Rational singularities for moment maps of totally negative quivers

TL;DR

This paper proves that the zero-fiber of the moment map for totally negative quivers has rational singularities, extending to related moduli spaces and exploring arithmetic implications such as jet count asymptotics and p-adic volumes.

Contribution

It generalizes dimension bounds on jet spaces for these fibers and transfers rational singularities to broader moduli spaces in 2-Calabi-Yau categories, with arithmetic applications.

Findings

Zero-fiber of the moment map has rational singularities.

Generalized jet space dimension bounds for these fibers.

Connected jet count asymptotics to p-adic volumes of moduli spaces.

Abstract

We prove that the zero-fiber of the moment map of a totally negative quiver has rational singularities. Our proof consists in generalizing dimension bounds on jet spaces of this fiber, which were introduced by Budur. We also transfer the rational singularities property to other moduli spaces of objects in 2-Calabi-Yau categories, based on recent work of Davison. This has interesting arithmetic applications on quiver moment maps and moduli spaces of objects in 2-Calabi-Yau categories. First, we generalize results of Wyss on the asymptotic behaviour of counts of jets of quiver moment maps over finite fields. Moreover, we interpret the limit of counts of jets on a given moduli space as its p-adic volume under a canonical measure analogous to the measure built by Carocci, Orecchia and Wyss on certain moduli spaces of coherent sheaves.