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Annales de la Faculte' des Sciences de Toulouse
Vol. XX, $\mathrm{n}^{\mathrm{o}}$ Sp\'{e}cial, 2011
pp. 123-135
A converse to the Andreotti-Grauert theorem
Jean-Pierre DemAillY(${}^{\text{1}}$)
{\it Dedicated to Professor Nguyen Thanh Van}
ABSTRACT. --The goal of this paper is to show that there are strong rela-
tions between certain $\mathrm{Monge}-\mathrm{Amp}\grave{\mathrm{e}}$ re integrals appearing in holomorphic
Morse inequalities, and asymptotic cohomology estimates for tensor pow-
ers of holomorphic line bundles. Especially, we prove that these relations
hold without restriction for projective surfaces, and in the special case of
the volume, i.e. of asymptotic 0-cohomology, for all projective manifolds.
These results can be seen as a partial converse to the Andreotti-Grauert
vanishing theorem.
RE'SUME'. -- Le but de ce travail est de montrer qu il $\mathrm{y}$ a des relations
fortes entre certaines inte'grales de $\mathrm{Monge}-\mathrm{Amp}\grave{\mathrm{e}}$ re apparaissant dans les
ine'galite's de Morse holomorphes, et les estimations asymptotiques de co-
homologie pour les fibre's holomorphes en droites. En particulier, nous
montrons que ces relations sont satisfaites sans restriction pour toutes les
surfaces projectives, et dans le cas particulier du volume, c'est-\`{a}-dire de
la 0-cohomologie asymptotique, pour toutes les varie'te's projectives. Ces
re'sultats peuvent \^{e}tre vus comme une re'ciproque partielle au th\'{e}or\`{e}me
d'annulation d'Andreotti-Grauert.
1. Main results
Throughout this paper, $X$ denotes a compact complex manifold,
$n=\dim_{\mathbb{C}}X$ its complex dimension and $L\rightarrow X$ a holomorphic line bun-
dle. In order to estimate the growth of cohomology groups, it is interest-
ing to consider appropriate ``asymptotic cohomology functions''. Following
(1) Universite' de Grenoble I, De'partement de Mathe'matiques, Institut Fourier, 38402
Saint-Martin $\mathrm{dHe}_{\mathrm{r}\mathrm{e}\mathrm{s}}$, France
demailly@fourier.uj f-grenoble.fr
-123-

Jean-Pierre Demailly
partly notation and concepts introduced by A. K\"{u}ronya [K\"{u}r06, FKL07],
we introduce
Definition 1.1. --
(i) {\it The q-th asymptotic cohomology functional is defined as}
$$
\hat{h}^{q}(X, L)\ :=\lim_{k\rightarrow+}\sup_{\infty}\frac{n!}{k^{n}}h^{q}(X, L^{\otimes k}).
$$
(ii) {\it The q-th asymptotic holomorphic Morse sum of} $L$ {\it is}
$$
\hat{h}^{\leq q}(X, L):=\lim_{k\rightarrow+}\sup_{\infty}\frac{n!}{k^{n}}\sum_{0\leq j\leq q}(-1)^{q-j}h^{j}(X, L^{\otimes k}).
$$
When the $\displaystyle \lim\sup$'s are limits, we have the obvious relation
$$
\hat{h}^{\leq q}(X, L)=\sum_{0\leq j\leq q}(-1)^{q-j}\hat{h}^{j}(X, L).
$$
Clearly, Defnition 1.1 can also be given for a $\mathbb{Q}$-line bundle $L$ or a $\mathbb{Q}$-divisor
$D$, and in the case $q=0$ one gets what is usually called the volume of $L$,
namely
\begin{center}
$\displaystyle \mathrm{Vol}(X, L)=\hat{h}^{0}(X, L)=\lim_{k\rightarrow+}\sup_{\infty}\frac{n!}{k^{n}}h^{0}(X, L^{\otimes k})$.   (1.1)
\end{center}
(see also [DEL00], [Bou02], [Laz04]). It has been shown in [K\"{u}r06] for the
projective case and in [DemlO] in general that the $\hat{h}^{q}$ functional induces a
continuous map
DNSR (X) $\ni\alpha\rightarrow\hat{h}^{q}(X, \alpha)$
defned on the ``divisorial Neron-Severi space'' $\mathrm{DNS}_{\mathbb{R}}(X)\subset H_{\mathrm{B}\mathrm{C}}^{1,1}(X, \mathbb{R})$ con-
sisting of real linear combinations of classes of divisors in the real Bott-
Chern cohomology group of bidegree (1, 1). Here $H_{\mathrm{B}\mathrm{C}}^{p,q}(X, \mathbb{C})$ is defned as
the quotient of {\it d}-closed $(p, q)$-forms by $\partial\overline{\partial}$-exact $(p, q)$-forms, and there is
a natural conjugation $H_{\mathrm{B}\mathrm{C}}^{p,q}(X, \mathbb{C})\rightarrow H_{\mathrm{B}\mathrm{C}}^{q,p}(X, \mathbb{C})$ which allows us to speak
of real classes when $q=p$. The $\hat{h}^{q}$ functional is in fact locally Lipschitz
continous on $\mathrm{DNS}_{\mathbb{R}}(X)$, and can be obtained as a limit (not just a limsup)
on all those classes. Notice that $H_{\mathrm{B}\mathrm{C}}^{p,q}(X, \mathbb{C})$ coincides with the usual Dol-
beault cohomology group $H^{p,q}(X, \mathbb{C})$ when $X$ is K\"{a}hler, and that DNSR (X)
coincides with the usual Ne'ron-Severi space
$$
\mathrm{NS}_{\mathbb{R}}(X)=\mathbb{R}\otimes \mathbb{Q}(H^{2}(X, \mathbb{Q})\cap H^{1,1}(X, \mathbb{C}))
$$
$$
-124-
$$
A converse to the Andreotti-Grauert theorem
when $X$ is projective. It follows from holomorphic {\it Morse} inequalities (cf.
