A non-Archimedean theory of complex spaces and the cscK problem

TL;DR

This paper introduces a non-Archimedean framework for complex spaces and uses it to extend K-stability results, establishing a link to the existence of constant scalar curvature K"ahler metrics on compact K"ahler manifolds.

Contribution

It develops a non-Archimedean theory for complex spaces and generalizes K-stability results to broader classes of K"ahler manifolds.

Findings

A new non-Archimedean theory for complex spaces

Generalization of K-stability implications for cscK metrics

Proof of existence and uniqueness of cscK metrics under stronger K-stability

Abstract

In this paper we develop an analogue of the Berkovich analytification for non-necessarily algebraic complex spaces. We apply this theory to generalize to arbitrary compact K\"ahler manifolds a result of Chi Li, proving that a stronger version of K-stability implies the existence of a unique constant scalar curvature K\"ahler metric.