Moduli of $G$-constellations and crepant resolutions I: the abelian case

TL;DR

This paper investigates when crepant resolutions of quotient varieties by finite abelian groups can be realized as moduli spaces of $G$-constellations, establishing conditions for such an isomorphism.

Contribution

It provides criteria under which a crepant resolution is isomorphic to a moduli space of $G$-constellations, extending understanding in the abelian case.

Findings

If a crepant resolution admits a natural $G$-constellation family with indecomposable fibers, it is isomorphic to the normalization of a moduli space.

The paper characterizes when crepant resolutions arise as moduli spaces in the abelian setting.

Abstract

For a finite abelian subgroup $G\subset SL_n(\mathbb{C})$, we study whether a given crepant resolution $X$ of the quotient variety $\mathbb{C}^n/G$ is obtained as a moduli space of $G$-constellations. In particular we show that, if $X$ admits a natural $G$-constellation family in the sense of Logvinenko over it with all fibers being indecomposable as $\mathbb{C}[\mathbb{C}^n]$-modules, then $X$ is isomorphic to the normalization of a fine moduli space of $G$-constellations.