Stability conditions for polarised varieties

TL;DR

This paper develops a new stability framework for polarised varieties, linking algebraic and analytic notions, and proves existence results for special K"ahler metrics under stability conditions.

Contribution

It introduces Z-stability as an analogue of Bridgeland stability for polarised varieties and establishes a connection with Z-critical K"ahler metrics.

Findings

Existence of Z-critical K"ahler metrics under K-semistability and Z-stability.

Introduction of a local wall crossing perspective.

Special case relating to deformed Hermitian Yang-Mills equation.

Abstract

We introduce an analogue of Bridgeland's stability conditions for polarised varieties. Much as Bridgeland stability is modelled on slope stability of coherent sheaves, our notion of Z-stability is modelled on the notion of K-stability of polarised varieties. We then introduce an analytic counterpart to stability, through the notion of a Z-critical K\"ahler metric, modelled on the constant scalar curvature K\"ahler condition. Our main result shows that a polarised variety which is analytically K-semistable and asymptotically Z-stable admits Z-critical K\"ahler metrics in the large volume regime. We also prove a local converse, and explain how these results can be viewed in terms of local wall crossing. A special case of our framework gives a manifold analogue of the deformed Hermitian Yang-Mills equation.