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

TL;DR

This paper introduces a regularization method for canonical theta-psh functions on complex manifolds using complex Monge-Ampere equations, proving convergence as a parameter tends to infinity, with applications in Kähler geometry and statistical mechanics.

Contribution

It develops a new regularization approach for envelope functions via Monge-Ampere equations, extending to nef and big classes, and provides PDE proofs for envelope regularity.

Findings

Regularizations converge to the envelope in the strongest Holder sense.

Extension of results to nef and big cohomology classes.

Applications to regularization of quasi-psh functions and geodesic rays.

Abstract

Let X be a compact complex manifold equipped with a smooth (but not necessarily positive) closed form theta of one-one type. By a well-known envelope construction this data determines a canonical theta-psh function u which is not two times differentiable, in general. We introduce a family of regularizations of u, parametrized by a positive number beta, defined as the smooth solutions of complex Monge-Ampere equations of Aubin-Yau type. It is shown that, as beta tends to infinity, the regularizations converge to the envelope u in the strongest possible Holder sense. A generalization of this result to the case of a nef and big cohomology class is also obtained. As a consequence new PDE proofs are obtained for the regularity results for envelopes in [14] (which, however, are weaker than the results in [14] in the case of a non-nef big class). Applications to the regularization problem for quasi-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 the regularizations as transcendental Bergman metrics.