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Euler characteristics of affine ADE Nakajima quiver varieties via collapsing fibers

Lukas Bertsch, Ádám Gyenge, Balázs Szendrői

TL;DR

The paper proves a universal substitution formula that relates Euler-characteristic generating series of affine ADE Nakajima quiver varieties across generic and nongeneric stability regions by analyzing collapsing fibers of the variation of GIT map. This framework yields modular (theta-function) expressions for Quot schemes on Kleinian orbifolds and provides a Type A combinatorial second proof via torus localization, connecting to both affine and finite Lie-algebra representations. The results unify and extend previous computations, illuminate fiber-collapsing geometry, and reveal deep links to representation theory and modular forms. The higher-rank framing discussion shows the method's generality, while Type A concretely demonstrates how partitions and Fock-space techniques realize the generating functions as graded dimensions.

Abstract

We prove a universal substitution formula that compares generating series of Euler characteristics of Nakajima quiver varieties associated with affine ADE diagrams at generic and at certain nongeneric stability conditions via a study of collapsing fibres in the associated variation of GIT map, unifying and generalising earlier results of the last two authors with Némethi and of Nakajima. As a special case, we compute generating series of Euler characteristics of noncommutative Quot schemes of Kleinian orbifolds. In type A and rank 1, we give a second, combinatorial proof of our substitution formula, using torus localisation and partition enumeration. This gives a combinatorial model of the fibers of the variation of GIT map, and also leads to relations between our results and the representation theory of the affine and finite Lie algebras in type A.

Euler characteristics of affine ADE Nakajima quiver varieties via collapsing fibers

TL;DR

The paper proves a universal substitution formula that relates Euler-characteristic generating series of affine ADE Nakajima quiver varieties across generic and nongeneric stability regions by analyzing collapsing fibers of the variation of GIT map. This framework yields modular (theta-function) expressions for Quot schemes on Kleinian orbifolds and provides a Type A combinatorial second proof via torus localization, connecting to both affine and finite Lie-algebra representations. The results unify and extend previous computations, illuminate fiber-collapsing geometry, and reveal deep links to representation theory and modular forms. The higher-rank framing discussion shows the method's generality, while Type A concretely demonstrates how partitions and Fock-space techniques realize the generating functions as graded dimensions.

Abstract

We prove a universal substitution formula that compares generating series of Euler characteristics of Nakajima quiver varieties associated with affine ADE diagrams at generic and at certain nongeneric stability conditions via a study of collapsing fibres in the associated variation of GIT map, unifying and generalising earlier results of the last two authors with Némethi and of Nakajima. As a special case, we compute generating series of Euler characteristics of noncommutative Quot schemes of Kleinian orbifolds. In type A and rank 1, we give a second, combinatorial proof of our substitution formula, using torus localisation and partition enumeration. This gives a combinatorial model of the fibers of the variation of GIT map, and also leads to relations between our results and the representation theory of the affine and finite Lie algebras in type A.
Paper Structure (16 sections, 25 theorems, 100 equations)

This paper contains 16 sections, 25 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nakajima quiver varieties
  4. Nakajima quiver varieties for affine ADE Dynkin diagrams
  5. Root systems
  6. The substitution formula and its applications
  7. The substitution rule and its properties
  8. Proof of the universal substitution formula
  9. Quot schemes of Kleinian orbifolds
  10. Modularity
  11. Higher rank framings
  12. Type A: combinatorics and representation theory
  13. Torus fixed points
  14. Combinatorics of the substitution
  15. Representation theory
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $I^0 \subset I$ be a nonempty, proper subset of the vertex set $I$ with complement $I^+=I\setminus I^0$. Let $\zeta^{\bullet}\in F_{I^+}$ be a degenerate stability condition in the stratum of the boundary of the open chamber $F_I$ corresponding to $I^+$; see subsec:Nak_q_ADE. Let where, for every $v^+ \in \mathop{\mathrm{\mathbb{N}}}\nolimits^{I^+}$, $v^{\oplus} \in \mathop{\mathrm{\mathbb{N}

Theorems & Definitions (44)

  • Theorem 1.1: Theorem \ref{['thm_sub']}
  • Theorem 1.2: Corollary \ref{['cor_quot']})
  • Theorem 1.3: Theorem \ref{['thm:ZIplusmodular']})
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Example 3.3
  • Proposition 3.4
  • proof
  • ...and 34 more