Uniform stability of twisted constant scalar curvature K\"ahler metrics

TL;DR

This paper introduces a new norm on test configurations and explores its relation to uniform K-stability and the existence of constant scalar curvature K"ahler metrics, providing algebraic and geometric insights.

Contribution

It defines the minimum norm on test configurations and establishes its equivalence with triviality, linking uniform K-stability to the existence of twisted constant scalar curvature K"ahler metrics.

Findings

Uniform K-stability with respect to the minimum norm implies existence of twisted cscK metrics.

Algebro-geometric proofs of uniform K-stability in general type, Calabi-Yau, and Fano cases.

Log K-stability implies twisted K-stability and mild singularities.

Abstract

We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature K\"ahler metric. This notion of uniform K-stability is analogous to coercivity of the Mabuchi functional. We characterise the triviality of test configurations, by showing that a test configuration has zero minimum norm if and only if it has zero $L^2$-norm, if and only if it is almost trivial. We prove that the existence of a twisted constant scalar curvature K\"ahler metric implies uniform twisted K-stability with respect to the minimum norm, when the twisting is ample. We give algebro-geometric proofs of uniform K-stability in the general type and Calabi-Yau cases, as well as in the Fano case under an alpha invariant condition. Our results hold for line bundles sufficiently close to the (anti)-canonical line bundle, and also in the twisted setting. We show that log K-stability implies twisted K-stability, and also that twisted K-semistability of a variety implies that the variety has mild singularities.