% Complex Analytic and Differential Geometry, Chapter VI
% J.-P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France

\input analgeom.mac

\def\BC{{\rm BC}}
\def\DR{{\rm DR}}
\def\prim{{\rm prim}}

\def\Ll{\langle\!\langle}
\def\Gg{\rangle\!\rangle}

\def\rank{\mathop{\rm rank}\nolimits}
\def\Jac{\mathop{\rm Jac}\nolimits}
\def\Alb{\mathop{\rm Alb}\nolimits}
\def\Prim{\mathop{\rm Prim}\nolimits}

\titlea{Chapter VI}{\newline Hodge Theory}
\begpet
The goal of this chapter is to prove a number of basic facts in the Hodge 
theory of real or complex manifolds. The theory rests essentially on the fact 
that the De Rham (or Dolbeault) cohomology groups of a compact manifold can 
be represented by means of spaces of harmonic forms, once a Riemannian 
metric has been chosen. At this point, some knowledge of basic
results about elliptic differential operators is required. The special
properties of compact K\"ahler manifolds are then investigated in detail: 
Hodge decomposition theorem, hard Lefschetz theorem, Jacobian and Albanese
variety, $\ldots\,$; the example of curves is treated in detail. Finally,
the Hodge-Fr\"olicher spectral sequence is applied to get some 
results on general compact complex manifolds, and it is shown that
Hodge decomposition still holds for manifolds in the Fujiki class
$(\cC)$.
\endpet

\titleb{\S 1.}{Differential Operators on Vector Bundles}
We first describe some basic concepts concerning differential operators
(symbol, composition, adjunction, ellipticity), in the general setting
of vector bundles. Let $M$ be a $\ci$ differentiable manifold,
$\dim_{\bbbr}M=m$, and let $E$, $F$ be $\bbbk$-vector bundles over $M$,
with $\bbbk=\bbbr$ or $\bbbk=\bbbc$, $\rank E=r$, $\rank F=r'$.

\begstat{(1.1) Definition} A $($linear$)$ differential operator of degree
$\delta$ from $E$ to $F$ is a $\bbbk$-linear operator
$P:C^\infty(M,E)\to C^\infty(M,F)$, $u\mapsto Pu$ of the form
$$Pu(x)=\sum_{|\alpha|\le\delta}a_\alpha(x)D^\alpha u(x),$$
where $E_{\restriction\Omega}\simeq\Omega\times\bbbk^r$,
$F_{\restriction\Omega}\simeq\Omega\times\bbbk^{r'}$ are trivialized
locally on some open chart \hbox{$\Omega\subset M$} equipped with local
coordinates $(x_1\ld x_m)$, and where
\hbox{$a_\alpha(x)=\big(a_{\alpha\lambda\mu}
(x)\big)_{1\le\lambda\le r',\,1\le\mu\le r}$} are $r'\times r$-matrices
with $\ci$ coefficients on $\Omega$. Here \hbox{$D^\alpha=(\partial/
\partial x_1)^{\alpha_1}\ldots(\partial/\partial x_m)^{\alpha_m}$}
as usual, and \hbox{$u=(u_\mu)_{1\le\mu\le r}$},
\hbox{$D^\alpha u=(D^\alpha u_\mu)_{1\le\mu\le r}$} are viewed as column
matrices.
\endstat

If $t\in\bbbk$ is a parameter, a simple calculation shows that
$e^{-tu(x)}P(e^{tu(x)})$ is a polynomial of degree $\delta$ in $t$, of
the form
$$e^{-tu(x)}P(e^{tu(x)})=t^\delta\sigma_P(x,du(x))+\hbox{\rm lower order
terms $c_j(x)t^j$, $j<\delta$},$$
where $\sigma_P$ is the polynomial map from $T^\star_M\to\Hom(E,F)$
defined by
$$T^\star_{M,x}\ni\xi\mapsto\sigma_P(x,\xi)\in\Hom(E_x,F_x),\qquad
\sigma_P(x,\xi)=\sum_{|\alpha|=\delta}a_\alpha(x)\xi^\alpha.\leqno(1.2)$$
The formula involving $e^{-tu}P(e^{tu})$ shows that $\sigma_P(x,\xi)$
actually does not depend on the choice of coordinates nor on the
trivializations used for $E$, $F$. It is clear that $\sigma_P(x,\xi)$ is
smooth on $T^\star_M$ as a function of $(x,\xi)$, and is a homogeneous
polynomial of degree $\delta$ in~$\xi$. We say that $\sigma_P$ is {\it the
principal symbol} of~$P$. Now, if $E$, $F$, $G$ are vector bundles and
$$P:C^\infty(M,E)\to C^\infty(M,F),\qquad
Q:C^\infty(M,F)\to C^\infty(M,G)$$
are differential operators of respective degrees $\delta_P$, $\delta_Q$, it
is easy to check that $Q\circ P:C^\infty(M,E)\to C^\infty(M,G)$ is a
differential operator of degree $\delta_P+\delta_Q$ and that
$$\sigma_{Q\circ P}(x,\xi)=\sigma_{Q}(x,\xi)\sigma_{P}(x,\xi).\leqno(1.3)$$
Here the product of symbols is computed as a product of matrices.

Now, assume that $M$ is oriented and is equipped with a smooth volume
form $dV(x)=\gamma(x)dx_1\wedge\ldots dx_m$, where $\gamma(x)>0$ is a
smooth density. If $E$ is a euclidean or hermitian vector bundle,
we have a Hilbert space $L^2(M,E)$ of global sections $u$ of~$E$ with
measurable coefficients, satisfying the $L^2$ estimate
$$\|u\|^2=\int_M|u(x)|^2\,dV(x)<+\infty.\leqno(1.4)$$
We denote by
$$\Ll u,v\Gg=\int_M\langle u(x),v(x)\rangle\,dV(x),\qquad
u,v\in L^2(M,E)\leqno(1.4')$$
the corresponding $L^2$ inner product.

\begstat{(1.5) Definition} If $P:\ci(M,E)\to\ci(M,F)$
is a differential operator and both $E$, $F$ are euclidean or hermitian,
there exists a unique differential operator
$$P^\star:\ci(M,F)\to\ci(M,E),$$
called the {\it formal adjoint} of $P$, such that for all sections
$u\in\ci(M,E)$ and $v\in\ci(M,F)$ there is an identity
$$\Ll Pu,v\Gg=\Ll u,P^\star v\Gg,\qquad\hbox{\rm whenever
$\Supp u\cap\Supp v\compact M$}.$$
\endstat

\begproof{} The uniqueness is easy, using the density of the set
of elements $u\in\ci(M,E)$ with compact support in $L^2(M,E)$.
Since uniqueness is clear, it is enough, by a partition of unity argument,
to show the existence of $P^\star$ locally. Now, let
$Pu(x)=\sum_{|\alpha|\le\delta}a_\alpha(x)D^\alpha u(x)$
be the expansion of $P$ with respect to trivializations of $E$, $F$ given
by orthonormal frames over some coordinate open set $\Omega\subset M$. When
\hbox{$\Supp u\cap\Supp v\compact\Omega$} an integration by parts
yields
$$\eqalign{
\Ll Pu,v\Gg&=\int_\Omega\sum_{|\alpha|\le\delta,\lambda,\mu}
a_{\alpha\lambda\mu}D^\alpha u_\mu(x)\ovl v_\lambda(x)\,\gamma(x)\,
dx_1\ld dx_m\cr
&=\int_\Omega\sum_{|\alpha|\le\delta,\lambda,\mu}(-1)^{|\alpha|}
u_\mu(x)\ovl{D^\alpha(\gamma(x)\,\ovl a_{\alpha\lambda\mu}v_\lambda(x)}
\,dx_1\ld dx_m\cr
&=\int_\Omega\langle u,\sum_{|\alpha|\le\delta}(-1)^{|\alpha|}\gamma(x)^{-1}
D^\alpha\big(\gamma(x)\,{}^t\ovl a_\alpha v(x)\big)\rangle\,dV(x).\cr}$$
Hence we see that $P^\star$ exists and is uniquely defined by
$$P^\star v(x)=\sum_{|\alpha|\le\delta}(-1)^{|\alpha|}\gamma(x)^{-1}D^\alpha
\big(\gamma(x)\,{}^t\ovl a_\alpha v(x)\big).\hfill\square\leqno(1.6)$$
\endproof

It follows immediately from (1.6) that the principal symbol of $P^\star$ is
$$\sigma_{P^\star}(x,\xi)=(-1)^\delta\sum_{|\alpha|=\delta}
{}^t\ovl a_\alpha\xi^\alpha=(-1)^\delta\sigma_P(x,\xi)^\star.\leqno(1.7)$$

\begstat{(1.8) Definition} A differential operator $P$ is said to be
elliptic if
$$\sigma_P(x,\xi)\in\Hom(E_x,F_x)$$
is injective for every $x\in M$ and $\xi\in T^\star_{M,x}\ssm\{0\}$.
\endstat

\titleb{\S 2.}{Formalism of PseudoDifferential Operators}

We assume throughout this section that $(M,g)$ is a compact Riemannian
manifold. For any positive integer $k$ and any hermitian bundle $F\to
M$, we denote by $W^k(M,F)$ the Sobolev space of sections $s:M\to F$
whose derivatives up to order $k$ are in $L^2$. Let $\|~~\|_k$ be the
norm of the Hilbert space $W^k(M,F)$. Let $P$ be an elliptic
differential operator of order $d$ acting on $C^\infty(M,F)$. We need
the following basic facts of elliptic $PDE$ theory, see e.g.\
(H\"ormander 1963).

\begstat{(2.1) Sobolev lemma} For
$k>l+{m\over 2}$, $W^k(M,F)\subset C^l(M,F)$.
\endstat

\begstat{(2.2) Rellich lemma} For every integer $k$, the inclusion 
$$W^{k+1}(M,F)\lhra W^k(M,F)$$ 
is a compact linear operator.
\endstat

\begstat{(2.3) G\aa rding's inequality} Let $\wt P$ be the extension of 
$P$ to sections with distribution coefficients. For any $u\in W^0 (M,F)$
such that $\wt P u\in W^k(M,F)$, then $u\in W^{k+d}(M,F)$ and
$$\|u\|_{k+d}\le C_k(\|\wt Pu\|_k+\|u\|_0),$$
where $C_k$ is a positive constant depending only on $k$.
\endstat

\begstat{(2.4) Corollary} The operator $P:C^\infty(M,F)\to C^\infty (M,F)$
has the following properties:
\smallskip
\item{\rm i)} $\ker P$ is finite dimensional.
\smallskip
\item{\rm ii)} $P\big(\ci(M,F)\big)$ is closed and of finite codimension; 
furthermore, if $P^\star$ is the formal adjoint of $P$, there is a 
decomposition
$$C^\infty(M,F)=P\big(\ci(M,F)\big)\oplus\ker P^\star$$
as an orthogonal direct sum in $W^0(M,F)=L^2(M,F)$.\smallskip
\endstat

\begproof{} (i) G\aa rding's inequality shows that $\|u\|_{k+d}\le C_k\|u\|_0$
for any $u$ in $\ker P$. Thanks to the Sobolev  lemma, this implies that 
$\ker P$ is closed in $W^0(M,F)$. Moreover, the unit closed 
$\|~~\|_0$-ball of $\ker P$ is contained in the $\|~~\|_d$-ball of radius
$C_0$, thus compact by the Rellich lemma. Riesz' theorem implies that 
$\dim\ker P<+\infty$.
\medskip\noindent(ii) We first show that the extension
$$\wt P:W^{k+d}(M,F)\to W^k(M,F)$$
has a closed range for any $k$. For every $\varepsilon>0$, there  exists
a finite number of elements $v_1\ld v_N\in W^{k+d}(M,F)$, $N=N(\varepsilon)$,
such that
$$\|u\|_0\le\varepsilon\|u\|_{k+d}+\sum^N_{j=1}|\Ll u,v_j\Gg_0|~;
\leqno(2.5)$$
indeed the set
$$K_{(v_j)}=\Big\{ u\in W^{k+d}(M,F)~ ;~\varepsilon\|u\|_{k+d}+
\sum^N_{j=1} |\Ll u,v_j\Gg_0|\le 1\Big\}$$
is relatively compact in $W^0(M,F)$ and $\bigcap_{(v_j)}\ovl K_{(v_j)}=\{
0\}$. It follows that there exist elements $(v_j)$ such that 
$\ovl K_{(v_j)}$ is contained in the unit ball of $W^0(M,F)$,~$QED$. 
Substitute $||u||_0$ by the upper bound (2.5) in G\aa rding's inequality;
we get
$$(1-C_k\varepsilon)\|u\|_{k+d}\le C_k\Big(\|\wt P u\|_k+\sum^N_{j=1}
|\Ll u,v_j\Gg_0|\Big).$$
Define $G=\big\{u\in W^{k+d}(M,F)~;~u\perp v_j,~1\le j\le n\}$ and choose 
$\varepsilon=1/2C_k$.  We obtain
$$\|u\|_{k+d}\le 2C_k\|\wt P u\|_k,~~\forall u\in G.$$
This implies that $\wt P(G)$ is closed. Therefore
$$\wt P\big(W^{k+d}(M,F)\big)=\wt P(G)+{\rm Vect}
\big(\wt P(v_1)\ld\wt P(v_N)\big)$$ 
is closed in $W^k(M,F)$. Take in particular $k=0$. Since $\ci(M,F)$ is dense
in $W^d(M,F)$, we see that in $W^0(M,F)$ 
$$\left(\wt P\big(W^d(M,F)\big)\right)^\perp=
\Big(P\big(\ci(M,F)\big)\Big)^\perp=\ker\wt{P^\star}.$$
We have proved that
$$W^0(M,F)=\wt P\big(W^d(M,F)\big)\oplus\ker\wt{P^\star}.
\leqno(2.6)$$
Since $P^\star$ is also elliptic, it follows that $\ker\wt{P^\star}$
is finite dimensional and that $\ker\wt{P^\star}=\ker P^\star$ is
contained in $\ci(M,F)$. Thanks to G\aa rding's inequality, the
decomposition formula (2.6) yields
$$\leqalignno{
W^k(M,F)&=\wt P\big(W^{k+d}(M,F)\big)\oplus\ker P^\star,&(2.7)\cr
\ci(M,F)&=P\big(\ci(M,F)\big)\oplus\ker P^\star.&(2.8)\cr}$$
\endproof

\titleb{\S 3.}{Hodge Theory of Compact Riemannian Manifolds}
\titlec{\S 3.1.}{Euclidean Structure of the Exterior Algebra}
Let $(M,g)$ be an oriented Riemannian $\ci$-manifold, $\dim_{\bbbr} M=m$,
and $E\to M$ a hermitian vector bundle of rank $r$ over $M$. We denote 
respectively by $(\xi_1\ld\xi_m)$ and $(e_1\ld e_r)$ orthonormal frames of
$T_M$ and $E$ over an open subset $\Omega\subset M$, and by 
$(\xi^\star_1\ld\xi^\star_m)$,~$(e^\star_1\ld e^\star_r)$ the corresponding 
dual frames of $T^\star_M,~E^\star$. Let $dV$ stand for the Riemannian volume
form on $M$. The exterior algebra $\Lambda T^\star_M$ has a natural inner 
product $\langle\bu,\bu\rangle$ such that
$$\langle u_1\wedge\ldots\wedge u_p,v_1\wedge\ldots\wedge v_p\rangle=
\det(\langle u_j,v_k\rangle)_{1\le j,k\le p},~~~~u_j,v_k\in T^\star_M
\leqno(3.1)$$
for all $p$, with $\Lambda T^\star_M=\bigoplus\Lambda^p T^\star_M$ as an
orthogonal sum. Then the covectors $\xi^\star_I=\xi^\star_{i_1}\wedge
\cdots\wedge\xi^\star_{i_p},~
i_1<i_2<\cdots<i_p$, provide an orthonormal basis of $\Lambda T^\star_M$.
We also denote by $\langle\bu,\bu\rangle$ the corresponding inner product on
$\Lambda T^\star_M\otimes E$.

\begstat{(3.2) Hodge Star Operator} The Hodge-Poincar\'e-De Rham 
operator $\star$ is the collection of linear maps defined by
$$\star{}:\Lambda^pT^\star_M\to\Lambda^{m-p}T^\star_M,\qquad
u\wedge{}\star v=\langle u,v\rangle\,dV,\qquad
\forall u,v\in\Lambda^pT^\star_M.$$
\endstat

The existence and uniqueness of this operator is easily seen by using
the duality pairing
$$\leqalignno{
\Lambda^pT^\star_M\times\Lambda^{m-p}T^\star_M&{}\lra\bbbr\cr
(u,v)&{}\longmapsto u\wedge v/dV=\sum\varepsilon(I,\complement I)\,
u_Iv_{\complement I},&(3.3)\cr}$$
where $u=\sum_{|I|=p}u_I\,\xi^\star_I$, $v=\sum_{|J|=m-p}v_J\,\xi^\star_J$,
where $\complement I$ stands for the (ordered) complementary multi-index of
$I$ and $\varepsilon(I,\complement I)$ for the signature of the permutation
$(1,2\ld m)\longmapsto (I,\complement I)$. From this, we find
$${}\star v=\sum_{|I|=p}\varepsilon(I,\complement I)v_I\,
\xi^\star_{\complement I}.\leqno(3.4)$$
More generally, the sesquilinear pairing $\{\bu,\bu\}$ defined in
(V-7.1) yields an operator $\star$ on vector valued forms, such that
$$\leqalignno{
&\star{}:\Lambda^pT^\star_M\otimes E\to\Lambda^{m-p}T^\star_M\otimes E,\qquad   
\{s,{}\star t\}=\langle s,t\rangle\,dV,\qquad
s,t\in\Lambda^pT^\star_M\otimes E,&(3.3')\cr
&\star t=\sum_{|I|=p,\lambda}\varepsilon(I,\complement I)\,
t_{I,\lambda}\,\xi^\star_{\complement I}\otimes e_\lambda&(3.4')\cr}$$
for $t=\sum t_{I,\lambda}\,\xi^\star_I\otimes e_\lambda$. 
Since $\varepsilon(I,\complement I)\varepsilon(\complement I,I)=(-1)^{p(m-p)}=(-1)^{p(m-1)}$, 
we get immediately
$$\star\star t=(-1)^{p(m-1)}t~~~~{\rm on}~~\Lambda^pT^\star_M\otimes E. 
\leqno(3.5)$$
It is clear that $\star$ is an isometry of $\Lambda^\bu T^\star_M\otimes E$. 

