Equivariant localisation in the theory of $Z$-stability for K\"ahler manifolds

Abstract

We apply equivariant localisation to the theory of $Z$-stability and $Z$-critical metrics on a K\"ahler manifold $(X,\alpha)$, where $\alpha$ is a K\"ahler class. We show that the invariants used to determine $Z$-stability of the manifold, which are integrals over test configurations, can be written as a product of equivariant classes, hence equivariant localisation can be applied. We also study the existence of $Z$-critical K\"ahler metrics in $\alpha$, whose existence is conjectured to be equivalent to $Z$-stability of $(X,\alpha)$. In particular, we study a class of invariants that give an obstruction to the existence of such metrics. Then we show that these invariants can also be written as a product of equivariant classes. From this we give a new, more direct proof of an existing result: the former invariants determining $Z$-stability on a test configuration are equal to the latter invariants related to the existence of $Z$-critical metrics on the central fibre of the test configuration. This provides a new approach from which to derive the $Z$-critical equation.