lambda)_{1\le\lambda\le r}$ and $(\zeta_j)_{1\le j\le n}$ denote orthonormal frames of $E$, $G$ and $(T^\star X,\omega_\varepsilon)$ respectively. For $$u=\sum_{|J|=q,\lambda}u_{J,\lambda}\,\zeta_1\wedge\ldots\wedge\zeta_n \wedge\ovl\zeta_J\otimes e^k\otimes g_\lambda \in\Lambda^{n,q}T^\star X\otimes E^k\otimes G,$$ inequality (3.2) yields $$\langle[\ii\Theta(E),\Lambda_\varepsilon]u,u\rangle_\varepsilon=\sum_{J,\lambda} \Big(\sum_{j\in J}\gamma_{j,\varepsilon}\Big)|u_{J,\lambda}|^2\ge \big(q-s+1-(s-1)\varepsilon\big)|u|^2.$$ Choosing $\varepsilon=1/s$ and $q\ge s$, the right hand side becomes $\ge (1/s)|u|^2$. Since $\Theta(E^k\otimes G)=k\Theta(E)\otimes\Id_G+\Theta(G)$, there exists an integer $k_0$ such that $$\big[\ii\Theta(E^k\otimes G),\Lambda_\varepsilon\big]+T_{\omega_\varepsilon}~~~~ \hbox{\rm acting~on~~}\Lambda^{n,q}T^\star X\otimes E^k\otimes G$$ is positive definite for $q\ge s$ and $k\ge k_0$. The proof is complete.\qed \endproof \titleb{6.}{Positivity Concepts for Vector Bundles} Let $E$ be a hermitian holomorphic vector bundle of rank $r$ over $X$, where $\dim_{\bbbc}X = n$. Denote by $(e_1\ld e_r)$ an orthonormal frame of $E$ over a coordinate patch $\Omega \subset X$ with complex coordinates $(z_1\ld z_n)$, and $$\ii\Theta(E) = \ii\sum_{1\le j,k\le n,~1\le \lambda,\mu\le r} c_{jk\lambda\mu}\,dz_j \wedge d\ovl z_k \otimes e^\star_\lambda \otimes e_\mu, ~~~~\ovl c_{jk\lambda\mu} = c_{kj\mu\lambda} \leqno (6.1)$$ the Chern curvature tensor. To $\ii\Theta(E)$ corresponds a natural hermitian form $\theta_E$ on $TX\otimes E$ defined by $$\theta_E = \sum_{j,k,\lambda,\mu} c_{jk\lambda\mu}(dz_j \otimes e^\star_\lambda) \otimes (\ovl{dz_k \otimes e^\star_\mu}),$$ and such that $$\theta_E(u,u) = \sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}(x)\,u_{j\lambda} \ovl u_{k\mu},~~~~u\in T_xX\otimes E_x.(6.2)$$ \begstat{(6.3) Definition (Nakano 1955)} $E$ is said to be Nakano positive $($resp. Nakano semi-negative$)$ if $\theta_E$ is positive definite $($resp. semi-negative$)$ as a hermitian form on $TX\otimes E$, i.e.\ if for every $u\in TX\otimes E,~u\ne 0,$ we have $$\theta_E(u,u) > 0\quad(\hbox{\it resp.}\le 0).$$ We write $>_\Nak$~~$($resp.~$\le_\Nak)$ for Nakano positivity $($resp. semi-negativity$)$. \endstat \begstat{(6.4) Definition (Griffiths 1969)} $E$ is said to be Griffiths positive $($resp. Griffiths semi-negative$)$ if for all $\xi \in T_xX$, $\xi\ne 0$ and $s\in E_x$, $s\ne 0$ we have $$\theta_E(\xi\otimes s,\xi\otimes s) > 0\quad (\hbox{\it resp.}\le 0).$$ We write $>_\Grif$ $($resp. $\le_\Grif)$ for Griffiths positivity $($resp. semi-negativity$)$. \endstat It is clear that Nakano positivity implies Griffiths positivity and that both concepts coincide if $r=1$ (in the case of a line bundle, $E$ is merely said to be positive). One can generalize further by introducing additional concepts of positivity which interpolate between Griffiths positivity and Nakano positivity. \begstat{(6.5) Definition} Let $T$ and $E$ be complex vector spaces of dimensions $n,r$ respectively, and let $\Theta$ be a hermitian form on $T\otimes E$. \smallskip \item{\rm a)} A tensor $u\in T\otimes E$ is said to be of rank $m$ if $m$ is the smallest $\ge 0$ integer such that $u$ can be written $$u = \sum_{j=1}^m \xi_j \otimes s_j,~~~~\xi_j\in T,~s_j\in E.$$ \smallskip \item{\rm b)} $\Theta$ is said to be $m$-positive $($resp. $m$-semi-negative$)$ if $\Theta(u,u)>0$ $($resp. $\Theta(u,u)\le 0)$ for every tensor $u\in T\otimes E$ of rank $\le m$, $u\ne 0$. In this case, we write $$\Theta >_m 0~~~~(\hbox{\it resp.}~~\Theta \le_m 0).$$\smallskip \endstat We say that the bundle $E$ is $m$-positive if $\theta_E >_m 0$. Griffiths positivity corresponds to $m=1$ and Nakano positivity to $m\ge \min(n,r)$. \begstat{(6.6) Proposition} A bundle $E$ is Griffiths positive if and only if $E^\star$ is Griffiths negative. \endstat \begproof{} By (V-4.$3'$) we get $\ii\Theta(E^\star) = -\ii\Theta(E)^\dagger$, hence $$\theta_{E^\star}(\xi_1\otimes\ovl s_2,\xi_2\otimes\ovl s_1)= -\theta_E(\xi_1\otimes s_1,\xi_2\otimes s_2),~~~~\forall\xi_1,\xi_2\in TX,~ \forall s_1,s_2\in E,$$ where $\ovl s_j=\langle\bu,s_j\rangle\in E^\star$. Proposition 6.6 follows immediately.\qed \endproof It should be observed that the corresponding duality property for Nakano positive bundles is {\it not true}. In fact, using (6.1) we get $$\ii\Theta(E^\star) = -\ii\sum_{j,k,\lambda,\mu} c_{jk\mu\lambda} dz_j\wedge d\ovl z_k \otimes e^{\star\star}_\lambda \otimes e^\star_\mu,$$ $$\theta_{E^\star}(v,v)=-\sum_{j,k,\mu,\lambda}c_{jk\mu\lambda}v_{j\lambda} \ovl v_{k\mu},\leqno(6.7)$$ for any $v=\sum v_{j\lambda}\,(\partial/\partial z_j)\otimes e^\star_\lambda \in TX\otimes E^\star$. The following example shows that Nakano positivity or negativity of $\theta_E$ and $\theta_{E^\star}$ are unrelated. \begstat{(6.8) Example} \rm Let $H$ be the rank $n$ bundle over $\bbbp^n$ defined in \S~V-15. For any $u=\sum u_{j\lambda}(\partial/\partial z_j)\otimes \wt e_\lambda\in TX\otimes H$, $v=\sum v_{j\lambda}(\partial/\partial z_j) \otimes\wt e_\lambda^\star\in TX\otimes H^\star$, $1\le j,\lambda\le n$, formula (V-15.9) implies $$\left\{ \eqalign{ \theta_H(u,u)&=\sum u_{j\lambda}\ovl u_{\lambda j}\cr \theta_{H^\star}(v,v)&=\sum v_{jj}\ovl v_{\lambda\lambda}= \big|\sum v_{jj}\big|^2.\cr}\right.\leqno(6.9)$$ It is then clear that $H\ge_\Grif 0$ and $H^\star\le_\Nak 0$ , but $H$ is neither $\ge_\Nak0$ nor $\le_\Nak0$. \endstat \begstat{(6.10) Proposition} Let $0\to S \to E \to Q \to 0$ be an exact sequence of hermitian vector bundles. Then \medskip \item{\rm a)} $E\ge_\Grif 0~~\Longrightarrow~~Q\ge_\Grif 0,$ \smallskip \item{\rm b)} $E\le_\Grif 0~~\Longrightarrow~~S\le_\Grif 0,$ \smallskip \item{\rm c)} $E\le_\Nak0~~\Longrightarrow~~S\le_\Nak 0,$ \smallskip \noindent and analogous implications hold true for strict positivity. \endstat \begproof{} If $\beta$ is written $\sum dz_j \otimes \beta_j$,~$\beta_j \in \hom(S,Q)$, then formulas (V-14.6) and (V-14.7) yield $$\eqalign{ \ii\Theta(S) &= \ii\Theta(E)_{\restriction S} - \sum dz_j \wedge d\ovl z_k \otimes \beta^\star_k \beta_j,\cr \ii\Theta(Q) &= \ii\Theta(E)_{\restriction Q} + \sum dz_j \wedge d\ovl z_k \otimes \beta_j\beta^\star_k.