We shall also need a variant of the $\star$ operator, namely the 
conjugate-linear operator
$$\#~:~~\Lambda^pT^\star_M\otimes E\lra\Lambda^{m-p}T^\star_M\otimes E^\star$$
defined by $s\wedge\#\,t=\langle s,t\rangle\,dV,$
where the wedge product $\wedge$ is combined with the canonical pairing
\hbox{$E\times E^\star\to\bbbc$}. We have
$$\#\,t=\sum_{|I|=p,\lambda}\varepsilon(I,\complement I)\,\ovl t_{I,\lambda}\,
\xi^\star_{\complement I}\otimes e^\star_\lambda.\leqno(3.6)$$

\begstat{(3.7) Contraction by a Vector Field.} Given a tangent vector
$\theta\in T_M$ and a form $u\in\Lambda^pT^\star_M$, the
contraction $\theta\ort u\in\Lambda^{p-1}T^\star_M$ is defined by
$$\theta\ort u\,(\eta_1\ld\eta_{p-1})=u(\theta,\eta_1\ld\eta_{p-1}),~~~~
\eta_j\in T_M.$$
\endstat

In terms of the basis $(\xi_j)$, $\bu\ort\bu$ is the bilinear operation 
characterized by
$$\xi_l\ort(\xi^\star_{i_1}\wedge\ldots\wedge\xi^\star_{i_p})=
\cases{0&if~~$l\notin\{i_1\ld i_p\}$,\cr
(-1)^{k-1}\xi^\star_{i_1}\wedge\ldots\wh{\xi^\star_{i_k}}\ldots\wedge
\xi^\star_{i_p}&if~~$l=i_k$.\cr}$$
This formula is in fact valid even when $(\xi_j)$ is non orthonormal.
A rather easy computation shows that $\theta\ort\bu$ is a {\it derivation}
of the exterior algebra, i.e.\ that
$$\theta\ort(u\wedge v)=(\theta\ort u)\wedge v+(-1)^{{\rm deg}\,u}u\wedge
(\theta\ort v).$$
Moreover, if $\wt\theta=\langle\bu,\theta\rangle\in T^\star_M$, the operator
$\theta\ort\bu$ is the adjoint map of $\wt\theta\wedge\bu$, that is,
$$\langle\theta\ort u,v\rangle=\langle u,\wt\theta\wedge v\rangle,~~~~
u,v\in\Lambda T^\star_M.\leqno(3.8)$$
Indeed, this property is immediately checked when $\theta=\xi_l$,
$u=\xi^\star_I$, $v=\xi^\star_J$.

\titlec{\S 3.2.}{Laplace-Beltrami Operators}
Let us consider the Hilbert space $L^2(M,\Lambda^pT^\star_M)$
of $p$-forms $u$ on $M$ with
measurable coefficients such that
$$\|u\|^2=\int_M |u|^2\,dV<+\infty.$$
We denote by $\Ll~,~\Gg$ the global inner product on $L^2$-forms. The Hilbert 
space  $L^2(M,\Lambda^pT^\star_M\otimes E)$ is defined similarly.

\begstat{(3.9) Theorem} The operator $d^\star=(-1)^{mp+1}\star d\,\star $
is the formal adjoint of the exterior derivative $d$ acting on
$C^\infty(M,\Lambda^pT^\star_M\otimes E)$.
\endstat

\begproof{} If $u\in C^\infty(M,\Lambda^pT^\star_M),~v\in C^\infty(M,
\Lambda^{p+1}T^\star_M\otimes)$ are compactly supported we get
$$\eqalign{
\Ll du,v\Gg&=\int_M\langle du,v\rangle\,dV=\int_M du\wedge{}\star v\cr
&=\int_M  d(u\wedge{}\star v)-(-1)^p u\wedge d\star v=-(-1)^p\int_M
u\wedge d\star v\cr}$$
by Stokes' formula. Therefore (3.4) implies
$$\Ll du,v\Gg=-(-1)^p(-1)^{p(m-1)}\int_M u\wedge\star\star d\star v
=(-1)^{mp+1}\Ll u,\star\,d\star v\Gg.\eqno{\square}$$
\endproof

\begstat{(3.10) Remark} \rm If $m$ is even, the formula reduces to 
$d^\star=-\star d\,\star $.
\endstat

\begstat{(3.11) Definition} The operator $\Delta=dd^\star+d^\star d$ is called the
Laplace-Beltrami operator of $M$.
\endstat

Since $d^\star$ is the adjoint of $d$, the Laplace operator $\Delta$ is
formally self-adjoint, i.e.\ $\Ll\Delta u,v\Gg=\Ll u,\Delta v\Gg$ when
the forms $u,v$ are of class $\ci$ and compactly supported.

\begstat{(3.12) Example} \rm Let $M$ be an open subset of $\bbbr^m$ and 
$g=\sum^m_{i=1} dx_i^2$. In that case we get
$$\eqalign{
u&=\sum_{|I|=p} u_I dx_I,~~~~
du=\sum_{|I|=p,j} {\partial u_I\over\partial x_j} dx_j\wedge dx_I,\cr
\Ll u,v\Gg&=\int_M\langle u,v\rangle\,dV=\int_M\sum_I u_Iv_I\,dV\cr}$$
One can write $dv=\sum dx_j\wedge(\partial v/\partial x_j)$ where
$\partial v/\partial x_j$ denotes the form $v$ in which all coefficients
$v_I$ are differentiated as $\partial v_I/\partial x_j$.
An integration by parts combined with contraction gives
$$\eqalign{
\Ll d^\star u,v\Gg&=\Ll u,dv\Gg=\int_M\langle u,\sum_j dx_j\wedge 
{\partial v\over\partial x_j}\rangle\,dV\cr
&=\int_M\sum_j\langle {\partial\over\partial x_j}\ort u,~{\partial v\over
\partial x_j}\rangle\,dV=-\int_M\langle\sum_j {\partial\over\partial x_j} 
\ort {\partial u\over\partial x_j},v\rangle\,dV ,\cr
d^\star u&=-\sum_j {\partial\over\partial x_j}\ort
{\partial u\over\partial x_j}=-\sum_{I,j}{\partial u_I\over\partial x_j} 
{\partial\over\partial x_j}\ort dx_I.\cr}$$
We get therefore
$$\eqalign{
dd^\star u&=-\sum_{I,j,k} {\partial^2u_I\over\partial x_j\partial x_k}
dx_k\wedge\Big( {\partial\over\partial x_j}\ort dx_I\Big),\cr 
d^\star du&=-\sum_{I,j,k} {\partial^2u_I\over\partial x_j\partial x_k}
{\partial\over\partial x_j}\ort (dx_k\wedge dx_I).\cr}$$
Since 
$${\partial\over\partial x_j}\ort (dx_k\wedge dx_I)=\Big({\partial\over
\partial x_j}\ort dx_k\Big) dx_I-dx_k\wedge\Big({\partial\over\partial
x_j}\ort dx_I\Big),$$
we obtain
$$\Delta u=-\sum_I\Big(\sum_j {\partial^2u_I\over\partial x^2_j}\Big)
dx_I.$$
In the case of an arbitrary riemannian manifold $(M,g)$ we have
$$\eqalign{
u&=\sum u_I\,\xi^\star_I,\cr
du&=\sum_{I,j}(\xi_j\cdot u_I)\,\xi^\star_j\wedge\xi^\star_I
    +\sum_I u_I\,d\xi^\star_I,\cr
d^\star u&=-\sum_{I,j}(\xi_j\cdot u_I)\,\xi_j\ort\xi^\star_I
    +\sum_{I,K}\alpha_{I,K} u_I\,\xi^\star_K\,,\cr}$$
for some $\ci$ coefficients $\alpha_{I,K}$,~$|I|=p$,~$|K|=p-1$. It follows
that the principal part of  $\Delta$ is the same as that of the second order
operator
$$u\longmapsto-\sum_I\big(\sum_j\xi^2_j\cdot u_I\big)\xi^\star_I.$$
As a consequence, $\Delta$ is {\it elliptic}.
\endstat

Assume now that $D_E$ is a hermitian connection on $E$. The formal adjoint
operator of  $D_E$ acting on $C^\infty(M,\Lambda^pT^\star_M\otimes E)$ is
$$D_E^\star=(-1)^{mp+1}\star  D_E\star  .\leqno (3.13)$$
Indeed, if $s\in C^\infty(M,\Lambda^pT^\star_M\otimes E)$,~
$t\in C^\infty(M,\Lambda^{p+1}T^\star_M\otimes E)$ have compact 
support, we get
$$\eqalign{
\Ll D_Es,t\Gg&=\int_M\langle D_Es,t\rangle\,dV=\int_M\{D_Es,{}\star t\}\cr
&=\int_M d\{s,{}\star t\}-(-1)^p\{s,D_E\star t\}
=(-1)^{mp+1}\Ll s,{}\star D_E\star t\Gg.\cr}$$

\begstat{(3.14) Definition} The Laplace-Beltrami operator associated to
$D_E$ is the second order operator $\Delta_E=D_ED_E^\star+D_E^\star D_E$.
\endstat

$\Delta_E$ is a self-adjoint elliptic operator with principal part
$$s\longmapsto-\sum_{I,\lambda}\Big(\sum_j\xi^2_j\cdot s_{I,\lambda}
\Big)\xi^\star_I\otimes e_\lambda.$$

\titlec{\S 3.3.}{Harmonic Forms and Hodge Isomorphism}
Let $E$ be a hermitian vector bundle over a {\it compact} Riemannian manifold
$(M,g)$. We assume that $E$ possesses a {\it flat} hermitian connection $D_E$
(this means that $\Theta(D_E)=D^2_E=0$, or equivalently, that $E$ is given
by a representation $\pi_1(M)\to U(r)$, cf.\ \S$\,$V-6). A fundamental
example is of course the trivial bundle $E=M\times\bbbc$ with the connection
$D_E=d$. Thanks to our flatness assumption, $D_E$ defines a generalized
De Rham complex
$$D_E:C^\infty(M,\Lambda^pT^\star_M\otimes E)\lra
C^\infty(M,\Lambda^{p+1}T^\star_M\otimes E).$$
The cohomology groups of this complex will be denoted by $H^p_{DR}(M,E)$.

The space of {\it harmonic forms of degree $p$} with respect to
the Laplace-Beltrami operator $\Delta_E=D_ED_E^\star+D_E^\star D_E$ is
defined by
$$\cH^p(M,E)=\big\{ s\in C^\infty(M,\Lambda^pT^\star_M\otimes E)~;~
\Delta_E s=0\big\}.\leqno(3.15)$$
Since $\Ll\Delta_Es,s\Gg=||D_Es||^2+||D_E^\star s||^2$, we see that 
$s\in\cH^p(M,E)$ if and only if $D_Es=D_E^\star s=0$.

\begstat{(3.16) Theorem} For any $p$, there exists an orthogonal decomposition
$$\eqalign{
&C^\infty(M,\Lambda^pT^\star_M\otimes E)=\cH^p(M,E)\oplus\Im D_E
\oplus\Im D_E^\star,\cr
&\Im D_E=D_E\big(C^\infty(M,\Lambda^{p-1}T^\star_M\otimes E)\big),\cr
&\Im D_E^\star=D_E^\star\big(C^\infty(M,\Lambda^{p+1}T^\star_M\otimes E)\big).
\cr}$$
\endstat

\begproof{} It is immediate that $\cH^p(M,E)$ is orthogonal to both
subspaces $\Im D_E$ and $\Im D_E^\star$. The orthogonality of these two
subspaces is also clear, thanks to the assumption $D^2_E=0\,$:
$$\Ll D_Es,D_E^\star t\Gg=\Ll D^2_Es,t\Gg=0.$$
We apply now Cor.~2.4 to the elliptic operator $\Delta_E=\Delta^\star_E$
acting on \hbox{$p$-forms,} i.e.\ on the bundle $F=\Lambda^pT^\star_M
\otimes E$. We get
$$\eqalignno{
&C^\infty(M,\Lambda^pT^\star_M\otimes E)=\cH^p(M,E)\oplus
\Delta_E\big(\ci(M,\Lambda^pT^\star_M\otimes E)\big),\cr
&\Im\Delta_E=\Im(D_ED_E^\star+D_E^\star D_E)
\subset \Im D_E+\Im D_E^\star.&\square\cr}$$
\endproof

\begstat{(3.17) Hodge isomorphism theorem} The De Rham cohomology group
$H^p_{DR}(M,E)$ is finite dimensional and $H^p_{DR}(M,E)\simeq\cH^p(M,E)$.
\endstat

\begproof{} According to decomposition 3.16, we get
$$\eqalign{
B^p_{DR}(M,E)&=D_E\big(C^\infty(M,\Lambda^{p-1}T^\star_M\otimes E)\big),\cr
Z^p_{DR}(M,E)&=\ker D_E=(\Im D_E^\star)^\perp=\cH^p(M,E)\oplus\Im D_E.\cr}$$
This shows that every De Rham cohomology class contains a unique harmonic
representative.\qed
\endproof

\begstat{(3.18) Poincar\'e duality} The bilinear pairing
$$H^p_{DR}(M,E)\times H^{m-p}_{DR}(M,E^\star)\lra\bbbc,~~~~
(s,t)\longmapsto\int_M s\wedge t$$
is a non degenerate duality.
\endstat

\begproof{} First note that there exists a naturally defined flat connection
$D_{E^\star}$ such that for any $s_1\in C^\infty_\bu(M,E)$,
$s_2\in C^\infty_\bu(M,E^\star)$ we have
$$d(s_1\wedge s_2)=(D_Es_1)\wedge s_2+(-1)^{\deg s_1}s_1\wedge D_{E^\star}s_2.
\leqno(3.19)$$
It is then a consequence of Stokes' formula that the map $(s,t)\mapsto\int_M
s\wedge t$ can be factorized through cohomology groups. Let $s\in 
C^\infty(M,\Lambda^pT^\star_M\otimes E)$. We leave to the reader the proof
of the following formulas (use (3.19) in analogy with the proof of Th.~3.9):
$$\leqalignno{
D_{E^\star}(\#\,s)&=(-1)^p\#\,D_E^\star s,\cr
\delta_{E^\star}(\#\,s)&=(-1)^{p+1}\#\,D_E s,&(3.20)\cr
\Delta_{E^\star}(\#\,s)&=\#\,\Delta_E s,\cr}$$
Consequently $\#s\in\cH^{m-p}(M,E^\star)$ if and only if $s\in\cH^p(M,E)$. Since
$$\int_M s\wedge\#\,s=\int_M |s|^2\,dV=\|s\|^2,$$
we see that the Poincar\'e pairing has zero kernel in the left hand factor
$\cH^p(M,E)\simeq H^p_{DR}(M,E)$. By symmetry, it has also zero kernel on
the right. The proof is achieved.\qed
\endproof

\titleb{\S 4.}{Hermitian and K\"ahler Manifolds}
Let $X$ be a complex $n$-dimensional manifold. A {\it hermitian metric}
on $X$ is a positive definite hermitian form of class $\ci$ on $T_X\,$;
in a coordinate system $(z_1\ld z_n)$, such a form can be written
\hbox{$h(z)=\sum_{1\le j,k\le n} h_{jk}(z)\,dz_j\otimes d\ovl z_k$,}
where $(h_{jk})$ is a positive hermitian matrix with $\ci$ coefficients.
According to~(III-1.8), the {\it fundamental $(1,1)$-form} associated to $h$ is
the positive form of type $(1,1)$
$$\omega=-{\rm Im~} h={\ii\over 2}\sum~ h_{jk} dz_j\wedge d\ovl z_k,~~~~
1\le j,k\le n.$$

\begstat{(4.1) Definition} \smallskip
\item{\rm a)} A hermitian manifold is a pair  $(X,\omega)$ where $\omega$
is a $\ci$ positive definite $(1,1)$-form on $X$.
\smallskip
\item{\rm b)} The metric $\omega$ is said to be k\"ahler if $d\omega=0$.
\smallskip
\item{\rm c)} $X$ is said to be a K\"ahler manifold if $X$ carries at
least one K\"ahler metric.\smallskip
\endstat

Since $\omega$ is real, the conditions $d\omega=0$, $d'\omega=0$,
$d''\omega=0$ are all equivalent. In local coordinates we see that
$d'\omega=0$ if and only if
$${\partial h_{jk}\over\partial z_l}={\partial h_{lk}\over
\partial z_j}\quad,\quad 1\le j,k,l\le n.$$
A simple computation gives
$${\omega^n\over n!}
=\det(h_{jk})\bigwedge_{1\le j\le n}\Big({\ii\over 2} dz_j\wedge d\ovl z_j\Big)
=\det(h_{jk})\,dx_1\wedge dy_1\wedge\cdots\wedge dx_n\wedge dy_n,$$
where $z_n=x_n+\ii y_n$. Therefore the $(n,n)$-form
$$dV={1\over n!}\omega^n\leqno(4.2)$$
is positive and coincides with the hermitian volume element of $X$. If $X$ is
compact, then $\int_X\omega^n=n!\,\Vol_\omega(X)>0$. This simple remark
already implies that compact K\"ahler manifolds must satisfy some restrictive
topological conditions:

\begstat{(4.3) Consequence} \smallskip
\item{\rm a)} If $(X,\omega)$ is compact K\"ahler and if $\{\omega\}$
denotes the  cohomology class of $\omega$ in $H^2(X,\bbbr)$, then
$\{\omega\}^n\ne 0$.
\smallskip
\item{\rm b)} If $X$ is compact K\"ahler, then $H^{2k}(X,\bbbr)\ne 0$ for
$0\le k\le n$. In fact, $\{\omega\}^k$ is a non zero class in
$H^{2k}(X,\bbbr)$.
\endstat

