Equivariant localization in the theory of Z-stability for Kähler manifolds

Abstract

We apply equivariant localization to the theory of $Z$-stability and $Z$-critical metrics on a Kähler manifold $(X,\alpha )$, where $\alpha$ is a Kähler 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 localization can be applied. We also study the existence of $Z$-critical Kähler 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.