A lower bound on the Calabi functional for a degeneration of polarized varieties

TL;DR

This paper establishes a lower bound on the Calabi functional for degenerations of polarized varieties using non-Archimedean geometry and GIT height, generalizing Donaldson's earlier results.

Contribution

It introduces a new lower bound involving CM degrees, extending Donaldson's work to degenerations and developing GIT height theory for this purpose.

Findings

Derived a lower bound on the Calabi functional for degenerations.

Applied GIT height to prove separatedness of GIT quotients.

Generalized Donaldson's result from a single variety to degenerations.

Abstract

We prove a lower bound on the Calabi functional for degenerations of polarized varieties, involving the difference of CM degrees between generically isomorphic families. This may be viewed as a discretely valued version of Donaldson's lower bound for models, in the sense of non-Archimedean geometry. In particular, this generalizes a result of Donaldson, who considered a single polarized variety. As a main tool, we develop the theory of GIT height, introduced by Wang, and apply it to the family GIT problem of the Chow variety. Using the GIT height, we also give a numerical proof of separatedness of GIT quotients for general and special linear actions, strengthening prior work of Wang--Xu.