\cr}$$ Since $\beta\cdot(\xi\otimes s)= \sum \xi_j\beta_j\cdot s$ and $\beta^\star \cdot(\xi\otimes s)= \sum \ovl \xi_k\beta^\star_k\cdot s$ we get $$\theta_S(\xi\otimes s,\xi'\otimes s') = \theta_E(\xi\otimes s,\xi'\otimes s') - \sum_{j,k} \xi_j \ovl \xi'_k \langle \beta_j\cdot s,\beta_k\cdot s'\rangle,$$ $$\theta_S(u,u) = \theta_E(u,u) - |\beta\cdot u|^2,$$ $$\theta_Q(\xi\otimes s,\xi'\otimes s') = \theta_E(\xi\otimes s,\xi'\otimes s') + \sum_{j,k} \xi_j \ovl \xi'_k \langle \beta^\star_k\cdot s,\beta^\star_j\cdot s'\rangle,$$ $$\theta_Q(\xi\otimes s,\xi\otimes s) = \theta_E(\xi\otimes s,\xi\otimes s) +|\beta^\star\cdot(\xi\otimes s)|^2.\eqno{\square}$$ \endproof Since $H$ is a quotient bundle of the trivial bundle $\soul V$, Example~6.8 shows that $E\ge_\Nak 0$ {\it does not imply} $Q\ge_\Nak 0$. \titleb{7.}{Nakano Vanishing Theorem} Let $(X,\omega)$ be a compact K\"ahler manifold, $\dim_\bbbc X=n$, and $E\lra X$ a hermitian vector bundle of rank $r$. We are going to compute explicitly the hermitian operator $[\ii\Theta(E),\Lambda]$ acting on $\Lambda^{p,q}T^\star X \otimes E$. Let $x_0\in X$ and $(z_1\ld z_n)$ be local coordinates such that $(\partial/\partial z_1\ld\partial/\partial z_n)$ is an orthonormal basis of $(TX,\omega)$ at $x_0$. One can write $$\eqalign{ \omega_{x_0}&=\ii\sum_{1\le j\le n}dz_j\wedge d\ovl z_j,\cr \ii\Theta(E)_{x_0}&=\ii\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}\,dz_j\wedge d\ovl z_k \otimes e_\lambda^\star\otimes e_\mu\cr}$$ where $(e_1\ld e_r)$ is an orthonormal basis of $E_{x_0}$. Let $$u=\sum_{|J|=p,\,|K|=q,\,\lambda}u_{J,K,\lambda}\,dz_J\wedge d\ovl z_K \otimes e_\lambda\in\big(\Lambda^{p,q}T^\star X\otimes E\big)_{x_0}.$$ A simple computation as in the proof of Prop.~VI-8.3 gives $$\eqalign{ \Lambda u&=\ii(-1)^p\sum_{J,K,\lambda,s}u_{J,K,\lambda}\, \Big({\partial\over\partial z_s}\ort dz_J\Big)\wedge \Big({\partial\over\partial\ovl z_s}\ort d\ovl z_K\Big)\otimes e_\lambda,\cr \ii\Theta(E)\wedge u&=\ii(-1)^p\sum_{j,k,\lambda,\mu,J,K}c_{jk\lambda\mu}\, u_{J,K,\lambda}\,dz_j\wedge dz_J\wedge d\ovl z_k\wedge d\ovl z_K\otimes e_\mu,\cr [\ii\Theta(E),\Lambda]u&=\sum_{j,k,\lambda,\mu,J,K} c_{jk\lambda\mu}\,u_{J,K,\lambda}\,dz_j\wedge\Big({\partial\over\partial z_k} \ort dz_J\Big)\wedge d\ovl z_K\otimes e_\mu\cr &+\sum_{j,k,\lambda,\mu,J,K}c_{jk\lambda\mu}\,u_{J,K,\lambda}\,dz_J\wedge d\ovl z_k\wedge\Big({\partial\over\partial\ovl z_j}\ort d\ovl z_K\Big)\otimes e_\mu\cr &-\sum_{j,\lambda,\mu,J,K}c_{jj\lambda\mu}\,u_{J,K,\lambda}\,dz_J\wedge d\ovl z_K \otimes e_\mu.\cr}$$ We extend the definition of $u_{J,K,\lambda}$ to non increasing multi-indices $J=(j_s)$, $K=(k_s)$ by deciding that $u_{J,K,\lambda}=0$ if $J$ or $K$ contains identical components repeated and that $u_{J,K,\lambda}$ is alternate in the indices $(j_s)$, $(k_s)$. Then the above equality can be written $$\eqalign{ \langle[\ii\Theta(E),\Lambda]u,u\rangle&= \sum c_{jk\lambda\mu}\,u_{J,jS,\lambda}{\ovl u_{J,kS,\mu}}\cr &+\sum c_{jk\lambda\mu}\,u_{kR,K,\lambda}{\ovl u_{jR,K,\mu}}\cr &-\sum c_{jj\lambda\mu}\,u_{J,K,\lambda}{\ovl u_{J,K,\mu}},\cr}$$ extended over all indices $j,k,\lambda,\mu,J,K,R,S$ with $|R|=p-1$, $|S|=q-1$. This hermitian form appears rather difficult to handle for general $(p,q)$ because of sign compensation. Two interesting cases are $p=n$ and $q=n$. \medskip \noindent$\bu$ For $u=\sum u_{K,\lambda}\,dz_1\wedge\ldots\wedge dz_n\wedge d\ovl z_K\otimes e_\lambda$ of type $(n,q)$, we get $$\langle[\ii\Theta(E),\Lambda]u,u\rangle=\sum_{|S|=q-1}\sum_{j,k,\lambda,\mu} c_{jk\lambda\mu}\,u_{jS,\lambda}\ovl u_{kS,\mu},\leqno(7.1)$$ because of the equality of the second and third summations in the general formula. Since $u_{jS,\lambda}=0$ for $j\in S$, the rank of the tensor $(u_{jS,\lambda})_{j,\lambda}\in\bbbc^n\otimes\bbbc^r$ is in fact $\le\min\{n-q+1,r\}$. We obtain therefore: \begstat{(7.2) Lemma} Assume that $E>_m0$ in the sense of Def.~$6.5$. Then the hermitian operator $[\ii\Theta(E),\Lambda]$ is positive definite on $\Lambda^{n,q}T^\star X\otimes E$ for $q\ge 1$ and $m\ge\min\{n-q+1,r\}.$ \endstat \begstat{(7.3) Theorem} Let $X$ be a compact connected K\"ahler manifold of dimension $n$ and $E$ a hermitian vector bundle of rank $r$. If $\theta_E\ge_m 0$ on $X$ and $\theta_E>_m0$ in at least one point, then $$H^{n,q}(X,E)=H^q(X,K_X\otimes E)=0~~~~\hbox{\rm for}~~q\ge 1~~\hbox{\rm and}~~ m\ge\min\{n-q+1,r\}.$$ \endstat \vskip-6pt\noindent$\bu$ Similarly, for $u=\sum u_{J,\lambda}\,dz_J\wedge d\ovl z_1\wedge\ldots\wedge d\ovl z_n\otimes e_\lambda$ of type $(p,n)$, we get $$\langle[\ii\Theta(E),\Lambda]u,u\rangle=\sum_{|R|=p-1}\sum_{j,k,\lambda,\mu} c_{jk\lambda\mu}\,u_{kR,\lambda}\ovl u_{jR,\mu},$$ because of the equality of the first and third summations in the general formula. The indices $j,k$ are twisted, thus $[\ii\Theta(E),\Lambda]$ defines a positive hermitian form under the assumption $\ii\Theta(E)^\dagger>_m 0$, i.e.\ $\ii\Theta(E^\star)<_m 0$, with \hbox{$m\ge\min\{n-p+1,r\}$.} Serre duality $\big(H^{p,0}(X,E)\big)^\star=H^{n-p,n}(X,E^\star)$~gives: \begstat{(7.4) Theorem} Let $X$ and $E$ be as above. If $\theta_E\le_m 0$ on $X$ and $\theta_E<_m 0$ in at least one point, then $$H^{p,0}(X,E)=H^0(X,\Omega^p_X\otimes E)=0~~~~\hbox{\rm for}~~p_\Grif 0~~\Longrightarrow~~E\otimes \det E >_\Nak 0.$$ \endstat To prove this result, we first use (V-4.$2'$) and (V-4.6). If $\End(E\otimes\det E)$ is identified to $\hom(E,E)$, one can write $$\Theta(E\otimes \det E) = \Theta(E)+\Tr_E(\Theta(E)) \otimes \Id_E,$$ $$\theta_{E\otimes \det E} = \theta_E+\Tr_E\theta_E \otimes h,$$ where $h$ denotes the hermitian metric on $E$ and where $\Tr_E\theta_E$ is the hermitian form on $TX$ defined by $$\Tr_E \theta_E(\xi,\xi) = \sum_{1\le \lambda\le r} \theta_E(\xi\otimes e_\lambda,\xi\otimes e_\lambda),~\xi\in TX,$$ for any orthonormal frame $(e_1\ld e_r)$ of $E$. Theorem 8.1 is now a consequence of the following simple property of hermitian forms on a tensor product of complex vector spaces. \begstat{(8.2) Proposition} Let $T,E$ be complex vector spaces of respective dimensions $n,r,\,$ and $h$ a hermitian metric on $E$. Then for every hermitian form $\Theta$ on $T\otimes E$ $$\Theta >_\Grif 0~~\Longrightarrow~~ \Theta + \Tr_E\Theta \otimes h >_\Nak 0.$$ \endstat We first need a lemma analogous to Fourier inversion formula for discrete Fourier transforms. \begstat{(8.3) Lemma} Let $q$ be an integer $\ge 3$, and $x_\lambda,~y_\mu,~1\le \lambda,\mu \le r$, be complex numbers. Let $\sigma$ describe the set $U^r_q$ of $r$-tuples of $q$-th roots of unity and put $$x'_\sigma = \sum_{1\le \lambda\le r} x_\lambda \ovl \sigma_\lambda,~~~~ y'_\sigma = \sum_{1\le \mu\le r} y_\mu \ovl \sigma_\mu,~~~~ \sigma\in U^r_q.