[Dem85], [Dem91] $)$ that asymptotic cohomology can be compared with cer-
tain $\mathrm{Monge}-\mathrm{Amp}\grave{\mathrm{e}}$ re integrals.
Theorem ([Dem85]) 1.2. -- {\it For every holomorphic line bundle} $L$ {\it on}
{\it a compact complex manifold} $X$, {\it one has the} ``{\it weak Morse inequality}''
(i) $\displaystyle \hat{h}^{q}(X, L)\leq\inf_{u\in c_{1}(L)}\int_{X(u,q)}(-1)^{q}u^{n}$
{\it where} $u$ {\it runs over all smooth d-closed} (1, 1)-{\it forms which belong to the co}-
{\it homology class} $c_{1}(L)\in H_{\mathrm{B}\mathrm{C}}^{1,1}(X, \mathbb{R})$, {\it and} $X(u, q)$ {\it is the open set}
$X(u, q);=$\{ $z\in X;u(z)$ {\it has signature} $(n-q, q)$\}.
{\it Moreover}, $ifX(u, \leq q) :=\displaystyle \bigcup_{0\leq j\leq q}X(u, j)$, {\it one has the} ``{\it strong Morse inequa}-
{\it lity}''
(ii) $\displaystyle \hat{h}^{\leq q}(X, L)\leq\inf_{u\in c_{1}(L)}\int_{X(u,\leq q)}(-1)^{q}u^{n}$.
It is a natural problem to ask whether the inequalities (1.2) (i) and
(1.2) (ii) might not always be equalities. These questions are strongly related
to the Andreotti-Grauert vanishing theorem [AG62]. A well-known variant
of this theorem says that if for some integer $q$ and some $u\in c_{1}(L)$ the
form $u(z)$ has at least $n-q+1$ positive eigenvalues everywhere (so that
$X(u, \displaystyle \geq q)=\bigcup_{j\geq q}X(u, j)=\emptyset)$, then $H^{j}(X, L^{\otimes k})=0$ for $j\geq q$ and $ k\gg$
1. We are asking here whether conversely the knowledge that cohomology
groups are asymptotically small in a certain degree $q$ implies the existence
of a hermitian metric on $L$ with suitable curvature, i.e. no {\it q}-index points or
only a very small amount of such.
The frst goal of this note is to prove that the answer is positive in the
case of the volume functional (i.e. in the case of degree $q=0$), at least when
$X$ is projective algebraic.
Theorem 1.3. -- {\it Let} $L$ {\it be a holomorphic line bundle on a projective}
{\it algebraic manifold. then}
$$
\mathrm{Vol}(X, L)=\inf_{u\in c_{1}(L)}\int_{X(u,0)}u^{n}.
$$
The proof relies mainly on fve ingredients: (a) approximate Zariski de-
composition for a K\"{a}hler current $T\in c_{1}(L)$ (when $L$ is big), i.e. a decompo-
$$
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$$
Jean-Pierre Demailly
sition $\mu^{*}T=[E]+\beta$ where $\mu$ : $\overline{X}\rightarrow X-$ is a modification, $E$ an exceptional
divisor and $\beta$ a K\"{a}hler metric on $X;(\mathrm{b})$ the characterization of the pseu-
doeffective cone ([BDPP04]), and the orthogonality estimate
$$
E\cdot\beta^{n-1}\leq C(\mathrm{Vol}(X, L)-\beta^{n})^{1/2}
$$
proved as an intermediate step of that characterization; (c) properties of
solutions of Laplace equations to get smooth approximations of $[E];(\mathrm{d})\log$
concavity of the $\mathrm{Monge}-\mathrm{Amp}\grave{\mathrm{e}}$ re operator; and finally (e) birational invari-
ance of the Morse inflmums. In the case of higher cohomology groups, we
have been able to treat only the case of projective surfaces.
Theorem 1.4. -- {\it Let} $L\rightarrow X$ {\it be a holomorphic line bundle on a com}-
{\it plex projective surface. Then both weak and strong inequalities} $(1.3\mathrm{J}_{\backslash }(\mathrm{i})$ {\it and}
(1.3) (ii) {\it are equalities for} $q=0,1,2$, {\it and the} $\displaystyle \lim\sup$'{\it s involved in} $h^{q}(X, L)$
{\it and} $\hat{h}\leq q(X, L)$ {\it are limits}.
Thanks to Serre duality and the Riemann-Roch formula, the (in)equality
for a given $q$ is equivalent to the (in)equality for $n-q$. Therefore, on surfaces,
the only substantial case which remains to be proved is the case $q=1$;
our statements are of course trivial on curves since the curvature of any
holomorphic line bundle can be taken to be constant with respect to any
given hermitian metric.
