Stability conditions in geometric invariant theory

TL;DR

This paper develops an axiomatic framework linking geometric invariant theory and stability conditions, unifying concepts like K-stability and stability in complex geometry, and clarifies their analytic aspects.

Contribution

It introduces a new axiomatic notion of stability conditions on schemes and stacks, connecting GIT, K-stability, and complex geometric structures.

Findings

Unified axiomatic framework for stability conditions

Connection between GIT stability and K-stability

Analytic interpretation of stability concepts

Abstract

We explain how structures analogous to those appearing in the theory of stability conditions on abelian and triangulated categories arise in geometric invariant theory. This leads to an axiomatic notion of a central charge on a scheme with a group action, and ultimately to a notion of a stability condition on a stack analogous to that on an abelian category. We use these ideas to introduce an axiomatic notion of a stability condition for polarised schemes, defined in such a way that K-stability is a special case. In the setting of axiomatic geometric invariant theory on a smooth projective variety, we produce an analytic counterpart to stability and explain the role of the Kempf-Ness theorem. This clarifies many of the structures involved in the study of deformed Hermitian Yang-Mills connections, Z-critical connections and Z-critical K\"ahler metrics.