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

Abstract

We apply equivariant localization to the theory of [Formula: see text]-stability and [Formula: see text]-critical metrics on a Kähler manifold [Formula: see text], where [Formula: see text] is a Kähler class. We show that the invariants used to determine [Formula: see text]-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 [Formula: see text]-critical Kähler metrics in [Formula: see text], whose existence is conjectured to be equivalent to [Formula: see text]-stability of [Formula: see text]. 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 [Formula: see text]-stability on a test configuration are equal to the latter invariants related to the existence of [Formula: see text]-critical metrics on the central fibre of the test configuration. This provides a new approach from which to derive the [Formula: see text]-critical equation.