(1.5) Remark. -- It is interesting to put these results in perspec-
tive with the algebraic version of holomorphic Morse inequalities proved
in [Dem94] (see also [Siu93] and [Tra95] for related ideas, and [Ang94] for
an algebraic proof). When $X$ is projective, the algebraic Morse inequalities
used in combination with the birational invariance of the Morse integrals
(cf. section 2) imply the inequalities
(i) $\displaystyle \inf_{u\in c_{1}(L)}\int_{X(u,q)}(-1)^{q}u^{n}\leq\inf_{\mu^{*}(L)\simeq \mathrm{O}(A-B)} \left(\begin{array}{l}
n\\
q
\end{array}\right) A^{n-q}B^{q}$,
(ii) $\displaystyle \inf_{u\in c_{1}(L)}\int_{X(u,\leq q)}(-1)^{q}u^{n}\leq\inf_{\mu^{*}(L)\simeq \mathrm{O}(A-B)}\sum_{0\leq j\leq q}(-1)^{q-j} \left(\begin{array}{l}
n\\
j
\end{array}\right) A^{n-j}B^{j}$,
where the inflmums on the right hand side are taken over all modifications
$\mu$ : $X\rightarrow X$ and all $\mathrm{decompo}\underline{\mathrm{s}\mathrm{i}}\mathrm{tions}\mu^{*}L=\mathcal{O}(A-B)$ of $\mu^{*}L$ as a difference of
two nef $\mathbb{Q}$-divisors $A,\ B$ on $X$. In case $A$ and $B$ are ample, the proof simply
consists of taking positive curvature forms $\Theta_{\mathcal{O}(A),h_{A}},\ \Theta_{\mathcal{O}(B),h_{B}}$ on $\mathcal{O}(A)$ and
$\mathcal{O}(B)$, and evaluating the Morse integrals with $u=\Theta_{\mathcal{O}(A),h_{A}}-\Theta_{\mathcal{O}(B),h_{B}}$;
$$
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$$
A converse to the Andreotti-Grauert theorem
the general case follows by approximating the nef divisors $A$ and $B$ by ample
divisors $A+\xi:H$ and $B+\xi:H$ with $H$ ample and $\xi j >0$, see [Dem94]. Again, a
natural question is to know whether these infmums derived from algebraic
intersection numbers are equal to the asymptotic cohomology functionals
$\hat{h}^{q}(X, L)$ and $\hat{h}\leq q(X, L)$. A positive answer would of course automatically
yield a positive answer to the equality cases in 1.3 (i) and 1.3 (ii). However,
the Zariski decompositions involved in our proofs of the ``analytic equality
case'' produces certain effective exceptional divisors which are not nef. It
is unclear how to write those effective divisors as a difference of nef divi-
sors. This fact raises a lot of doubts upon the sufficiency of taking merely
differences of nef divisors in the infimums 1.6 (i) and 1.6 (ii).
$$
\square 
$$
I warmly thank Burt Totaro for stimulating discussions in connection
with his recent work [Tot10].
2. Invariance by modification
It is easy to check that the asymptotic cohomology function is invariant
by modification, namely that for every modification $\mu$ : $X\rightarrow X$ and every
line bundle $L$ we have
\begin{center}
$\hat{h}^{q}(X, L)=\hat{h}^{q}(\overline{X}, \mu^{*}L)$.   (2.1)
\end{center}
In fact the Leray's spectral sequence provides an $E_{2}$ term
$E_{2}^{p,q}=H^{p}(X, R^{q}\mu_{*}\mathcal{O}_{\overline{X}}(\mu^{*}L^{\otimes k}))=H^{p}(X, \mathcal{O}_{X}(L^{\otimes k})\otimes R^{q}\mu_{*}\mathcal{O}_{\overline{X}})$.
Since $R^{q}\mu_{*}\mathcal{O}_{\overline{X}}$ is equal to $\mathcal{O}_{X}$ for $q=0$ and is supported on a proper analytic
subset of $X$ for $q\geq 1$, one infers that $h^{p}(X, \mathcal{O}_{X}(L^{\otimes k}\otimes R^{q}\mu_{*}\mathcal{O}_{\overline{X}})=O(k^{n-1})$
for all $q\geq 1$. The spectral sequence implies that
$$
h^{q}(X, L^{\otimes k})-\hat{h}^{q}(\overline{X}, \mu^{*}L^{\otimes k})=O(k^{n-1}).
$$
We claim that the Morse integral infimums are also invariant by modifica-
tion.
PROPOSITION 2.1. -- {\it Let} $(X, \omega)$ {\it be a compact Kdhler manifold}, $\alpha\in$
$H^{1,1}(X, \mathbb{R})$ {\it a real cohomology class and} $\mu$ : $\overline{X}\rightarrow X$ {\it a modification. Then}
(i) $\displaystyle \inf_{u\in\alpha}\int_{X(u,q)}(-1)^{q}u^{n}=\inf_{v\in\mu^{*}\alpha}\int_{X(v,q)}(-1)^{q}v^{n}$,
(ii) $\displaystyle \inf_{u\in\alpha}\int_{X(u,\leq q)}(-1)^{q}u^{n}=\inf_{v\in\mu^{*}\alpha}\int_{X(v,\leq q)}(-1)^{q}v^{n}$.
