Regularity of plurisubharmonic upper envelopes in big cohomology classes

Abstract

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.

27 pages, no figures

References in corpus (6)

• Monge-Ampère equations in big cohomology classes
• The Dirichlet problem for degenerate complex Monge-Ampere equations
• Growth of balls of holomorphic sections and energy at equilibrium
• Canonical singular hermitian metrics on relative logcanonical bundles
• Bergman kernels and equilibrium measures for line bundles over projective manifolds
• Canonical volume forms on compact Kähler manifolds

Cited by in corpus (5)

• A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem
• A variational approach to complex Monge-Ampere equations
• A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry
• Hölder continuous solutions of the Monge-Ampère equation on compact Hermitian manifolds
• Restricted Bergman kernel asymptotics