$$ Then for every pair $(\alpha,\beta),~1\le \alpha,\beta\le r$, the following identity holds: $$q^{-r} \sum_{\sigma\in U^r_q} x'_\sigma \ovl y'_\sigma \sigma_\alpha \ovl \sigma_\beta=\cases{ x_\alpha \ovl y_\beta&if~~$\alpha\ne \beta,$\cr \displaystyle\sum_{1\le \mu\le r} x_\mu \ovl y_\mu&if~~$\alpha=\beta.$\cr}$$ \endstat \begproof{} The coefficient of $x_\lambda\ovl y_\mu$ in the summation $q^{-r} \sum_{\sigma\in U^r_q} x'_\sigma \ovl y'_\sigma \sigma_\alpha \ovl \sigma_\beta$ is given by $$q^{-r} \sum_{\sigma\in U^r_q}\sigma_\alpha \ovl \sigma_\beta\ovl \sigma_\lambda \sigma_\mu.$$ This coefficient equals 1 when the pairs $\{ \alpha,\mu\}$ and $\{\beta,\lambda\}$ are equal (in which case $\sigma_\alpha \ovl \sigma_\beta \ovl \sigma_\lambda \sigma_\mu = 1$ for any one of the $q^r$ elements of $U^r_q$). Hence, it is sufficient to prove that $$\sum_{\sigma\in U^r_q} \sigma_\alpha \ovl \sigma_\beta \ovl \sigma_\lambda \sigma_\mu = 0$$ when the pairs $\{ \alpha,\mu\}$ and $\{ \beta,\lambda\}$ are distinct. If $\{ \alpha,\mu\} \ne \{\beta,\lambda\}$, then one of the elements of one of the pairs does not belong to the other pair. As the four indices $\alpha,\beta,\lambda,\mu$ play the same role, we may suppose for example that $\alpha\notin \{\beta,\lambda\}$. Let us apply to $\sigma$ the substitution $\sigma\mapsto \tau$, where $\tau$ is defined by $$\tau_\alpha = e^{2\pi\ii/q}\sigma_\alpha,~\tau_\nu = \sigma_\nu \quad \hbox{\rm for}\quad \nu\ne \alpha.$$ We get $$\sum_\sigma\sigma_\alpha\ovl\sigma_\beta\ovl\sigma_\lambda\sigma_\mu=\sum_\tau =\cases{ \displaystyle e^{2\pi\ii/q}\sum_\sigma&if~~$\alpha\ne\mu,$\cr \displaystyle e^{4\pi\ii/q}\sum_\sigma&if~~$\alpha=\mu,$\cr}$$ Since $q\ge 3$ by hypothesis, it follows that $$\sum_\sigma\sigma_\alpha\ovl\sigma_\beta\ovl\sigma_\lambda\sigma_\mu=0.$$ \endproof \begproof{of Proposition 8.2.} Let $(t_j)_{1\le j\le n}$ be a basis of $T$, $(e_\lambda)_{1\le \lambda\le r}$ an orthonormal basis of $E$ and $\xi=\sum_j\xi_jt_j\in T$, $u=\sum_{j,\lambda}u_{j\lambda}\,t_j\otimes e_\lambda \in T\otimes E$. The coefficients $c_{jk\lambda\mu}$ of $\Theta$ with respect to the basis $t_j\otimes e_\lambda$ satisfy the symmetry relation $\ovl c_{jk\lambda\mu}=c_{kj\mu\lambda}$, and we have the formulas $$\eqalign{\Theta(u,u) &=\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}u_{j\lambda}\ovl u_{k\mu},\cr \Tr_E\Theta(\xi,\xi)&=\sum_{j,k,\lambda}c_{jk\lambda\lambda}\xi_j\ovl\xi_k,\cr (\Theta+\Tr_E\Theta\otimes h)(u,u)&=\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu} u_{j\lambda} \ovl u_{k\mu}+ c_{jk\lambda\lambda}u_{j\mu}\ovl u_{k\mu}.\cr}$$ For every $\sigma\in U^r_q$ (cf.\ Lemma~8.3), put $$\eqalign{ u'_{j\sigma}&=\sum_{1\le\lambda\le r}u_{j\lambda}\ovl\sigma_\lambda\in\bbbc,\cr \wh u_\sigma&=\sum_j u'_{j\sigma}t_j\in T\quad,\quad \wh e_\sigma=\sum_\lambda\sigma_\lambda e_\lambda\in E.\cr}$$ Lemma 8.3 implies $$\eqalign{ q^{-r} \sum_{\sigma\in U^r_q}\Theta(\wh u_\sigma\otimes\wh e_\sigma, \wh u_\sigma\otimes\wh e_\sigma)&= q^{-r} \sum_{\sigma\in U^r_q} c_{jk\lambda\mu} u'_{j\sigma} \ovl u'_{k\sigma}\sigma_\lambda \ovl \sigma_\mu\cr &= \sum_{j,k,\lambda\ne \mu} c_{jk\lambda\mu} u_{j\lambda} \ovl u_{k\mu} + \sum_{j,k,\lambda,\mu} c_{jk\lambda\lambda} u_{j\mu} \ovl u_{k\mu}.\cr}$$ The Griffiths positivity assumption shows that the left hand side is $\ge 0$, hence $$(\Theta + \Tr_E \Theta \otimes h)(u,u) \ge \sum_{j,k,\lambda} c_{jk\lambda\lambda} u_{j\lambda} \ovl u_{k\lambda} \ge 0$$ with strict positivity if $\Theta >_\Grif 0$ and $u\ne 0$.\qed \endproof \begstat{(8.4) Example} \rm Take $E=H$ over $\bbbp^n=P(V)$. The exact sequence $$0 \lra \cO(-1) \lra \soul V \lra H \lra 0$$ implies $\det \soul V=\det H \otimes \cO(-1)$. Since $\det \soul V$ is a trivial bundle, we get (non canonical) isomorphisms $$\eqalign{ \det H &\simeq \cO(1),\cr T\bbbp^n &= H\otimes \cO(1) \simeq H\otimes \det H.\cr}$$ We already know that $H\ge_\Grif 0$, hence $T\bbbp^n \ge_\Nak 0$. A direct computation based on (6.9) shows that $$\eqalign{ \theta_{T\bbbp^n}(u,u) &= (\theta_H + \Tr_H \theta_H \otimes h)(u,u)\cr &= \sum_{1\le j,k\le n} u_{jk} \ovl u_{kj} + u_{jk} \ovl u_{jk} = {1\over 2} \sum_{1\le j,k\le n} |u_{jk} + u_{kj}|^2.\cr}$$ In addition, we have $T\bbbp^n>_\Grif 0$. However, the Serre duality theorem gives $$\eqalign{ H^q(\bbbp^n,K_{\bbbp^n}\otimes T\bbbp^n)^\star &\simeq H^{n-q}(\bbbp^n,T^\star\bbbp^n)\cr &=H^{1,n-q}(\bbbp^n,\bbbc)=\cases{ \bbbc&if~~$q=n-1$,\cr 0 &if~~$q\ne n-1$.\cr}\cr}$$ For $n\ge 2$, Th.~7.3 implies that $T\bbbp^n$ has no hermitian metric such that $\theta_{T\bbbp^n}\ge_20$ on $\bbbp^n$ and $\theta_{T\bbbp^n}>_20$ in one point. This shows that the notion of $2$-positivity is actually stronger than $1$-positivity (i.e.\ Griffiths positivity). \endstat \begstat{(8.5) Remark} \rm Since $\Tr_H\theta_H = \theta_{\cO(1)}$ is positive and $\theta_{T\bbbp^n}$ is not $>_\Nak 0$ when $n\ge 2$, we see that Prop.~8.2 is best possible in the sense that there cannot exist any constant $c<1$ such that $$\Theta>_\Grif0~~~\Longrightarrow~~~\Theta + c \Tr_E \Theta\otimes h \ge_\Nak 0.$$ \endstat \titleb{9.}{Applications to Griffiths Positive Bundles} We first need a preliminary result. \begstat{(9.1) Proposition} Let $T$ be a complex vector space and $(E,h)$ a hermitian vector space of respective dimensions $n,r$ with $r\ge 2$. Then for any hermitian form $\Theta$ on $T\otimes E$ and any integer $m\ge 1$ $$\Theta>_\Grif0~~~\Longrightarrow~~~m\Tr_E\Theta\otimes h-\Theta>_m0.$$ \endstat \begproof{} Let us distinguish two cases. \medskip \noindent a)~ $m=1$.~ Let $u\in T \otimes E$ be a tensor of rank 1. Then $u$ can be written $u = \xi_1 \otimes e_1$ with $\xi_1\in T,~\xi_1 \ne 0$, and $e_1 \in E,~|e_1| = 1$. Complete $e_1$ into an orthonormal basis $(e_1\ld e_r)$ of $E$. One gets immediately $$\eqalign{ (\Tr_E \Theta \otimes h)(u,u) &= \Tr_E \Theta (\xi_1,\xi_1) = \sum_{1\le \lambda\le r} \Theta(\xi_1\otimes e_\lambda , \xi_1 \otimes e_\lambda)\cr &> \Theta (\xi_1\otimes e_1,\xi_1\otimes e_1) = \Theta (u,u).\cr}$$ \medskip \noindent b)~ $m\ge 2$.~ Every tensor $u\in T\otimes E$ of $\hbox{\rm rank~} \le m$ can be written $$u=\sum_{1\le \lambda\le q} \xi_\lambda \otimes e_\lambda\quad, \quad \xi_\lambda \in T,$$ with $q = \min(m,r)$ and $(e_\lambda)_{1\le \lambda\le r}$ an orthonormal basis of $E$. Let $F$ be the vector subspace of $E$ generated by $(e_1\ld e_q)$ and $\Theta_F$ the restriction of $\Theta$ to $T\otimes F$. The first part shows that $$\Theta':= \Tr_F \Theta_F \otimes h - \Theta_F >_\Grif 0.$$ Proposition 9.2 applied to $\Theta'$ on $T\otimes F$ yields $$\Theta'+\Tr_F \Theta'\otimes h=q\Tr_F\Theta_F\otimes h-\Theta_F>_q 0.