$$
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$$
Jean-Pierre Demailly
{\it Proof}.--Given $ u\in\alpha$ on $X$, we obtain Morse integrals with the same
$$
-
$$
values by taking $v=\mu^{*}u$ on $X$, hence the infimum (resp. supremum) on $X$
is smaller (resp. larger) than what is on $X$, or it is equal. $\underline{\mathrm{C}}\mathrm{onversely}$, we
have to show that given a smooth representative $ v\in\mu^{*}\alpha$ on $X$, one can fnd
a smooth representative $u\in X$ such that the Morse integrals do not differ
much. We can always assume that $X$ itself is K\"{a}hler, since by Hironaka
[Hir64] any modifcation $X$ is dominated by a composition of blow-ups of
{\it X}. Let us fx some $ u_{0}\in\alpha$ and write
$$
 v=\mu^{*}u_{0}+dd^{c}\varphi
$$
where $\varphi$ is a smooth function on $\overline{X}$. We adjust $\varphi$ by a constant in such
a $\mathrm{wa}\underline{\mathrm{y}}$ that $\varphi\geq 1$ on $X$. There exists an analytic set $S\subset X$ such that
$\mu$ : $X\backslash \mu^{-1}(S) \rightarrow X\backslash S$ is a biholomorphism, and a quasi-psh function
$\psi_{S}$ which is smooth on $X\backslash S$ and $\mathrm{has}-\infty$ logarithmic poles on $S$ (see e.g.
[Dem82] $)$. We defne
$\displaystyle \overline{u}=\mu^{*}u_{0}+dd^{c}\max_{\in 0}(\varphi+\delta\psi_{S}\circ\mu, 0)=v+dd^{c}\max_{\in 0}(\delta\psi_{S}\circ\mu, -\varphi)$ (2.3)
where $\displaystyle \max_{\in 0},0<\in 0 <1$, is a regularized $\displaystyle \max$ function and $\delta>0$ is
very small. By construction $\overline{u}$ coincides with $\mu^{*}u_{0}$ in a neighborhood of
$\mu^{-1}(S)$ and therefore $\overline{u}$ descends to a smooth closed (1, 1)-form $u$ on $X$ which
coincides with $u_{0}$ near $S$, so that $\overline{u}=\mu^{*}u$. Clearly $\overline{u}$ converges uniformly
to $v$ on every compact subset of $X\backslash \mu^{-1}(S)$ as $\delta\rightarrow 0$, so we only have to
show that the Morse integrals are small (uniformly in $\delta$) when restricted to
a suitable small neighborhood of the exceptional set $E=\mu^{-1}(S)$. Take a
sufficiently large K\"{a}hler metric $\overline{\omega}$ on $\overline{X}$ such that
$$
-\frac{1}{2}\overline{\omega}\leq v\leq\frac{1}{2}\overline{\omega},\ -\frac{1}{2}\overline{\omega}\leq dd^{c}\varphi\leq\frac{1}{2}\overline{\omega},\ -\overline{\omega}\leq dd^{c}\psi_{S}\circ\mu.
$$
Then $\overline{u}\geq-\overline{\omega}$ and $\overline{u}\leq\overline{\omega}+\delta dd^{c}\psi_{S}\circ\mu$ everywhere on $\overline{X}$. As a consequence
$|\overline{u}^{n}|\leq(\overline{\omega}+\delta(\overline{\omega}+dd^{c}\psi_{S}\circ\mu))^{n}\leq\overline{\omega}^{n}+n\delta(\overline{\omega}+dd^{c}\psi_{S}\circ\mu)\wedge(\overline{\omega}+\delta(\overline{\omega}+dd^{c}\psi_{S}\circ\mu))^{n-1}$
thanks to the inequality $(a+b)^{n}\leq a^{n}+nb(a+b)^{n-1}$. For any neighborhood
$V$ of $\mu^{-1}(S)$ this implies
$$
\int_{V}|\overline{u}^{n}|\leq\int_{V}\overline{\omega}^{n}+n\delta(1+\delta)^{n-1}\int_{\overline{X}}\overline{\omega}^{n}
$$
by Stokes formula. We thus see that the integrals are small if $V$ and $\delta$
are small. The reader may be concerned that $\mathrm{Monge}-\mathrm{Amp}\grave{\mathrm{e}}$ re integrals were
used with an unbounded potential $\psi_{S}$, but in fact, for any given {\it 5}, all
the above formulas and estimates are still valid when we replace $\psi_{S}$ by
$\displaystyle \max_{\in 0}(\psi_{S}, -(M+2)/\delta)$ with $ M=\displaystyle \max_{\overline{X}}\varphi$, especially formula (2.3) shows
that the form $\overline{u}$ is unchanged. Therefore our calculations can be handled by
using merely smooth potentials.
$$
\square 
$$
-128-

A converse to the Andreotti-Grauert theorem
3. Proof of the infimum formula for the volume
We have to show here that
\begin{center}
$\displaystyle \inf_{u\in c_{1}(L)}\int_{X(u,0)}u^{n}\leq \mathrm{Vol}(X, L)$   (3.1)
\end{center}
Let us frst assume that $L$ is a big line bundle, i.e. that $\mathrm{Vol}(X, L)>0$.
