Regularity of Plurisubharmonic Upper Envelopes in Big Cohomology Classes

Kapitel i bok, 2012

The goal of this work is to prove the regularity of certain quasiplurisubharmonic 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, that generalize a well-known construction of Narasimhan and Simha.