Smooth correspondences between quiver varieties

TL;DR

This paper introduces split parabolic quiver varieties as smooth correspondences, providing explicit resolutions of singularities in quiver Brill--Noether loci and establishing their geometric properties, extending previous results on Hilbert schemes.

Contribution

It presents a new class of smooth correspondences between Nakajima quiver varieties and applies them to resolve singularities of quiver Brill--Noether loci, generalizing prior work.

Findings

Explicit resolution of singularities for quiver Brill--Noether loci

Proved irreducibility and Cohen-Macaulay property of these loci

Extended results from Hilbert schemes to broader quiver varieties

Abstract

We introduce a new class of smooth correspondences between Nakajima quiver varieties called split parabolic quiver varieties, and study their properties. We use these correspondences to construct an explicit resolution of singularities of quiver Brill--Noether loci and prove that the latter are irreducible and Cohen-Macaulay of expected dimension (if non-empty). This generalizes the results of Nakajima--Yoshioka and Bayer--Chen--Jiang for Hilbert schemes of points on surfaces.