Tits buildings and K-stability

TL;DR

This paper explores the geometric structure of test configurations for polarized varieties using Tits buildings, establishing continuity of the Donaldson-Futaki invariant and introducing a pseudo-metric to analyze K-stability.

Contribution

It models the space of test configurations as a limit of Tits buildings and connects admissible filtrations with Cauchy sequences in this space.

Findings

The space of test configurations can be described as a limit of Tits buildings.

The Donaldson-Futaki invariant is continuous on this space.

Admissible filtrations form Cauchy sequences with respect to a new pseudo-metric.

Abstract

A polarized variety is K-stable if, for any test configuration, the Donaldson-Futaki invariant is positive. In this paper, inspired by classical geometric invariant theory, we describe the space of test configurations as a limit of a direct system of Tits buildings. We show that the Donaldson-Futaki invariant, conveniently normalized, is a continuous function on this space. We also introduce a pseudo-metric on the space of test configurations. Recall that K-stability can be enhanced by requiring that the Donaldson-Futaki invariant is positive on any admissible filtration of the co-ordinate ring. We show that admissible filtrations give rise to Cauchy sequences of test configurations with respect to the above mentioned pseudo-metric.