A converse to the Andreotti-Grauert theorem

Jean-Pierre Demailly (1)

1 IF - Institut Fourier

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.

Mots clés: Holomorphic Morse inequalities; Monge-Ampère integrals; asymptotic cohomology groups; Riemann-Roch formula; hermitian metrics; Chern curvature tensor; approximate Zariski decomposition

Domaines: Géométrie algébrique [math.AG]

Jean-Pierre Demailly. A converse to the Andreotti-Grauert theorem. 2010. ⟨hal-00536317v2⟩