Regularity of Plurisubharmonic Upper Envelopes in Big Cohomology Classes

Abstract

AbstractThe 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–Ampère operator; applications to geodesics in the space of Kähler 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.KeywordsPlurisubharmonic functionUpper envelopeHermitian line bundleSingularmetricLogarithmic polesLegendre–Kiselman transformPseudoeffective coneVolumeMonge–Ampère measureSupercanonical metricOhsawa–Takegoshi theorem