Quasitoric manifolds over a product of simplices

Abstract

A quasitoric manifold (resp.a small cover) is a 2n-dimensional (resp.an ndimensional) smooth closed manifold with an effective locally standard action of (S 1 ) n (resp.( 2 ) 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 , where or , is a sequence of projective bundles of the Whitney sum of -line bundles starting with a point.Each stage of the tower over , which we call a generalized Bott manifold, provides an example of quasitoric manifolds (when ) and small covers (when ) 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 coefficients.