$$ Since $u\in T\otimes F$ is of $\hbox{\rm rank~} \le q \le m$, we get (for $u\ne 0$) $$\eqalignno{ \Theta(u,u) = \Theta_F(u,u) &< q(\Tr_F \Theta_F \otimes h)(u,u)\cr &=q\sum_{1\le j,\lambda\le q}\Theta(\xi_j\otimes e_\lambda, \xi_j\otimes e_\lambda)\le m\Tr_E\Theta\otimes h(u,u).&\square\cr}$$ \endproof Proposition 9.1 is of course also true in the semi-positive case. From these facts, we deduce \begstat{(9.2) Theorem} Let $E$ be a Griffiths $($semi-$)$positive bundle of rank $r\ge 2$. Then for any integer $m\ge 1$ $$E^\star \otimes (\det E)^m >_m 0~~~~(\hbox{\it resp.}~~\ge_m 0).$$ \endstat \begproof{} Apply Prop.~8.1 to $\Theta = -\theta_{E^\star} >_\Grif 0$ and observe that $$\theta_{\det E} = - \theta_{\det E^\star} = \Tr_{E^\star} \Theta.$$ \endproof \begstat{(9.3) Theorem} Let $0\to S \to E \to Q \to 0$ be an exact sequence of hermitian vector bundles. Then for any $m\ge 1$ $$E>_m0~~~\Longrightarrow~~~S\otimes (\det Q)^m >_m 0.$$ \endstat \begproof{} Formulas (V-14.6) and (V-14.7) imply $$\ii\Theta(S) >_m \ii\beta^\star \wedge \beta\quad,\quad \ii\Theta(Q) >_m \ii\beta \wedge \beta^\star,$$ $$\ii\Theta(\det Q) = \Tr_Q(\ii\Theta(Q)) > \Tr_Q(\ii\beta \wedge \beta^\star).$$ If we write $\beta = \sum dz_j \otimes \beta_j$ as in the proof of Prop.~6.10, then $$\eqalign{ \Tr_Q(\ii\beta \wedge \beta^\star) &= \sum idz_j \wedge d\ovl z_k \Tr_Q(\beta_j\beta^\star_k)\cr &=\sum idz_j \wedge d\ovl z_k \Tr_S(\beta^\star_k\beta_j) = \Tr_S(-\ii\beta^\star \wedge \beta).\cr}$$ Furthermore, it has been already proved that $-\ii\beta^\star \wedge \beta \ge_\Nak 0$. By Prop.~8.1 applied to the corresponding hermitian form $\Theta$ on $TX \otimes S$, we get $$m \Tr_S(-\ii\beta^\star \wedge \beta) \otimes \Id_S + \ii\beta^\star\wedge \beta \ge_m 0,$$ and Th.~9.3 follows. \endproof \begstat{(9.4) Corollary} Let $X$ be a compact $n$-dimensional complex manifold, $E$ a vector bundle of rank $r\ge 2$ and $m\ge 1$ an integer. Then \smallskip \item{\rm a)} $E>_\Grif0 \Longrightarrow H^{n,q}(X,E\otimes \det\,E)=0$~~~for~~$q\ge 1\,;$ \smallskip \item{\rm b)} $E>_\Grif0\Longrightarrow H^{n,q}\big (X,E^\star\otimes(\det\,E)^m\big)=0$~~~for~~$q\ge 1$\hfill\break and~~$m\ge\min\{n-q+1,r\}\,;$ \smallskip \item{\rm c)} Let~ $0\to S\to E\to Q\to 0$ be an exact sequence of vector bundles and $m=\min\{n-q+1,\rk\,S\}$, $q\ge 1$. If $E>_m0$ and if $L$ is a line bundle such that $L\otimes(\det Q)^{-m}\ge 0$, then $$H^{n,q}(X,S\otimes L)=0.$$ \endstat \begproof{} Immediate consequence of Theorems 7.3, 8.1, 9.2 and 9.3.\qed \endproof Note that under our hypotheses $\omega=\ii\,\Tr_E\,\Theta(E)= \ii\Theta(\Lambda^rE)$ is a K\"ahler metric on $X$. Corollary 2.5 shows that it is enough in a), b), c) to assume semi-positivity and strict positivity in one point (this is true a priori only if $X$ is supposed in addition to be K\"ahler, but this hypothesis can be removed by means of Siu's result mentioned after~(4.5). a) is in fact a special case of a result of (Griffiths 1969), which we will prove in full generality in volume II (see the chapter on vanishing theorems for ample vector bundles); property b) will be also considerably strengthened there. Property c) is due to (Skoda 1978) for $q=0$ and to (Demailly 1982c) in general. Let us take the tensor product of the exact sequence in c) with $(\det\,Q)^l$. The corresponding long cohomology exact sequence implies that the natural morphism $$H^{n,q}\big(X,E\otimes(\det\,Q)^l\big)\lra H^{n,q}\big(X,Q\otimes(\det\,Q)^l\big)$$ is surjective for $q\ge 0$ and $l,m\ge\min\{n-q,\rk\,S\}$, bijective for $q\ge 1$ and $l,m\ge\min\{n-q+1,\rk\,S\}$. \titleb{10.}{Cohomology Groups of $\cO(k)$ over $\bbbp^n$} As an illustration of the above results, we compute now the cohomology groups of all line bundles $\cO(k)\to\bbbp^n$. This precise evaluation will be needed in the proof of a general vanishing theorem for vector bundles, due to Le Potier (see volume II). As in \S V-15, we consider a complex vector space $V$ of dimension $n+1$ and the exact sequence $$0\lra\cO(-1)\lra\soul V\lra H\lra 0\leqno(10.1)$$ of vector bundles over $\bbbp^n=P(V)$. We thus have \hbox{ $\det\soul V=\det H\otimes\cO(-1)$}, and as $TP(V)=H\otimes\cO(1)$ by Th.~V-15.7, we find $$K_{P(V)}=\det T^\star P(V)=\det H^\star\otimes\cO(-n)=\det\soul V^\star \otimes\cO(-n-1)\leqno(10.2)$$ where~ $\det\soul V$ is a trivial line bundle. Before going further, we need some notations. For every integer $k\in\bbbn$, we consider the homological complex $C^{\bu,k} (V^\star)$ with differential $\gamma$ such that $$\left\{\eqalign{ C^{p,k}(V^\star) =&~\Lambda^pV^\star\otimes S^{k-p}V^\star,~~~~0\le p\le k,\cr =&~0~~~~~\hbox{\rm otherwise},\cr \gamma~:~~\Lambda^pV^\star&\otimes S^{k-p}V^\star\lra\Lambda^{p-1}V^\star \otimes S^{k-p+1}V^\star,\cr}\right.\leqno(10.3)$$ where $\gamma$ is the linear map obtained by contraction with the Euler vector field $\Id_V\in V\otimes V^\star$, through the obvious maps $V\otimes\Lambda^pV^\star\lra\Lambda^{p-1}V^\star$ and $V^\star\otimes S^{k-p}V^\star\lra S^{k-p+1}V^\star$. If $(z_0\ld z_n)$ are coordinates on $V$, the module $C^{p,k}(V^\star)$ can be identified with the space of $p$-forms $$\alpha(z)=\sum_{|I|=p}\alpha_I(z)\,dz_I$$ where the $\alpha_I$ 's are homogeneous polynomials of degree $k-p$. The differential $\gamma$ is given by contraction with the Euler vector field $\xi=\sum_{0\le j\le n}z_j\,\partial/\partial z_j$. Let us denote by $Z^{p,k}(V^\star)$ the space of $p$-cycles of $C^{\bu,k}(V^\star)$, i.e.\ the space of forms $\alpha\in C^{p,k}(V^\star)$ such that $\xi\ort\alpha=0$. The exterior derivative $d$ also acts on $C^{\bu,k}(V^\star)\,;$ we have $$d~:~~C^{p,k}(V^\star)\lra C^{p+1,k}(V^\star),$$ and a trivial computation shows that $d\gamma+\gamma d=k\cdot\Id_ {C^{\bu,k}(V^\star)}.$ \begstat{(10.4) Theorem} For $k\ne 0$, $C^{\bu,k}(V^\star)$ is exact and there exist canonical isomorphisms $$C^{\bu,k}(V^\star)=\Lambda^pV^\star\otimes S^{k-p}V^\star\simeq Z^{p,k}(V^\star)\oplus Z^{p-1,k}(V^\star).$$ \endstat \begproof{} The identity $d\gamma+\gamma d=k\cdot\Id$ implies the exactness. The isomorphism is given by ${1\over k}\gamma d\oplus\gamma$ and its inverse by $\cP_1+{1\over k}d\circ\cP_2$.