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 $\mathcal K$0 (E (X)) of algebraic vector bundles on E (X) as a $\mathcal K$0(B)-algebra.
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An erratum to this article is available at http://dx.doi.org/10.1007/s00014-004-0817-x.
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Sankaran, P., Uma, V. Cohomology of toric bundles . Comment. Math. Helv. 78, 540–554 (2003). https://doi.org/10.1007/s00014-003-0761-1
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DOI: https://doi.org/10.1007/s00014-003-0761-1