Cohomological rigidity of manifolds with torus actions: I

Abstract

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.