\begstat{(4.4) Example} \rm The complex projective space $\bbbp^n$ is K\"ahler.
A natural K\"ahler metric $\omega$ on $\bbbp^n$, called the {\it Fubini-Study
metric}, is defined by
$$p^\star\omega={\ii\over 2\pi} d'd''\log\big(|\zeta_0|^2+|\zeta_1|^2
+\cdots+|\zeta_n|^2\big)$$
where $\zeta_0,\zeta_1\ld\zeta_n$ are coordinates of $\bbbc^{n+1}$ and
where \hbox{$p:\bbbc^{n+1}\setminus\{0\}\to\bbbp^n$} is the projection.
Let $z=(\zeta_1/\zeta_0\ld\zeta_n/\zeta_0)$ be non homogeneous coordinates
on $\bbbc^n\subset\bbbp^n$. Then (V-15.8) and (V-15.12) show that
$$\omega={\ii\over 2\pi}d'd''\log(1+|z|^2)={\ii\over 2\pi}c\big(\cO(1)\big),
~~~~\int_{\bbbp^n}\omega^n=1.$$
Furthermore $\{\omega\}\in H^2(\bbbp^n,\bbbz)$ is a generator of the
cohomology algebra $H^\bu(\bbbp^n,\bbbz)$ in virtue of Th.~V-15.10.
\endstat

\begstat{(4.5) Example} \rm A {\it complex torus} is a quotient
$X=\bbbc^n/\Gamma$ by a lattice $\Gamma$ of rank~$2n$. Then $X$ is a
compact complex manifold. Any positive definite hermitian form
$\omega=\ii\sum h_{jk}dz_j\wedge d\ovl z_k$ with constant coefficients
defines a K\"ahler metric on~$X$.
\endstat

\begstat{(4.6) Example} \rm Every (complex) submanifold $Y$ of a
K\"ahler manifold  $(X,\omega)$ is K\"ahler with metric
$\omega_{\restriction Y}$.  Especially, all submanifolds of $\bbbp^n$
are K\"ahler.
\endstat

\begstat{(4.7) Example} \rm Consider the complex surface
$$X=(\bbbc^2\setminus\{ 0\})/\Gamma$$
where $\Gamma=\{\lambda^n~;~n\in\bbbz\}$, $\lambda<1$, acts as a
group of homotheties. Since $\bbbc^2\setminus\{ 0\}$ is diffeomorphic
to  $\bbbr^\star_+\times S^3$, we have $X\simeq S^1\times S^3$. 
Therefore $H^2(X,\bbbr)=0$ by K\"unneth's formula IV-15.10, and
property 4.3 b) shows that $X$ is not K\"ahler. More generally,
one can obtaintake $\Gamma$ to be an infinite cyclic group generated by
a holomorphic contraction of $\bbbc^2$, of the form
$$\pmatrix{z_1\cr z_2\cr}\longmapsto\pmatrix{\lambda_1z_1\cr\lambda_2z_2\cr},
\qquad\hbox{\rm resp.}\quad\pmatrix{z_1\cr z_2\cr}\longmapsto
\pmatrix{\lambda z_1\cr\lambda z_2+z_1^p\cr},$$
where $\lambda,\lambda_1,\lambda_2$ are complex numbers such that
$0<|\lambda_1|\le|\lambda_2|<1$, \hbox{$0<|\lambda|<1$}, and $p$ a positive
integer. These non K\"ahler surfaces are called {\it Hopf surfaces}.
\endstat

The following Theorem shows that a hermitian metric $\omega$ on $X$
is K\"ahler if and only if the metric $\omega$ is tangent at order $2$ to a 
hermitian metric with constant coefficients at every point of~$X$.

\begstat{(4.8) Theorem} Let $\omega$ be a $\ci$ positive definite $(1,1)$-form
on $X$. In order that $\omega$ be K\"ahler, it is necessary and sufficient
that to every point $x_0\in X$ corresponds a holomorphic coordinate system
$(z_1\ld z_n)$ centered at $x_0$ such that
$$\omega=\ii\sum_{1\le l,m\le n}\omega_{lm}\,dz_l\wedge d\ovl z_m,~~~~
\omega_{lm}=\delta_{lm}+O(|z|^2).\leqno(4.9)$$
If $\omega$ is K\"ahler, the coordinates $(z_j)_{1\le j\le n}$ can be
chosen such that
$$\omega_{lm}=\langle{\partial\over\partial z_l},{\partial\over\partial z_m}
\rangle=\delta_{lm}-\sum_{1\le j,k\le n}c_{jklm}\,z_j\ovl z_k+O(|z|^3)
,\leqno(4.10)$$
where $(c_{jklm})$ are the coefficients of the Chern curvature tensor
$$\Theta(T_X)_{x_0}=\sum_{j,k,l,m} c_{jklm}\,dz_j\wedge d\ovl z_k\otimes
\Big({\partial\over\partial z_l}\Big)^\star
\otimes{\partial\over\partial z_m}\leqno(4.11)$$
associated to $(T_X,\omega)$ at $x_0$. Such a system $(z_j)$
will be called a geodesic coordinate system at $x_0$.
\endstat

\begproof{} It is clear that (4.9) implies $d_{x_0}\omega=0$, so the
condition is sufficient. Assume now that $\omega$ is K\"ahler. Then one can
choose local coordinates $(x_1\ld x_n)$ such that $(dx_1\ld dx_n)$ is an
$\omega$-orthonormal basis of $T_{x_0}^\star X$. Therefore
$$\leqalignno{
\omega&=\ii\sum_{1\le l,m\le n}\wt\omega_{lm}\,dx_l\wedge d\ovl x_m,
~~~~{\rm where}\cr
\qquad\wt\omega_{lm}&=\delta_{lm}+O(|x|)=\delta_{lm}+\sum_{1\le j\le n}
(a_{jlm}x_j+a'_{jlm}\ovl x_j)+O(|x|^2).&(4.12)\cr}$$
Since $\omega$ is real, we have $a'_{jlm}=\ovl a_{jml}\,$; on the other
hand the K\"ahler condition $\partial\omega_{lm}/\partial x_j=
\partial\omega_{jm}/\partial x_l$ at $x_0$ implies $a_{jlm}=a_{ljm}$. Set now
$$z_m=x_m+{1\over 2}\sum_{j,l}a_{jlm}x_jx_l,~~~~1\le m\le n.$$
Then $(z_m)$ is a coordinate system at $x_0$, and
$$\eqalign{
dz_m&=dx_m+\sum_{j,l}a_{jlm}x_jdx_l,\cr
\ii\sum_mdz_m\wedge d\ovl z_m&=\ii\sum_m dx_m\wedge d\ovl x_m
+\ii\sum_{j,l,m}a_{jlm}x_j\,dx_l\wedge d\ovl x_m\cr
&\phantom{=\ii\sum_m dx_m~\wedge d\ovl x_m}
+\ii\sum_{j,l,m}\ovl a_{jlm}\ovl x_j\,dx_m\wedge d\ovl x_l+O(|x|^2)\cr
&=\ii\sum_{l,m}\wt\omega_{lm}\,dx_l\wedge\ovl dx_m+O(|x|^2)=
\omega+O(|z|^2).\cr}$$
Condition (4.9) is proved. Suppose the coordinates $(x_m)$
chosen from the beginning so that (4.9) holds with respect to $(x_m)$.
Then the Taylor expansion (4.12) can be refined into
$$\leqalignno{
\qquad\wt\omega_{lm}&=\delta_{lm}+O(|x|^2)&(4.13)\cr
&=\delta_{lm}+\sum_{j,k}
\big(a_{jklm}x_j\ovl x_k+a'_{jklm}x_jx_k+a''_{jklm}\ovl x_j\ovl x_k\big)+
O(|x|^3).}$$
These new coefficients satisfy the relations
$$a'_{jklm}=a'_{kjlm},~~~~a''_{jklm}=\ovl a'_{jkml},~~~~\ovl a_{jklm}=a_{kjml}.$$
The K\"ahler condition $\partial\omega_{lm}/\partial x_j=\partial
\omega_{jm}/\partial x_l$ at $x=0$ gives the equality
$a'_{jklm}=a'_{lkjm}\,;$ in particular $a'_{jklm}$ is invariant under all
permutations of~$j,k,l$. If we set
$$z_m=x_m+{1\over 3}\sum_{j,k,l}a'_{jklm}\,x_jx_kx_l,~~~~1\le m\le n,$$
then by (4.13) we find
$$\leqalignno{
\qquad dz_m&=dx_m+\sum_{j,k,l}a'_{jklm}\,x_jx_k\,dx_l,~~~~1\le m\le n,\cr
\omega&=\ii\sum_{1\le m\le n}dz_m\wedge d\ovl z_m+\ii\sum_{j,k,l,m}a_{jklm}\,x_j
\ovl x_k\,dx_l\wedge d\ovl x_m+O(|x|^3),\cr
\omega&=\ii\sum_{1\le m\le n}dz_m\wedge d\ovl z_m+\ii\sum_{j,k,l,m}a_{jklm}\,z_j
\ovl z_k\,dz_l\wedge d\ovl z_m+O(|z|^3).&(4.14)\cr}$$
It is now easy to compute the Chern curvature tensor $\Theta(T_X)_{x_0}$ in terms of
the coefficients $a_{jklm}$. Indeed
$$\eqalign{
\langle{\partial\over\partial z_l},{\partial\over\partial z_m}\rangle&=
\delta_{lm}+\sum_{j,k}a_{jklm}\,z_j\ovl z_k+O(|z|^3),\cr
d'\langle{\partial\over\partial z_l},{\partial\over\partial z_m}\rangle&=
\Big\{D'{\partial\over\partial z_l},{\partial\over\partial z_m}\Big\}=
\sum_{j,k}a_{jklm}\,\ovl z_k\,dz_j+O(|z|^2),\cr
\Theta(T_X)\cdot{\partial\over\partial z_l}&=D''D'\Big({\partial\over\partial z_l}\Big)
=-\sum_{j,k,m}a_{jklm}\,dz_j\wedge d\ovl z_k\otimes{\partial\over\partial z_m}
+O(|z|),\cr}$$
therefore $c_{jklm}=-a_{jklm}$ and the expansion (4.10) follows from 
(4.14).\qed
\endproof

\begstat{(4.15) Remark} \rm As a by-product of our computations, we
find that on a K\"ahler manifold the coefficients of $\Theta(T_X)$ satisfy
the symmetry relations
$$\ovl c_{jklm}=c_{kjml},~~~~c_{jklm}=c_{lkjm}=c_{jmlk}=c_{lmjk}.$$
\endstat

\titleb{\S 5.}{Basic Results of K\"ahler Geometry}
\titlec{\S 5.1.}{Operators of Hermitian Geometry}
Let $(X,\omega)$ be a hermitian manifold and let $z_j=x_j+\ii y_j$,
$1\le j\le n$, be analytic coordinates at a point $x\in X$ such that
$\omega(x)=\ii\sum dz_j\wedge d\ovl z_j$ is diagonalized at this point.
The associated hermitian form is the $h(x)=2\sum dz_j\otimes d\ovl z_j$
and its real part is the euclidean metric $2\sum (dx_j)^2+(dy_j)^2$. It
follows from this that $|dx_j|=|dy_j|=1/\sqrt{2}$, $|dz_j|=|d\ovl z_j|=1$,
and that $(\partial/\partial z_1\ld\partial/\partial z_n)$ is an orthonormal
basis of $(T^\star_x X,\omega)$. Formula (3.1) with $u_j,v_k$ in the
orthogonal sum $(\bbbc\otimes T_X)^\star=T^\star_X\oplus\ovl{T^\star_X}$
defines a natural inner product on the exterior algebra
$\Lambda^\bu(\bbbc\otimes T_X)^\star$. The norm of a form
$$u=\sum_{I,J}~~u_{I,J} dz_I\wedge d\ovl z_J\in\Lambda
(\bbbc\otimes T_X)^\star$$
at the given point $x$ is then equal to
$$|u(x)|^2=\sum_{I,J} ~|u_{I,J}(x)|^2.\leqno(5.1)$$

The Hodge $\star $ operator (3.2) can be extended to $\bbbc$-valued forms by 
the formula
$$u\wedge{}\star\ovl v=\langle u,v\rangle\,dV.\leqno(5.2)$$
It follows that $\star $ is a $\bbbc$-linear isometry
$$\star~:~~\Lambda^{p,q} T^\star_X\lra\Lambda^{n-q,n-p} T^\star_X.$$
The usual operators of hermitian geometry are the operators $d,~\delta=
-\star d\,\star ,~\Delta=d\delta+\delta d$ already defined, and their
complex counterparts
$$\left\{\eqalign{
d&=d'+d'',\cr
\delta&=d^{\prime\star}+d^{\prime\prime\star},~~~~d^{\prime\star}=(d')^\star=-\star d''\star,~~~~
d^{\prime\prime\star}=(d'')^\star=-\star d'\star,\cr
\Delta'&=d'd^{\prime\star}+d^{\prime\star}d',~~~~
\Delta''=d''d^{\prime\prime\star}+d^{\prime\prime\star}d''.\cr}\right.\leqno(5.3)$$
Another important operator is the operator $L$ of type (1,1) defined by
$$Lu=\omega\wedge u\leqno(5.4)$$
and its adjoint  $\Lambda=\star ^{-1}L\,\star ~:$
$$\langle u,\Lambda v\rangle=\langle Lu,v\rangle.\leqno(5.5)$$

\titlec{\S 5.2.}{Commutation Identities}
If $A,B$ are endomorphisms of the algebra $C^\infty_{\bu,\bu}(X,\bbbc)$,
their graded commutator (or graded Lie bracket) is defined by
$$[A,B]=AB-(-1)^{ab} BA\leqno(5.6)$$
where $a,b$ are the degrees of $A$ and $B$ respectively. If $C$ is another
endomorphism of degree $c$, the following {\it Jacobi identity} is easy to
check:
$$(-1)^{ca}\big[A,[B,C]\big]+(-1)^{ab}\big[B,[C,A]\big]
+(-1)^{bc}\big[C,[A,B]\big]=0.\leqno(5.7)$$
For any $\alpha\in\Lambda^{p,q}T^\star_X$, we still denote by $\alpha$
the endomorphism of type $(p,q)$ on $\Lambda^{\bu,\bu} T^\star_X$ defined by
$u\mapsto\alpha\wedge u$.

Let $\gamma\in\Lambda^{1,1} T^\star_X$ be a real (1,1)-form. There exists an
$\omega$-orthogonal basis $(\zeta_1,\zeta_2\ld\zeta_n)$ in $T_X$ which
diagonalizes both forms $\omega$ and $\gamma\,$:
$$\omega=\ii\sum_{1\le j\le n}\zeta^\star_j\wedge\ovl\zeta^\star_j,~~~~
\gamma=\ii\sum_{1\le j\le n}\gamma_j\,\zeta^\star_j\wedge\ovl\zeta^\star_j,~~
\gamma_j\in\bbbr.$$

\begstat{(5.8) Proposition} For every form $u=\sum u_{J,K}\,\zeta^\star_J 
\wedge\ovl\zeta^\star_K$, one has
$$[\gamma,\Lambda]u=\sum_{J,K}\Big(\sum_{j\in J}\gamma_j+\sum_{j\in K}
\gamma_j-\sum_{1\le j\le n}\gamma_j\Big) u_{J,K}\,\zeta^\star_J\wedge\ovl\zeta^\star_K.$$
\endstat

\begproof{} If $u$ is of type $(p,q)$, a brute-force computation yields
$$\eqalignno{
\Lambda u&=\ii(-1)^p\sum_{J,K,l} u_{J,K}\,(\zeta_l\ort\zeta^\star_J)
\wedge(\ovl\zeta_l\ort\ovl\zeta^\star_K),~~~~1\le l\le n,\cr 
\gamma\wedge u&=\ii(-1)^p\sum_{J,K,m}\gamma_m u_{J,K}\,\zeta^\star_m\wedge
\zeta^\star_J\wedge\ovl\zeta^\star_m\wedge\ovl\zeta^\star_K,~~~~1\le m\le n,\cr 
[\gamma,\Lambda] u&=\sum_{J,K,l,m}\gamma_m\,u_{J,K}\Big(
\big(\zeta^\star_l\wedge(\zeta_m\ort\zeta^\star_J)\big)\wedge
\big(\ovl\zeta^\star_l\wedge(\ovl\zeta_m\ort\ovl\zeta^\star_K)\big)\cr
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-\big(\zeta_m\ort(\zeta^\star_l\wedge\zeta^\star_J)\big)\wedge
\big(\ovl\zeta_m\ort(\ovl\zeta^\star_l\wedge\ovl\zeta^\star_K)\big)\Big)\cr
&=\sum_{J,K,m}\gamma_m\,u_{J,K}\Big(
\zeta^\star_m\wedge(\zeta_m\ort\zeta^\star_J)\wedge\ovl\zeta^\star_K\cr
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+\zeta^\star_J\wedge\ovl\zeta^\star_m\wedge(\ovl\zeta_m\ort\ovl\zeta^\star_K)-
\zeta^\star_J\wedge\ovl\zeta^\star_K\Big)\cr
&=\sum_{J,K}\bigg(\sum_{m\in J}\gamma_m+\sum_{m\in K}\gamma_m-\sum_{1\le 
m\le n}\gamma_m\bigg) u_{J,K}\,\zeta^\star_J\wedge\ovl\zeta^\star_K.
&\square\cr}$$
\endproof

\begstat{(5.9) Corollary} For every $u\in\Lambda^{p,q}T^\star_X$, we have
$$[L,\Lambda]u=(p+q-n) u.$$
\endstat

\begproof{} Indeed, if $\gamma=\omega$, we have $\gamma_1=\cdots=\gamma_n=1$.
\qed
\endproof

This result can be generalized as follows: for every
$u\in\Lambda^k(\bbbc\otimes T_X)^\star$, we have
$$[L^r,\Lambda]u=r(k-n+r-1)\,L^{r-1}u.\leqno(5.10)$$
In fact, it is clear that
$$\eqalign{
[L^r,\Lambda]u&=\sum_{0\le m\le r-1}\kern-8pt L^{r-1-m}[L,\Lambda]L^m u\cr
&=\sum_{0\le m\le r-1}\kern-8pt (2m+k-n)L^{r-1-m}L^m u
=\big(r(r-1)+r(k-n)\big) L^{r-1} u.\cr}$$

\titlec{\S 5.3.}{Primitive Elements and Hard Lefschetz Theorem}
In this subsection, we prove a fundamental decomposition theorem for the
representation of the unitary group $U(T_X)\simeq U(n)$ acting on
the spaces $\Lambda^{p,q}T^\star_X$ of $(p,q)$-forms. It turns out that
the representation is never irreducible if $0<p,q<n$.

