Large Deviations for Gibbs Measures with Singular Hamiltonians and Emergence of Kähler–Einstein Metrics

Abstract

Abstract:In the present paper and the companion paper (Berman, Kähler-Einstein metrics, canonical random point processes and birational geometry. arXiv:1307.3634, 2015) a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on complex algebraic varieties X is introduced by sampling "temperature deformed" determinantal point processes. The main new ingredient is a large deviation principle for Gibbs measures with singular Hamiltonians, which is proved in the present paper. As an application we show that the unique Kähler-Einstein metric with negative Ricci curvature on a canonically polarized algebraic manifold X emerges in the many particle limit of the canonical point processes on X. In the companion paper (Berman in 2015) the extension to algebraic varieties X with positive Kodaira dimension is given and a conjectural picture relating negative temperature states to the existence problem for Kähler-Einstein metrics with positive Ricci curvature is developed.

Statistical mechanics of interpolation nodes, pluripotential theory and complex geometry

“…Comparing with the one-dimensional situation it is tempting to think of Fekete points on a compact subset K of C n (and more generally, nearly optimal interpolation nodes) as interacting particles confined to K, forming a microscopic equilibrium state. In [18,21] this analogy is pushed further by introducing temperature (and thus randomness) into the picture. To explain this let us first recall the general statistical mechanical setup.…”

“…However, in the present setting E (N ) (as defined by formula 2.14) is both very non-linear (when n > 1) and singular. What saves the situation is that E (N ) is superharmonic so that the following key result in [18] can be applied:…”

“…The results in [27,19] apply to general big line bundles, but here we will only consider the simpler case when L is positive, i.e. the case of a polarized manifold (X, L) ( as in [15,18]). There is natural correspondence between the space H 0 (X, L) and the space of all 1-homogeneous holomorphic functions on the complex manifold defined by the total space of the dual line bundle L * → X, endowed with its standard C * −action.…”

“…See [18] for the proof (the existence of a recovery sequence is based on an application of a general approximation result of Brøndsted-Rockafellar, which applies in certain topological vector spaces).…”

“…The main purpose of the present work is to explain how the probabilistic (statistical mechanical) construction of Kähler-Einstein metrics on compact complex manifolds -which is the subject of the series of works [19,18,15,21] -naturally arises from classical approximation and interpolation problems in C n . In particular, the connection to Siciak's conjecture [104] (proved in [27]) about the convergence of Fekete points towards the pluripotential equilibrium measure is stressed.…”