Friday, July 12, 2013

1307.3008 (Robert J. Berman)

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

Robert J. Berman
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. More generally, a generalization of this result to the case of a nef and big cohomology class is obtained. Application to the regularization problem for geodesic rays in 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.
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