Cohomology of toric bundles (Q1417427)

Let \(T\) be an algebraic torus \(T\), and let \(E \to B\) be a topological \(T\)-principal bundle. For a smooth projective \(T\)-toric variety \(X\), consider the fibre bundle \(E(X) := E \times_T X\) over \(B\). The authors describe the following data of \(E(X)\) in terms of the base \(B\) and the fan of the fibre \(X\): the singular cohomology ring; the \(K\)-ring of topological complex vector bundles, provided that \(B\) is compact; the Chow ring of cycles and the Grothendieck ring of algebraic vector bundles, provided that \(E \to B\) is algebraic with \(B\) irreducible and smooth. In fact, the results are proven for smooth complete toric varieties \(X\) the fan of which satisfies certain shellability conditions. For a more general construction of algebraic bundles with a compact \(\mathbb{Q}\)-factorial toric variety as typical fibre, and descriptions of the respective cohomology and Chow rings, see \textit{M. Halic}, Math. Nachr. 261--262, 60--84 (2003; Zbl 1040.14014).