\state 18.1. Theorem|
Let $(L,h)$ be a pseudo-effective line bundle on a
compact K\"ahler manifold $(X,\omega)$ of dimension $n$, let
$\Theta_{L,h}\ge 0$ be its curvature current and $\cI(h)$ the
associated multiplier ideal sheaf. Then, the wedge multiplication
operator $\omega^q\wedge\bu$ induces a surjective morphism
$$
\Phi^q_{\omega,h}:
H^0(X,\Omega_X^{n-q}\otimes L\otimes\cI(h))\longrightarrow
H^q(X,\Omega_X^n\otimes L\otimes\cI(h)).
$$
\endstate

The special case when $L$ is nef is due to Takegoshi [Tak97]. An even
more special case is when $L$ is semipositive, i.e.\ $L$ possesses a
smooth metric with semipositive curvature.
In that case the multiple ideal sheaf $\cI(h)$ coincides with $\cO_X$
and we get the following consequence already observed by Enoki [Eno93]
and Mourougane [Mou95].

\state 18.2. Corollary|
Let $(L,h)$ be a semipositive line bundle
on a compact K\"ahler mani\-fold $(X,\omega)$ of dimension~$n$. Then, the wedge
multiplication operator $\omega^q\wedge\bu$ induces a surjective morphism
$$
\Phi^q_\omega:H^0(X,\Omega_X^{n-q}\otimes L)\longrightarrow
H^q(X,\Omega_X^n\otimes L).
$$
\endstate

It should be observed that although all objects involved in Theorem~18.1
are algebraic when $X$ is a projective manifold, there is no known
algebraic proof of the statement!

\subsection \Smark 18.B. A variant of the Bochner formula|

We first recall a variation of the Bochner formula that is required in the proof of the Hard Lefschetz Theorem with values in a positively curved (and therefore non flat) line bundle $L$ equipped with a hermitian metric written locally as $h=e^{-\varphi}$. Here the base manifold is a K\"ahler (non necessarily compact) manifold $(Y,\omega)$.  We denote by
$|~~|=|~~|_{\omega,h}$ the pointwise Hermitian norm on
\hbox{$\Lambda^{p,q}T^*_Y\otimes L$} associated with $\omega$ and $h$,
and by $\Vert~~\Vert=\Vert~~\Vert_{\omega,h}$ the global $L^2$ norm
$$
\Vert u\Vert^2 = \int_Y |u|^2 dV_\omega \qquad
\hbox{where}\quad  dV_\omega={\omega^n\over n!}
$$
We consider the $\dbar$ operator acting on $(p,q)$-forms with values in
$L$, its adjoint $\dbarhstar$ with respect to $h$ and the complex
Laplace-Beltrami operator $\Delta''_h=\dbar\dbarhstar+\dbarhstar\dbar$.
Let $v$ be a smooth $(n-q,0)$-form with compact support
in~$Y$. Then $u=\omega^q\wedge v$ satisfies
$$\Vert\dbar u\Vert^2+\Vert\dbarhstar u\Vert^2=
\Vert\dbar v\Vert^2+\int_Y\sum_{I,J}\Big(\sum_{j\in J}\lambda_j\Big)|u_{IJ}|^2
\leqno(18.3)$$
where $\lambda_1\le\cdots\le\lambda_n$ are the curvature
eigenvalues of $\Theta_{L,h}$ expressed in an orthonormal frame
$(\partial/\partial z_1,\ldots,\partial/\partial z_n)$ (at some fixed
point $x_0\in Y$), in such a way that
$$
\omega_{x_0}=\ii\sum_{1\le j\le n}dz_j\wedge d\overline z_j,\qquad
(\Theta_{L,h})_{x_0}=dd^c\varphi_{x_0}=
\ii\sum_{1\le j\le n} \lambda_jdz_j\wedge d\overline z_j.
$$
Formula (18.3) follows from the more or less
straightforward identity
$$
(\dbar^*_\varphi\,\dbar+\dbar\,\dbar^*_\varphi)(v\wedge\omega^q)-
(\dbar^*_\varphi\,\dbar v)\wedge\omega^q=
q\,\ii\ddbar\varphi\wedge\omega^{q-1}\wedge v,
$$
by taking the inner product with $u=\omega^q\wedge v$ and integrating
by parts in the left hand side (we leave the easy details to the reader). 
Our formula is thus established when 
$v$ is smooth and compactly supported. In general, we have:

\state 18.4. Proposition|
Let $(Y,\omega)$ be a {\rm complete} K\"ahler
manifold and $(L,h)$ a smooth Hermitian line bundle such that the
curvature possesses a uniform lower bound $\Theta_{L,h}\ge -C\omega$.
For every measurable $(n-q,0)$-form $v$ with $L^2$ coefficients and
values in $L$ such that $u=\omega^q\wedge v$ has differentials
$\dbar u$, $\dbar^* u$ also in $L^2$, we have
$$
\Vert\dbar u\Vert^2+\Vert\dbarhstar u\Vert^2=
\Vert\dbar v\Vert^2+\int_Y\sum_{I,J}\Big(\sum_{j\in J}\lambda_j\Big)|u_{IJ}|^2
$$
$($here, all differentials are computed in the sense of distributions$)$.
\endstate

\preuve. Since $(Y,\omega)$ is assumed to be complete,
there exists a sequence of smooth forms $v_\nu$ with compact support in $Y$
(obtained by truncating $v$ and taking the convolution with a regularizing
kernel) such that $v_\nu\to v$ in $L^2$ and such that
$u_\nu=\omega^q\wedge v_\nu$ satisfies $u_\nu\to u$, $\dbar u_\nu\to\dbar u$,
$\dbar^* u_\nu\to\dbar^* u$ in~$L^2$. By the curvature assumption,
the final integral in the right hand side of (18.3) must be under control
(i.e.\ the integrand becomes nonnegative if we add a term $C\Vert u\Vert^2$
on both sides, $C\gg 0$). We thus get the equality by passing to the limit
and using Lebesgue's monotone convergence theorem.\QED

\subsection \Smark 18.C. Proof of Theorem 18.1 in the special case
where $(L,h)$ is Hermitian semipositive (and smooth)|

To fix the ideas, we first indicate the~proof in the
much simpler case when $(L,h)$ has a smooth metric $h$ (so that
$\cI(h)=\cO_X$), and then treat the general case.
Let $\{\beta\}\in H^q(X,\Omega^n_X\otimes L)$ be an arbitrary cohomology class.
By standard $L^2$ Hodge theory, $\{\beta\}$ can be represented by a smooth
harmonic $(0,q)$-form $\beta$ with values in $\Omega^n_X\otimes L$. We can
also view $\beta$ as a $(n,q)$-form with values in $L$. The pointwise
Lefschetz isomorphism produces a unique $(n-q,0)$-form $\alpha$ such
that $\beta=\omega^q\wedge\alpha$. Proposition~18.4 then yields
$$
\Vert\dbar\alpha\Vert^2+\int_Y\sum_{I,J}\Big(\sum_{j\in J}\lambda_j\Big)
|\alpha_{IJ}|^2=\Vert\dbar\beta\Vert^2+\Vert\dbarhstar \beta\Vert^2=0,
$$
and the curvature eigenvalues $\lambda_j$ are nonnegative by our assumption.
Hence $\dbar\alpha=0$ and $\{\alpha\}\in H^0(X,\Omega^{n-q}_X\otimes L)$
is mapped to $\{\beta\}$ by $\Phi^q_{\omega,h}=\omega^q\wedge\bu~$.

\subsection \Smark 18.D. Proof of Theorem 18.1 in the general case|

The main difficulty is that the metric $h=e^{-\varphi}$ is not
necessarily smooth, and
in that case we cannot directly represent cohomology classes by
harmonic forms. We circumvent this problem by regularizing the metric
on an (analytic) Zariski open subset, and then avoid the remaining poles
by working on this noncompact set, equipped with a complete K\"ahler metric.
We refer to [DPS01] or [Dem13] for a proof of the following

\state 18.5. Lemma|Let $T=\alpha+i\ddbar\varphi$ be a closed
$(1,1)$-current on a compact Hermitian manifold $(X,\omega)$, where
$\alpha$ is a smooth closed $(1,1)$-form and $\varphi$ a quasi-psh
function. Let $\gamma$ be a continuous real $(1,1)$-form such that
$T\ge\gamma$. Then one can write
\hbox{$\varphi=\lim_{\nu\to+\infty}\widetilde\varphi_\nu$} where
\vskip3pt
\item{\rm a)} $\widetilde\varphi_\nu$ is smooth in the complement $X\ssm Z_\nu$
of an analytic set $Z_\nu\subset X\,;$
\vskip3pt
\item{\rm b)} $\{\widetilde\varphi_\nu\}$ is a non-increasing sequence, and $Z_\nu\subset Z_{\nu+1}$ for all~$\nu\,;$
\vskip3pt
\item{\rm c)} $\int_X(e^{-\varphi}-e^{-\widetilde\varphi_\nu})dV_\omega$
is finite for every $\nu$ and converges to $0$ as $\nu\to+\infty\,;$
\vskip3pt
\item{\rm d)} $($``equisingularity''$)$ $\cI(\widetilde\varphi_\nu)=\cI(\varphi)$ for all $\nu\,;$
\vskip3pt
\item{\rm e)} $T_\nu=\alpha+dd^c\widetilde\varphi_\nu$ satisfies $T_\nu\ge
\gamma-\varepsilon_\nu\omega$, where $\lim_{m\to+\infty}\varepsilon_\nu=0$.\QED
\endstate