\qed \endproof Let us consider now the canonical mappings $$\pi~:~~V\setminus\{0\}\lra P(V),~~~~\mu'~:~~V\setminus\{0\}\lra\cO(-1)$$ defined in \S V-15. As $T_{[z]}P(V)\simeq V/\bbbc\xi(z)$ for all $z\in V\setminus\{0\}$, every form $\alpha\in Z^{p,k}(V^\star)$ defines a holomorphic section of $\pi^\star\big(\Lambda^pT^\star P(V)\big)$, $\alpha(z)$ being homogeneous of degree $k$ with respect to $z$. Hence $\alpha(z)\otimes\mu'(z)^{-k}$ is a holomorphic section of $\pi^\star\big(\Lambda^pT^\star P(V)\otimes\cO(k)\big)$, and since its homogeneity degree is $0$, it induces a holomorphic section of $\Lambda^pT^\star P(V)\otimes\cO(k)$. We thus have an injective morphism $$Z^{p,k}(V^\star)\lra H^{p,0}\big(P(V),\cO(k)\big).\leqno(10.5)$$ \begstat{(10.6) Theorem} The groups $H^{p,0}\big(P(V),\cO(k)\big)$ are given by \smallskip \item{\rm a)} $H^{p,0}\big(P(V),\cO(k)\big)\simeq Z^{p,k}(V^\star)$ ~~for~~$k\ge p\ge 0,$ \smallskip \item{\rm b)} $H^{p,0}\big(P(V),\cO(k)\big)=0$ ~~for~~$k\le p$~~and~~$(k,p)\ne(0,0)$. \endstat \begproof{} Let $s$ be a holomorphic section of $\Lambda^pT^\star P(V) \otimes\cO(k)$. Set $$\alpha(z)=(d\pi_z)^\star\big(s([z])\otimes\mu'(z)^k\big),~~~~z\in V\setminus\{0\}.$$ Then $\alpha$ is a holomorphic $p$-form on $V\setminus\{0\}$ such that $\xi\ort\alpha=0$, and the coefficients of $\alpha$ are homogeneous of degree $k-p$ on $V\setminus\{0\}$ (recall that $d\pi_{\lambda z}=\lambda^{-1} d\pi_z$). It follows that $\alpha=0$ if $kp\,;$ \smallskip \item{${\rm \ovl c)}$} $H^{p,n}\big(P(V),\cO(k)\big)\simeq Z^{n-p,-k}(V)$ ~~for~~$k<-n+p\,;$ \smallskip \item{${\rm \ovl d)}$} $H^{p,p}\big(P(V),\bbbc\big)=\bbbc,~~~~0\le p\le n.$ \endstat \begproof{} $\bullet$ ${\rm \ovl d)}$ is already known, and so is a) when $k=0$ (Th.~VI-13.3). \medskip \noindent$\bullet$ b) and ${\rm \ovl b)}$ follow from Th.~10.6, and c), ${\rm \ovl c)}$ are equivalent to b), ${\rm \ovl b)}$ via Serre duality: $$H^{p,q}\big(P(V),\cO(k)\big)^\star=H^{n-p,n-q}\big(P(V),\cO(-k)\big),$$ thanks to the canonical isomorphism $\big(Z^{p,k}(V)\big)^\star=Z^{p,k} (V^\star)$. \medskip \noindent$\bullet$ Let us prove now property a) when $k\ne 0$ and property d). By Serre duality, we may assume $k>0$. Then $$\Lambda^pT^\star P(V)\simeq K_{P(V)}\otimes\Lambda^{n-p}TP(V).$$ It is very easy to verify that $E\ge_\Nak0$ implies $\Lambda^sE \ge_\Nak0$ for every integer $s$. Since $TP(V)\ge_\Nak0$, we get therefore $$F=\Lambda^{n-p}TP(V)\otimes\cO(k)>_\Nak0~~~~\hbox{\rm for}~~k>0,$$ and the Nakano vanishing theorem implies $$\eqalignno{ H^{p,q}\big(P(V),\cO(k)\big)&=H^q\big(P(V),\Lambda^pT^\star P(V)\otimes \cO(k)\big)\cr &=H^q\big(P(V),K_{P(V)}\otimes F\big)=0,~~~~q\ge 1.&\square\cr}$$ \endproof \titleb{11.}{Ample Vector Bundles} \titlec{11.A.}{Globally Generated Vector Bundles} All definitions concerning ampleness are purely algebraic and do not involve differential geometry. We shall see however that ampleness is intimately connected with the differential geometric notion of positivity. For a general discussion of properties of ample vector bundles in arbitrary characteristic, we refer to (Hartshorne 1966). \begstat{(11.1) Definition} Let $E\to X$ be a holomorphic vector bundle over an arbitrary complex manifold~$X$. \smallskip \item{\rm a)} $E$ is said to be globally generated if for every $x\in X$ the evaluation map $H^0(X,E) \to E_x$ is onto. \smallskip \item{\rm b)} $E$ is said to be semi-ample if there exists an integer $k_0$ such that $S^kE$ is globally generated for $k\ge k_0$. \smallskip \endstat Any quotient of a trivial vector bundle is globally generated, for example the tautological quotient vector bundle $Q$ over the Grassmannian $G_r(V)$ is globally generated. Conversely, every globally generated vector bundle $E$ of rank $r$ is isomorphic to the quotient of a trivial vector bundle of rank $\le n+r$, as shown by the following result. \begstat{(11.2) Proposition} If a vector bundle $E$ of rank $r$ is globally generated, then there exists a finite dimensional subspace $V\subset H^0(X,E)$, $\dim V\le n+r$, such that $V$ generates all fibers $E_x$, $x\in X$. \endstat \begproof{} Put an arbitrary hermitian metric on $E$ and consider the Fr\'echet space ${\cal F}=\big(H^0(X,E)\big)^{n+r}$ of $(n+r)$-tuples of holomorphic sections of $E$, endowed with the topology of uniform convergence on compact subsets of~$X$. For every compact set $K\subset X$, we set $$A(K)= \{(s_1\ld s_{n+r})\in{\cal F}~{\rm which~do~not~generate~}E~{\rm on}~K\}.$$ It is enough to prove that $A(K)$ is of first category in ${\cal F}\,:$ indeed, Baire's theorem will imply that $A(X)= \bigcup A(K_\nu)$ is also of first category, if $(K_\nu)$ is an exhaustive sequence of compact subsets of $X$. It is clear that $A(K)$ is closed, because $A(K)$ is characterized by the closed condition $$\min_K\sum_{i_1<\cdotsd ~~{\rm if}~~r\ge 2.$$ \endstat \begstat{(11.9) Proposition} If $E$ is very ample of rank $r$, there exists a subspace $V$ of $H^0(X,E)$, $\dim V\le\max\big(nr+n+r,2(n+r)\big)$, such that all the evaluation maps $V\to E_x\oplus E_y$, $x\ne y$, and $V\to (J^1E)_x$, $x\in X$, are surjective. \endstat \begproof{} The arguments are exactly the same as in the proof of Prop.~11.4, if we consider instead the bundles $J^1E\lra X$ and $E\times E\lra X\times X\setminus\Delta_X$ of respective ranks $r(n+1)$ and $2r$, and sections $s'_1\ld s'_N\in H^0(X,E)$ generating these bundles.\qed \endproof \begstat{(11.10) Proposition} Let $E\to X$ be a holomorphic vector bundle. \smallskip \item{\rm a)} If $V\subset H^0(X,E)$ generates $J^1E\lra X$ and $E\times E\lra X\times X\setminus\Delta_X$, then $\psi_V$ is an embedding. \smallskip \item{\rm b)} Conversely, if rank$\,E=1$ and if there exists $V\subset H^0(X,E)$ generating $E$ such that $\psi_V$ is an embedding, then $E$ is very ample.\smallskip \endstat \begproof{} b) is immediate, because $E=\psi_V^\star(\cO(1))$ and $\cO(1)$ is very ample. Note that the result is false for $r\ge 2$ as shown by the example $E=Q$ over $X=G_r(V)$. \medskip \noindent a) Under the assumption of a), it is clear since $S_x=\{s\in V~;~ s(x)=0\}$ that $S_x=S_y$ implies $x=y$, hence $\psi_V$ is injective. Therefore, it is enough to prove that the map $x\mapsto S_x$ has an injective differential. Let $x\in X$ and $W\subset V$ such that $S_x\oplus W=V$. Choose a coordinate system in a neighborhood of $x$ in $X$ and a small tangent vector $h\in T_xX$. The element $S_{x+h}\in G_r(V)$ is the graph of a small linear map $u=O(|h|):S_x\to W$. Thus we have $$S_{x+h}=\{s'=s+t\in V~;~s\in S_x,~t=u(s)\in W,~s'(x+h)=0\}.$$ Since $s(x)=0$ and $|t|=O(|h|)$, we find $$s'(x+h)=s'(x)+d_xs'\cdot h+O(|s'|\cdot|h|^2)=t(x)+d_xs\cdot h+ O(|s|\cdot|h|^2),$$ thus $s'(x+h)=0$ if and only if $t(x)=-d_xs\cdot h+O(|s|\cdot|h|^2)$. Thanks to the fiber isomorphism $\Psi_V:E_x\lra V/S_x\simeq W$,~ $t(x)\longmapsto t$ mod $S_x$, we get: $$u(s)=t=\Psi_V(t(x))=-\Psi_V\big(d_xs\cdot h+O(|s|\cdot|h|^2)\big).