Regularity of plurisubharmonic upper envelopes in big cohomology classes

Details Regularity of plurisubharmonic upper envelopes in big cohomology classes Regularity of plurisubharmonic upper envelopes in big cohomology classes

2009-05-08

arxiv.org/abs/0905.1246v1

Mathematics

Complex Variables

27 pages, no figures

Scientific paper

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.

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