[Paper Review] A converse to the Andreotti-Grauert theorem

Jean-Pierre Demailly|Nov 16, 2010

Geometry and complex manifolds9 references4 citations

TL;DR

This paper establishes a converse to the Andreotti-Grauert vanishing theorem by proving that for projective manifolds, the volume of a holomorphic line bundle equals the infimum of Monge-Ampère integrals over all smooth representatives of its first Chern class. The proof relies on approximate Zariski decomposition, orthogonality estimates, and curvature current constructions to show equality in holomorphic Morse inequalities for the volume functional, even when the line bundle is not pseudo-effective.

ABSTRACT

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 vanishing theorem.

Motivation & Objective

• To establish a converse to the Andreotti-Grauert vanishing theorem in the context of asymptotic cohomology and holomorphic Morse inequalities.
• To investigate whether asymptotic cohomology vanishing implies the existence of a hermitian metric with controlled curvature, particularly in degree q=0.
• To prove that the volume functional of a holomorphic line bundle on a projective manifold equals the infimum of Monge-Ampère integrals over all smooth representatives of its first Chern class.
• To extend the equality between asymptotic cohomology and Monge-Ampère integrals beyond the big case, including when neither L nor -L are pseudo-effective.
• To demonstrate that the weak and strong Morse inequalities become equalities in the volume case via approximation techniques and curvature current analysis.

Proposed method

• Utilizes approximate Zariski decomposition for Kähler currents on a blow-up of the manifold, expressing the pullback of a current as a sum of an exceptional divisor and a Kähler form.
• Applies the characterization of the pseudoeffective cone and orthogonality estimates to control intersection numbers between the Kähler form and the exceptional divisor.
• Constructs a family of smooth $(1,1)$-forms $u_\varepsilon$ in the first Chern class of $L$ by combining curvature terms from line bundles on exceptional divisors with Kähler forms.
• Employs a limiting argument to show that the $L^2$-norm of the $0$-index set of $u_\varepsilon$ converges to $\beta^2 + 2\beta \cdot E$, where $\beta$ is a Kähler form and $E$ is an exceptional divisor.
• Uses Serre duality and Riemann-Roch to handle cases where $L$ or $-L$ is not pseudo-effective, by combining positive and negative parts via linear combinations.
• Analyzes the weak limit of $u_\varepsilon^2$ and shows that the mass of the Monge-Ampère integral concentrates on the $1$-index set, implying the infimum is achieved.

Experimental results

Research questions

• RQ1 Does the vanishing of asymptotic $0$-cohomology imply the existence of a smooth representative in $c_1(L)$ with non-positive curvature?
• RQ2 Can the volume of a holomorphic line bundle on a projective manifold be expressed as the infimum of Monge-Ampère integrals over all smooth $(1,1)$-forms in $c_1(L)$?
• RQ3 Is equality achieved in the holomorphic Morse inequalities for the volume functional on projective surfaces and higher-dimensional projective manifolds?
• RQ4 What happens to the Morse inequality infimum when neither $L$ nor $-L$ are pseudo-effective, and can curvature current approximations still yield equality?
• RQ5 To what extent do approximate Zariski decompositions and orthogonality estimates allow control over the asymptotic cohomology in terms of curvature integrals?

Key findings

• For any holomorphic line bundle $L$ on a projective manifold $X$, the volume satisfies $\mathop{\rm Vol}(X,L) = \inf_{u \in c_1(L)} \int_{X(u,0)} u^n$, establishing equality in the weak Morse inequality for $q=0$.
• The equality holds even when $L$ is not pseudo-effective, by constructing a suitable limit of curvature currents $u_\varepsilon$ in $c_1(L)$ with controlled $0$-index sets.
• The limit of the Monge-Ampère integrals $\int_{X(u_\varepsilon,0)} u_\varepsilon^n$ converges to $\beta^2 + 2\beta \cdot E$, where $\beta$ is a Kähler form and $E$ is an exceptional divisor from Zariski decomposition.
• The construction shows that the essential mass of the Monge-Ampère measure is concentrated on the $1$-index set, confirming that the infimum is not achieved on the $0$-index set but through curvature approximation.
• The method extends to all projective manifolds, not just surfaces, and provides a partial converse to the Andreotti-Grauert vanishing theorem via curvature control.
• The result confirms that asymptotic cohomology estimates for the volume functional are precisely captured by Monge-Ampère integrals, with no loss in the infimum.