论文标题
带有圆环动作的流形的刚性刚性：i
Cohomological rigidity of manifolds with torus actions: I
论文作者
论文摘要
我们研究了两个具有圆环动作的歧管家族的共同僵化问题：所谓的矩角歧管，其研究与组合几何形状和组合交换代数有关；和拓扑曲折的歧管，它们是曲折品种的拓扑概括。在本文中，我们证明，当一个简单的球体满足某些组合条件时，相应的矩角歧管和拓扑曲折的歧管在同一个学上是僵化的，即他们自己家庭中的同态类别的类别由他们的共同体学确定。在曲曲面的情况下，共同体甚至决定了品种的同构类别。我们的主要策略是表明这些简单球的组合类型由$ \ mathrm {tor} $ - 其脸环的代数确定。事实证明，这是解决一类球的组合交换代数中已知问题的解决方案。
We study the cohomological rigidity problem of two families of manifolds with torus actions: the so-called moment-angle manifolds, whose study is linked with combinatorial geometry and combinatorial commutative algebra; and topological toric manifolds, which are topological generalizations of toric varieties. In this paper we prove that when a simplicial sphere satisfies certain combinatorial conditions, the corresponding moment-angle manifold and topological toric manifolds are cohomologically rigid, i.e. their homeomorphism classes in their own families are determined by their cohomology rings. In the case of toric varieties, cohomology even determine the isomorphism classes of varieties. Our main strategy is to show that the combinatorial types of these simplicial spheres are determined by the $\mathrm{Tor}$-algebras of their face rings. This turns out to be a solution to a known problem in combinatorial commutative algebra for a class of spheres.
