Quasitoric Manifolds over a Product of Simplices

A quasitoric manifold (resp. a small cover) is a 2ndimensional (resp. an n-dimensional) smooth closed manifold with an effective locally standard action of (S) (resp. (Z2)n) whose orbit space is combinatorially an n-dimensional simple convex polytope P . In this paper we study them when P is a product of simplices. A generalized Bott tower over F, where F = C or R, is a sequence of projective bundles of the Whitney sum of F-line bundles starting with a point. Each stage of the tower over F, which we call a generalized Bott manifold, provides an example of quasitoric manifolds (when F = C) and small covers (when F = R) over a product of simplices. It turns out that every small cover over a product of simplices is equivalent (in the sense of Davis and Januszkiewicz [5]) to a generalized Bott manifold. But this is not the case for quasitoric manifolds and we show that a quasitoric manifold over a product of simplices is equivalent to a generalized Bott manifold if and only if it admits an almost complex structure left invariant under the action. Finally, we show that a quasitoric manifold M over a product of simplices is homeomorphic to a generalized Bott manifold if M has the same cohomology ring as a product of complex projective spaces with Q coefficients.