Regularity of Plurisubharmonic Upper Envelopes in Big Cohomology Classes
Paper i proceeding, 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.

dirichlet problem

Hermitian line bundle

Upper envelope

Singular metric



Plurisubharmonic function


Robert Berman

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

J. P. Demailly

Perspectives in Analysis, Geometry, and Topology: On the Occasion of the 60th Birthday of Oleg Viro. Marcus Wallenberg Symposium on Perspectives in Analysis, Geometry and Topology. Stockholm, Sweden, May 19-25, 2008

0743-1643 (ISSN)

Vol. 296 39-66







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