Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the intersection theory of moduli spaces of parabolic bundles

Miguel Moreira

TL;DR

This work develops a comprehensive framework to compute intersection numbers of tautological classes on moduli spaces of parabolic bundles by stitching together wall-crossing in Joyce’s vertex-algebra setting, flag-bundle geometry, and Hecke modifications. The central result is a reconstruction theorem: integrals are determined by rank-1 data, wall-crossing coefficients, flag-bundle formulas with Weyl anti-symmetry, and Hecke compatibility, and extend to non-regular (semistable) weights. The paper then applies this framework to prove Newstead–Earl–Kirwan vanishing with a Chern-filtration viewpoint, and to establish Virasoro constraints for parabolic bundles, with proofs that remain valid beyond the stable-bundle case. Rank-3 examples illustrate the wall-chamber structure and the interplay of the various symmetries. Overall, the approach offers an algebraic, recursion-based route to intersection theory on parabolic moduli spaces with broad consequences for related enumerative problems on orbifolds and root stacks.

Abstract

This paper concerns the intersection numbers of tautological classes on moduli spaces of parabolic bundles on a smooth projective curve. We show that such intersection numbers are completely determined by wall-crossing formulas, Hecke isomorphisms, and flag bundle structures and resulting Weyl symmetry. As applications of these ideas, we prove the Newstead--Earl--Kirwan vanishing -- and a natural strengthening in terms of Chern filtrations -- and the Virasoro constraints for parabolic bundles. Both of these results were already known for moduli of stable bundles without parabolic structure, but even in those cases our proofs are new and independent of the existing ones. We use a Joyce style vertex algebra formulation of wall-crossing, and define intersection numbers even in the presence of strictly semistable parabolic bundles; all of our results hold in that setting as well.

On the intersection theory of moduli spaces of parabolic bundles

TL;DR

This work develops a comprehensive framework to compute intersection numbers of tautological classes on moduli spaces of parabolic bundles by stitching together wall-crossing in Joyce’s vertex-algebra setting, flag-bundle geometry, and Hecke modifications. The central result is a reconstruction theorem: integrals are determined by rank-1 data, wall-crossing coefficients, flag-bundle formulas with Weyl anti-symmetry, and Hecke compatibility, and extend to non-regular (semistable) weights. The paper then applies this framework to prove Newstead–Earl–Kirwan vanishing with a Chern-filtration viewpoint, and to establish Virasoro constraints for parabolic bundles, with proofs that remain valid beyond the stable-bundle case. Rank-3 examples illustrate the wall-chamber structure and the interplay of the various symmetries. Overall, the approach offers an algebraic, recursion-based route to intersection theory on parabolic moduli spaces with broad consequences for related enumerative problems on orbifolds and root stacks.

Abstract

This paper concerns the intersection numbers of tautological classes on moduli spaces of parabolic bundles on a smooth projective curve. We show that such intersection numbers are completely determined by wall-crossing formulas, Hecke isomorphisms, and flag bundle structures and resulting Weyl symmetry. As applications of these ideas, we prove the Newstead--Earl--Kirwan vanishing -- and a natural strengthening in terms of Chern filtrations -- and the Virasoro constraints for parabolic bundles. Both of these results were already known for moduli of stable bundles without parabolic structure, but even in those cases our proofs are new and independent of the existing ones. We use a Joyce style vertex algebra formulation of wall-crossing, and define intersection numbers even in the presence of strictly semistable parabolic bundles; all of our results hold in that setting as well.

Paper Structure

This paper contains 55 sections, 36 theorems, 285 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Wall-crossing, Joyce style vertex algebras, and stable$\neq$semistable
  4. Flag bundles, Hecke isomorphisms and (affine) Weyl symmetry
  5. Flag bundles
  6. Weyl symmetry
  7. Hecke isomorphisms
  8. Affine Weyl symmetry
  9. Proofs of main results
  10. Rank 3 example
  11. Acknowledgements
  12. Table of notation
  13. Preliminaries on parabolic bundles
  14. The category of parabolic bundles
  15. Stack of parabolic bundles and stability
  16. ...and 40 more sections

Key Result

Theorem A

Integrals of descendents eq: integrals over moduli spaces of semistable parabolic bundles are completely determined by the following properties:

Figures (2)

  • Figure 1: The spaces of stability conditions $S_{3, d}$ for $d=1, 0, -1$.
  • Figure 2: An example of the graphs of functions $f$ (on the left) and $\widetilde{f}$ (on the right). In this example, $r=7$ and $\tau$ is between $c_2$ and $c_3$. We have $h_\tau(7, d, f)=(7, d-4, \widetilde{f})\,.$

Theorems & Definitions (88)

  • Theorem A: Reconstruction theorem
  • Remark 1.1
  • Theorem B: =Theorem \ref{['thm: newstead']}
  • Theorem C: =Theorem \ref{['thm: virasoro']}
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • ...and 78 more