\begstat{(5.11) Definition} A homogeneous element $u\in\Lambda^k
(\bbbc\otimes T_X)^\star$
is called {\it primitive} if $\Lambda u=0$. The space of primitive elements
of total degree $k$ will be denoted
$$\Prim^kT^\star_X=\bigoplus_{p+q=k}\Prim^{p,q}T^\star_X.$$
\endstat

Let  $u\in\Prim^kT^\star_X$. Then
$$\Lambda^s L^r u=\Lambda^{s-1}(\Lambda L^r-L^r\Lambda)u
=r(n-k-r+1)\Lambda^{s-1}L^{r-1}u.$$
By induction, we get for $r\ge s$
$$\Lambda^sL^ru=r(r-1)\cdots (r-s+1)\cdot (n-k-r+1) 
\cdots (n-k-r+s)L^{r-s}u.\leqno(5.12)$$
Apply (5.12) for $r=n+1$. Then $L^{n+1}u$ is of degree $>2n$ and
therefore we have $L^{n+1}u=0$. This gives
$$(n+1)\cdots\big(n+1-(s-1)\big)\cdot (-k)(-k+1)\cdots(-k+s-1)L^{n+1-s}u=0.$$
The integral coefficient is $\ne 0$ if $s\le k$, hence:

\begstat{(5.13) Corollary} If $u\in\Prim^kT^\star_X$, then $L^su=0$ for
$s\ge (n+1-k)_+$.
\endstat

\begstat{(5.14) Corollary} $\Prim^kT^\star_X=0$ for $n+1\le k\le 2n$.
\endstat

\begproof{} Apply Corollary 5.13 with $s=0$.\qed
\endproof

\begstat{(5.15) Primitive decomposition formula} For every
$u\in\Lambda^k(\bbbc\otimes T_X)^\star$, there is a unique decomposition
$$u=\sum_{r\ge (k-n)_+} L^ru_r,~~~~u_r\in\Prim^{k-2r}T^\star_X.$$
Furthermore~ $u_r=\Phi_{k,r} (L,\Lambda)u$~ where $\,\Phi_{k,r}\,$ is a non
commutative polynomial in $L,\Lambda$ with rational coefficients. As
a consequence, there are direct sum decompositions of $U(n)$-representations
$$\eqalign{
&\Lambda^k(\bbbc\otimes T_X)^\star=\bigoplus_{r\ge (k-n)_+}
L^r\Prim^{k-2r}T^\star_X,\cr
&\Lambda^{p,q}T_X^\star=\bigoplus_{r\ge(p+q-n)_+}
L^r\Prim^{p-r,q-r}T^\star_X.\cr}$$
\endstat

\begproof{of the uniqueness of the decomposition}
Assume that $u=0$ and that $u_r\ne 0$ 
for some $r$. Let $s$ be the largest integer such that $u_s\ne 0$. Then
$$\Lambda^s u=0=\sum_{(k-n)_+\le r\le s}\Lambda^s L^r u_r=
\sum_{(k-n)_+\le r\le s}\Lambda^{s-r}\Lambda^r L^r u_r.$$
But formula (5.12) shows that $\Lambda^r L^r u_r=c_{k,r} u_r$ for
some non zero integral coefficient $c_{k,r}=r!(n-k+r+1)\cdots (n-k+2r)$.
Since $u_r$ is primitive we get $\Lambda^sL^ru_r=0$ when $r<s$, hence
$u_s=0$, a contradiction.
\endproof

\begproof{of the existence of the decomposition} We prove by induction on
$s\ge(k-n)_+$ that $\Lambda^s u=0$ implies
$$u=\sum_{(k-n)_+\le r<s} L^ru_r,~~~~
u_r=\Phi_{k,r,s}(L,\Lambda)u\in\Prim^{k-2r}T^\star_X.\leqno(5.16)$$
The Theorem will follow from the step $s=n+1$.

Assume that the
result is true for $s$ and that $\Lambda^{s+1}u=0$. Then $\Lambda^s u$ is
in $\Prim^{k-2s}T^\star_X$. Since $s\ge (k-n)_+$ we have $c_{k,s}\ne 0$
and we set
$$\eqalign{
u_s&={1\over c_{k,s}}\Lambda^s u\in\Prim^{k-2s}T^\star_X,\cr
u'&=u-L^s u_s=\Big(1-{1\over c_{k,s}}L^s\Lambda^s\Big)u.\cr}$$
By formula (5.12), we get
$$\Lambda^s u'=\Lambda^su-\Lambda^s L^s u_s=\Lambda^s u-c_{k,s} u_s=0.$$
The induction hypothesis implies
$$u'=\sum_{(k-n)_+\le r<s} L^ru'_r,~u'_r=\Phi_{k,r,s} (L,\Lambda)
u'\in\Prim^{k-2r}T^\star_X,$$
hence
$u=\sum_{(k-n)_+\le r\le s} L^ru_r$ with
$$\cases{
u_r=u'_r=\Phi_{k,r,s}(L,\Lambda)\Big(1-{\displaystyle 1\over\displaystyle 
c_{k,s}} L^s\Lambda^s\Big)u,~r<s,\cr
u_s={\displaystyle 1\over\displaystyle c_{k,s}}\Lambda^su.\cr}$$
It remains to prove the validity of the decomposition 5.16) for the initial 
step $s=(k-n)_+$, i.e.\ that $\Lambda^su=0$ implies $u=0$. If $k\le n$, then $s=0$ and there is nothing to prove.
We are left with the
case  $k>n$, $\Lambda^{k-n}u=0$. Then $v={}\star u\in\Lambda^{2n-k}
(\bbbc\otimes T_X)^\star$ and $2n-k<n$. Since the decomposition exists in degree
$\le n$ by what we have just proved, we get
$$\eqalign{
v&={}\star u=\sum_{r\ge 0} L^rv_r,~ v_r\in\Prim^{2n-k-2r}T^\star_X,\cr
0&={}\star\Lambda^{k-n}u=L^{k-n}\star  u=\sum_{r\ge 0} L^{r+k-n} v_r,\cr}$$
with degree $(L^{r+k-n}v_r)=2n-k+2(k-n)=k$. The uniqueness part shows that
$v_r=0$ for~all~$r\,$, hence $u=0$. The Theorem is proved.\qed
\endproof

\begstat{(5.17) Corollary} The linear operators
$$\eqalign{
L^{n-k}:\Lambda^k (\bbbc\otimes T_X)^\star
&\lra\Lambda^{2n-k}(\bbbc\otimes T_X)^\star,\cr
L^{n-p-q}:\Lambda^{p,q}T^\star_X&\lra\Lambda^{n-q,n-p}T^\star_X,\cr}$$
are isomorphisms for all integers $k\le n$, $p+q\le n$.
\endstat

\begproof{} For every $u\in\Lambda^k_\bbbc T^\star_X$, the primitive
decomposition $u=\sum_{r\ge 0} L^ru_r$ is mapped bijectively onto
that of $L^{n-k}u\,$:
$$L^{n-k}u=\sum_{r\ge 0} L^{r+n-k}u_r.\eqno{\square}$$
\endproof

\titleb{\S 6.}{Commutation Relations}
\titlec{\S 6.1.}{Commutation Relations on a K\"ahler Manifold}
Assume first that $X=\Omega\subset\bbbc^n$ is an open subset and that
$\omega$ is the standard K\"ahler metric
$$\omega=\ii\sum_{1\le j\le n} dz_j\wedge d\ovl z_j.$$
For any form $u\in C^\infty(\Omega,\Lambda^{p,q}T^\star_X)$ we have
$$\leqalignno{
d'u&=\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
dz_k\wedge dz_I\wedge d\ovl z_J ,&(6.1')\cr
d''u&=\sum_{I,J,k} {\partial u_{I,J}\over\partial\ovl z_k}
d\ovl z_k\wedge dz_I\wedge d\ovl z_J.&(6.1'')\cr}$$
Since the global $L^2$ inner product is given by
$$\Ll u,v\Gg=\int_\Omega\sum_{I,J} u_{I,J}\ovl v_{I,J}\,dV,$$
easy computations analogous to those of Example 3.12 show that
$$\leqalignno{
d^{\prime\star}u&=-\sum_{I,J,k} {\partial u_{I,J}\over\partial\ovl z_k}
{\partial\over\partial z_k}\ort (dz_I\wedge d\ovl z_J),&(6.2')\cr
d^{\prime\prime\star}u&=-\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
{\partial\over\partial\ovl z_k}\ort (dz_I\wedge d\ovl z_J).&(6.2'')\cr}$$
We first prove a lemma due to (Akizuki and Nakano 1954).

\begstat{(6.3) Lemma} In $\bbbc^n$, we have $[d^{\prime\prime\star},L]=\ii d'$.
\endstat

\begproof{} Formula (6.$2''$) can be written more briefly
$$d^{\prime\prime\star}u=-\sum_k {\partial\over\partial\ovl z_k}\ort\Big(
{\partial u\over\partial z_k}\Big).$$
Then we get
$$[d^{\prime\prime\star},L]u=-\sum_k {\partial\over\partial\ovl z_k}\ort\Big(
{\partial\over\partial z_k}(\omega\wedge u)\Big)+\omega\wedge\sum_k 
{\partial\over\partial\ovl z_k}\ort\Big({\partial u\over\partial z_k}\Big).$$
Since $\omega$ has constant coefficients, we have $\displaystyle{\partial\over
\partial z_k}(\omega\wedge u)=\omega\wedge{\partial u\over\partial z_k}$
and therefore
$$\eqalign{
[d^{\prime\prime\star},L]\,u&=-\sum_k\bigg({\partial\over\partial\ovl z_k}\ort
\Big(\omega\wedge {\partial u\over\partial z_k}\Big)-\omega\wedge\Big(
{\partial\over\partial\ovl z_k}\ort{\partial u\over\partial z_k}\Big)\bigg)\cr
&=-\sum_k\Big({\partial\over\partial\ovl z_k}\ort\omega\Big)\wedge
{\partial u\over\partial z_k}.\cr}$$
Clearly $\displaystyle{\partial\over\partial\ovl z_k}\ort\omega=-\ii dz_k$, so
$$[d^{\prime\prime\star},L]\,u=\ii\sum_k dz_k\wedge{\partial u\over\partial z_k}=\ii d'u.
\eqno{\square}$$
\endproof

We are now ready to derive the basic commutation relations in the case of an
arbitrary K\"ahler manifold $(X,\omega)$.

\begstat{(6.4) Theorem} If $(X,\omega)$ is K\"ahler, then
$$\cmalign{&[d^{\prime\prime\star},L]&=&\ii d',~~~~~&[d^{\prime\star},L]&=-&\ii d'',\cr
&[\Lambda,d'']&=-&\ii d^{\prime\star},~~~~~&[\Lambda,d']&=&\ii d^{\prime\prime\star}.\cr}$$
\endstat

\begproof{} It is sufficient to verify the first relation, because the
second one is the conjugate of the first, and the relations of the second line
are the adjoint of those of the first line. If $(z_j)$ is a geodesic
coordinate system at a point $x_0\in X$, then for any $(p,q)$-forms $u,v$
with compact support in a neighborhood of $x_0$, (4.9) implies
$$\Ll u,v\Gg=\int_M\Big(\sum_{I,J}u_{IJ}\ovl v_{IJ}+\sum_{I,J,K,L}a_{IJKL}\,
u_{IJ}\ovl v_{KL}\Big)\,dV,$$
with $a_{IJKL}(z)=O(|z|^2)$ at $x_0$. An integration by parts as in 
(3.12) and (6.$2''$) yields
$$d^{\prime\prime\star}u=-\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
{\partial\over\partial\ovl z_k}\ort(dz_I\wedge d\ovl z_J)+
\sum_{I,J,K,L}b_{IJKL}\,u_{IJ}\,dz_K\wedge d\ovl z_L,$$
where the coefficients~ $b_{IJKL}$~ are obtained by derivation of the~
$a_{IJKL}$'s. Therefore $b_{IJKL}=O(|z|)$. Since
$\partial\omega/\partial z_k=O(|z|)$, the proof of Lemma~6.3 implies
here  $[d^{\prime\prime\star},L]u=\ii d'u+O(|z|)$, in particular
both terms coincide at every given point $x_0\in X$.\qed
\endproof

\begstat{(6.5) Corollary} If $(X,\omega)$ is K\"ahler, the complex
Laplace-Beltrami operators satisfy 
$$\Delta'=\Delta''={1\over 2}\Delta.$$
\endstat

\begproof{} It will be first shown that $\Delta''=\Delta'$. We have
$$\Delta''=[d'',d^{\prime\prime\star}]=-\ii\big[d'',[\Lambda,d']\big].$$
Since $[d',d'']=0$, Jacobi's identity (5.7) implies 
$$-\big[d'',[\Lambda,d']\big]+\big[d',[d'',\Lambda]\big]=0,$$
hence $\Delta''=\big[d',-\ii[d'',\Lambda]\big]=[d',d^{\prime\star}]=\Delta'$.
On the other hand
$$\Delta=[d'+d'',d^{\prime\star}+d^{\prime\prime\star}]=\Delta'+\Delta''+[d',d^{\prime\prime\star}]
+[d'',d^{\prime\star}].$$
Thus, it is enough to prove:
\endproof

\begstat{(6.6) Lemma} $[d',d^{\prime\prime\star}]=0,~[d'',d^{\prime\star}]=0$.
\endstat

\begproof{} We have $[d',d^{\prime\prime\star}]=-\ii\big[d',[\Lambda,d']\big]$ and 
(5.7) implies
$$-\big[d',[\Lambda,d']\big]+\big[\Lambda,[d',d']\big]+
\big[d',[d',\Lambda]\big]=0,$$
hence $-2\big[d',[\Lambda,d']\big]=0$ and $[d',d^{\prime\prime\star}]=0$. The second
relation 
$[d'',d^{\prime\star}]=0$ is the adjoint of the first.\qed
\endproof

\begstat{(6.7) Theorem} $\Delta$ commutes with all operators 
$\star ,d',d'',d^{\prime\star},d^{\prime\prime\star},L,\Lambda$.
\endstat

\begproof{} The identities $[d',\Delta']=[d^{\prime\star},\Delta']=0$,
$[d'',\Delta'']=[d^{\prime\prime\star},\Delta'']=0$ and $[\Delta,\star]=0$
are immediate. Furthermore, the equality $[d',L]=d'\omega=0$\break together
with the Jacobi identity implies
$$[L,\Delta']=\big[L,[d',d^{\prime\star}]\big]=
-\big[d',[d^{\prime\star},L]\big]=\ii[d',d'']=0.$$
By adjunction, we also get $[\Delta',\Lambda]=0$.\qed
\endproof
 
\titlec{\S 6.2}{Commutation Relations on Hermitian Manifolds}
We are going to extend the commutation relations of \S$\,$6.1 to an arbitrary
hermitian manifold $(X,\omega)$. In that case $\omega$ is no longer tangent
to a constant metric, and the commutation relations involve extra terms
arising from the {\it torsion} of $\omega$. Theorem~6.8 below is taken
from (Demailly~1984), but the idea was already contained in (Griffiths~1966).

\begstat{(6.8) Theorem}  Let $\tau$ be the operator of type $(1,0)$ and order
$0$ defined by $\tau=[\Lambda, d'\omega]$. Then
\medskip\noindent
$\cmalign{
&{\rm a)}\qquad&[d^{\prime\prime\star},L]&=&\ii(d'+\tau),\cr
&{\rm b)}\qquad&[d^{\prime\star},L]&=-&\ii(d''+\ovl\tau),\cr
&{\rm c)}\qquad&[\Lambda,d'']&=-&\ii(d^{\prime\star}+\tau^\star),\cr
&{\rm d)}\qquad&[\Lambda,d']&=&\ii(d^{\prime\prime\star}+\ovl\tau^\star)~;\cr}$
\medskip\noindent
$d'\omega$ will be called the torsion form of $\omega$, 
and $\tau$ the torsion operator.
\endstat

\begproof{} b) follows from a) by conjugation, whereas c), d) follow
from a), b)  by adjunction. It is therefore enough to prove relation a).

Let  $(z_j)_{1\le j\le n}$ be complex coordinates centered at a point
$x_0\in X$, such that  $(\partial/\partial z_1\ld\partial/\partial z_n)$ 
is an orthonormal basis of  $T_{x_0}X$ for the metric
$\omega(x_0)$. Consider the metric with constant coefficients
$$\omega_0=\ii\sum_{1\le j\le n} dz_j\wedge d\ovl z_j.$$
The metric $\omega$ can then be written
$$\omega=\omega_0+\gamma~~{\rm with}~~\gamma=O(|z|).$$
Denote by $\langle~,~\rangle_0,~L_0,~\Lambda_0,~d^{\prime\star}_0,~d^{\prime\prime\star}_0$
the inner product and the operators asso\-ciated to the constant metric
$\omega_0$, and let $dV_0=\omega^n_0/2^nn!$. The proof of relation~a)
is based on a Taylor expansion of $L,~\Lambda,~d^{\prime\star},~d^{\prime\prime\star}$
in terms of the operators with constant coefficients $L_0,~\Lambda_0,
~d^{\prime\star}_0,~d^{\prime\prime\star}_0$.
\endproof

\begstat{(6.9) Lemma} Let $u,v\in C^\infty(X,\Lambda^{p,q}T^\star_X)$. Then in a
neighborhood of $x_0$
$$\langle u,v\rangle\,dV=\langle u-[\gamma,\Lambda_0]u,v\rangle_0\,dV_0
+O(|z|^2).$$
\endstat

