Quantization in geometric pluripotential theory

TL;DR

This paper establishes a connection between finite-dimensional algebraic metrics and infinite-dimensional pluripotential spaces in K"ahler geometry through quantization, enabling new insights and approximation techniques.

Contribution

It demonstrates that Finsler structures on K"ahler potentials can be quantized, linking algebraic and pluripotential geometric frameworks.

Findings

Path length metric spaces recover metric completions of K"ahler potentials.

Introduces a new approach to rooftop envelopes and Pythagorean formulas.

Provides approximation of finite energy potentials and geodesic segments by algebraic objects.

Abstract

The space of K\"ahler metrics can, on the one hand, be approximated by subspaces of algebraic metrics, while, on the other hand, can be enlarged to finite-energy spaces arising in pluripotential theory. The latter spaces are realized as metric completions of Finsler structures on the space of K\"ahler metrics. The former spaces are the finite-dimensional spaces of Fubini--Study metrics of K\"ahler quantization. The goal of this article is to draw a connection between the two. We show that the Finsler structures on the space of K\"ahler potentials can be quantized. More precisely, given a K\"ahler manifold polarized by an ample line bundle we endow the space of Hermitian metrics on powers of that line bundle with Finsler structures and show that the resulting path length metric spaces recover the corresponding metric completions of the Finsler structures on the space of K\"ahler potentials. This has a number of applications, among them a new approach to the rooftop envelopes and Pythagorean formulas of K\"ahler geometry, a new Lidskii type inequality on the space of K\"ahler metrics, and approximation of finite energy potentials, as well as geodesic segments by the corresponding smooth algebraic objects.