Bott Towers, Crosspolytopes and Torus Actions

Abstract

We study the geometry and topology of Bott towers in the context of toric geometry. We show that any kth stage of a Bott tower is a smooth projective toric variety associated to a fan arising from a crosspolytope; conversely, we prove that any toric variety associated to a fan obtained from a crosspolytope actually gives rise to a Bott tower. The former leads us to a description of the tangent bundle of the kth stage of the tower, considered as a complex manifold, which splits into a sum of complex line bundles. Applying Danilov–Jurkiewicz theorem, we compute the cohomology ring of any kth stage, and by way of construction, we provide all the monomial identities defining the related affine toric varieties.