The goal of this paper is to show that there are strong relations between certain Monge-Ampère integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers 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...

Let $E$ be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form $\mathrm{id}+K$ where $K$ is compact. Assume that the characteristic fiber of $E$ has the compact approximation property. Let $n$ be the complex dimension of $X$ and $0\le q\le n$. Then: If $V\to X$ is a holomorphic vector bundle (of finite rank) with $H^q(X,V)=0$, then $dim{H}^q(X,V\otimes E)\<\infty$. In particular, if $dim{H}^q(X,𝒪)=0$, then $dim{H}^q(X,E)\<\infty$.

Let $X$ be a Banach space with a countable unconditional basis (e.g., $X={\ell }_2$), $\Omega \subset X$ an open set and $f_1,...,f_k$ complex-valued holomorphic functions on $\Omega$, such that the Fréchet differentials $d{f}_1\left(x\right),...,d{f}_k\left(x\right)$ are linearly independant over $ℂ$ at each $x\in \Omega$. We suppose that $M=\{x\in \Omega :f_1\left(x\right)=...=f_k\left(x\right)=0\}$ is a complete intersection and we consider a holomorphic Banach vector bundle $E\to M$. If $I$ (resp.${𝒪}^E$) denote the ideal of germs of holomorphic functions on $\Omega$ that vanish on $M$ (resp. the sheaf of germs of holomorphic sections of $E$), then the sheaf cohomology groups $H^q(\Omega ,I)$, $H^q(M,{𝒪}^E)$ vanish...

In this paper we obtain the full asymptotic expansion of the Bergman-Hodge kernel associated to a high power of a holomorphic line bundle with non-degenerate curvature. We also explore some relations with asymptotic holomorphic sections on symplectic manifolds.