Then it is known by [Bou02] that $\mathrm{Vol}(X, L)$ is obtained as the supremum of
$\displaystyle \int_{X\backslash \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}(T)}T^{n}$ for K\"{a}hler currents $T=-\displaystyle \frac{i}{2\pi}\partial\overline{\partial}h$ with analytic singularities
in $c_{1}(L)$; this means that locally $h=e^{-\varphi}$ where $\varphi$ is a strictly plurisubhar-
monic function which has the same singularities as $c\displaystyle \log\sum|g_{j}|^{2}$ where $c>0$
and the $g_{j}$ are holomorphic functions. By [Dem92], there exists a blow-up
$\mu$ : $\overline{X}\rightarrow X$ such that $\mu^{*}T=[E]+\beta$ where $E$ is a normal crossing divisor
on $\overline{X}$ and $\beta\geq 0$ smooth. Moreover, by [BDPP04] we have the orthogonality
estimate
\begin{center}
$[E]\displaystyle \cdot\beta^{n-1}=\int_{E}\beta^{n-1}\leq C(\mathrm{Vol}(X, L)-\beta^{n})^{1/2}$,   (3.2)
\end{center}
while
\begin{center}
$\displaystyle \beta^{n}=\int_{\overline{X}}\beta^{n}=\int_{X\backslash }$ sing $(\tau)^{T^{n}}$ approaches $\mathrm{Vol}(X, L)$.   (3.3)
\end{center}
In other words, $E$ and $\beta$ become ``more and more orthogonal'' as $\beta^{n}$ ap-
proaches the volume (approximate Zariski decomposition, cf. [Fuj94]). By
subtracting to $\beta$ a small linear combination of the exceptional divisors and
increasing accordingly the coefficients of $E$, we can even achieve that the
cohomology class $\{\beta\}$ contains a positive defnite form $\beta'$ on $X$ (i.e. is the
fundamental form of a K\"{a}hler metric); we refer e.g. to ([DP04], proof of
Lemma 3.5) for details. This means that we can replace $T$ by a cohomolo-
gous current such that the corresponding form $\beta$ is actually a K\"{a}hler metric,
and we will assume for simplicity of notation that this situation occurs right
away for $T$. Under this assumption, there exists a smooth closed (1, 1)-form
$v$ belonging to the Bott-Chern cohomology class as $[E]$, such that we have
identically $(v-\delta\beta)\wedge\beta^{n-1}=0$ where
\begin{center}
$\displaystyle \delta=\frac{[E]\cdot\beta^{n-1}}{\beta^{n}}\leq C'(\mathrm{Vol}(X, L)-\beta^{n})^{1/2}$   (3.4)
\end{center}
for some constant $C'>0$. In fact, given an arbitrary smooth representative
$v_{0}\in\{[E]\}$, the existence {\it of} $ v=v_{0}+i\partial\overline{\partial}\psi$ amounts to solving a Laplace
equation $\triangle\psi=f$ with respect to the K\"{a}hler metric $\beta$, and the choice of $\delta$
ensures that we have $\displaystyle \int_{X}f\beta^{n}=0$ and hence that the equation is solvable.
Then $\overline{u};=v+\beta$ is a smooth closed (1, 1)-form in the cohomology class
$\mu^{*}c_{1}(L)$, and its eigenvalues with respect to $\beta$ are of the form $1+\lambda_{j}$ where
$$
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$$
Jean-Pierre Demailly
$\lambda_{j}$ are the eigenvalues of $v$. The Laplace equation is equivalent to the identity
$\displaystyle \sum\lambda_{j}=n\delta$. Therefore
\begin{center}
$\displaystyle \sum_{1\leq j\leq n}\lambda_{j}\leq C''(\mathrm{Vol}(X, L)-\beta^{n})^{1/2}$   (3.5)
\end{center}
The inequality between arithmetic means and geometry means implies
$\displaystyle \prod_{1\leq j\leq n}(1+\lambda_{j})\leq(1+\frac{1}{n}\sum_{1\leq j\leq n}\lambda_{j})^{n}\leq 1+C_{3}(\mathrm{Vol}(X, L)-\beta^{n})^{1/2}$
whenever all factors $(1+\lambda_{j})$ are nonnegative. By 2.2 (i) we get
$$
\inf_{u\in c_{1}(L)}\int_{X(u,0)}u^{n}\ \leq\ \int_{\overline{X}(\overline{u},0)}\overline{u}^{n}
$$
$$
\leq\ \int_{\overline{X}}\beta^{n}(1+C_{3}(\mathrm{Vol}(X, L)-\beta^{n})^{1/2})
$$
$$
\leq\ \mathrm{Vol}(X, L)+C_{4}(\mathrm{Vol}(X, L)-\beta^{n})^{1/2}
$$
As $\beta^{n}$ approches $\mathrm{Vol}(X, L)$, this implies inequality (3.1).
