Masuda–Suh cohomological rigidity for toric manifolds

Determine whether every pair of smooth toric varieties (toric manifolds) whose integral cohomology rings are isomorphic are diffeomorphic; that is, prove or refute the Masuda–Suh conjecture asserting diffeomorphism of toric manifolds from isomorphism of their integral cohomology rings.

Background

The paper situates its main results within broader questions about cohomological rigidity. For toric manifolds, Masuda and Suh formulated a non-equivariant cohomological rigidity conjecture asserting that isomorphic integral cohomology rings imply diffeomorphism. This conjecture has been verified in many special cases, but a general resolution is not stated.

In contrast, equivariant cohomological rigidity is known for compact toric manifolds (and quasitoric manifolds), where isomorphism of equivariant cohomology rings implies equivariant diffeomorphism. The present work proves strong equivariant cohomological rigidity for compact, connected, four-dimensional Hamiltonian S^1-manifolds, highlighting the open status of the non-equivariant toric manifold conjecture by Masuda and Suh.