\begproof{} In a neighborhood of $x_0$, let
$$\gamma=\ii\sum_{1\le j\le n}\gamma_j\,\zeta^\star_j\wedge\ovl\zeta^\star_j,
~~~~\gamma_1\le\gamma_2\le\cdots\le\gamma_n,$$
be a diagonalization of the (1,1)-form $\gamma(z)$ with respect to an
orthonormal basis $(\zeta_j)_{1\le j\le n}$ of $T_zX$ for $\omega_0(z)$.
We thus have
$$\omega=\omega_0+\gamma=\ii\sum\lambda_j\,\zeta^\star_j\wedge
\ovl\zeta^\star_j$$
with $\lambda_j=1+\gamma_j$ and $\gamma_j=O(|z|)$. Set now
$$J=\{ j_1\ld j_p\},\qquad\zeta^\star_J=\zeta^\star_{j_1}\wedge\cdots\wedge
\zeta^\star_{j_p},\qquad\lambda_J=\lambda_{j_1}\cdots\lambda_{j_p},$$
$$u=\sum u_{J,K}\,\zeta^\star_J\wedge\ovl\zeta^\star_K,\qquad
v=\sum v_{J,K}\,\zeta^\star_J\wedge\ovl\zeta^\star_K$$
where summations are extended to increasing multi-indices~ $J$, $K$~ such 
that 
$|J|=p$, $|K|=q$. With respect to  $\omega$ we have
$\langle\zeta^\star_j,\zeta^\star_j\rangle=\lambda^{-1}_j$,
hence
$$\eqalign{
\langle u,v\rangle\,dV&=\sum_{J,K}\lambda^{-1}_J\lambda^{-1}_K\,
u_{J,K}\ovl v_{J,K}\,\lambda_1\cdots\lambda_n\, dV_0\cr
&=\sum_{J,K}\bigg(1-\sum_{j\in J}\gamma_j-\sum_{j\in K}\gamma_j
+\sum_{1\le j\le n}\gamma_j\bigg)
u_{J,K}\ovl v_{J,K}\,dV_0+O(|z|^2).\cr}$$
Lemma 6.9 follows if we take Prop.~5.8 into account.\qed
\endproof

\begstat{(6.10) Lemma} $d^{\prime\prime\star}=d^{\prime\prime\star}_0+
\big[\Lambda_0,[d^{\prime\prime\star}_0,\gamma]\big]$ at point $x_0$, i.e.\ at this
point both operators have the same formal expansion.
\endstat

\begproof{} Since $d^{\prime\prime\star}$ is an operator of order 1, Lemma~6.9
shows that $d^{\prime\prime\star}$ coincides at point $x_0$ with the formal
adjoint of $d''$ for the metric
$$\Ll u,v\Gg_1=\int_X\langle u-[\gamma,\Lambda_0]u,v\rangle_0\,dV_0.$$
For any compactly supported  $u\in C^\infty(X,\Lambda^{p,q}T^\star_X)$,
$v\in C^\infty(X,\Lambda^{p,q-1}T^\star_X)$ we get by definition
$$\Ll u,d''v\Gg_1=\int_X\langle u-[\gamma,\Lambda_0]u,d''v\rangle_0\,dV_0
=\int_X\langle d^{\prime\prime\star}_0u-d^{\prime\prime\star}_0[\gamma,\Lambda_0]u,v\rangle_0\,dV_0.$$
Since $\omega$ and $\omega_0$ coincide at point $x_0$ and since $\gamma(x_0)=0$
we obtain at this point
$$\eqalign{
d^{\prime\prime\star}u&=d^{\prime\prime\star}_0u-d^{\prime\prime\star}_0[\gamma,\Lambda_0]u=d^{\prime\prime\star}_0u-\big[d^{\prime\prime\star}_0,[\gamma,\Lambda_0]\big]u~;\cr
d^{\prime\prime\star}&=d^{\prime\prime\star}_0-\big[d^{\prime\prime\star}_0,[\gamma,\Lambda_0]\big].\cr}$$
We have $[\Lambda_0,d^{\prime\prime\star}_0]=[d'',L_0]^\star=0$ since $d''\omega_0=0$.
The Jacobi identity (5.7) implies
$$\big[d^{\prime\prime\star}_0,[\gamma,\Lambda_0]\big]+\big[\Lambda_0,[d^{\prime\prime\star}_0,\gamma]
\big]=0,$$
and Lemma~6.10 follows.\qed
\endproof

\begproof{Proof of formula {\rm 6.8 a)}} The equality
$L=L_0+\gamma$ and Lemma~6.10 yield
$$[L,d^{\prime\prime\star}]=[L_0,d^{\prime\prime\star}_0]+\Big[L_0,\big[\Lambda_0,[d^{\prime\prime\star}_0,\gamma]
\big]\Big]+[\gamma,d^{\prime\prime\star}_0]\leqno(6.11)$$
at point $x_0$, because the triple bracket involving $\gamma$ twice vanishes
at~$x_0$. From the Jacobi identity applied to $C=[d^{\prime\prime\star}_0,\gamma]$, we get
$$\cases{\phantom{\Big(}
\big[L_0,[\Lambda_0,C]\big] 
=-[\Lambda_0,[C,L_0]\big]-\big[C,[L_0,\Lambda_0]\big],\cr
\phantom{\Big(}
[C,L_0]=\big[L_0,[d^{\prime\prime\star}_0,\gamma]\big]=\big[\gamma,[L_0,d^{\prime\prime\star}_0]\big]
~~~{\rm(since}~~[\gamma,L_0]=0).\cr}\leqno(6.12)$$
Lemma 6.3 yields $[L_0,d^{\prime\prime\star}_0]=-\ii d'$, hence
$$[C,L_0]=-[\gamma,\ii d']=\ii d'\gamma=\ii d'\omega .\leqno(6.13)$$
On the other hand, $C$ is of type $(1,0)$ and Cor.~5.9 gives
$$\big[ C,[L_0,\Lambda_0]\big]=-C=-[d^{\prime\prime\star}_0,\gamma].
\leqno(6.14)$$
From (6.12), (6.13), (6.14) we get
$$\Big[L_0,\big[\Lambda_0,[d^{\prime\prime\star}_0,\gamma]\big]\Big]=-[\Lambda_0,\ii d'\omega]+[d^{\prime\prime\star}_0,\gamma].$$
This last equality combined with (6.11) implies
$$[L,d^{\prime\prime\star}]=[L_0,d^{\prime\prime\star}_0]-[\Lambda_0,\ii d'\omega]=-\ii(d'+\tau)$$
at point $x_0$. Formula 6.8 a) is proved.\qed
\endproof

\begstat{(6.15) Corollary} The complex Laplace-Beltrami operators satisfy
$$\eqalign{\Delta''&=\Delta'+[d',\tau^\star]-[d'',\ovl\tau^\star],\cr
[d',d^{\prime\prime\star}]&=-[d',\ovl\tau^\star],~~~~[d'',d^{\prime\star}]=-[d'',\tau^\star],\cr
\Delta&=\Delta'+\Delta''-[d',\ovl\tau^\star]-[d'',\tau^\star].\cr}$$
Therefore $\Delta'$, $\Delta''$ and ${1\over 2}\Delta$ no longer
coincide, but they differ by linear differential operators of order 1 only.
\endstat

\begproof{} As in the K\"ahler case (Cor.~6.5 and Lemma~6.6), 
we find
$$\eqalign{\Delta''
&=[d'',d^{\prime\prime\star}]=\big[d'',-i[\Lambda,d']-\ovl\tau^\star]\cr
&=\big[d',-\ii[d'',\Lambda]\big]-[d'',\ovl\tau^\star\big]=
\Delta'+[d',\tau^\star]-[d'',\ovl\tau^\star],\cr
[d',d^{\prime\prime\star}+\ovl\tau^\star]
&=-\ii\big[d',[\Lambda,d']\big]=0,\cr}$$
and the first two lines are proved. The third one is an immediate
consequence of the second.\qed
\endproof

\titleb{\S 7.}{Groups $\cH^{p,q}(X,E)$ and Serre Duality}
Let $(X,\omega)$ be a {\it compact hermitian} manifold and $E$ a holomorphic
hermitian vector bundle of rank $r$ over $X$. We denote by  $D_E$ the Chern
connection of $E$, by $D_E^\star=-\star D_E\,\star$ the formal adjoint of 
$D_E$, and by $D^{\prime\star}_E\,,~D^{\prime\prime\star}_E$ the components
of $D_E^\star$ of type $(-1,0)$ and $(0,-1)$.

Corollary 6.8 implies that the principal part of the operator
$\Delta''_E=D''D^{\prime\prime\star}_E+D^{\prime\prime\star}_E D''$ is one
half that of $\Delta_E$. Consequently, the operator $\Delta''_E$ acting
on each space $C^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)$
is a self-adjoint elliptic operator. Since  $D^{\prime\prime 2}=0$,
the following results can be obtained in a way similar to those of \S$\,$3.3.

\begstat{(7.1) Theorem} For every bidegree $(p,q)$, there exists an orthogonal 
decomposition
$$C^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)=\cH^{p,q}(X,E)\oplus\Im D''_E
\oplus\Im D^{\prime\prime\star}_E$$
where $\cH^{p,q}(X,E)$ is the space of $\Delta''_E$-harmonic forms in
$C^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)$.
\endstat

The above decomposition shows that the subspace of $d''$-cocycles in
$\ci(X,\Lambda^{p,q}T^\star_X\otimes E)$ is $\cH^{p,q}(X,E)\oplus
\Im D''_E$. From this, we infer

\begstat{(7.2) Hodge isomorphism theorem} The Dolbeault cohomology group
$H^{p,q}(X,E)$ is finite dimensional, and there is an isomorphism
$$H^{p,q}(X,E)\simeq\cH^{p,q}(X,E).$$
\endstat

\begstat{(7.3) Serre duality theorem} The bilinear pairing
$$H^{p,q}(X,E)\times H^{n-p,n-q}(X,E^\star)\lra\bbbc,\qquad
(s,t)\longmapsto\int_M s\wedge t$$
is a non degenerate duality.
\endstat

\begproof{} Let $s_1\in C^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)$, $s_2\in C^\infty
(X,\Lambda^{n-p,n-q-1}T^\star_X\otimes E)$.
Since $s_1\wedge s_2$ is of bidegree $(n,n-1)$, we have
$$d(s_1\wedge s_2)=d''(s_1\wedge s_2)=d''s_1\wedge s_2+(-1)^{p+q} s_1
\wedge d''s_2.\leqno(7.4)$$
Stokes' formula implies that the above bilinear pairing can be
factorized through Dolbeault cohomology groups. The $\#$ operator defined
in \S$\,$3.1 is such that
$$\#:C^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)\lra C^\infty(X,
\Lambda^{n-p,n-q}T^\star_X\otimes E^\star).$$
Furthermore, (3.20) implies
$$d''(\#\,s)=(-1)^{\deg s}\#\,D^{\prime\prime\star}_Es,~~~~
D^{\prime\prime\star}_{E^\star}(\#\,s)=(-1)^{\deg s+1}\#\,D^{\prime\prime\star}_Es,$$
$$\Delta''_{E^\star}(\#\,s)=\#\,\Delta''_Es,$$
where $D_{E^\star}$ is the Chern connection of $E^\star$. Consequently, 
$s\in\cH^{p,q}(X,E)$ if and only if
$\#\,s\in\cH^{n-p,n-q}(X,E^\star)$. Theorem~7.3
is then a consequence of the fact that the integral
$\|s\|^2=\int_X s\wedge\#\,s$ does not vanish unless $s=0$.\qed
\endproof

\titleb{\S 8.}{Cohomology of Compact K\"ahler Manifolds}
\titlec{\S 8.1.}{Bott-Chern Cohomology Groups}
Let $X$ be for the moment an arbitrary complex manifold. The following
``cohomology'' groups are helpful to describe Hodge theory on
compact complex manifolds which are not necessarily K\"ahler.

\begstat{(8.1) Definition} We define the Bott-Chern cohomology groups
of $X$ to be
$$H_\BC^{p,q}(X,\bbbc)=\big(\ci(X,\Lambda^{p,q}T^\star_X)\cap\ker d\big)/
d'd''\ci(X,\Lambda^{p-1,q-1}T^\star_X).$$
Then $H_\BC^{\bu,\bu}(X,\bbbc)$ has the structure of a bigraded algebra,
which we call the Bott-Chern cohomology algebra of~$X$.
\endstat

As the group $d'd''\ci(X,\Lambda^{p-1,q-1}T^\star_X)$ is contained in the
coboundary groups $d''\ci(X,\Lambda^{p,q-1}T^\star_X)$ or $d\ci(X,
\Lambda^{p+q-1}(\bbbc\otimes T_X)^\star)$, there are 
cano\-nical morphisms
$$\leqalignno{
&H_\BC^{p,q}(X,\bbbc)\lra H^{p,q}(X,\bbbc),&(8.2)\cr
&H_\BC^{p,q}(X,\bbbc)\lra H^{p+q}_\DR(X,\bbbc),&(8.3)\cr}$$
of the Bott-Chern cohomology to the Dolbeault or De Rham cohomology.
These morphisms are homomorphisms of $\bbbc$-algebras. 
It is also clear from the definition that we have the symmetry
property $H_\BC^{q,p}(X,\bbbc)=\ovl{H_\BC^{p,q}(X,\bbbc)}$.
It can be shown from the Hodge-Fr\"olicher spectral sequence (see \S$\,$11
and Exercise 13.??) that $H_\BC^{p,q}(X,\bbbc)$ is always
finite dimensional if $X$ is compact.

\titlec{\S 8.2.}{Hodge Decomposition Theorem}
We suppose from now on that $(X,\omega)$ is a {\it compact K\"ahler}
manifold. The equality $\Delta=2\Delta''$ shows that $\Delta$ is homogeneous
with respect to bidegree and that there is an orthogonal decomposition
$$\cH^k(X,\bbbc)=\bigoplus_{p+q=k}\cH^{p,q}(X,\bbbc).\leqno(8.4)$$
As $\ovl{\Delta''}=\Delta'=\Delta''$, we also have $\cH^{q,p}(X,\bbbc)=
\ovl{\cH^{p,q}(X,\bbbc)}$. Using the Hodge isomorphism theorems
for the De Rham and Dolbeault cohomology, we get:

\begstat{(8.5) Hodge decomposition theorem} On a compact K\"ahler manifold,
there are canonical isomorphisms
$$\matrix{\displaystyle
H^k(X,\bbbc)\simeq\bigoplus_{p+q=k}H^{p,q}(X,\bbbc)\hfill&\quad
\hbox{$($Hodge decomposition$)$},\hfill\cr
H^{q,p}(X,\bbbc)\simeq\ovl{H^{p,q}(X,\bbbc)}\hfill&\quad
\hbox{$($Hodge symmetry$)$}.\hfill\cr}$$
\endstat

The only point which is not a priori completely clear is that this
decomposition is independent of the K\"ahler metric. In order to show that
this is the case, one can use the following Lemma, which allows us to
compare all three types of cohomology groups considered in~\S$\,$8.1.

\begstat{(8.6) Lemma} Let $u$ be a $d$-closed $(p,q)$-form. The following 
properties are equivalent:
\smallskip
\itemitem{\llap{\hbox{${\rm a})$\hfill}}} $u$ is $d$-exact$\,;$
\smallskip
\itemitem{\llap{\hbox{${\rm b'})$\hfill}}} $u$ is $d'$-exact$\,;$
\smallskip
\itemitem{\llap{\hbox{${\rm b''})$\hfill}}} $u$ is $d''$-exact$\,;$
\smallskip
\itemitem{\llap{\hbox{${\rm c})$\hfill}}} $u$ is $d'd''$-exact, i.e.\
$u$ can be written $u=d'd''v$.
\smallskip
\itemitem{\llap{\hbox{${\rm d})$\hfill}}} $u$ is orthogonal to
$\cH^{p,q}(X,\bbbc)$.
\endstat

\begproof{} It is obvious that $c)$ implies a), ${\rm b')}$, ${\rm b'')}$ and
that a) or ${\rm b')}$ or ${\rm b'')}$ implies $d)$. It is thus sufficient
to prove that d) implies~c).  As $du=0$, we have $d'u=d''u=0$, and 
as $u$ is supposed to be orthogonal to $\cH^{p,q}(X,\bbbc)$,
Th.~7.1 implies $u=d''s$, $s\in\ci(X,\Lambda^{p,q-1}T^\star_X)$. By the
analogue of Th.~7.1 for $d'$, we have $s=h+d'v+d^{\prime\star}w$, with
$h\in\cH^{p,q-1}(X,\bbbc)$, $v\in\ci(X,\Lambda^{p-1,q-1}T^\star_X)$ and 
$w\in\ci(X,\Lambda^{p+1,q-1}T^\star_X)$. Therefore 
$$u=d''d'v+d''d^{\prime\star}w=-d'd''v-d^{\prime\star}d''w$$
in view of Lemma~6.6. As $d'u=0$, the component $d^{\prime\star}d''w$
orthogonal to $\ker d'$ must be zero.\qed
\endproof

From Lemma~8.6 we infer the following Corollary, which in turn implies
that the Hodge decomposition does not depend on the K\"ahler metric.

\begstat{(8.7) Corollary} Let $X$ be a compact K\"ahler manifold. Then
the natural morphisms
$$H^{p,q}_\BC(X,\bbbc)\lra H^{p,q}(X,\bbbc),\qquad
\bigoplus_{p+q=k}H^{p,q}_\BC(X,\bbbc)\lra H^k_\DR(X,\bbbc)$$
are isomorphisms.
\endstat

\begproof{} The surjectivity of $H^{p,q}_\BC(X,\bbbc)\to H^{p,q}(X,\bbbc)$
comes from the fact that every class in $H^{p,q}(X,\bbbc)$ can be represented
by a harmonic $(p,q)$-form, thus by a $d$-closed $(p,q)$-form; the
injectivity means nothing more than the equivalence (8.5$\,{\rm b}'')
\Leftrightarrow(8.5\,$c). Hence $H^{p,q}_\BC(X,\bbbc)\simeq H^{p,q}(X,\bbbc)
\simeq\cH^{p,q}(X,\bbbc)$, and the isomorphism
$\bigoplus_{p+q=k}H^{p,q}_\BC(X,\bbbc)\lra H^k_\DR(X,\bbbc)$
follows from (8.4).\qed
\endproof

Let us quote now two simple applications of Hodge theory. The first of these
is a computation of the Dolbeault cohomology groups of $\bbbp^n$.
As $H^{2p}_\DR(\bbbp^n,\bbbc)=\bbbc$ and
$H^{p,p}(\bbbp^n,\bbbc)\ni\{\omega^p\}\ne 0$, the Hodge decomposition
formula implies:

\begstat{(8.8) Application} The Dolbeault cohomology groups of $\bbbp^n$ are 
$$H^{p,p}(\bbbp^n,\bbbc)=\bbbc\quad\hbox{for}~0\le p\le n,\qquad
H^{p,q}(\bbbp^n,\bbbc)=0\quad\hbox{for}~p\ne q.\eqno\square$$
\endstat

\begstat{(8.9) Proposition} Every holomorphic $p$-form on a
compact K\"ahler mani\-fold $X$ is $d$-closed.
\endstat

\begproof{} If $u$ is a holomorphic form of type $(p,0)$ then
$d''u=0$. Furthermore $d^{\prime\prime\star}u$ is of type $(p,-1)$, hence
$d^{\prime\prime\star}u=0$. Therefore $\Delta u=2\Delta''u=0$, which implies 
$du=0$.\qed
\endproof

\begstat{(8.10) Example} \rm Consider the {\it Heisenberg group}
$G\subset{\rm Gl}_3(\bbbc)$, defined as the subgroup of matrices
$$M=\pmatrix{
1~~&\hfill x~~&\hfill z\hfill\cr
0~~&\hfill 1~~&\hfill y\hfill\cr
0~~&\hfill 0~~&\hfill 1\hfill\cr}\quad,\quad (x,y,z)\in\bbbc^3.$$
Let $\Gamma$ be the discrete subgroup of matrices with entries 
$x,y,z\in\bbbz[\ii]$ (or more generally in the ring of integers of an
imaginary quadratic field). Then $X=G/\Gamma$ is a compact complex
3-fold, known as the {\it Iwasawa manifold}. The equality
$$M^{-1}dM=\pmatrix{0&dx&dz-xdy\cr 0&0&dy\cr 0&0&0\cr}$$
shows that $dx,~dy,~dz-xdy$ are left invariant holomorphic $1$-forms on
$G$. These forms induce holomorphic $1$-forms on the quotient $X=G/\Gamma$.
Since $dz-xdy$ is not $d$-closed, we see that $X$ cannot be K\"ahler.
\endstat

\titlec{\S 8.3.}{Primitive Decomposition and Hard Lefschetz Theorem}
We first introduce some standard notation. The {\it Betti numbers} and
{\it Hodge numbers} of $X$ are by definition
$$b_k=\dim_\bbbc H^k(X,\bbbc),~~~~h^{p,q}=\dim_\bbbc H^{p,q}(X,\bbbc).
\leqno(8.11)$$
Thanks to Hodge decomposition, these numbers satisfy the relations
$$b_k=\sum_{p+q=k} h^{p,q},~~~~h^{q,p}=h^{p,q}.\leqno(8.12)$$
As a consequence, the Betti numbers $b_{2k+1}$ of a compact K\"ahler manifold
are even. Note that the Serre duality theorem gives the additional
relation \hbox{$h^{p,q}=h^{n-p,n-q}$}, which holds as soon as $X$ is
compact. The existence of primitive decomposition implies other interesting
specific features of the cohomology algebra of compact K\"ahler manifolds.

