Equivariant $K$-theory of cellular toric varieties
V. Uma
TL;DR
This work computes the topological T_comp-equivariant K-theory of complete T-cellular toric varieties X(Δ). Using a Bialynicki-Birula cellular decomposition and GKM-type localization, the authors show K^0_{T_comp}(X) is isomorphic to the ring PLP(Δ) of piecewise Laurent polynomial functions on the fan, extending prior results for divisive weighted projective spaces and cellular simplicial toric varieties to a broader class. They construct canonical basis elements f_i indexed by the BB-cells, with explicit restrictions to fixed points, and provide a closed, iterative formula for the structure constants of multiplication in this basis. The results bridge combinatorial data of the fan with topological equivariant K-theory, generalizing Vezzosi–Vistoli and related algebraic results to the topological setting and enabling explicit computations even in singular or non-simplicial cases. The concluding remarks outline extensions to non-complete and non-smooth settings and potential future applications to broader classes of cellular toric objects. Key formulas involve the PLP(Δ) presentation and the basis-restriction pattern: for each cell i, f_i|_{x_i} = ∏_{j=1}^{n−k_i} (1 − e^{−u_{i_j}}) and f_i|_{x_l}=0 for l>i, with structure constants given by a^l_{i,j} = ([f_i f_j − ∑_{p>l} a^p_{i,j} f_p]|_{x_l}) / ∏_{r=1}^{n−k_l} (1 − e^{−u_{l_r}}).
Abstract
In this article we describe the $T_{comp}$-equivariant topological $K$-ring of a $T$-{\it cellular} complete toric variety. We further show that $K_{T_{comp}}^0(X)$ is isomorphic as an $R(T_{comp})$-algebra to the ring of piecewise Laurent polynomial functions on the associated fan denoted $PLP(Δ)$. Furthermore, we compute a basis for $K_{T_{comp}}^0(X)$ as a $R(T_{comp})$-module and multiplicative structure constants with respect to this basis.