Moscow Mathematical Journal

Volume 24, Issue 4, October–December 2024 pp. 529–585.

Cohomological Rigidity of Manifolds with Torus Actions: I

Authors: Feifei Fan (1), Jun Ma (2), and Xiangjun Wang (3)
Author institution:(1) School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China
(2) College of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China
(3) School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Summary:

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.

2020 Math. Subj. Class. 57R19, 57R91, 57S25, 52B10, 14M25, 13F55.

Keywords: Cohomological rigidity, moment-angle manifolds, topological toric manifolds, Gorenstein* complexes, Stanley–Reisner rings.

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