Regularity of plurisubharmonic upper envelopes in big cohomology classes

Understand

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.