We still have to treat the case when $L$ is not big, i.e. $\mathrm{Vol}(X, L)=0$. Let
$A$ be an ample line bundle and let $t_{0}\geq 0$ be the infmum of real numbers
such that $L+tA$ is a big $\mathbb{Q}$-line bundle for $t$ rational, $t>t_{0}$. The continuity
of the volume function implies that $0<\mathrm{Vol}(X, L+tA)\leq e$ for $t>t_{0}$
sufficiently close to $t_{0}$. By what we have just proved, there exists a smooth
form $u_{t}\in c_{1}(L+tA)$ such that $\displaystyle \int_{X(u_{t},0)}u_{t}^{n}\leq 2\in$. Take a K\"{a}hler metric
$\omega \in c_{1}(A)$ and define $ u=u_{t}-t\omega$. Then clearly
$$
\int_{X(u,0)}u^{n}\leq\int_{X(u_{t},0)}u_{t}^{n}\leq 2\in,
$$
hence
$$
\inf_{u\in c_{1}(L)}\int_{X(u,0)}u^{n}=0.
$$
Inequality (3.1) is now proved in all cases.
$$
\square 
$$
4. Estimate of the first cohomology group on a projective surface
We start with a projective non singular variety $X$ of arbitrary dimension
$n$, and will later restrict ourselves to the case when $X$ is a surface. The
proof again consists of using (approximate) Zariski decomposition, but now
we try to compute more explicitly the resulting curvature forms and Morse
integrals; this will turn out to be much easier on surfaces.
$$
-130-
$$
A converse to the Andreotti-Grauert theorem
Assume frst that $L$ is a {\it big} line bundle on $X$. As in section 3, we can
ind an approximate Zariski decomposition, i.e. a blow-up $\mu$ : $X\rightarrow X$ and
a current $T\in c_{1}(L)$ such $\mu^{*}T=[E]+\beta$, where $E$ an effective divisor and
$\beta$ a K\"{a}hler metric on $\overline{X}$ such that
\begin{center}
$\mathrm{Vol}(X, L)-\eta<\beta^{n}<\mathrm{Vol}(X, L),\ \eta\ll 1$.   (4.1)
\end{center}
(On a projective surface, one can even get exact Zariski decomposition, but
we want to remain general as long as possible). By blowing-up further, we
may even assume that $E$ is a normal crossing divisor. We select a hermitian
metric $h$ on $\mathcal{O}(E)$ and take
\begin{center}
$ u_{\in}=\displaystyle \frac{i}{2\pi}\partial\overline{\partial}\log(|\sigma_{E}|_{h}^{2}+\in^{2})+\Theta_{\mathcal{O}(E),h}+\beta \in \mu^{*}c_{1}(L)$   (4.2)
\end{center}
where $\sigma_{E}\in H^{0}(\overline{X}, \mathcal{O}(E))$ is the canonical section and $\Theta_{\mathcal{O}(E),h}$ the Chern
curvature form. Clearly, by the Lelong-Poincare' equation, $u_{\in}$ converges to
$[E]+\beta$ in the weak topology as $\xi j \rightarrow 0$. Straightforward calculations yield
$$
 u_{\in}=\frac{i}{2\pi}\frac{\in^{2}D_{h}^{1,0}\sigma_{E}\wedge\overline{D_{h}^{1,0}\sigma_{E}}}{(\epsilon^{2}+|\sigma_{E}|^{2})^{2}}+\overline{\epsilon^{2}+|\sigma_{E}|^{2^{\Theta_{E,h}}}}\in^{2}+\beta.
$$
The frst term converges to $[E]$ in the weak topology, while the $\mathrm{seco}\underline{\mathrm{n}}\mathrm{d}$, which
is close to $\Theta_{E,h}$ near $E$, converges pointwise everywhere to $0$ on $X\backslash E$. A
simple asymptotic analysis shows that
$$
(\frac{i}{2\pi}\frac{\in^{2}D_{h}^{1,0}\sigma_{E}\wedge\overline{D_{h}^{1,0}\sigma_{E}}}{(\epsilon^{2}+|\sigma_{E}|^{2})^{2}}+\overline{\epsilon^{2}+|\sigma_{E}|^{2^{\Theta_{E,h})^{p}}}}\in^{2}\rightarrow[E]\wedge\Theta_{E,h}^{p-1}
$$
in the weak topology for $p\geq 1$, hence
\begin{center}
$\displaystyle \lim_{\in\rightarrow 0}u_{\in}^{n}=\beta^{n}+\sum_{p=1}^{n} \left(\begin{array}{l}
n\\
p
\end{array}\right) [E]\wedge\Theta_{E,h}^{p-1}\wedge\beta^{n-p}$.   (4.3)
\end{center}
In arbitrary dimension, the signature of $u_{\in}$ is hard to evaluate, and it is also
non trivial to decide the sign of the limiting measure lim $u_{\in}^{n}$. However, when
$n=2$, we get the simpler formula
$$
\lim_{\in\rightarrow 0}u_{\in}^{2}=\beta^{2}+2[E]\wedge\beta+[E]\wedge\Theta_{E,h}.
$$
In this case, $E$ can be assumed to be an exceptional divisor (otherwise
some part of it would be nef and could be removed from the poles of $T$).
Hence the matrix $(E_{j}\cdot E_{k})$ is negative defnite and we can fnd a smooth
hermitian metric $h$ on $\mathcal{O}(E)$ such that $(\Theta_{E,h})_{|E}<0$, i.e. $\Theta_{E,h}$ has one
negative eigenvalue everywhere along $E$.
$$
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$$
Jean-Pierre Demailly
Lemma 4.1. -- {\it One can adjust the metric} $h$ {\it of} $\mathcal{O}(E)$ {\it in such a way}
{\it that} $\Theta_{E,h}$ {\it is negative definite on a neighborhood of the support} $|E|$ {\it of the}
{\it exceptional divisor, and} $\Theta_{E,h}+\beta$ {\it has signature} (1, 1) {\it there}. ({\it We do not care}
{\it about the signature far away from} $|E|)$.
