论文标题
$ z $稳定性的Kähler歧管理论中的本地化
Equivariant localisation in the theory of $Z$-stability for Kähler manifolds
论文作者
论文摘要
我们将均等本地化应用于$ z $稳定性的理论和$ z $ - 批评指标上的kähler歧管$（x，α）$，其中$α$是kähler类。我们表明，用于确定歧管的$ z $稳定性的不变性，即在测试配置上是积分，可以写成均等类的产物，因此可以应用equivariant本地化。我们还研究了$ Z $ to $α$中的$ z $ - calliticalkähler指标的存在，其存在的存在与$ z $ - 稳定性相当于$（x，α）$。特别是，我们研究了一类不变的人，这些不变性会阻碍这种指标的存在。然后，我们证明这些不变性也可以写成等效类的产物。由此，我们提供了现有结果的新的，更直接的证明：确定测试配置上的$ z $稳定性的前不变性等于与测试配置中央光纤上存在$ z $ - 临界指标有关的后一个不变性。这提供了一种新的方法，可以从中得出$ z $ - 临界方程。
We apply equivariant localisation to the theory of $Z$-stability and $Z$-critical metrics on a Kähler manifold $(X,α)$, where $α$ 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 localisation can be applied. We also study the existence of $Z$-critical Kähler metrics in $α$, whose existence is conjectured to be equivalent to $Z$-stability of $(X,α)$. 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.