$$ Recall that $T_yG_r(V)=\hom(S_y,Q_y)=\hom(y,V/y)$ (see V-16.5) and use these identifications at $y=S_x$. It follows that $$(d_x\psi_V)\cdot h=u=\big(S_x\lra V/S_x,~s\longmapsto -\Psi_V (d_xs\cdot h)\big),\leqno(11.11)$$ Now hypothesis a) implies that $S_x\ni s\longmapsto d_xs\in\hom(T_xX,E_x)$ is onto, hence $d_x\psi_V$ is injective.\qed \endproof \begstat{(11.12) Corollary} If $E$ is an ample line bundle, then $E>0$. \endstat \begproof{} If $E$ is very ample, diagram $(11.4')$ shows that $E$ is the pull-back of $\cO(1)$ by the embedding $\psi_V$, hence $i\Theta(E)= \psi_V^\star\big(\ii\Theta(\cO(1))\big)>0$ with the induced metric. The ample case follows by extracting roots.\qed \endproof \begstat{(11.13) Corollary} If $E$ is a very ample vector bundle, then $E$ carries a hermitian metric such that $E^\star<_\Nak 0$, in particular $E>_\Grif 0$. \endstat \begproof{} Choose $V$ as in Prop.~11.9 and select an arbitrary hermitian metric on $V$. Then diagram 11.4 yields $E=\psi_V^\star Q$, hence $\theta_E=\Psi_V^\star\theta_Q$. By formula (V-16.9) we have for every $\xi\in TG_r(V)=\hom(S,Q)$ and $t\in Q\,:$ $$\theta_Q(\xi\otimes t,\xi\otimes t)=\sum_{j,k,l} \xi_{jk}\ovl\xi_{lk}t_l\ovl t_j =\sum_k\Big|\sum_j\ovl t_j\xi_{jk}\Big|^2=|\langle\bu,t\rangle\circ\xi|^2.$$ Let $h\in T_xX$, $t\in E_x$. Thanks to formula (11.11), we get $$\eqalign{ \theta_E(h\otimes t&,h\otimes t)=\theta_Q\big((d_x\psi_V\cdot h)\otimes \Psi_V(t),(d_x\psi_V\cdot h)\otimes\Psi_V(t)\big)\cr &=\big|\langle\bu,\Psi_V(t)\rangle\circ(d_x\psi_V\cdot h)\big|^2 =\big|S_x\ni s\longmapsto\langle\Psi_V(d_xs\cdot h),\Psi_V(t)\rangle\big|^2\cr &=\big|S_x\ni s\longmapsto\langle d_xs\cdot h,t\rangle\big|^2\ge 0.\cr}$$ As $S_x\ni s\mapsto d_xs\in T^\star X\otimes E$ is surjective, it follows that \hbox{$\theta_E(h\otimes t,h\otimes t)\ne 0$} when $h\ne 0$, $t\ne 0$. Now, $d_xs$ defines a linear form on $TX\otimes E^\star$ and the above formula for the curvature of $E$ clearly yields $$\theta_{E^\star}(u,u)=-|S_x\ni s\longmapsto d_xs\cdot u|^2<0~~~~{\rm if}~ u\ne 0.\eqno{\square}$$ \endproof \begstat{(11.14) Problem (Griffiths 1969)} If $E$ is an ample vector bundle over a compact manifold~$X$, then is $E>_\Grif 0$~? \endstat Griffiths' problem has been solved in the affirmative when $X$ is a curve (Umemura~1973), see also (Campana-Flenner~1990), but the general case is still unclear and seems very deep. The next sections will be concerned with the important result of Kodaira asserting the equivalence between positivity and ampleness for line bundles. \titleb{12.}{Blowing-up along a Submanifold} Here we generalize the blowing-up process already considered in Remark 4.5 to arbitrary manifolds. Let $X$ be a complex $n$-dimensional manifold and $Y$ a closed submanifold with $\codim_XY=s$. \begstat{(12.1) Notations} The normal bundle of $Y$ in $X$ is the vector bundle over $Y$ defined as the quotient $NY=(TX)_{\restriction Y}/TY$. The fibers of $NY$ are thus given by $N_yY=T_yX/T_yY$ at every point $y\in Y$. We also consider the projectivized normal bundle $P(NY)\lra Y$ whose fibers are the projective spaces $P(N_yY)$ associated to the fibers of $NY$. \endstat The {\it blow-up of $X$ with center $Y$} (to be constructed later) is a complex $n$-dimensional manifold $\wt X$ together with a holomorphic map $\sigma:\wt X\lra X$ such that: \smallskip \item{i)} $E:=\sigma^{-1}(Y)$ is a smooth {\it hypersurface} in $\wt X$, and the restriction \hbox{$\sigma:E\to Y$} is a holomorphic fiber bundle isomorphic to the projec\-ti\-vized normal bundle $P(NY)\to Y$. \smallskip \item{ii)} $\sigma:\wt X\setminus E\lra X\setminus Y$ is a biholomorphism. \smallskip \noindent In order to construct $\wt X$ and $\sigma$, we first define the set-theoretic underlying objects as the disjoint sums $$\cmalign{ \wt X&=(X\setminus Y)\amalg E,~~~~ &{\rm where}~~E:=P(NY),\cr \sigma&=\Id_{X\setminus Y}\amalg{}\,\,\pi,~~~~ &{\rm where}~~\pi:E\lra Y.\cr}$$ \Input epsfiles/fig_7_1.tex \vskip16mm \centerline{{\bf VII-1} Blow-up of one point in $X$.} \vskip6mm \noindent This means intuitively that we have replaced each point $y\in Y$ by the projective space of all directions normal to $Y$. When $Y$ is reduced to a single point, the geometric picture is given by Fig.~1 below. In general, the picture is obtained by slicing $X$ transversally to $Y$ near each point and by blowing-up each slice at the intersection point with~$Y$. It remains to construct the manifold structure on $\wt X$ and in particular to describe what are the holomorphic functions near a point of~$E$. Let $f,g$ be holomorphic functions on an open set $U\subset X$ such that \hbox{$f=g=0$} on $Y\cap U$. Then $df$ and $dg$ vanish on $TY_{\restriction Y\cap U}$, hence $df$ and $dg$ induce linear forms on $NY_{\restriction Y\cap U}$. The holomorphic function $h(z)=f(z)/g(z)$ on the open set $$U_g:=\big\{z\in U~;~g(z)\ne 0\big\}\subset U\setminus Y$$ can be extended in a natural way to a function $\wt h$ on the set $$\wt U_g=U_g\cup\big\{(z,[\xi])\in P(NY)_{\restriction Y\cap U}~;~dg_z(\xi) \ne 0\big\}\subset\wt X$$ by letting $$\wt h(z,[\xi])={df_z(\xi)\over dg_z(\xi)},~~~~(z,[\xi])\in P(NY)_{\restriction Y\cap U}.$$ Using this observation, we now define the manifold structure on $\wt X$ by giving explicitly an atlas. Every coordinate chart of $X\setminus Y$ is taken to be also a coordinate chart of $\wt X$. Furthermore, for every point $y_0\in Y$, there exists a neighborhood $U$ of $y_0$ in $X$ and a coordinate chart $\tau(z)=(z_1\ld z_n):U\to\bbbc^n$ centered at $y_0$ such that $\tau(U)=B'\times B''$ for some balls \hbox{$B'\subset\bbbc^s$}, $B''\subset\bbbc^{n-s}$, and such that \hbox{$Y\cap U=\tau^{-1}(\{0\}\times B'')=\{z_1{=}\ldots{=}z_s{=}0\}$.} It~follows that $(z_{s+1}\ld z_n)$ are local coordinates on $Y\cap U$ and that the vector fields $(\partial/\partial z_1\ld\partial/\partial z_s)$ yield a holomorphic frame of $NY_{\restriction Y\cap U}$. Let us denote by $(\xi_1\ld\xi_s)$ the corresponding coordinates along the fibers of $NY$. Then $(\xi_1\ld\xi_s,z_{s+1}\ld z_n)$ are coordinates on the total space $NY$. For every $j=1\ld s$, we set $$\wt U_j=\wt U_{z_j}=\big\{z\in U\setminus Y~;~z_j\ne 0\big\}\cup \big\{(z,[\xi])\in P(NY)_{\restriction Y\cap U}~;~\xi_j\ne 0\big\}.$$ Then $(\wt U_j)_{1\le j\le s}$ is a covering of $\wt U=\sigma^{-1}(U)$ and for each $j$ we define a coordinate chart $\wt\tau_j=(w_1\ld w_n): \wt U_j\lra\bbbc^n$ by $$w_k:=\Big({z_k\over z_j}\Big)^\sim~~~~{\rm for}~~1\le k\le s,k\ne j\,;~~~~ w_k:=z_k~~~{\rm for}~~k>s~~{\rm or}~~k=j.