Strong cohomological rigidity of toric varieties

TL;DR

This paper proves that for certain toric varieties and quasitoric manifolds with second Betti number 2, any isomorphism of their cohomology rings corresponds to an actual geometric equivalence.

Contribution

It establishes strong cohomological rigidity results, showing cohomology ring isomorphisms imply diffeomorphisms or homeomorphisms for these manifolds.

Findings

Cohomology ring isomorphisms are realizable by diffeomorphisms for toric varieties.

Cohomology ring isomorphisms are realizable by homeomorphisms for quasitoric manifolds.

Results apply specifically to manifolds with second Betti number 2.

Abstract

Every cohomology ring isomorphism between two non-singular complete toric varieties and quasitoric manifolds, respectively, with second Betti number $2$ is realizable by a diffeomorphism and homeomorphism, respectively.