From Monge-Ampere equations to envelopes and geodesic rays in the zero temperature limit

摘要

Let (X, theta) be a compact complex manifold X equipped with a smooth (but not necessarily positive) closed (1, 1)-form theta. By a well-known envelope construction this data determines, in the case when the cohomology class [theta] is pseudoeffective, a canonical theta-psh function u(theta). When the class [theta] is Kahler we introduce a family u(beta) of regularizations of u(theta), parametrized by a large positive number beta, where u(beta) is defined as the unique smooth solution of a complex Monge-Ampere equation of Aubin-Yau type. It is shown that, as beta -> infinity, the functions u(beta) converge to the envelope u(theta) uniformly on X in the Holder space C-1 alpha(X) for any alpha is an element of]0, 1[(which is optimal in terms of Holder exponents). A generalization of this result to the case of a nef and big cohomology class is also obtained and a weaker version of the result is obtained for big cohomology classes. The proofs of the convergence results do not assume any a priori regularity of u(theta). Applications to the regularization of omega-psh functions and geodesic rays in the closure of the space of Kahler metrics are given. As briefly explained there is a statistical mechanical motivation for this regularization procedure, where beta appears as the inverse temperature. This point of view also leads to an interpretation of u(beta) as a "transcendental" Bergman metric.