Strong cohomological rigidity of a product of projective spaces

TL;DR

This paper proves that for certain toric manifolds, any isomorphism of their cohomology rings with that of a product of complex projective spaces is realized by a diffeomorphism, demonstrating strong cohomological rigidity.

Contribution

It establishes that cohomology ring isomorphisms between these manifolds are always induced by diffeomorphisms, confirming a rigidity property for products of projective spaces.

Findings

Cohomology ring isomorphisms correspond to diffeomorphisms.

Strong cohomological rigidity holds for products of projective spaces.

The result applies to toric manifolds with specific cohomological structures.

Abstract

We prove that for a toric manifold $M$, any graded ring isomorphism $H^\ast(M) \to H^\ast(\prod_{i=1}^{m}\CP^{n_i})$ is induced by a diffeomorphism $\prod_{i=1}^m \CP^{n_i} \to M$.