\begstat{(8.13) Lemma} If $u=\sum_{r\ge (k-n)_+}L^ru_r$ is the primitive
decomposition of a harmonic $k$-form $u$, then all components
$u_r$ are harmonic.
\endstat

\begproof{} Since  $[\Delta,L]=0$, we get $0=\Delta u=\sum_r L^r\Delta
u_r$, hence $\Delta u_r=0$ by uniqueness.\qed
\endproof

Let us denote by $\Prim\cH^k(X,\bbbc)=\bigoplus_{p+q=k}\Prim\cH^{p,q}(X,\bbbc)$
the spaces of primitive harmonic $k$-forms and let $b_{k,\prim}$,
$h_\prim^{p,q}$ be 
their respective dimensions. Lemma 8.13 yields
$$\leqalignno{
\cH^{p,q}(X,\bbbc)&=\bigoplus_{r\ge (p+q-n)_+}L^r\Prim\cH^{p-r,q-r}(X,\bbbc),
&(8.14)\cr
h^{p,q}&=\sum_{r\ge (p+q-n)_+}h_\prim^{p-r,q-r}.&(8.15)\cr}$$
Formula (8.15) can be rewritten
$$\cases{\phantom{\Big(}
{\rm if}~~p+q\le n,~~h^{p,q}=h_\prim^{p,q}+h_\prim^{p-1,q-1}+\cdots\cr
\phantom{\Big(}
{\rm if}~~p+q\ge n,~~h^{p,q}=h_\prim^{n-q,n-p}+h_\prim^{n-q-1,n-p-1}+\cdots.
\cr}\leqno(8.15')$$

\begstat{(8.16) Corollary} The Hodge and Betti numbers satisfy the inequalities
\smallskip
\item{\rm a)} if $k=p+q\le n$,
then~ $h^{p,q}\ge h^{p-1,q-1},~~b_k\ge b_{k-2}$,
\smallskip
\item{\rm b)} if $k=p+q\ge n$,
then~ $h^{p,q}\ge h^{p+1,q+1},~~b_k\ge b_{k+2}$.\qed
\endstat

Another important result of Hodge theory (which is in fact a direct
consequence of Cor.~5.17) is the

\begstat{(8.17) Hard Lefschetz theorem} The mappings
$$\cmalign{
&L^{n-k}~:~~\hfill H^k(X,\bbbc)&\lra H^{2n-k}(X,\bbbc),~~~~&k\le n,\cr
&L^{n-p-q}~:~~H^{p,q}(X,\bbbc)&\lra H^{n-q,n-p}(X,\bbbc),~~~~&p+q\le n,\cr}$$
are isomorphisms.\qed
\endstat

\titleb{\S 9.}{Jacobian and Albanese Varieties}
\titlec{\S 9.1.}{Description of the Picard Group}
An important application of Hodge theory is a description of the Picard 
group $H^1(X,\cO^\star)$ of a compact K\"ahler manifold. We assume here that
$X$ is connected. The exponential exact
sequence $0\to\bbbz\to\cO\to\cO^\star\to 1$ gives
$$\leqalignno{
0\lra&H^1(X,\bbbz)\lra H^1(X,\cO)\lra H^1(X,\cO^\star)&(9.1)\cr
\buildo c_1\over\lra&H^2(X,\bbbz)\lra H^2(X,\cO)\cr}$$
because the map $\exp(2\pi i\bu):H^0(X,\cO)=\bbbc\lra H^0(X,\cO^\star)=\bbbc^\star$
is onto. 
We have $H^1(X,\cO)\simeq H^{0,1}(X,\bbbc)$ by (V-11.6). The dimension
of this group is called the {\it irregularity of $X$} and is usually denoted
$$q=q(X)=h^{0,1}=h^{1,0}.\leqno(9.2)$$
Therefore we have $b_1=2q$ and
$$H^1(X,\cO)\simeq\bbbc^q,~~~~H^0(X,\Omega^1_X)=H^{1,0}(X,\bbbc)
\simeq\bbbc^q.\leqno(9.3)$$

\begstat{(9.4) Lemma} The image of $H^1(X,\bbbz)$ in $H^1(X,\cO)$ is a lattice.
\endstat

\begproof{} Consider the morphisms
$$H^1(X,\bbbz)\lra H^1(X,\bbbr)\lra H^1(X,\bbbc)\lra H^1(X,\cO)$$
induced by the inclusions $\bbbz\subset\bbbr\subset\bbbc\subset\cO$. Since the
\v Cech cohomology groups with values in $\bbbz$, $\bbbr$ can be computed
by finite acyclic coverings, we see that $H^1(X,\bbbz)$ is a finitely 
generated $\bbbz$-module and that the image of $H^1(X,\bbbz)$ in $H^1(X,\bbbr)$
is a lattice. It is enough to check that the map $H^1(X,\bbbr)\lra H^1(X,\cO)$
is an isomorphism. However, the commutative \hbox{diagram}
$$\cmalign{
&0&\lra&\bbbc&\lra&~\cE^0&\buildo d\over\lra&~\cE^1&\buildo d\over\lra&~\cE^2
&\lra&\cdots\cr
&&&\big\downarrow&&~\big\downarrow&&~\big\downarrow&&~\big\downarrow&&\cr
&0&\lra&\cO&\lra&\cE^{0,0}&\buildo d''\over\lra&\cE^{0,1}&\buildo d''\over\lra&
\cE^{0,2}&\lra&\cdots\cr}$$
shows that the map $H^1(X,\bbbr)\lra H^1(X,\cO)$ corresponds in De Rham and
Dolbeault cohomologies to the composite mapping
$$H^1_{DR}(X,\bbbr)\subset H^1_{DR}(X,\bbbc)\lra H^{0,1}(X,\bbbc).$$
Since~ $H^{1,0}(X,\bbbc)$ and~ $H^{0,1}(X,\bbbc)$ are complex conjugate
subspaces in\break
$H^1_{DR}(X,\bbbc)$, we conclude that $H^1_{DR}(X,\bbbr)\lra H^{0,1}(X,\bbbc)$
is an isomorphism.\qed
\endproof

As a consequence of this lemma, $H^1(X,\bbbz)\simeq\bbbz^{2q}$.
The $q$-dimensional complex torus
$$\Jac(X)=H^1(X,\cO)/H^1(X,\bbbz)\leqno(9.5)$$
is called the {\it Jacobian variety of $X$} and is isomorphic to the
subgroup of $H^1(X,\cO^\star)$ corresponding to line bundles of zero
first Chern class. On the other hand, the kernel of
$$H^2(X,\bbbz)\lra H^2(X,\cO)=H^{0,2}(X,\bbbc)$$
which consists of integral cohomology classes of type~$(1,1)$,
is equal to the image of $c_1$ in $H^2(X,\bbbz)$. This subgroup is called
the {\it Neron-Severi group} of~$X$, and is denoted $NS(X)$. The exact 
sequence (9.1) yields
$$0\lra\Jac(X)\lra H^1(X,\cO^\star)\buildo c_1\over\lra NS(X)\lra 0.\leqno(9.6)$$
The Picard group $H^1(X,\cO^\star)$ is thus an extension of the complex torus
$\Jac(X)$ by the finitely generated $\bbbz$-module $NS(X)$.

\begstat{(9.7) Corollary} The Picard group of $\bbbp^n$ is $H^1(\bbbp^n,\cO^\star)
\simeq\bbbz$, and every line bundle over $\bbbp^n$ is isomorphic to one of the 
line bundles $\cO(k)$, $k\in\bbbz$.
\endstat

\begproof{} We have $H^k(\bbbp^n,\cO)=H^{0,k}(\bbbp^n,\bbbc)=0$ for
$k\ge 1$ by Appl.~8.8, thus $\Jac({\bbbp^n})=0$ and
$NS(\bbbp^n)=H^2(\bbbp^n,\bbbz)\simeq\bbbz$. Moreover,
$c_1\big(\cO(1)\big)$ is a generator of $H^2(\bbbp^n,\bbbz)$ in virtue 
of Th.~V-15.10.\qed
\endproof

\titlec{\S 9.2.}{Albanese Variety}
A proof similar to that of Lemma~9.4 shows that the image of 
$H^{2n-1}(X,\bbbz)$ in $H^{n-1,n}(X,\bbbc)$ via the composite map
$$H^{2n-1}(X,\bbbz)\to H^{2n-1}(X,\bbbr)\to H^{2n-1}(X,\bbbc)\to
H^{n-1,n}(X,\bbbc)\leqno(9.8)$$
is a lattice. The $q$-dimensional complex torus
$$\Alb(X)=H^{n-1,n}(X,\bbbc)/\Im H^{2n-1}(X,\bbbz)\leqno(9.9)$$
is called the {\it Albanese variety of $X$}. We first give a slightly
different description of $\Alb(X)$, based on the Serre duality
isomorphism
$$H^{n-1,n}(X,\bbbc)\simeq\big(H^{1,0}(X,\bbbc)\big)^\star
\simeq\big(H^0(X,\Omega^1_X)\big)^\star.$$

\begstat{(9.10) Lemma} For any compact oriented differentiable
manifold $M$ with $\dim_\bbbr M=m$, there is a natural isomorphism
$$H_1(M,\bbbz)\to H^{m-1}(M,\bbbz)$$
where $H_1(M,\bbbz)$ is the first homology group of~$M$, that is, the
abelianization of~$\pi_1(M)$.
\endstat

\begproof{} This is a well known consequence of Poincar\'e duality,
see e.g.\ (Spanier 1966). We will content ourselves with a description
of the morphism. Fix a base point $a\in M$. Every homotopy class
$[\gamma]\in\pi_1(M,a)$ can be represented by as a composition of
closed loops diffeomorphic to~$S^1$. Let~$\gamma$ be such a loop.
As every oriented vector bundle over
$S^1$ is trivial, the normal bundle to $\gamma$ is trivial. Hence
$\gamma(S^1)$ has a neighborhood $U$ diffeomorphic to $S^1\times\bbbr^{m-1}$,
and there is a diffeomorphism $\varphi:S^1\times\bbbr^{m-1}\to U$ with 
$\varphi_{\restriction S^1\times\{0\}}=\gamma$. Let 
$\{\delta_0\}\in H^{m-1}_c(\bbbr^{m-1},\bbbz)$ be the fundamental class 
represented by the Dirac measure $\delta_0\in\cD'_0(\bbbr^{m-1})$
in De Rham cohomology. Then the cartesian product $1\times\{\delta_0\}\in
H^{m-1}_c(S^1\times\bbbr^{m-1},\bbbz)$ is 
represented by the current \hbox{$[S^1]\otimes\{\delta_0\}\in\cD'_1(S^1\times
\bbbr^{m-1})$} and the current of integration over~$\gamma$ is precisely
the direct image current
$$I_\gamma:=\varphi_\star([S^1]\otimes\delta_0)=(\varphi^{-1})^\star
([S^1]\otimes\delta_0).$$
Its cohomology class $\{I_\gamma\}\in H^{m-1}_c(U,\bbbr)$ is thus the
image of the class \hbox{$(\varphi^{-1})^\star\big(1\times\{\delta_0\}\big)
\in H^{m-1}_c(U,\bbbz)$}. Therefore, we have obtained a well defined morphism
$$\pi_1(M,a)\lra H^{m-1}_c(U,\bbbz)\lra H^{m-1}(M,\bbbz),~~~~[\gamma]
\longmapsto(\varphi^{-1})^\star\big(1\times\{\delta_0\}\big)$$
and the image of $[\gamma]$ in $H^{m-1}(M,\bbbr)$ is the De Rham
cohomology class of the integration current $I_\gamma$.\qed
\endproof

Thanks to Lemma~9.10, we can reformulate the definition of the
Albanese variety as
$$\Alb(X)=\big(H^0(X,\Omega^1_X)\big)^\star/\Im H_1(X,\bbbz)\leqno(9.11)$$
where $H_1(X,\bbbz)$ is mapped to $\big(H^0(X,\Omega^1_X)\big)^\star$ by
$$[\gamma]\longmapsto\wt I_\gamma=\Big(u\mapsto\int_\gamma u\Big).$$
Observe that the integral only depends on the homotopy class $[\gamma]$
because all holomorphic $1$-forms $u$ on $X$ are closed by Prop.~8.9.

We are going to show that there exists a canonical holomorphic map 
$\alpha:X\to\Alb(X)$. Let $a$ be a base point in $X$. For any
$x\in X$, we select a path $\xi$ from $a$ to $x$ and associate to
$x$ the linear form in $\big(H^0(X,\Omega^1_X)\big)^\star$ defined
by~$\wt I_\xi$. By construction the class of this linear form
mod $\Im H_1(X,\bbbz)$ does not depend
on~$\xi$, since $\wt I_{\xi^{\prime\,-1}\xi}$ is in the
image of $H_1(X,\bbbz)$ for any other path~$\xi'$. It is thus
legitimate to define the {\it Albanese map} as
$$\alpha:X\lra\Alb(X),~~~~x\longmapsto
\Big(u\mapsto\int_a^xu\Big)~~\hbox{\rm mod}~~\Im\,H_1(X,\bbbz).
\leqno(9.12)$$
Of course, if we fix a basis $(u_1\ld u_q)$ of $H^0(X,\Omega^1_X)$,
the Albanese map can be seen in coordinates as the map
$$\alpha:X\lra\bbbc^q/\Lambda,~~~~x\longmapsto
\Big(\int_a^xu_1\ld\int_a^xu_q\Big)~~\hbox{\rm mod}~~\Lambda,
\leqno(9.13)$$
where $\Lambda\subset\bbbc^q$ is the {\it group of periods} of $(u_1\ld u_q)\,$:
$$\Lambda=\Big\{\Big(\int_\gamma u_1\ld\int_\gamma u_q\Big)~;~
[\gamma]\in\pi_1(X,a)\,\Big\}.\leqno(9.13')$$
It is then clear that $\alpha$ is a holomorphic map. With the
original definition (9.9) of the Albanese variety, it is not
difficult to see that $\alpha$ is the map given by
$$\alpha:X\lra\Alb(X),~~~~x\longmapsto\{I^{n-1,n}_\xi\}~~{\rm mod}~~
H^{2n-1}(X,\bbbz),\leqno(9.14)$$
where $\{I^{n-1,n}_\xi\}\in H^{n-1,n}(X,\bbbc)$ denotes the
$(n-1,n)$-component of the De Rham cohomology class $\{I_\xi\}$.

\titleb{\S 10.}{Complex Curves}
We show here how Hodge theory can be used to derive quickly the basic properties
of compact manifolds of complex dimension $1$ (also called {\it complex
curves} or {\it Riemann surfaces}). Let $X$ be such a curve. We shall always 
assume in this section that $X$ is compact and connected. Since every positive 
$(1,1)$-form on a curve defines a K\"ahler metric, the results of \S$\,$8
and \S$\,$9 can be applied.

\titlec{\S 10.1.}{Riemann-Roch Formula}
Denoting $g=h^1(X,\cO)$, we find
$$\leqalignno{
&H^1(X,\cO)\simeq\bbbc^g,~~~~H^0(X,\Omega^1_X)\simeq\bbbc^g,
&(10.1)\cr
&H^0(X,\bbbz)=\bbbz,~~~~H^1(X,\bbbz)=\bbbz^{2g},~~~~H^2(X,\bbbz)=\bbbz.&(10.2)}$$
The classification of oriented topological surfaces shows that $X$
is homeomorphic to a sphere with $g$ handles ( $=$ torus
with $g$ holes), but this property will not be used in the sequel.
The number $g$ is called the {\it genus} of~$X$.

Any divisor on $X$ can be written $\Delta=\sum m_ja_j$ where $(a_j)$ is a
finite sequence of points and $m_j\in\bbbz$. Let $E$ be a line bundle
over $X$. We shall identify $E$ and the associated locally free sheaf 
$\cO(E)$. According to V-13.2, we denote by $E(\Delta)$ the sheaf of germs of
meromorphic sections $f$ of $E$ such that ${\rm div}\,f+\Delta\ge 0$, 
i.e.\ which have a pole of order $\le m_j$ at $a_j$ if $m_j>0$, and which 
have a zero of order $\ge|m_j|$ at $a_j$ if $m_j<0$. Clearly
$$E(\Delta)=E\otimes\cO(\Delta),~~~~\cO(\Delta+\Delta')=\cO(\Delta)\otimes
\cO(\Delta').\leqno(10.3)$$
For any point $a\in X$ and any integer $m>0$, there is an exact sequence
$$0\lra E\lra E(m[a])\lra\cS\lra 0$$
where $\cS=E(m[a])/E$ is a sheaf with only one non zero stalk $\cS_a$
isomorphic to $\bbbc^m$. Indeed, if $z$ is a holomorphic coordinate near
$a$, the stalk $\cS_a$ corresponds to the polar parts
$\sum_{-m\le k<0} c_kz^k$ in the power series expansions of germs
of meromorphic sections at point $a$. We get an exact sequence
$$H^0\big(X,E(m[a])\big)\lra\bbbc^m\lra H^1(X,E).$$
When $m$ is chosen larger than $\dim H^1(X,E)$, we see that
$E(m[a])$ has a non zero section and conclude:

\begstat{(10.4) Theorem} Let $a$ be a given point on a curve. Then every line 
bundle $E$ has non zero meromorphic sections $f$ with a pole at $a$ and
no other poles.
\endstat

If $\Delta$ is the divisor of a meromorphic section $f$ of $E$, we have
$E\simeq\cO(\Delta)$, so the map 
$${\rm Div}(X)\lra H^1(X,\cO^\star),~~~~\Delta\longmapsto\cO(\Delta)$$ 
is onto (cf.\ (V-13.8)). On the other hand,
${\rm Div}$ is clearly a soft sheaf, thus $H^1(X,{\rm Div})=0$.
The long cohomology sequence associated to the exact sequence
$1\to\cO^\star\to\cM^\star\to{\rm Div}\to 0$ implies:

\begstat{(10.5) Corollary} On any complex curve, one has $H^1(X,\cM^\star)=0$
and there is an exact sequence
$$0\lra\bbbc^\star\lra\cM^\star(X)\lra{\rm Div}(X)\lra H^1(X,\cO^\star)\lra 0.$$
\endstat

The first Chern class $c_1(E)\in H^2(X,\bbbz)$ can be interpreted
as an integer. This integer is called the {\it degree} of~$E$.
If $E\simeq\cO(\Delta)$ with $\Delta=\sum m_ja_j$, formula V-13.6
shows that the image of $c_1(E)$ in $H^2(X,\bbbr)$ is the De Rham
cohomology class of the associated current $[\Delta]=\sum m_j\delta_{a_j}$,
hence
$$c_1(E)=\int_X[\Delta]=\sum m_j.\leqno(10.6)$$
If $\sum m_ja_j$ is the divisor of a meromorphic
function, we have $\sum m_j=0$ because the associated bundle $E=\cO(\sum
m_ja_j)$ is trivial.

