Complex projective towers and their cohomological rigidity up to dimension six

Abstract

Abstract. A complex projective tower or simply a CP -tower is an iterated complex projective fibrations starting from a point. In this paper we classify all 6-dimensional CP -towers up to diffeomorphism, and as a consequence, we show that all such manifolds are cohomologically rigid, i.e., they are completely determined up to diffeomorphism by their cohomology rings. We also show that cohomological rigidity is not valid for 8-dimensional CP -towers by classifying some CP 1 -fibrations over CP 3 up to diffeomorphism. As a corollary we show that such CP -towers are diffeomorphic if they are homotopy equivalent.

Cohomological non-rigidity of eight-dimensional complex projective towers

“…CP towers include many interesting classes of manifolds. In a previous paper [7], we showed that generalized Bott manifolds and the Milnor hypersurface admit a CP tower structure. We first introduce two other examples of CP towers.…”

“…This leads us to the following proposition. 3 Some preliminaries 3A Preliminaries from [7] We first recall some basic facts from [7,Section 2].…”

“…If we forget the tower structure, then we call C i an (i -stage) CP manifold. In [7], we show that the diffeomorphism types of 6-dimensional CP manifolds are determined by their cohomology rings; ie the collection of 6-dimensional CP manifolds CPM 6 is cohomologically rigid. This is a generalization of the fact, due to Choi, Masuda and Suh [5], that the collection GBM 6 of 6-dimensional generalized Bott manifolds is cohomologically rigid.…”

“…H M is bijective or not is called the cohomological rigidity problem. In this paper, we study the cohomological rigidity problem for complex projective towers (or simply CP towers), which we introduced in [7].…”

“…In Section 2, as examples of CP towers, we explain when a flag manifold admits the structure of a CP tower. In Section 3, we recall some basic facts from [7]. In Section 4, we show that N satisfies cohomological rigidity.…”