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Smooth correspondences between quiver varieties

Nicolle González, Eugene Gorsky, José Simental

TL;DR

The paper introduces split parabolic quiver varieties as new smooth correspondences between Nakajima quiver varieties, enabling explicit resolutions of singularities for quiver Brill-Noether loci. It develops two equivalent definitions, proves smoothness and dimension formulas, and shows that these correspondences yield Grassmannian- or flag-bundle fiber structures that resolve Brill-Noether loci while preserving irreducibility and Cohen–Macaulayness. The results generalize known statements for Hilbert schemes of points and connect to broader geometric representation theory through projections, torus actions, and tautological bundles. This framework sets the stage for further exploration in equivariant $K$-theory and algebra actions on quiver varieties (as pursued in follow-up work).

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.

Smooth correspondences between quiver varieties

TL;DR

The paper introduces split parabolic quiver varieties as new smooth correspondences between Nakajima quiver varieties, enabling explicit resolutions of singularities for quiver Brill-Noether loci. It develops two equivalent definitions, proves smoothness and dimension formulas, and shows that these correspondences yield Grassmannian- or flag-bundle fiber structures that resolve Brill-Noether loci while preserving irreducibility and Cohen–Macaulayness. The results generalize known statements for Hilbert schemes of points and connect to broader geometric representation theory through projections, torus actions, and tautological bundles. This framework sets the stage for further exploration in equivariant -theory and algebra actions on quiver varieties (as pursued in follow-up work).

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.
Paper Structure (22 sections, 38 theorems, 168 equations)

This paper contains 22 sections, 38 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Quiver varieties
  3. Brill-Noether theory for quiver varieties
  4. Split parabolic quiver varieties
  5. Organization of the paper
  6. Future work
  7. Split parabolic quiver varieties
  8. Quiver varieties
  9. Split parabolic quiver varieties
  10. Alternative definition
  11. Smoothness and dimension
  12. Maps between split parabolic quiver varieties
  13. Isotropic property
  14. Torus action
  15. Tautological bundles
  16. ...and 7 more sections

Key Result

Theorem 1.3

Assume that $k_v\ge k_v^0$ for all $v$ and $\mathrm{BN}_Q^{\underline{k}}(\underline{d};\underline{f})$ is not empty. Then it is an irreducible Cohen-Macaulay variety of (expected) dimension

Theorems & Definitions (104)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Corollary 1.10
  • ...and 94 more