Equivariant $K$-theory of cellular toric bundles and related spaces

TL;DR

The paper addresses computing the equivariant and ordinary topological $K$-theory rings for toric bundles whose fiber is a complete $T$-cellular toric variety, and applies the framework to toroidal horospherical embeddings. It introduces a relative Kunneth-type decomposition and a GKM/PLP-based description that expresses $K^0_{G_{comp}}(\mathcal{E}(X))$ in terms of the base $K$-theory and the fan data of the fiber, i.e. $K^0_{G_{comp}}(\mathcal{E}(X))\cong K^0_{G_{comp}}(\mathcal{B})\otimes_{R(T_{comp})}PLP(\Delta)$. For horospherical varieties, the authors obtain a parallel formula $K^0_{G_{comp}}(X)\cong K^0_{G_{comp}}(G/P)\otimes_{R((P/H)_{comp})}PLP(\mathbb{F}(X))$, with smooth cases reducing to Stanley–Reisner presentations. The results extend previous work on smooth toric fibers to singular fibers and provide explicit algebraic descriptions tied to the fan data, enabling concrete computations in equivariant Grothendieck rings and their cohomological analogues.

Abstract

In this article we describe the equivariant and ordinary topological $K$-ring of a toric bundle with fiber a $T$-{\it cellular} toric variety. This generalizes the results in \cite{su} on $K$-theory of smooth projective toric bundles. We apply our results to describe the equivariant topological $K$-ring of a toroidal horospherical embedding.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 3.1
• Definition 3.2
• Theorem 4.2
• proof
• Corollary 4.3
• Corollary 4.4
• Proposition 4.5