Valuative stability of polarised varieties

TL;DR

This paper introduces a new notion of valuative stability for polarized varieties, extending K-stability concepts, and demonstrates its equivalence to K-stability with specific test configurations, supported by examples and invariant analysis.

Contribution

It defines valuative stability for polarized varieties, linking it to K-stability and expanding the theoretical framework with new invariants and examples.

Findings

Valuative stability is equivalent to K-stability for certain test configurations.

Examples include stable and unstable toric varieties.

The delta invariant's role in stability is discussed.

Abstract

Fujita and Li have given a characterisation of K-stability of a Fano variety in terms of quantities associated to valuations, which has been essential to all recent progress in the area. We introduce a notion of valuative stability for arbitrary polarised varieties, and show that it is equivalent to K-stability with respect to test configurations with integral central fibre. The numerical invariant governing valuative stability is modelled on Fujita's beta invariant, but includes a term involving the derivative of the volume. We give several examples of valuatively stable and unstable varieties, including the toric case. We also discuss the role that the delta invariant plays in the study of valuative stability and K-stability of polarised varieties.