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Generalized intersection pairings on moduli spaces of vector bundles over a curve

Chenjing Bu, Young-Hoon Kiem

TL;DR

The paper develops and compares several constructions of generalized intersection pairings on Artin stacks with proper good moduli spaces, focusing on moduli of semistable vector bundles over curves. It unifies four approaches—partial desingularization, parabolic bundles, stable pairs, and Joyce wall-crossing—via a common wall-crossing framework and computes explicit low-rank cases to illuminate their relationships. Central tools include JK-type residue computations, the Joyce vertex algebra, and projection maps between parabolic and non-parabolic data, tied together through detailed comparisons and a derived understanding of IH-pairings on singular moduli. The results clarify when different generalized pairings coincide (notably rank 2) and when they diverge (higher ranks), with concrete computations and a structured pathway for extending wall-crossing analyses to broader moduli problems.

Abstract

We introduce the notion of a generalized intersection pairing for an Artin stack with a proper good moduli space and nonempty stable part. For the moduli stack of semistable bundles over a smooth projective curve, there are four known constructions by partial desingularization, parabolic bundles, stable pairs and wall crossing. In this paper, we compare all these generalized intersection pairings by establishing wall crossing formulas between them. Explicit computations for low rank cases are included.

Generalized intersection pairings on moduli spaces of vector bundles over a curve

TL;DR

The paper develops and compares several constructions of generalized intersection pairings on Artin stacks with proper good moduli spaces, focusing on moduli of semistable vector bundles over curves. It unifies four approaches—partial desingularization, parabolic bundles, stable pairs, and Joyce wall-crossing—via a common wall-crossing framework and computes explicit low-rank cases to illuminate their relationships. Central tools include JK-type residue computations, the Joyce vertex algebra, and projection maps between parabolic and non-parabolic data, tied together through detailed comparisons and a derived understanding of IH-pairings on singular moduli. The results clarify when different generalized pairings coincide (notably rank 2) and when they diverge (higher ranks), with concrete computations and a structured pathway for extending wall-crossing analyses to broader moduli problems.

Abstract

We introduce the notion of a generalized intersection pairing for an Artin stack with a proper good moduli space and nonempty stable part. For the moduli stack of semistable bundles over a smooth projective curve, there are four known constructions by partial desingularization, parabolic bundles, stable pairs and wall crossing. In this paper, we compare all these generalized intersection pairings by establishing wall crossing formulas between them. Explicit computations for low rank cases are included.

Paper Structure

This paper contains 44 sections, 8 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Generalized intersection pairings on Artin stacks
  3. Intersection pairings on partial desingularizations
  4. Intersection pairings on smooth fibrations
  5. Enumerative invariants by wall crossings
  6. Intersection cohomology pairings
  7. GIT moduli space of n points on projective line
  8. Intersection pairings for n odd
  9. Generalized intersection pairings for n even
  10. Intersection pairings by a smooth resolution
  11. Intersection pairings by partial desingularization
  12. A comparison theorem
  13. Intersection cohomology pairings
  14. Moduli spaces of vector bundles with extra structure
  15. Parabolic bundles
  16. ...and 29 more sections

Key Result

Theorem 1

Let $X$ be a smooth projective curve over $\mathbb{C}$ of genus $g > 1$, and let $\mathcal{M}^{{\mathrm{rig}}}_{k}$ be the moduli stack of semistable vector bundles on $X$ of a fixed slope $k \in \mathbb{Q}$. Consider Joyce's joyce-wall-crossing Lie bracket on the homology $\mathrm{H}_{\bullet + 2 \

Theorems & Definitions (26)

  • Theorem
  • Definition 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 7
  • Remark 8
  • Proposition 9
  • Example 10
  • Remark 11
  • ...and 16 more