A first step towards Non-Archimedean Geometric Quantization

Calabi--Yau manifolds have long attracted interest from both mathematics and physics, particularly in the context of mirror symmetry, and form an important class of compact Kähler manifolds. A compact Kähler manifold is Calabi--Yau if and only if it admits a Ricci-flat Kähler metric, which we shall call a CY metric. Such a metric is highly analytic in nature, as it is given as the solution to a second-order PDE on the manifold, namely the complex Monge--Ampère equation.
When the Calabi--Yau manifold is a complex projective variety, one algebraic approach to understanding this analytically defined CY metric is to approximate it by algebraically defined metrics called balanced metrics. This framework was initiated by Donaldson and is now known as geometric quantization.
In this talk, following the spirit of this theory, we consider a non-Archimedean analogue of this approximation theory. More precisely, for a non-Archimedean analytic space associated with a maximally degenerating family of Calabi--Yau manifolds, we study the approximation of the NACY metric, a non-Archimedean analogue of the CY metric, by algebraically defined metrics. In particular, we introduce NA balanced metrics, which are expected to provide such an approximation, and explain that, for totally degenerating families of abelian varieties, NA balanced metrics indeed approximate the NACY metric.