\begstat{(10.7) Theorem} Let $E$ be a line bundle on a complex curve~$X$. Then
\smallskip
\item{\rm a)} $H^0(X,E)=0$~~if~~$c_1(E)<0$ or if~~$c_1(E)=0$ and
$E$ is non trivial$\,;$
\smallskip
\item{\rm b)} For every positive $(1,1)$-form $\omega$ on $X$ with
$\int_X\omega=1$, $E$ has a hermitian metric such that ${\ii\over 2\pi}
\Theta(E)=c_1(E)\,\omega$.  In particular, $E$ has a metric of positive
$($resp.  negative$)$ curvature if and only if $c_1(E)>0$ $($resp. 
if and only if $c_1(E)<0)$.\smallskip
\endstat

\begproof{} a) If $E$ has a non zero holomorphic section $f$, then its degree
is $c_1(E)=\int_X{\rm div}\,f\ge 0$. In fact, we even have $c_1(E)>0$
unless $f$ does not vanish, in which case $E$ is trivial.

b) Select an arbitrary hermitian metric $h$ on $E$. Then
$c_1(E)\,\omega-{\ii\over 2\pi}\Theta_h(E)$ is a real $(1,1)$-form
cohomologous to zero (the integral over $X$ is zero), so Lemma~8.6~c) implies
$$c_1(E)\,\omega-{\ii\over 2\pi}\Theta_h(E)=\ii d'd''\varphi$$
for some real function $\varphi\in\ci(X,\bbbr)$. If we replace the initial
metric of $E$ by $h'=h\,e^{-\varphi}$, we get a metric of constant
curvature $c_1(E)\,\omega$.\qed
\endproof
          
\begstat{(10.8) Riemann-Roch formula} Let $E$ be a holomorphic
line bundle and let $h^q(E)=\dim H^q(X,E)$. Then
$$h^0(E)-h^1(E)=c_1(E)-g+1.$$
Moreover $h^1(E)=h^0(K\otimes E^\star)$, where $K=\Omega^1_X$ is
the canonical line bundle of~$X$.
\endstat

\begproof{} We claim that for every line bundle $F$ and every divisor $\Delta$ 
we have the equality
$$h^0\big(F(\Delta)\big)-h^1\big(F(\Delta)\big)=h^0(F)-h^1(F)+\int_X[\Delta].
\leqno(10.9)$$
If we write $E=\cO(\Delta)$ and apply the above equality with $F=\cO$, the
Riemann-Roch formula results from (10.6), (10.9) and from the equalities
$$h^0(\cO)=\dim H^0(X,\cO)=1,~~~~h^1(\cO)=\dim H^1(X,\cO)=g.$$
However, (10.9) need only be proved when $\Delta\ge 0\,$: otherwise we
are reduced to this case by writing $\Delta=\Delta_1-\Delta_2$ with
$\Delta_1,\Delta_2\ge 0$ and by applying the result to the pairs
$(F,\Delta_1)$ and $\big(F(\Delta),\Delta_2\big)$. If $\Delta=\sum m_ja_j
\ge 0$, there is an exact sequence
$$0\lra F\lra F(\Delta)\lra\cS\lra 0$$
where $\cS_{a_j}\simeq\bbbc^{m_j}$ and the other stalks are zero. 
Let $m=\sum m_j=\int_X[\Delta]$. The sheaf $\cS$ is acyclic, because
its support $\{a_j\}$ is of dimension $0$. Hence there is
an exact sequence
$$0\lra H^0(F)\lra H^0\big(F(\Delta)\big)\lra\bbbc^m\lra H^1(F)\lra
H^1\big(F(\Delta)\big)\lra 0$$
and (10.9) follows. The equality $h^1(E)=h^0(K\otimes E^\star)$ is a
consequence of the Serre duality theorem
$$\big(H^{0,1}(X,E)\big)^\star\simeq H^{1,0}(X,E^\star),~~~~\hbox{\rm i.e.}~~
\big(H^1(X,E)\big)^\star\simeq H^0(X,K\otimes E^\star).\eqno{\square}$$
\endproof

\begstat{(10.10) Corollary (Hurwitz' formula)} $c_1(K)=2g-2$.
\endstat

\begproof{} Apply Riemann-Roch to $E=K$ and observe that
$$\left.\eqalign{
h^0(K)&=\dim H^0(X,\Omega^1_X)=g\cr
h^1(K)&=\dim H^1(X,\Omega^1_X)=h^{1,1}=b_2=1\cr}\right.\leqno(10.11)$$
\endproof

\begstat{(10.12) Corollary} For every $a\in X$ and every $m\in\bbbz$
$$h^0\big(K(-m[a])\big)=h^1\big(\cO(m[a])\big)=h^0\big(\cO(m[a])\big)-m+g-1.$$
\endstat

\titlec{\S 10.2.}{Jacobian of a Curve}
By the Neron-Severi sequence (9.6), there is an exact sequence
$$0\lra\Jac(X)\lra H^1(X,\cO^\star)\buildo c_1\over\lra\bbbz\lra 0,
\leqno(10.13)$$
where the Jacobian $\Jac(X)$ is a $g$-dimensional torus. Choose a base point
$a\in X$. For every point $x\in X$, the line bundle $\cO([x]-[a])$
has zero first Chern class, so we have a well-defined map
$$\Phi_a:X\lra\Jac(X),~~~~x\longmapsto\cO([x]-[a]).\leqno(10.14)$$
Observe that the Jacobian $\Jac(X)$ of a curve coincides by definition
with the Albanese variety $\Alb(X)$.

\begstat{(10.15) Lemma} The above map $\Phi_a$ coincides with the Albanese
map \hbox{$\alpha:X\lra\Alb(X)$} defined in $(9.12)$.
\endstat

\begproof{} By holomorphic continuation, it is enough to prove that
\hbox{$\Phi_a(x)=\alpha(x)$} when $x$ is near $a$. Let $z$ be a complex 
coordinate and let $D'\subset\!\subset D$ be open disks centered at~$a$.
Relatively to the covering
$$U_1=D,~~~~U_2=X\setminus\ovl{D'},$$
the line bundle $\cO([x]-[a])$ is defined by the \v Cech cocycle $c\in 
C^1({\cal U},\cO^\star)$ such that
$$c_{12}(z)={z-x\over z-a}~~~{\rm on}~~U_{12}=D\setminus\ovl{D'}.$$
On the other hand, we compute $\alpha(x)$ by Formula~(9.14). The path
integral current $I_{[a,x]}\in\cD'_1(X)$ is equal to $0$ on~$U_2$.
Lemma~I-2.10 implies \hbox{$d''(dz/2\pi\ii z)=\delta_0\,d\ovl z
\wedge dz/2i=\delta_0$} according to the usual identification of
distributions and currents of degree $0$, thus
$$I^{0,1}_{[a,x]}=d''\Big({dz\over2\pi iz}\star I^{0,1}_{[a,x]}\Big)~~~
{\rm on}~~U_1.$$
Therefore $\{I^{0,1}_{[a,x]}\}\in H^{0,1}(X,\bbbc)$ is equal to the \v Cech
cohomology class $\c['\}$ in $H^1(X,\cO)$ represented by the cocycle
$$c'_{12}(z)={dw\over2\pi iw}\star I^{0,1}_{[a,x]}={1\over 2\pi i}\int_a^x 
{dw\over w-z}={1\over 2\pi\ii}\log{z-x\over z-a}~~~{\rm on}~~U_{12}$$
and we have $c=\exp(2\pi\ii c')$ in $H^1(X,\cO^\star)$.\qed
\endproof

The nature of $\Phi_a$ depends on the value of the genus $g$. A careful
examination of $\Phi_a$ will enable us to determine all curves of genus $0$
and $1$.  

\begstat{(10.16) Theorem} The following properties are equivalent:
\smallskip
\item{\rm a)} $g=0\,;$
\smallskip
\item{\rm b)} $X$ has a meromorphic function $f$ having only one 
simple pole $p\,;$
\smallskip
\item{\rm c)} $X$ is biholomorphic to $\bbbp^1$.
\endstat

\begproof{} c) $\Longrightarrow$ a) is clear.
\medskip
\noindent a) $\Longrightarrow$ b). Since $g=0$, we have $\Jac(X)=0$. If
$p,p'\in X$ are distinct points, the bundle $\cO([p']-[p])$ has zero first
Chern class, therefore it is trivial and there exists a meromorphic function
$f$ with ${\rm div}\,f=[p']-[p]$. In particular $p$ is the only pole
of $f$, and this pole is simple.
\medskip
\noindent b) $\Longrightarrow$ c). We may consider $f$ as a map $X\lra\bbbp^1=
\bbbc\cup\{\infty\}$. For every value $w\in\bbbc$, the function $f-w$ must have
exactly one simple zero  $x\in X$ because $\int_X{\rm div}(f-w)=0$ and $p$ is 
a simple pole. Therefore $f:X\to\bbbp^1$ is bijective and $X$ is biholomorphic
to $\bbbp^1$.\qed
\endproof

\begstat{(10.17) Theorem} The map $\Phi_a$ is always injective for $g\ge 1$. 
\smallskip
\item{\rm a)} If $g=1$, $\Phi_a$ is a biholomorphism. In particular 
every curve of genus $1$ is biholomorphic to a complex torus $\bbbc/\Gamma$. 
\smallskip
\item{\rm b)} If $g\ge 2$, $\Phi_a$ is an embedding.
\endstat

\begproof{} If $\Phi_a$ is not injective, there exist points $x_1\ne x_2$ such 
that $\cO([x_1]-[x_2])$ is trivial; then there is
a meromorphic function $f$ such that ${\rm div}\,f=[x_1]-[x_2]$
and Th.~10.16 implies that $g=0$.

When $g=1$, $\Phi_a$ is an injective map $X\lra\Jac(X)\simeq\bbbc/\Gamma$,
thus $\Phi_a$ is open. It follows that $\Phi_a(X)$ is a compact open subset
of $\bbbc/\Gamma$, so $\Phi_a(X)=\bbbc/\Gamma$ and $\Phi_a$ is a
biholomorphism of $X$ onto $\bbbc/\Gamma$.

In order to prove that $\Phi_a$ is an embedding when $g\ge 2$, it is
sufficient to show that the holomorphic $1$-forms $u_1\ld u_g$ do not all
vanish at a given point $x\in X$. In fact, $X$ has no non constant
meromorphic function having only a simple pole at $x$, thus 
$h^0\big(\cO([x])\big)=1$ and Cor.~10.12 implies
$$h^0\big(K(-[x])\big)=g-1<h^0(K)=g.$$
Hence $K$ has a section $u$ which does not vanish at $x$.\qed
\endproof

\titlec{\S 10.3.}{Weierstrass Points of a Curve}
We want to study how many meromorphic functions have a unique pole of
multiplicity $\le m$ at a given point $a\in X$, i.e.\ we want to compute
$h^0\big(\cO(m[a])\big)$. As we shall see soon, these numbers may
depend on $a$ only if $m$ is small. We have
$c_1\big(K(-m[a])\big)=2g-2-m$, so the degree is $<0$ and
$h^0\big(K(-m[a])\big)=0$ for $m\ge 2g-1$ by~10.7~a).
Cor.~10.12 implies
$$h^0\big(\cO(m[a])\big)=m-g+1~~~{\rm for}~~m\ge 2g-1.\leqno(10.18)$$
It remains to compute $h^0\big(K(-m[a])\big)$ for $0\le m\le 2g-2$ and
\hbox{$g\ge 1$.} Let $u_1\ld u_g$ be a basis of $H^0(X,K)$ and let $z$
be a complex coordinate centered at~$a$. Any germ $u\in\cO(K)_a$
can be written \hbox{$u=U(z)\,dz$} with \hbox{$U(z)=\sum_{m\in\bbbn}{1\over m!}
U^{(m)}(a)z^m\,dz$.} The unique non zero stalk of the quotient sheaf
\hbox{$\cO\big(K(-m[a])\big)/\cO\big(K(-(m+1)[a])\big)$}
is canonically isomorphic to $K^{m+1}_a$ via the map
$u\mapsto U^{(m)}(a)(dz)^{m+1}$, which is independant of the choice of~$z$.
Hence \hbox{$\bigwedge^g\big(\cO(K)/\cO(K-g[a])\big)\simeq K_a^{1+2+\ldots+g}$}
and the {\it Wronskian}
$$W(u_1\ld u_g)=\left|\matrix{
~U_1(z)&\ldots&U_g(z)\cr
~U'_1(z)&\ldots&U'_g(z)\cr
\vdots&&\vdots\cr
~U_1^{(g-1)}(z)\vphantom{\displaystyle\sum^a}&\ldots&U_g^{(g-1)}(z)~\cr}
\right|\,dz^{1+2+\ldots+g}\leqno(10.19)$$
defines a global section 
$W(u_1\ld u_g)\in H^0(X,K^{g(g+1)/2})$. At the given point $a$, 
we can find linear combinations $\wt u_1\ld\wt u_g$ of $u_1\ld u_g$ such that
$$\wt u_j(z)=\big(z^{s_j-1}+{\rm O}(z^{s_j})\big)dz,~~~~s_1<\ldots<s_g.$$
We know that not all sections of $K$ vanish at $a$ and that $c_1(K)=2g-2$,
thus  $s_1=1$ and $s_g\le 2g-1$. We have
$W(\wt u_1\ld\wt u_g)\sim W(z^{s_1-1}dz\ld z^{s_g-1}dz)$ at point $a$, and 
an easy induction on $\sum s_j$ combined with differentiation in $z$ yields
$$W(z^{s_1-1}dz\ld z^{s_g-1}dz)=C\,z^{s_1+\ldots+s_g-g(g+1)/2}\,
dz^{g(g+1)/2}$$
for some positive integer constant $C$. In particular, $W(u_1\ld u_g)$ is
not identically zero and vanishes at $a$ with multiplicity
$$\mu_a=s_1+\ldots+s_g-g(g+1)/2>0\leqno(10.20)$$
unless $s_1=1$, $s_2=2$, $\ldots$, $s_g=g$. Now, we have
$$h^0\big(K(-m[a])\big)={\rm card}\{j\,;\,s_j>m\}
=g-{\rm card}\{j\,;\,s_j\le m\}$$
and Cor.~10.12 gives
$$h^0\big(\cO(m[a])\big)=m+1-{\rm card}\{j\,;\,s_j\le m\}.\leqno(10.21)$$
If $a$ is not a zero of $W(u_1\ld u_g)$, we find
$$\cases{
h^0\big(\cO(m[a])\big)=1&for~~$m\le g$,\cr
h^0\big(\cO(m[a])\big)=m+1-g&for~~$m>g$.\cr}\leqno(10.22)$$
The zeroes of $W(u_1\ld u_g)$ are called the {\it Weierstrass points} of $X$,
and the associated {\it Weierstrass sequence} is the sequence 
$w_m=h^0\big(\cO(m[a])\big)$, $m\in\bbbn$. We have 
$w_{m-1}\le w_m\le w_{m-1}+1$ and $s_1<\ldots<s_g$ are precisely the integers
$m\ge 1$ such that $w_m=w_{m-1}$. The numbers $s_j\in\{1,2\ld 2g-1\}$ are 
called the {\it gaps} and $\mu_a$ the {\it weight} of the Weierstrass point 
$a$. Since $W(u_1\ld u_g)$ is a section of $K^{g(g+1)/2}$, 
Hurwitz' formula implies
$$\sum_{a\in X}\mu_a=c_1(K^{g(g+1)/2})=g(g+1)(g-1).\leqno(10.23)$$
In particular, a curve of genus $g$ has at most $g(g+1)(g-1)$ Weierstrass 
points.

\titleb{\S 11.}{Hodge-Fr\"olicher Spectral Sequence}
Let $X$ be a {\it compact} complex $n$-dimensional manifold. We consider the
double complex $K^{p,q}=\ci(X,\Lambda^{p,q}T^\star_X)$, $d=d'+d''$. The 
Hodge-Fr\"olicher spectral sequence is by definition the spectral
sequence associated to this double complex (cf.\ IV-11.9). It starts with
$$E^{p,q}_1=H^{p,q}(X,\bbbc)\leqno(11.1)$$
and the limit term $E^{p,q}_\infty$ is the graded module associated to a 
filtration of the De Rham cohomology group $H^k(X,\bbbc)$, $k=p+q$.
In particular, if the numbers $b_k$ and $h^{p,q}$ are still defined
as in (8.11), we have
$$b_k=\sum_{p+q=k}\dim E^{p,q}_\infty\le\sum_{p+q=k}\dim E^{p,q}_1=
\sum_{p+q=k}h^{p,q}.\leqno(11.2)$$
The equality is equivalent to the degeneration of the spectral sequence at 
$E^\bu_1$. As a consequence, the Hodge-Fr\"olicher spectral
sequence of a compact K\"ahler manifold degenerates in $E^\bu_1$.

\begstat{(11.3) Theorem and Definition} The existence of an isomorphism
$$H^k_\DR(X,\bbbc)\simeq\bigoplus_{p+q=k}H^{p,q}(X,\bbbc)$$
is equivalent to the degeneration of the Hodge-Fr\"olicher spectral
sequence at~$E_1$. In this case, the isomorphism is canonically defined
and we say that $X$ admits a Hodge decomposition.\qed
\endstat

In general, interesting informations can be deduced from the spectral
sequence. Theorem~IV-11.8 shows in particular that 
$$b_1\ge\dim E^{1,0}_2+(\dim E^{0,1}_2-\dim E^{2,0}_2)_+.\leqno(11.4)$$
However, $E^{1,0}_2$ is the central cohomology group in the sequence
$$d_1=d':E^{0,0}_1\lra E^{1,0}_1\lra E^{2,0}_1,$$
and as $E^{0,0}_1$ is the space of holomorphic functions on $X$, the
first map $d_1$ is zero (by the maximum principle, holomorphic functions
are constant on each connected component of $X\,$). Hence
$\dim E^{1,0}_2\ge h^{1,0}-h^{2,0}$. Similarly, $E^{0,1}_2$ is
the kernel of a map $E^{0,1}_1\to E^{1,1}_1$, thus
$\dim E^{0,1}_2\ge h^{0,1}-h^{1,1}$.\break By (11.4) we obtain
$$b_1\ge (h^{1,0}-h^{2,0})_++(h^{0,1}-h^{1,1}-h^{2,0})_+.\leqno(11.5)$$
Another interesting relation concerns the topological Euler-Poincar\'e
characteristic
$$\chi_{{\rm top}}(X)=b_0-b_1+\ldots-b_{2n-1}+b_{2n}.$$
We need the following simple lemma.