The application of Lemma~18.5 requires careful estimates in order to take
the error terms into account. To avoid multiple subscripts, write
$\varepsilon=\varepsilon_\nu$ and $h_\varepsilon=e^{-\varphi_\nu}$
the approximation of~$h$, so that $h_\varepsilon$ is smooth on
$X\ssm Z_\varepsilon$ ($Z_\varepsilon=Z_\nu$ being an analytic subset of $X$),
$\Theta_{L,h_\varepsilon}\ge -\varepsilon\omega$,
$h_\varepsilon\le h$ and $\cI(h_\varepsilon)=\cI(h)$.
Now, we can find a family
$$
\omega_{\varepsilon,\delta}=\omega+\delta(\ii\ddbar\psi_\varepsilon+\omega),
\qquad \delta>0
$$
of {\sl complete K\"ahler} metrics on $X\ssm Z_\varepsilon$, where
$\psi_\varepsilon$ is a quasi-psh function on $X$ with
$\psi_\varepsilon=-\infty$ on $Z_\varepsilon$,
$\psi_\varepsilon$ smooth on $X\ssm Z_\varepsilon$ and $\ii\ddbar
\psi_\varepsilon+\omega\ge 0$ (see e.g.\ [Dem82, Th\'eor\`eme~1.5]).
By construction, $\omega_{\varepsilon,\delta}\ge\omega$ and
$\lim_{\delta\to 0}\omega_{\varepsilon,\delta}=\omega$.
We look at the $L^2$ Dolbeault complex $K^\bu_{\varepsilon,\delta}$
of $(n,\bu)$-forms on $X\ssm Z_\varepsilon$, where the $L^2$ norms are
induced by $\omega_{\varepsilon,\delta}$ on differential forms and by
$h_\varepsilon$ on elements in~$L$. Specifically
$K^q_{\varepsilon,\delta}$ is the set of elements
$u:X\ssm Z_\varepsilon{\to}\Lambda^{n,q}T^*_X\otimes L$
such that
$$
\int_{X\ssm Z_\varepsilon}
(|u|^2_{\Lambda^{n,q}\omega_{\varepsilon,\delta}
\otimes h_\varepsilon}+|\dbar u|^2_{\Lambda^{n,q+1}\omega_{\varepsilon,\delta}
\otimes h_\varepsilon})dV_{\omega_{\varepsilon,\delta}}<\infty.
$$
Let $\cK^q_{\varepsilon,\delta}$ be the corresponding sheaf of germs
of locally $L^2$ sections on $X$ (the local $L^2$ condition
should hold on $X$, not only on $X\ssm Z_\varepsilon\,$!). Then,
for all $\varepsilon>0$ and $\delta\ge 0$,
$(\cK^q_{\varepsilon,\delta},\dbar)$ is a resolution of the sheaf
$\Omega^n_X\otimes L\otimes\cI(h_\varepsilon)=
\Omega^n_X\otimes L\otimes\cI(h)$. This is
because $L^2$ estimates hold locally on small Stein open sets, and the
$L^2$ condition on $X\ssm Z_\varepsilon$ forces holomorphic sections
to extend across~$Z_\varepsilon$ ([Dem82, Lemma 6.9]).

