K-Theoretic and Categorical Properties of Toric Deligne--Mumford Stacks

TL;DR

This paper establishes key K-theoretic and categorical properties of toric Deligne-Mumford stacks, including localization, Riemann-Roch, and derived equivalences under wall-crossing, advancing the understanding of their geometric and algebraic structures.

Contribution

It proves fundamental theorems like localization, Riemann-Roch, and derived equivalences for toric Deligne-Mumford stacks under minimal assumptions, extending prior results in the field.

Findings

Localization theorem in equivariant K-theory established

Equivariant Hirzebruch-Riemann-Roch theorem proved

Fourier--Mukai transforms induce derived equivalences during crepant wall-crossings

Abstract

We prove the following results for toric Deligne-Mumford stacks, under minimal compactness hypotheses: the Localization Theorem in equivariant K-theory; the equivariant Hirzebruch-Riemann-Roch theorem; the Fourier--Mukai transformation associated to a crepant toric wall-crossing gives an equivariant derived equivalence.