{\it Proof}--At a given point $x_{0}\in X$, let us fx coordinates and a positive
quadratic form $q$ on $\mathbb{C}^{2}$. If we put $\psi_{\in}(z)=\xi:\chi(z)\log(1+\in^{-1}q(z))$ with a
suitable cut-off function $\chi$, then the Hessian form of $\psi_{\in}$ is equal to $q$ at $x_{0}$ and
decays rapidly to $O(\xi:\log\in)|dz|^{2}$ away from $x_{0}$. In this way, after multiplying
$h$ with $e^{\pm\psi_{\Xi}(z)}$, we can replace the curvature $\Theta_{E,h}(x_{0})$ with $\Theta_{E,h}(x_{0})\pm$
$q$ without substantially modifying the form away from $x_{0}$. This allows to
adjust $\Theta_{E,h}$ to be equal to (say) $-\displaystyle \frac{1}{4}\beta(x_{0})$ at any singular point $x_{0}\in E_{j}\cap E_{k}$
in the support of $|E|$, while keeping $\Theta_{E,h}$ negative defnite along $E$. In
order to adjust the curvature at smooth points $x\in|E|$, we replace the
metric $h$ with $h'(z)=h(z)\exp(-c(z)|\sigma_{E}(z)|^{2})$. Then the curvature form
$\Theta_{E,h}$ is replaced by $\Theta_{E,h'}(x)=\Theta_{E_{h}}(x)+c(x)|d\sigma_{E}|^{2}$ at $x\in|E|$ (notice that
$d\sigma_{E}(x)=0$ if $ x\in$ sing $|E|)$, and we can always select a real function $c$ so
that $\Theta_{E,h'}$ is negative defnite with one negative eigenvalue between $-1/2$
and $0$ at any point of $|E|$. Then $\Theta_{E,h'}+\beta$ has signature (1, 1) near $|E|$.
$$
\square 
$$
With this choice of the metric, we see that for $\xi j >0$ small, the sum
$$
\overline{\epsilon^{2}+|\sigma_{E}|^{2^{\Theta_{E,h}}}}\in^{2}+\beta
$$
is of signature $($2, $0)$ or (1, 1) (or degenerate of signature (1, $0$)), the non posi-
tive defnite points being concentrated in a neighborhood of $E$. In particular
the index set $X(u_{\in}, 2)$ is empty, and also
$$
 u_{\in}\leq\frac{i}{2\pi}\frac{\in^{2}D_{h}^{1,0}\sigma_{E}\wedge\overline{D_{h}^{1,0}\sigma_{E}}}{(\epsilon^{2}+|\sigma_{E}|^{2})^{2}}+\beta
$$
on a neighborhood $V$ of $|E|$, while $u_{\in}$ converges uniformly to $\beta$ on $\overline{X}\backslash V$.
This implies that
$$
\beta^{2}\leq\lim_{\in\rightarrow}\inf_{0}\int_{X(u_{\Xi},0)}u_{\in}^{2}\leq\lim_{\in\rightarrow}\sup_{0}\int_{X(u_{\Xi},0)}u_{\in}^{2}\leq\beta^{2}+2\beta\cdot E.
$$
Since $\displaystyle \int_{\overline{X}}u_{\in}^{2}=L^{2}=\beta^{2}+2\beta\cdot E+E^{2}$ we conclude by taking the difference
that
$-E^{2}-2\displaystyle \beta\cdot E\leq\lim_{\in\rightarrow}\inf_{0}\int_{X(u_{\Xi},1)}-u_{\in}^{2}\leq\lim_{\in\rightarrow}\sup_{0}\int_{X(u_{\Xi},1)}-u_{\in}^{2}\leq-E^{2}$.
Let us recall that $\beta\cdot E\leq C(\mathrm{Vol}(X, L)-\beta^{2})^{1/2}=0(\eta^{1/2})$ is small by (4.1)
and the orthogonality estimate. The asymptotic cohomology is given here
$$
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$$
A converse to the Andreotti-Grauert theorem
by $\hat{h}^{2}(X, L)=0$ since $h^{2}(X, L^{\otimes k})=H^{0}(X, K_{X}\otimes L^{\otimes-k})=0$ for $k\geq k_{0}$,
and we have by Riemann-Roch
$\hat{h}^{1}(X, L)=\hat{h}^{0}(X, L)-L^{2}=\mathrm{Vol}(X, L)-L^{2}=-E^{2}-\beta\cdot E+O(\eta)$.
Here we use the fact that $\displaystyle \frac{n!}{k^{n}}h^{0}(X, L^{\otimes k})$ converges to the volume when $L$
is big. All this shows that equality occurs in the Morse inequalities (1.3)
when we pass to the infmum. By taking limits in the Neron-Severi space
$\mathrm{NS}_{\mathbb{R}}(X)\subset H^{1,1}(X, \mathbb{R})$, we further see that equality occurs as soon as $L$
is pseudo-effective, and the same is true if $-L$ is pseudo-effective by Serre
duality.
It remains to treat the case when neither $L\mathrm{nor}-L$ are pseudo-effective.
