Cohomology of toric bundles

Abstract

Let p:E→ B be a principal bundle with fibre and structure group the torus T≌(C^* )ⁿ over a topological space B. Let X be a nonsingular projective T-toric variety. One has the X-bundle π: E(X) → B where E(X)=E×_T X, π([e,x])=p(e). This is a Zariski locally trivial fibre bundle in case p:E →B is algebraic. The purpose of this note is to describe (i) the singular cohomology ring of E(X) as an H^*(B;Z)-algebra, (ii) the topological K-ring of K^*(E(X)) as a K^*(B)-algebra when B is compact. When p:E → B is algebraic over an irreducible, nonsingular, noetherian scheme over C, we describe (iii) the Chow ring of A^*(E(X)) as an A^*(B)-algebra, and (iv) the Grothendieck ring K⁰(E(X)) of algebraic vector bundles on E (X) as a K⁰(B)-algebra.