\begstat{(11.6) Lemma} Let $(C^\bu,d)$ a bounded complex of finite dimensional
vector spaces over some field. Then, the Euler characteristic 
$$\chi(C^\bu)=\sum(-1)^q\dim C^q$$
is equal to the Euler characteristic $\chi\big(H^\bu(C^\bu)\big)$ of the
cohomology module. 
\endstat

\begproof{} Set
$$c_q=\dim C^q,\qquad z_q=\dim Z^q(C^\bu),\qquad b_q=\dim B^q(C^\bu),\qquad
h_q=\dim H^q(C^\bu).$$
Then
$$c_q=z_q+b_{q+1},~~~~h_q=z_q-b_q.$$
Therefore we find
$$\sum(-1)^q\,c_q=\sum(-1)^q\,z_q-\sum(-1)^q\,b_q=\sum(-1)^q\,h_q.
\eqno{\square}$$
\endproof

In particular, if the term $E^\bu_r$ of the spectral sequence of a
filtered complex $K^\bu$ is a bounded and finite dimensional complex, we 
have
$$\chi(E^\bu_r)=\chi(E^\bu_{r+1})=\ldots=\chi(E^\bu_\infty)=
\chi\big(H^\bu(K^\bu)\big)$$
because $E^\bu_{r+1}=H^\bu(E^\bu_r)$ and $\dim E^l_\infty=\dim H^l(K^\bu)$.
In the Hodge-Fr\"olicher spectral sequence, we have
$\dim E^l_1=\sum_{p+q=l}h^{p,q}$, hence:

\begstat{(11.7) Theorem} For any compact complex manifold $X$, one has
$$\chi_{{\rm top}}(X)=\sum_{0\le k\le 2n}(-1)^kb_k=\sum_{0\le p,q\le n}
(-1)^{p+q}h^{p,q}.$$
\endstat

\titleb{\S 12.}{Effect of a Modification on Hodge Decomposition}
In this section, we show that the existence of a Hodge decomposition on
a compact complex manifold $X$ is guaranteed as soon as there exists
such a decomposition on a modification $\wt X$ of~$X$
(see II-??.?? for the Definition). This leads us to extend Hodge theory
to a class of manifolds which are non necessarily K\"ahler, the so called
Fujiki class $(\cC)$ of manifolds bimeromorphic to K\"ahler
manifolds.

\titlec{\S 12.1.}{Sheaf Cohomology Reinterpretation of
$H_\BC^{p,q}(X,\bbbc)$}
We first give a description of $H_\BC^{p,q}(X,\bbbc)$ in terms of
the hypercohomology of a suitable complex of sheaves. This
interpretation, combined with the analogue of the Hodge-Fr\"olicher
spectral sequence, will imply in particular that $H_\BC^{p,q}(X,\bbbc)$
is always finite dimensional when $X$ is compact.
Let us denote by $\cE^{p,q}$ the sheaf of germs of $\ci$ forms of
bidegree $(p,q)$, and by $\Omega^p$ the sheaf of germs of
holomorphic $p$-forms on $X$. 
For a fixed bidegree $(p_0,q_0)$, we let $k_0=p_0+q_0$ and we
introduce a complex of sheaves $(\cL^\bu_{p_0,q_0},\delta)$, also 
denoted $\cL^\bu$ for simplicity, such that
$$\eqalign{
\cL^k&=\bigoplus_{p+q=k,p<p_0,q<q_0}\cE^{p,q}~~~{\rm for}~~
k\le k_0-2,\cr
\cL^{k-1}&=\bigoplus_{p+q=k,p\ge p_0,q\ge q_0}\cE^{p,q}~~~{\rm for}~~
k\ge k_0.\cr}$$
The differential $\delta^k$ on $\cL^k$ is chosen equal
to the exterior derivative $d$ for $k\ne k_0-2$ (in the case
$k\le k_0-3$, we neglect the components which fall outside $\cL^{k+1}$), 
and we set
$$\delta^{k_0-2}=d'd'':\cL^{k_0-2}=\cE^{p_0-1,q_0-1}\lra\cL^{k_0-1}=
\cE^{p_0,q_0}.$$
We find in particular $H_\BC^{p_0,q_0}(X,\bbbc)=H^{k_0-1}\big(\cL^\bu(X)\big)$.
We observe that $\cL^\bu$ has subcomplexes $(\cS^{\prime\,\bu},d')$ and
$(\cS^{\prime\prime\,\bu},d'')$ defined by
$$\cmalign{
\cS^{\prime\, k}&=\Omega_X^k~~~{\rm for}~~0\le k\le p_0-1,~~~~
\cS^{\prime\, k}&=0~~~{\rm otherwise},\cr
\cS^{\prime\prime\, k}&=\ovl{\Omega_X^k}~~~{\rm for}~~0\le k\le q_0-1,~~~~
\cS^{\prime\prime\, k}&=0~~~{\rm otherwise}.\cr}$$
If $p_0=0$ or $q_0=0$ we set instead $\cS^{\prime\,0}=\bbbc$ or
$\cS^{\prime\prime\,0}=\bbbc$, and take the other components to be zero.
Finally, we let $\cS^\bu=\cS^{\prime\,\bu}+\cS^{\prime\prime\,\bu}
\subset\cL^\bu$ (the sum is direct except for $\cS^0$); we denote by 
$\cM^\bu$ the sheaf complex defined in the same way as $\cL^\bu$,
except that the sheaves $\cE^{p,q}$ are replaced by the sheaves of currents 
$\cD'_{n-p,n-q}$.

\begstat{(12.1) Lemma} The inclusions $\cS^\bu\subset\cL^\bu\subset\cM^\bu$
induce isomorphisms 
$$\cH^k(\cS^\bu)\simeq\cH^k(\cL^\bu)\simeq\cH^k(\cM^\bu),$$
and these cohomology sheaves vanish for $k\ne 0,p_0-1,q_0-1$.
\endstat

\begproof{} We will prove the result only for the inclusion $\cS^\bu\subset
\cL^\bu$,
the other case $\cS^\bu\subset\cM^\bu$ is identical. Let us denote by
$\cZ^{p,q}$ the sheaf of $d''$-closed differential forms of bidegree
$(p,q)$. We consider the filtration 
$$F_p(\cL^k)=\cL^k\cap\bigoplus_{r\ge p}\cE^{r,\bu}$$ and the induced 
filtration on $\cS^\bu$. In the case of $\cL^\bu$, the first spectral 
sequence has the following terms $E^\bu_0$ and $E^\bu_1\,$:
$$\cmalign{
&{\rm if}~~p<p_0~~~~&E^{p,\bu}_0~:~~~0\lra\cE^{p,0}\buildo d''\over\lra
\cE^{p,1}\lra\cdots\buildo d''\over\lra\cE^{p,q_0-1}\lra 0,\cr
&{\rm if}~~p\ge p_0~~~~&E^{p,\bu}_0~:~~~0\lra\cE^{p,q_0}\buildo d''\over\lra
\cE^{p,q_0+1}\lra\cdots\lra\cE^{p,q}\buildo d''\over\lra\cdots,\cr
&{\rm if}~~p<p_0~~~~&E^{p,0}_1=\Omega_X^p,~~E^{p,q_0-1}_1\simeq\cZ^{p,q_0},~~
E^{p,q}_1=0~~~{\rm for}~~q\ne 0,q_0-1,\cr
&{\rm if}~~p\ge p_0~~~~&E^{p,q_0-1}_1=\cZ^{p,q_0},~~
E^{p,q}_1=0~~~{\rm for}~~q\ne q_0-1.\cr}$$
The isomorphism in the third line is given by
$$\cE^{p,q_0-1}/d''\cE^{p,q_0-2}\simeq d''\cE^{p,q_0-1}\simeq\cZ^{p,q_0}.$$
The map $d_1:E^{p_0-1,q_0-1}_1\lra E^{p_0,q_0-1}_1$ is induced by
$d'd''$ acting on $\cE^{p_0-1,q_0-1}$, but thanks to the previous
identification, this map becomes $d'$ acting on $\cZ^{p_0-1,q_0}$.
Hence $E^\bu_1$ consists of two sequences
$$\eqalign{
&E_1^{\bu,0}~:~~0\lra\Omega_X^0\buildo d'\over\lra\Omega_X^1\lra\cdots\buildo d'\over
\lra\Omega_X^{p_0-1}\lra 0,\cr 
&E_1^{\bu,q_0-1}~:~~0\lra\cZ^{0,q_0}\buildo d'\over\lra\cZ^{1,q_0}\lra\cdots\lra
\cZ^{p,q_0}\buildo d'\over\lra\cdots~;\cr}$$
if these sequences overlap ($q_0=1$), only the second one has to be 
considered. The term $E^\bu_1$ in the spectral sequence of $\cS^\bu$ has the 
same first line, but the second is reduced to $E^{0,q_0-1}_1
=\ovl{d\Omega_X^{q_0-2}}$ (resp. $=\bbbc$ for $q_0=1$). Thanks to
Lemma~12.2 below, we see that the two spectral sequences coincide in
$E^\bu_2$, with at most three non zero terms:
$$E^{0,0}_2=\bbbc,~~~~E^{p_0-1,0}_2=d\Omega_X^{p_0-2}~~{\rm for}~~p_0\ge 2,~~~~
E^{0,q_0-1}_2=\ovl{d\Omega_X^{q_0-2}}~~{\rm for}~~q_0\ge 2.$$
Hence $\cH^k(\cS^\bu)\simeq\cH^k(\cL^\bu)$ and these sheaves vanish
for $k\ne 0,p_0-1,q_0-1$.\qed
\endproof

\begstat{(12.2) Lemma} The complex of sheaves
$$0\lra\cZ^{0,q_0}\buildo d'\over\lra\cZ^{1,q_0}\lra\cdots\lra\cZ^{p,q_0}
\buildo d'\over\lra\cdots$$
is a resolution of~ $\ovl{d\Omega_X^{q_0-1}}$ for $q_0\ge 1$, resp. of~ $\bbbc$
for $q_0=0$.
\endstat

\begproof{} Embed $\cZ^{\bu,q_0}$ in the double complex
$$K^{p,q}=\cE^{p,q}~~~{\rm for}~~q<q_0,~~~~
K^{p,q}=0~~~{\rm for}~~q\ge q_0.$$
For the first fitration of $K^\bu$, we find
$$E^{p,q_0-1}_1=\cZ^{p,q_0},~~~~E^{p,q}_1=0~~~{\rm for}~~q\ne q_0-1~$$
The second fitration gives $\wt E^{p,q}_1=0$ for $q\ge 1$ and
$$\wt E^{p,0}_1=H^0(K^{\bu,p})=
\cases{\ovl{H^0(\cE^{p,\bu})}=\ovl{\Omega_X^p}&for $p\le q_0-1$\cr
       0&for $p\ge q_0$,\cr}$$
thus the cohomology of $\cZ^{\bu,q_0}$ coincides with that of
$(\ovl{\Omega_X^p},d)_{0\le p<q_0}$.\qed
\endproof

Lemma IV-11.10 and formula (IV-12.9) imply
$$\leqalignno{
{\Bbb H}^k(X,\cS^\bu)&{}\simeq{\Bbb H}^k(X,\cL^\bu)\simeq
{\Bbb H}^k(X,\cM^\bu)&(12.3)\cr
&{}\simeq H^k\big(\cL^\bu(X)\big)\simeq H^k\big(\cM^\bu(X)\big)\cr}$$
because the sheaves $\cL^k$ and $\cM^k$ are soft. In particular,
the group $H_\BC^{p,q}(X,\bbbc)$ can be computed either by means
of $\ci$ differential forms or by means of currents. This property
also holds for the De Rham or Dolbeault groups $H^k(X,\bbbc)$,
$H^{p,q}(X,\bbbc)$, as was already remarked in \S IV-6. Another important 
consequence of (12.3) is:

\begstat{(12.4) Theorem} If $X$ is compact, then $\dim H_\BC^{p,q}(X,\bbbc)<+\infty$.
\endstat

\begproof{} We show more generally that the hypercohomology groups
${\Bbb H}^k(X,\cS^\bu)$ are finite dimensional. As there is an exact
sequence
$$0\lra\bbbc\lra\cS^{\prime\,\bu}\oplus\cS^{\prime\prime\,\bu}\lra
\cS^\bu\lra 0$$
and a corresponding long exact sequence for hypercohomology groups,
it is enough to show that the groups
${\Bbb H}^k(X,\cS^{\prime\,\bu})$ are finite
dimensional. This property is proved for $\cS^{\prime\,\bu}=
\cS^{\prime\,\bu}_{p_0}$ by induction on $p_0$. For $p_0=0$ or $1$, the
complex $\cS^{\prime\,\bu}$ is reduced to its term $\cS^{\prime\,0}$, thus
$${\Bbb H}^k(X,\cS^\bu)=H^k(X,\cS^{\prime\,0})=\cases{
H^k(X,\bbbc)&for $p_0=0$\cr
H^k(X,\cO)&for $p_0=1$\cr}$$
and this groups are finite dimensional. In general, we have an exact
sequence
$$0\lra\Omega_X^{p_0}\lra\cS^\bu_{p_0+1}\lra\cS^\bu_{p_0}\lra 0$$
where $\Omega_X^{p_0}$ denotes the subcomplex of $\cS^\bu_{p_0+1}$ reduced to 
one term in degree~$p_0$. As
$${\Bbb H}^k(X,\Omega_X^{p_0})=H^{k-p_0}(X,\Omega_X^{p_0})=
H^{p_0,k-p_0}(X,\bbbc)$$
is finite dimensional, the Theorem follows.\qed
\endproof

\begstat{(12.5) Definition} We say that a compact manifold admits a
strong Hodge decomposition if the natural maps
$$H_\BC^{p,q}(X,\bbbc)\lra H^{p,q}(X,\bbbc),~~~~
\bigoplus_{p+q=k}H_\BC^{p,q}(X,\bbbc)\lra H^k(X,\bbbc)$$
are isomorphisms.
\endstat

This implies of course that there are natural isomorphisms
$$H^k(X,\bbbc)\simeq\bigoplus_{p+q=k}H^{p,q}(X,\bbbc),~~~~H^{q,p}(X,\bbbc)\simeq
\ovl{H^{p,q}(X,\bbbc)}$$
and that the Hodge-Fr\"olicher spectral sequence degenerates in $E^\bu_1$.
It follows from \S$\,$8 that all K\"ahler manifolds admit a strong Hodge
decomposition.

\titlec{\S 12.2.}{Direct and Inverse Image Morphisms}
Let $F:X\lra Y$ be a holomorphic map between complex analytic manifolds of
respective dimensions $n,m$, and $r=n-m$. We have pull-back morphisms
$$\cmalign{
&F^\star~:~~&\hfill H^k(Y,\bbbc)&\lra H^k(X,\bbbc),\cr
&F^\star~:~~&\hfill H^{p,q}(Y,\bbbc)&\lra H^{p,q}(X,\bbbc),\cr
&F^\star~:~~&\hfill H_\BC^{p,q}(Y,\bbbc)&\lra H_\BC^{p,q}(X,\bbbc),\cr}
\leqno(12.6)$$
commuting with the natural morphisms (8.2), (8.3).

Assume now that $F$ is {\it proper}. Theorem~I-1.14 shows that one
can define direct image morphisms
$$F_\star~:~~\cD'_k(X)\lra\cD'_k(Y),~~~~
  F_\star~:~~\cD'_{p,q}(X)\lra\cD'_{p,q}(Y),$$
commuting with $d',d''$. To $F_\star$ therefore correspond cohomology 
morphisms
$$\cmalign{
&F_\star~:~~&\hfill H^k(X,\bbbc)&\lra H^{k-2r}(Y,\bbbc),\cr
&F_\star~:~~&\hfill H^{p,q}(X,\bbbc)&\lra H^{p-r,q-r}(Y,\bbbc),\cr
&F_\star~:~~&\hfill H_\BC^{p,q}(X,\bbbc)&\lra H_\BC^{p-r,q-r}(Y,\bbbc),\cr}
\leqno(12.7)$$
which commute also with (8.2), (8.3). In addition, I-1.14~c)
implies the {\it adjunction formula}
$$F_\star(\alpha\smallsmile F^\star\beta)=(F_\star\alpha)\smallsmile\beta
\leqno(12.8)$$
whenever $\alpha$ is a cohomology class (of any of the three above types) 
on $X$, and $\beta$ a cohomology class (of the same type) on $Y$.

\titlec{\S 12.3.}{Modifications and the Fujiki Class ($\cC$)}
Recall that a modification of a compact manifold $X$ is a
holomorphic map $\mu:\wt X\lra X$ such that
\smallskip
\item{\rm i)} $\wt X$ is a compact complex manifold of the
same dimension as $X\,$;
\smallskip
\item{\rm ii)} there exists an analytic subset $S\subset X$ of 
codimension $\ge 1$ such that $\mu:\wt X\setminus
\mu^{-1}(S)\lra X\setminus S$ is a biholomorphism.
\vskip0pt

\begstat{(12.9) Theorem} If $\wt X$ admits a strong Hodge decomposition, 
and if $\mu:\wt X\lra X$ is a modification, then $X$ also admits a
strong Hodge decomposition.
\endstat

\begproof{} We first observe that $\mu_\star\mu^\star f=f$ for every smooth form
$f$ on $Y$. In fact, this property is equivalent to the equality
$$\int_Y (\mu_\star\mu^\star f)\wedge g=\int_X\mu^\star(f\wedge g)=\int_Y
f\wedge g$$
for every smooth form $g$ on $Y$, and this equality is clear because 
$\mu$ is a biholomorphism outside sets of Lebesgue measure $0$. Consequently,
the induced cohomology morphism
$\mu_\star$ is surjective and $\mu^\star$ is injective (but these maps need 
not be isomorphisms).  Now, we have commutative diagrams
$$\cmalign{
&H_\BC^{p,q}(\wt X,\bbbc)\lra&H^{p,q}(\wt X,\bbbc),~~~~
&\smash{\displaystyle\bigoplus_{p+q=k}}
H_\BC^{p,q}(\wt X,\bbbc)\lra&H^k(\wt X,\bbbc)
\phantom{\raise-6pt\hbox{$\Big)$}}\cr
&~~~\mu_\star\big\downarrow\big\uparrow\mu^\star\hfill
&~~~\mu_\star\big\downarrow\big\uparrow\mu^\star\hfill
&\qquad~~\mu_\star\big\downarrow\big\uparrow\mu^\star\hfill
&~~\mu_\star\big\downarrow\big\uparrow\mu^\star\hfill\cr
&H_\BC^{p,q}(X,\bbbc)\lra&H^{p,q}(X,\bbbc),~~~~
&\displaystyle\bigoplus_{p+q=k}H_\BC^{p,q}(X,\bbbc)\lra&H^k(X,\bbbc)
\phantom{\Big)}\cr}$$
with either upward or downward vertical arrows. Hence the surjectivity or
injectivity of the top horizontal arrows implies that
of the bottom horizontal arrows.\qed
\endproof

\begstat{(12.10) Definition} A manifold $X$ is said to be in the
Fujiki class $(\cC)$ if $X$ admits a K\"ahler modification $\wt X$.
\endstat

By Th.~12.9, Hodge decomposition still holds for a manifold in the
class $(\cC)$. We will see later that there exist non-K\"ahler manifolds 
in $(\cC)$, for example all non projective Moi$\check{\rm s}$ezon 
manifolds (cf.\ \S ?.?). The class $(\cC)$ has been first introduced
in (Fujiki~1978).

\end