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Regularity of Plurisubharmonic Upper Envelopes in Big Cohomology Classes

Robert Berman (Institutionen för 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). Vol. 296 (2012), p. 39-66.
[Konferensbidrag, refereegranskat]

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.

Nyckelord: Plurisubharmonic function, Upper envelope, Hermitian line bundle, Singular metric, Logarithmic, dirichlet problem, equilibrium

Denna post skapades 2012-10-29.
CPL Pubid: 165171


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