$$ For $z\in U\setminus Y$, resp. $(z,[\xi])\in P(NY)_{\restriction Y\cap U}$, we get $$\eqalign{ \wt\tau_j(z)&=(w_1\ld w_n)=\Big({z_1\over z_j}\ld{z_{j-1}\over z_j},z_j, {z_{j+1}\over z_j}\ld{z_s\over z_j},z_{s+1}\ld z_n\Big),\cr \wt\tau_j(z,[\xi])&=(w_1\ld w_n)=\Big({\xi_1\over \xi_j}\ld{\xi_{j-1}\over \xi_j},\,0\,,{\xi_{j+1}\over\xi_j} \ld{\xi_s\over \xi_j},\xi_{s+1}\ld \xi_n\Big).\cr}$$ With respect to the coordinates $(w_k)$ on $\wt U_j$ and $(z_k)$ on $U$, the map $\sigma$ is given by $$\leqalignno{ \wt U_j&\buildo\sigma\over\lra U&(12.2)\cr w&\buildo\sigma_j\over\longmapsto (w_1w_j\ld w_{j-1}w_j\,;\,w_j\,;\,w_{j+1}w_j\ld w_sw_j\,;\,w_{s+1}\ld w_n)\cr}$$ where $\sigma_j=\tau\circ\sigma\circ\wt\tau_j^{-1}$, thus $\sigma$ is holomorphic. The range of the coordinate chart $\wt\tau_j$ is $\wt\tau_j(\wt U_j)=\sigma_j^{-1}\big(\tau(U)\big)$, so it is actually open in $\bbbc^n$. Furthermore $E\cap\wt U_j$ is defined by the single equation $w_j=0$, thus $E$ is a smooth hypersurface in $\wt X$. It remains only to verify that the coordinate changes $w\longmapsto w'$ associated to any coordinate change $z\longmapsto z'$ on $X$ are holomorphic. For that purpose, it is sufficient to verify that $(f/g)^\sim$ is holomorphic in $(w_1\ld w_n)$ on $\wt U_j\cap\wt U_g$. As $g$ vanishes on $Y\cap U$, we can write $g(z)=\sum_{1\le k\le s} z_kA_k(z)$ for some holomorphic functions $A_k$ on $U$. Therefore $${g(z)\over z_j}=A_j(\sigma_j(w))+\sum_{k\ne j}w_kA_k(\sigma_j(w))$$ has an extension $(g/z_j)^\sim$ to $\wt U_j$ which is a holomorphic function of the variables $(w_1\ld w_n)$. Since $(g/z_j)^\sim(z,[\xi])= dg_z(\xi)/\xi_j$ on $E\cap\wt U_j$, it is clear that $$\wt U_j\cap\wt U_g=\big\{w\in\wt U_j~;~(g/z_j)^\sim(w)\ne 0\big\}.$$ Hence $\wt U_j\cap\wt U_g$ is open in $\wt U_g$ and $(f/g)^\sim= (f/z_j)^\sim/(g/z_j)^\sim$ is holomorphic on $\wt U_j\cap\wt U_g$. \begstat{(12.3) Definition} The map $\sigma:\wt X\to X$ is called the blow-up of $X$ with center $Y$ and $E=\sigma^{-1}(Y)\simeq P(NY)$ is called the exceptional divisor of $\wt X$. \endstat According to (V-13.5), we denote by $\cO(E)$ the line bundle on $\wt X$ associated to the divisor $E$ and by $h\in H^0(\wt X,\cO(E))$ the canonical section such that ${\rm div}(h)=[E]$. On the other hand, we denote by $\cO_{P(NY)}(-1)\subset\pi^\star(NY)$ the tautological line subbundle over $E=P(NY)$ such that the fiber above the point $(z,[\xi])$ is $\bbbc\xi\subset N_zY$. \begstat{(12.4) Proposition} $\cO(E)$ enjoys the following properties: \smallskip \item{\rm a)} $\cO(E)_{\restriction E}$ is isomorphic to $\cO_{P(NY)}(-1)$. \smallskip \item{\rm b)} Assume that $X$ is compact. For every positive line bundle $L$ over $X$, the line bundle $\cO(-E)\otimes\sigma^\star(L^k)$ over $\wt X$ is positive for $k>0$ large enough.\smallskip \endstat \begproof{} a) The canonical section $h\in H^0(\wt X,\cO(E))$ vanishes at order $1$ along~$E$, hence the kernel of its differential $$dh:(T\wt X)_{\restriction E}\lra\cO(E)_{\restriction E}$$ is $TE$. We get therefore an isomorphism $NE\simeq \cO(E)_{\restriction E}$. Now, the map \hbox{$\sigma:\wt X\to X$} satisfies $\sigma(E)\subset Y$, so its differential $d\sigma:T\wt X\lra\sigma^\star(TX)$ is such that $d\sigma(TE)\subset\sigma^\star(TY)$. Therefore $d\sigma$ induces a morphism $$NE\lra\sigma^\star(NY)=\pi^\star(NY)\leqno(12.5)$$ of vector bundles over $E$. The vector field $\partial/ \partial w_j$ yields a non vanishing section of $NE$ on $\wt U_j$, and $(12.2)$ implies $$d\sigma_j\Big({\partial\over\partial w_j}\Big)= {\partial\over\partial z_j}+\sum_{1\le k\le s,k\ne j}w_k {\partial\over\partial z_k}~~~~/\!/~~~ \sum_{1\le k\le s}\xi_k{\partial\over\partial z_k}$$ at every point $(z,[\xi])\in E$. This shows that (12.5) is an isomorphism of $NE$ onto $\cO_{P(NY)}(-1)\subset\pi^\star(NY)$, hence $$\cO(E)_{\restriction E}\simeq NE\simeq\cO_{P(NY)}(-1).\leqno(12.6)$$ {\rm b)} Select an arbitrary hermitian metric on $TX$ and consider the induced metrics on $NY$ and on $\cO_{P(NY)}(1)\lra E=P(NY)$. The restriction of $\cO_{P(NY)}(1)$ to each fiber $P(N_zY)$ is the standard line bundle $\cO(1)$ over $\bbbp^{s-1}\,;$ thus by (V-15.10) this restriction has a positive definite curvature form. Extend now the metric of $\cO_{P(NY)}(1)$ on $E$ to a metric of $\cO(-E)$ on $X$ in an arbitrary way. If $F=\cO(-E)\otimes\sigma^\star(L^k)$, then $\Theta(F)=\Theta(\cO(-E))+ k\,\sigma^\star \Theta(L)$, thus for every $t\in T\wt X$ we have $$\theta_F(t,t)=\theta_{\cO(-E)}(t,t)+ k\,\theta_L\big(d\sigma(t),d\sigma(t)\big).$$ By the compactness of the unitary tangent bundle to $\wt X$ and the positivity of $\theta_L$, it is sufficient to verify that $\theta_{\cO(-E)}(t,t)>0$ for every unit vector $t\in T_z\wt X$ such that $d\sigma(t)=0$. However, from the computations of a), this can only happen when $z\in E$ and $t\in TE$, and in that case $d\sigma(t)=d\pi(t)=0$, so $t$ is tangent to the fiber $P(N_zY)$. Therefore $$\theta_{\cO(-E)}(t,t)=\theta_{\cO_{P(NY)}(1)}(t,t)>0.\eqno\square$$ \endproof \begstat{(12.7) Proposition} The canonical line bundle of $\wt X$ is given by $$K_{\wt X}=\cO\big((s-1)E\big)\otimes\sigma^\star K_X,~~~~\hbox{\it where}~~ s=\codim_XY.$$ \endstat \begproof{} $K_X$ is generated on $U$ by the holomorphic $n$-form $dz_1\wedge\ldots\wedge dz_n$. Using (12.2), we see that $\sigma^\star K_X$ is generated on $\wt U_j$ by $$\sigma^\star(dz_1\wedge\ldots\wedge dz_n)=w_j^{s-1}\,dw_1\wedge\ldots \wedge dw_n.$$ Since the divisor of the section $h\in H^0(\wt X,\cO(E))$ is the hypersurface $E$ defined by the equation $w_j=0$ in $\wt U_j$, we have a well defined line bundle isomorphism $$\sigma^\star K_X\lra \cO\big((1-s)E\big)\otimes K_{\wt X},~~~~~ \alpha\longmapsto h^{1-s}\sigma^\star(\alpha).\eqno\square$$ \endproof \titleb{13.}{Equivalence of Positivity and Ampleness for Line Bundles} We have seen in section 11 that every ample line bundle carries a hermitian metric of positive curvature. The converse will be a consequence of the following result. \begstat{(13.1) Theorem} Let $L\lra X$ be a positive line bundle and $L^k$ the \hbox{$k$-th} tensor power of $L$. For every $N$-tuple $(x_1\ld x_N)$ of distinct points of $X$, there exists a constant $C>0$ such that the evaluation maps $$H^0(X,L^k)\lra(J^mL^k)_{x_1}\oplus\cdots\oplus(J^mL^k)_{x_N}$$ are surjective for all integers $m\ge 0$, $k\ge C(m+1)$. \endstat \begstat{(13.2) Lemma} Let $\sigma:\wt X\lra X$ be the blow-up of $X$ with center the finite set $Y=\{x_1\ld x_N\}$, and let $\cO(E)$ be the line bundle associated to the exceptional divisor $E$. Then $$H^1(\wt X,\cO(-mE)\otimes\sigma^\star L^k)=0$$ for $m\ge 1$, $k\ge Cm$ and $C\ge 0$ large enough. \endstat \begproof{} By Prop.