Topology of Toric Varieties

Abstract

This thesis is devoted to a study of the cohomology and K-theory of complex toric bundles and the fundamental group of real toric varieties. Since the discovery of toric varieties in the early 1970's, the subject has developed immensely with various facts. Toric manifold - a quasitoric manifold, and its theory generalized to Unitary Toric manifolds are some of the interesting aspects of Toric Varieties. It is also of interest from the view point of topology and geometry to study the real valued points of the complex toric variety X. This thesis addresses two problems on Toric varieties: A locally trivial fibre bundle with 'fibre type' a smooth projective complex toric variety X is considered, and base an arbitrary topology space B associated to a principal T- bundle. It is aimed to describe the singular cohomology ring of E(X) as an H*(B;Z) algebra, and ii) the topological K-ring K* (E(X)) as a K* (B) algebra when B is compact and Hausdorff. When B is an irreducible, nonsingular, noetherian scheme over C and P : E --> B is algebraic, (iii) the Chow ring A*(E(X))described as an A*(B) algebra, (iv) the Grothendiek ring K^0(E(X)) of algebraic vector bundles on E(X) as a K^0(B)-algebra.