Let $\{\beta\}\in H^q(X,\Omega^n_X\otimes L\otimes\cI(h))$ be a
cohomology class represented by a smooth form with values in
$\Omega^n_X\otimes L\otimes\cI(h)$ (one can use a \v Cech cocycle
and convert it to an element in the $\cC^\infty$ Dolbeault complex by
means of a partition of unity, thanks to the usual De Rham-Weil
isomorphism, see also [DPS01] and [Dem13] for more details). Then
$$
\Vert\beta\Vert_{\varepsilon,\delta}^2\le \Vert\beta\Vert^2=
\int_X|\beta|^2_{\Lambda^{n,q}\omega\otimes h}dV_\omega<+\infty.
$$
The reason is that $|\beta|^2_{\Lambda^{n,q}\omega\otimes h}dV_\omega$
decreases as $\omega$ increases. This is just an easy calculation,
shown by comparing two metrics $\omega$, $\omega'$ which are
expressed in diagonal form in suitable coordinates; the norm
$|\beta|^2_{\Lambda^{n,q}\omega\otimes h}$ turns out to decrease
faster than the volume $dV_\omega$ increases; see e.g.\
[Dem82, Lemma 3.2];
a special case is $q=0$, then $|\beta|^2_{\Lambda^{n,q}\omega\otimes h}
dV_\omega=i^{n^2}\beta\wedge\overline\beta$ with the identification
$L\otimes\overline L\simeq\bC$ given by the metric $h$, hence the
integrand is even independent of $\omega$ in that case.

By the proof of the De Rham-Weil isomorphism, the map
$\alpha\mapsto\{\alpha\}$ from the cocycle space
$Z^q(\cK^\bu_{\varepsilon,\delta})$ equipped with its $L^2$ topology,
into $H^q(X,\Omega^n_X\otimes L\otimes\cI(h))$ equipped with its
finite vector space topology, is continuous.  Also, Banach's open
mapping theorem implies that the coboundary space
$B^q(\cK^\bu_{\varepsilon,\delta})$ is closed in
$Z^q(\cK^\bu_{\varepsilon,\delta})$. This is true for all $\delta\ge
0$ (the limit case $\delta=0$ yields the strongest $L^2$ topology in
bidegree $(n,q)$). Now, $\beta$ is a $\dbar$-closed form in the Hilbert space
defined by $\omega_{\varepsilon,\delta}$ on $X\ssm Z_\varepsilon$, so there
is a $\omega_{\varepsilon,\delta}$-harmonic form
$u_{\varepsilon,\delta}$ in the same cohomology class
as $\beta$, such that
$$\Vert u_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}\le
\Vert\beta\Vert_{\varepsilon,\delta}.
\leqno(18.6)$$
Let $v_{\varepsilon,\delta}$ be the unique $(n-q,0)$-form such that
$u_{\varepsilon,\delta}=v_{\varepsilon,\delta}\wedge
\omega_{\varepsilon,\delta}^q$ ($v_{\varepsilon,\delta}$ exists by the
pointwise Lefschetz isomorphism). Then
$$
\Vert v_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}=
\Vert u_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}\le
\Vert\beta\Vert_{\varepsilon,\delta}\le\Vert\beta\Vert.
$$
As $\sum_{j\in J}\lambda_j\ge -q\varepsilon$ by the assumption on
$\Theta_{L,h_\varepsilon}$, the Bochner formula yields
$$
\Vert\dbar v_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}^2\le
q\varepsilon\Vert u_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}^2
\le q\varepsilon\Vert\beta\Vert^2.
$$
These uniform bounds imply that there are subsequences $u_{\varepsilon,
\delta_\nu}$ and $v_{\varepsilon,\delta_\nu}$ with \hbox{$\delta_\nu\to 0$},
possessing weak-$L^2$ limits $u_\varepsilon=
\lim_{\nu\to+\infty}u_{\varepsilon,\delta_\nu}$
and $v_\varepsilon=\lim_{\nu\to+\infty}v_{\varepsilon,\delta_\nu}$.
The limit $v_\varepsilon=
\lim_{\nu\to+\infty}v_{\varepsilon,\delta_\nu}$ is with respect to
$L^2(\omega)=L^2(\omega_{\varepsilon,0})$. To check this, notice that
in bidegree $(n-q,0)$, the space $L^2(\omega)$ has the weakest topology
of all spaces~$L^2(\omega_{\varepsilon,\delta})$; indeed, an easy calculation
made in ([Dem82, Lemma 3.2]) yields
$$
|f|^2_{\Lambda^{n-q,0}\omega\otimes h}dV_\omega\le
|f|^2_{\Lambda^{n-q,0}\omega_{\varepsilon,\delta}\otimes h}
dV_{\omega_{\varepsilon,\delta}}\qquad
\hbox{if $f$ is of type $(n-q,0)$}.
$$
On the other hand, the limit
$u_\varepsilon=\lim_{\nu\to+\infty}u_{\varepsilon,\delta_\nu}$
takes place in all spaces $L^2(\omega_{\varepsilon,\delta})$, $\delta>0$,
since the topology gets stronger and stronger as $\delta\downarrow 0$
[$\,$possibly not in $L^2(\omega)$, though, because in bidegree $(n,q)$
the topology of $L^2(\omega)$ might be strictly stronger than that
of all spaces $L^2(\omega_{\varepsilon,\delta})\,$].
The above estimates yield
$$\eqalign{
&\Vert v_\varepsilon\Vert^2_{\varepsilon,0}=
\int_X|v_\varepsilon|^2_{\Lambda^{n-q,0}\omega\otimes h_\varepsilon}
dV_\omega\le\Vert\beta\Vert^2,\cr
&\Vert \dbar v_\varepsilon\Vert^2_{\varepsilon,0}\le q\varepsilon
\Vert\beta\Vert^2_{\varepsilon,0},\cr
\noalign{\vskip3pt}\cr
&u_\varepsilon=\omega^q\wedge v_\varepsilon\equiv\beta
\qquad\hbox{in}~~H^q(X,\Omega^n_X\otimes L\otimes\cI(h_\varepsilon)).\cr}
$$
Again, by arguing in a given Hilbert space $L^2(h_{\varepsilon_0})$,
we find $L^2$ convergent subsequences $u_\varepsilon\to u$,
$v_\varepsilon\to v$ as $\varepsilon\to 0$, and in this way get
$\dbar v=0$ and
$$\eqalign{
&\Vert v\Vert^2\le \Vert \beta\Vert^2,\cr
&u=\omega^q\wedge v\equiv \beta
\qquad\hbox{in}~~H^q(X,\Omega^n_X\otimes L\otimes\cI(h)).\cr}
$$
Theorem~18.1 is proved. Notice that the equisingularity property
$\cI(h_\varepsilon)=\cI(h)$ is crucial in the above proof,
otherwise we could not infer that $u\equiv \beta$ from the fact
that $u_\varepsilon\equiv \beta$. This is true only because
all cohomology classes $\{u_\varepsilon\}$ lie in the same fixed
cohomology group \hbox{$H^q(X,\Omega^n_X\otimes L\otimes\cI(h))$}, whose
topology is induced by the topology of $L^2(\omega)$ on $\dbar$-closed
forms (e.g.\ through the De Rham-Weil isomorphism).\QED

