Inertial Chow rings of toric stacks

TL;DR

This paper provides explicit descriptions of inertial products on the Chow rings of toric stacks and explores an asymptotic product as the bundle rank increases, extending previous orbifold Chow ring results.

Contribution

It offers explicit presentations for integral inertial Chow rings of toric stacks and introduces an asymptotic product for large bundle ranks, expanding prior orbifold results.

Findings

Explicit presentations for integral inertial Chow rings.

Extension of orbifold Chow ring results to more general toric stacks.

Definition of an asymptotic product as bundle rank tends to infinity.

Abstract

For any vector bundle $V$ on a toric Deligne-Mumford stack $\ix$ the formalism of \cite{EJK:16} defines two intertial products $\star_{V^{+}}$ and $\star_{V^{-}}$ on the Chow group of the inertia stack. We give an explicit presentation for the integral $\star_{V^+}$ and $\star_{V^-}$ Chow rings, extending earlier work of Boris-Chen-Smith \cite{BCS:05} and Jiang-Tsen \cite{JiTs:10} in the orbifold Chow ring case, which corresponds to $V = 0$. We also describe an {\em asymptotic} product on the rational Chow group of the inertia stack obtained by letting the rank of the bundle $V$ go to infinity.