Then $\hat{h}^{0}(X, L)=\hat{h}^{2}(X, L)=0$, and asymptotic cohomology appears only in
degree 1, with $\hat{h}^{1}(X, L)=-L^{2}$ by Riemann-Roch. Fix an ample line bundle
$A$ and let $t_{0}>0$ be the infimum of real numbers such that $L+tA$ is big for
$t$ rational, $t>t_{0}$, resp. let $t\text{\'{o}}>0$ be the infimum of real numbers $t'$ such
that $-L+t'A$ is big for $t' >${\it t}\'{o}. Then for $t>t_{0}$ and $t' >${\it t}\'{o}, we can find a
modification $\mu$ : $\overline{X}\rightarrow X$ and currents $T\in c_{1}(L+tA),\ T'\in c_{1}(-L+t'A)$
such that
$$
\mu^{*}T=[E]+\beta,\ \mu^{*}T'=[F]+\gamma
$$
where $\beta,\ \gamma$ are K\"{a}hler forms and $E,\ F$ normal crossing divisors. By taking
a suitable linear combination $t'(L+tA)-t(-L+t'A)$ the ample divisor $A$
disappears, and we get
$$
\frac{1}{t+t}(t'[E]+t'\beta-t[F]-t\gamma)\in\mu^{*}c_{1}(L).
$$
After replacing $E,\ F,\ \beta,\ \gamma$ by suitable multiples, we obtain an equality
$$
[E]-[F]+\beta-\gamma\in\mu^{*}c_{1}(L).
$$
We may further assume by subtracting that the divisors $E,\ F$ have no
common components. The construction shows that $\beta^{2}\leq \mathrm{Vol}(X, L+tA)$ can
be taken arbitrarily small (as well of course as $\gamma^{2}$), and the orthogonality
estimate implies that we can assume $\beta\cdot E$ and $\gamma\cdot F$ to be arbitrarily small.
Let us introduce metrics $h_{E}$ on $\mathcal{O}(E)$ and $h_{F}$ on $\mathcal{O}(F)$ as in Lemma 4.4,
and consider the forms
$u_{\in}=+\displaystyle \frac{i}{2\pi}\frac{\xi:^{2}D_{h_{E}}^{1,0}\sigma_{E}\wedge,\overline{D_{h_{E}}^{1,0}\sigma_{E}}}{\xi:^{2}D_{h_{F}}^{1,0}\sigma_{F}\wedge^{\frac{1^{2})^{2}}{D_{h_{F}}^{1,0}\sigma_{F}1^{2})^{2}}},(\xi:^{2}+|\sigma_{F}(\xi:^{2}+|\sigma_{E}}+_{\overline{\epsilon^{2}+|\sigma_{E}|^{2}}}\Theta_{E,h_{E}}+\beta-\frac{i}{2\pi}\frac{}{}-\Theta_{F,h_{F}}-\gamma\overline{\epsilon^{2}+|\sigma_{F}|^{2}}\in^{2}\in^{2} \in \mu^{*}c_{1}(L)$.
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Jean-Pierre Demailly
Observe that $u_{\in}$ converges uniformly to $\beta-\gamma$ outside of every neighborhood
of $|E|\cup|F|$. Assume that $\Theta_{E,h_{E}}<0$ on $V_{E}=\{|\sigma_{E}|<\in 0\}$ and $\Theta_{F,h_{F}}<0$
on $V_{F}=\{|\sigma_{F}|<\in 0\}$. On $V_{E}\cup V_{F}$ we have
$u_{\in}\displaystyle \leq\frac{i}{2\pi}\frac{\xi:^{2}D_{h_{E}}^{1,0}\sigma_{E}\wedge\overline{D_{h_{E}}^{1,0}\sigma_{E}}}{(\xi:^{2}+|\sigma_{E}|^{2})^{2}}-\Theta_{F,h_{F}}\overline{\epsilon^{2}+|\sigma_{F}|^{2}}\in^{2}+\beta+\overline{\epsilon_{0}^{2}}\in^{2_{\Theta_{E,h_{E}}^{+}}}$
where $\Theta_{E,h_{E}}^{+}$ is the positive part of $\Theta_{E,h_{E}}$ with respect to $\beta$. One sees im-
mediately that this term is negligible. The frst term is the only one which
is not uniformly bounded, and actually it converges weakly to the current
$[E]$. By squaring, we fnd
$$
\lim_{\in\rightarrow}\sup_{0}\int_{X(u_{\Xi},0)}u_{\in}^{2}\leq\int_{X(\beta-\gamma,0)}(\beta-\gamma)^{2}+2\beta\cdot E.
$$
Notice that the $\displaystyle \mathrm{term}-\frac{\in^{2}}{\in^{2}+|\sigma_{F}|^{2}}\Theta_{F,h_{F}}$ does not contribute to the limit as it
converges boundedly almost everywhere to $0$, the exceptions being points
of $|F|$, but this set is of measure zero with respect to the current $[E]$. Clearly
we have $\displaystyle \int_{X(\beta-\gamma,0)}(\beta-\gamma)^{2}\leq\beta^{2}$ and therefore
$$
\lim_{\in\rightarrow}\sup_{0}\int_{X(u_{\Xi},0)}u_{\in}^{2}\leq\beta^{2}+2\beta\cdot E.
$$
Similarly, by looking at $-u_{\in}$, we fnd
$$
\lim_{\in\rightarrow}\sup_{0}\int_{X(u_{\Xi},2)}u_{\in}^{2}\leq\gamma^{2}+2\gamma\cdot F.
$$
These $\displaystyle \lim\sup$'s are small and we conclude that the essential part of the
mass is concentrated on the 1-index set, as desired.
$$
\square 
$$
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\end{document}