\state 18.7. Remark| {\rm In (18.6), the existence of a harmonic 
representative holds true only for~$\omega_{\varepsilon,\delta}$, $\delta>0$,
because we need to have a complete K\"ahler metric on $X\ssm Z_\varepsilon$.
The trick of employing $\omega_{\varepsilon,\delta}$ instead of a fixed
metric $\omega$, however, is not needed when $Z_\varepsilon$ is (or can
be taken to be) empty. This is the case if $(L,h)$ is such that
$\cI(h)=\cO_X$ and $L$ is nef. Indeed, by definition, $L$ is nef iff
there exists a sequence of smooth metrics $h_\nu$ such that
$\ii\Theta_{L,h_\nu}\ge-\varepsilon_\nu\omega$, so we can take the 
$\varphi_\nu$'s to be everywhere smooth in Lemma 18.5.
However, multiplier ideal sheaves are needed in the surjectivity statement
even in case $L$ is nef, as it may happen that $\cI(h_{\min})\ne\cO_X$ even
then, and $h:=\lim h_\nu$ is anyway always more singular than $h_{\min}$.
Let us recall a standard example (see [DPS94], [DPS01]).
Let $B$ be an elliptic curve and let $V$ be the rank $2$ vector bundle
over $B$ which is defined as the (unique) non split
extension
$$0 \to \cO_B \to V \to \cO_B \to 0.$$
In particular, the bundle $V$ is numerically flat, i.e.\ $c_1(V)=0$,
$c_2(V)=0$. We consider the ruled surface $X=\bP(V)$. On that surface
there is a unique section $C=\bP(\cO_B) \subset X$ with $C^2=0$ and
$$\cO_X (C)= \cO_{\bP (V)} (1)$$
is a nef line bundle. One can check that $L=\cO_{\bP (V)}(3)$ leads
to a {\sl zero} Lefschetz map 
$$\omega\wedge\bu:~~H^0(X,\Omega^1_X\otimes L)\longrightarrow
H^1(X,K_X\otimes L)\simeq\bC,
$$
so this is a counterexample to Corollary 18.2 in
the nef case. Incidentally, this also shows (in a somewhat sophisticated
way) that $\cO_{\bP (V)}(1)$ is nef but not semipositive, a fact that 
was first observed in [DPS94].}
\endstate

