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.

我们将等变局域化应用于理论$Z$-稳定性和$Z$-凯勒流形的关键指标$（X,\alpha ）$， 在哪里$\alpha$是一个凯勒类。我们证明了用于确定的不变量$Z$-流形的稳定性是测试配置上的积分，可以写成等变类的乘积，因此可以应用等变局部化。我们还研究存在$Z$-关键的凯勒指标$\alpha$，其存在被推测等价于$Z$-稳定性$（X,\alpha ）$。特别是，我们研究了一类阻碍此类指标存在的不变量。然后我们证明这些不变量也可以写成等变类的乘积。由此，我们对现有结果给出了一个新的、更直接的证明：前面的不变量决定了$Z$- 测试配置的稳定性等于后面与存在相关的不变量$Z$- 测试配置的中心光纤的关键指标。这提供了一种新的方法来导出$Z$-临界方程。