The Mabuchi Geometry of Finite Energy Classes

TL;DR

This paper develops a geometric framework for finite energy classes of Kähler potentials using Finsler metrics, with applications to Kähler-Einstein metrics on Fano manifolds.

Contribution

It introduces new Finsler metrics on Kähler potential spaces and characterizes their topology, advancing understanding of geometric structures in complex geometry.

Findings

Characterization of topology via energy convergence

Introduction of Finsler metrics on energy classes

Applications to Kähler-Einstein metrics on Fano manifolds

Abstract

We introduce different Finsler metrics on the space of smooth K\"ahler potentials that will induce a natural geometry on various finite energy classes $\mathcal E_{\tilde \chi}(X,\omega)$. Motivated by questions raised by R. Berman, V. Guedj and Y. Rubinstein, we characterize the underlying topology of these spaces in terms of convergence in energy and give applications of our results to existence of K\"ahler-Einstein metrics on Fano manifolds.