K theory of smooth complete toric varieties and related spaces

The K-rings of non-singular complex pro jective varieties as well as quasi- toric manifolds were described in terms of generators and relations in an earlier work of the author with V. Uma. In this paper we obtain a similar description for complete non-singular toric varieties. Indeed, our approach enables us to obtain such a description for the more general class of torus manifolds with locally standard torus action and orbit space a homology polytope.

💡 Research Summary

The paper provides an explicit presentation of the topological K‑theory rings of complete smooth (non‑singular) toric varieties and, more generally, of torus manifolds whose torus actions are locally standard and whose orbit spaces are homology polytopes. Building on earlier work by the author together with V. Uma, which gave generators and relations for the K‑rings of nonsingular complex projective varieties and quasi‑toric manifolds, the author extends the method to the full class of complete smooth toric varieties and then to the broader class of torus manifolds.

The exposition begins with a concise review of the combinatorial data that defines a smooth toric variety: a complete fan Δ in a lattice N. Each one‑dimensional cone σ∈Δ(1) determines a T‑invariant divisor D_σ and an associated line bundle L_σ on the variety X_Δ. The fixed points of the torus action correspond bijectively to the maximal cones of Δ, and the structure sheaves of these fixed points provide a natural set of generators for K^0(X_Δ). By examining the exact sequences that relate the structure sheaves of fixed points, the line bundles L_σ, and their restrictions to invariant subvarieties, the author derives two families of relations:

1. Linear relations (R₁) coming from the fact that the sum of the divisor classes associated with the rays meeting at a given maximal cone vanishes in the Picard group. Algebraically these become ∑_{σ∈Δ(1)} ⟨m, v_σ⟩·x_σ = 0 for each lattice character m, where x_σ =