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

A complex projective tower or simply a $\mathbb CP$-tower is an iterated complex projective fibrations starting from a point. In this paper we classify all 6-dimensional $\mathbb CP$-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 $\mathbb CP$-towers by classifying $\mathbb CP^1$-fibrations over $\mathbb CP^3$ up to diffeomorphism. As a corollary we show that such $\mathbb CP$-towers are diffeomorphic if they are homotopy equivalent.