~12.7 we get $K_{\wt X}=\cO\big((n-1)E\big)\otimes \sigma^\star K_X$ and $$H^1\big(\wt X,\cO(-mE)\otimes\sigma^\star L^k\big) =H^{n,1}\big(\wt X,K_{\wt X}^{-1}\otimes\cO(-mE)\otimes\sigma^\star L^k\big) =H^{n,1}\big(\wt X,F\big)$$ where $F=\cO\big(-(m+n-1)E\big)\otimes\sigma^\star(K_X^{-1}\otimes L^k)$, so the conclusion will follow from the Kodaira-Nakano vanishing theorem if we can show that $F>0$ when $k$ is large enough. Fix an arbitrary hermitian metric on $K_X$. Then $$\Theta(F)=(m+n-1)\Theta(\cO(-E))+ \sigma^\star\big(k\Theta(L)-\Theta(K_X)\big).$$ There is $k_0\ge 0$ such that $\ii\big(k_0\Theta(L)-\Theta(K_X)\big)>0$ on~$X$, and Prop.~12.4 implies the existence of $C_0>0$ such that $\ii\big(\Theta(\cO(-E))+C_0\sigma^\star \Theta(L)\big)>0$ on~$\wt X$. Thus $\ii\Theta(F)>0$ for $m\ge 2-n$ and $k\ge k_0+C_0(m+n-1)$.\qed \endproof \begproof{of Theorem 13.1.} Let $v_j\in H^0(\Omega_j,L^k)$ be a holomorphic section of $L^k$ in a neighborhood $\Omega_j$ of $x_j$ having a prescribed $m$-jet at $x_j$. Set $$v(x)=\sum_j\psi_j(x)v_j(x)$$ where $\psi_j=1$ in a neighborhood of $x_j$ and $\psi_j$ has compact support in $\Omega_j$. Then $d''v=\sum d''\psi_j\cdot v_j$ vanishes in a neighborhood of $x_1\ld x_N$. Let $h$ be the canonical section of $\cO(E)^{-1}$ such that ${\rm div}(h)=[E]$. The $(0,1)$-form $\sigma^\star d''v$ vanishes in a neighborhood of $E=h^{-1}(0)$, hence $$w=h^{-(m+1)}\sigma^\star d''v\in\ci_{0,1}\big(\wt X,\cO(-(m+1)E)\otimes \sigma^\star L^k\big).$$ and $w$ is a $d''$-closed form. By Lemma~13.2 there exists a smooth section $u\in\ci_{0,0}\big(\wt X,\cO(-(m+1)E)\otimes\sigma^\star L^k\big)$ such that $d''u=w=h^{-(m+1)}\sigma^\star d''v$. This implies $$\sigma^\star v-h^{m+1}u\in H^0(\wt X,\sigma^\star L^k),$$ and since $\sigma^\star L$ is trivial near $E$, there exists a section $g\in H^0(X,L^k)$ such that $\sigma^\star g=\sigma^\star v-h^{m+1}u$. As $h$ vanishes at order 1 along $E$, the $m$-jet of $g$ at $x_j$ must be equal to that of $v$ (or $v_j$).\qed \endproof \begstat{(13.3) Corollary} For any holomorphic line bundle $L\lra X$, the following conditions are equivalent: \smallskip \item{\rm a)} $L$ is ample; \smallskip \item{\rm b)} $L>0$, i.e.\ $L$ possesses a hermitian metric such that $\ii\Theta(L)>0$. \endstat \begproof{} a) $\Longrightarrow$ b) is given by Cor.~11.12, whereas b) $\Longrightarrow$ a) is a consequence of Th.~13.1 for $m=1$.\qed \endproof \titleb{14.}{Kodaira's Projectivity Criterion} The following fundamental projectivity criterion is due to (Kodaira 1954). \begstat{(14.1) Theorem} Let $X$ be a compact complex manifold, $\dim_\bbbc X=n$. The following conditions are equivalent. \smallskip \item{\rm a)} $X$ is projective algebraic, i.e.\ $X$ can be embedded as an algebraic submanifold of the complex projective space $\bbbp^N$ for $N$ large. \smallskip \item{\rm b)} $X$ carries a positive line bundle $L$. \smallskip \item{\rm c)} $X$ carries a Hodge metric, i.e.\ a K\"ahler metric $\omega$ with rational cohomology class $\{\omega\}\in H^2(X,\bbbq)$.\smallskip \endstat \begproof{} a) $\Longrightarrow$ b). Take $L=\cO(1)_{\restriction X}$. \medskip \noindent b) $\Longrightarrow$ c). Take $\omega={\ii\over 2\pi}\Theta(L)\,;$ then $\{\omega\}$ is the image of $c_1(L)\in H^2(X,\bbbz)$. \medskip \noindent c) $\Longrightarrow$ b). We can multiply $\{\omega\}$ by a common denominator of its coefficients and suppose that $\{\omega\}$ is in the image of $H^2(X,\bbbz)$. Then Th.~V-13.9~b) shows that there exists a hermitian line bundle $L$ such that ${\ii\over 2\pi}\Theta(L)=\omega >0$. \medskip \noindent b) $\Longrightarrow$ a). Corollary 13.3 shows that $F=L^k$ is very ample for some integer $k>0$. Then Prop.~11.9 enables us to find a subspace $V$ of $H^0(X,F)$, $\dim V\le 2n+2$, such that $\psi_V:X\lra G_1(V)=P(V^\star)$ is an embedding. Thus $X$ can be embedded in $\bbbp^{2n+1}$ and Chow's theorem~II-7.10 shows that the image is an algebraic set in~$\bbbp^{2n+1}$.\qed \endproof \begstat{(14.2) Remark} \rm The above proof shows in particular that every $n$-dimen\-sional projective manifold $X$ can be embedded in $\bbbp^{2n+1}$. This can be shown directly by using generic projections $\bbbp^N\to\bbbp^{2n+1}$ and Whitney type arguments as in~11.2. \endstat \begstat{(14.3) Corollary} Every compact Riemann surface $X$ is isomorphic to an algebraic curve in $\bbbp^3$. \endstat \begproof{} Any positive smooth form $\omega$ of type $(1,1)$ is K\"ahler, and $\omega$ is in fact a Hodge metric if we normalize its volume so that $\int_X\omega=1$.\qed \endproof \noindent This example can be somewhat generalized as follows. \begstat{(14.4) Corollary} Every K\"ahler manifold $(X,\omega)$ such that $H^2(X,\cO)=0$ is projective. \endstat \begproof{} By hypothesis $H^{0,2}(X,\bbbc)=0=H^{2,0}(X,\bbbc)$, hence $$H^2(X,\bbbc)=H^{1,1}(X,\bbbc)$$ admits a basis $\{\alpha_1\}\ld\{\alpha_N\}\in H^2(X,\bbbq)$ where $\alpha_1\ld\alpha_N$ are harmonic real $(1,1)$-forms. Since $\{\omega\}$ is real, we have $\{\omega\}=\lambda_1\{\alpha_1\}+\ldots+\lambda_N\{\alpha_N\}$, $\lambda_j\in\bbbr$, thus $$\omega=\lambda_1\alpha_1+\ldots+\lambda_N\alpha_N$$ because $\omega$ itself is harmonic. If $\mu_1\ld\mu_N$ are rational numbers sufficiently close to $\lambda_1\ld\lambda_N$, then $\wt\omega:=\mu_1\alpha_1+\cdots\mu_N\alpha_N$ is close to $\omega$, so $\wt\omega$ is a positive definite $d$-closed $(1,1)$-form, and $\{\wt\omega\}\in H^2(X,\bbbq)$.\qed \endproof We obtain now as a consequence the celebrated Riemann criterion characterizing {\it abelian varieties} (${}={}$projective algebraic complex tori). \begstat{(14.5) Corollary} A complex torus $X=\bbbc^n/\Gamma$ $(\Gamma$ a lattice of $\bbbc^n)$ is an abelian variety if and only if there exists a positive definite hermitian form $h$ on $\bbbc^n$ such that $$\Im\big(h(\gamma_1,\gamma_2)\big)\in\bbbz~~~~{\rm for~all}~~\gamma_1,\gamma_2 \in\Gamma.$$ \endstat \begproof{(Sufficiency of the condition).} Set $\omega=-\Im h$. Then $\omega$ defines a constant K\"ahler metric on $\bbbc^n$, hence also on $X=\bbbc^n/\Gamma$. Let $(a_1\ld a_{2n})$ be an integral basis of the lattice $\Gamma$. We denote by $T_j$, $T_{jk}$ the real $1$- and $2$-tori $$T_{j}=(\bbbr/\bbbz)a_j,~~~1\le j\le n,~~~~T_{jk}=T_j\oplus T_k,~~~ 1\le j