181 research outputs found
A converse to the Andreotti-Grauert theorem
The goal of this paper is to show that there are strong relations between
certain Monge-Amp\`ere 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.Comment: 12 page
Regularity of plurisubharmonic upper envelopes in big cohomology classes
The goal of this work is to prove the regularity of certain
quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way
in the construction of hermitian metrics with minimal singularities on a big
line bundle over a compact complex manifold. We prove that the complex Hessian
forms of these envelopes are locally bounded outside an analytic set of
singularities. It is furthermore shown that a parametrized version of this
result yields a priori inequalities for the solution of the Dirichlet problem
for a degenerate Monge-Ampere operator; applications to geodesics in the space
of Kahler metrics are discussed. A similar technique provides a logarithmic
modulus of continuity for Tsuji's "supercanonical" metrics, which generalize a
well-known construction of Narasimhan-Simha.Comment: 27 pages, no figure
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