Projective bundles over toric surfaces (Q2807995)

Let \(E\) be a Whitney sum of complex line bundles over a smooth manifold \(X\). If \(X\) is a nonsingular complete toric variety, then so is the projectivization \(P(E)\) of \(E\). So an iteration of this procedure produces a nonsingular complete toric variety; some of these are known like the Hirzebruch surfaces and some of them are new nonsingular projective toric varieties. The authors proveNEWLINENEWLINE{Theorem 1.1}. A nonsingular projective toric variety is equivariantly diffeomorphic to a projective bundle \(P(E)\) over a nonsingular complete toric surface if and only if their cohomology rings are isomorphic as graded rings.NEWLINENEWLINEIn the sequel, the authors consider the cohomological rigidity problem: Given any two toric manifolds, if their cohomology rings are isomorphic, are they homeomorphic?NEWLINENEWLINEThere are many positive results obtained by \textit{S. Choi} et al. [Trans. Am. Math. Soc. 362, No. 2, 1097--1112 (2010; Zbl 1195.57060)] and the references therein. Applying Theorem 1.1 by using their previous results concerning generalized Bott manifolds and quasitoric manifolds, the authors showNEWLINENEWLINE{Theorem 1.2}. Let \(M,\, M'\) be projective bundles over \(4\)-dimensional quasitoric manifolds with fiber \({\mathbb CP}^1\) respectively. If \(H^*(M)\cong H^*(M')\) as graded rings, then \(M\) and \(M'\) are diffeomorphic.NEWLINENEWLINELet \(\mathcal B\) be a class of \(4\)-dimensional quasitoric manifolds but not described as either \({\mathbb CP}^2\# n\overline{{\mathbb CP}^2}\) or \(n{\mathbb CP}^2\#\overline{{\mathbb CP}^2}\) for \(n>9\). Take \(B,\,B'\in \mathcal B\).NEWLINENEWLINE{Theorem 1.3}. Let \(M,\, M'\) be projective bundles over \(B\) and \(B'\) respectively. If \(H^\ast(M)\cong H^\ast(M')\) as graded rings, then the ring isomorphism induces a diffeomorphism from \(M\) to \(M'\).