Classification of complex projective towers up to dimension 8 and cohomological rigidity

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.