STRONG COHOMOLOGICAL RIGIDITY OF A PRODUCT OF PROJECTIVE SPACES

Abstract

Abstract. We prove that for a toric manifold (respectively, a quasitoricmanifold) M, any graded ring isomorphism H ∗ (M) → H ∗ (Q mi=1 CP n i )can be realized by a diffeomorphism (respectively, a homeomoQrphism) mi=1 CP n i →M. 1. IntroductionThe cohomological rigidity problem for toric manifolds asks whether the inte-gral cohomology ring of a toric manifold determines its topological type or not.So far, there is no negative answer to the question but some positive results.In [2], the authors with M. Masuda show that if Mis a toric manifold whosecohomology ring is isomorphic to that ofQ mi=1 CP n i , a product of complexprojective spaces, then Mis actually diffeomorphic toQ mi=1 CP n i , which givesa positive result to the cohomological rigidity problem.On the other hand, one might aska strongerquestion as follows. Throughoutthis paper, H ∗ (X) denotes the integral cohomology ring of a topological spaceX.Problem 1.1. Let M and N be toric manifolds, and ϕ: H ∗ (N) → H ∗ (M)a graded ring isomorphism. Then, does there exist a homeomorphism or adiffeomorphism f: M→ Nsuch that f