Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Intersection theory on singular moduli spaces of vector bundles: a parabolic approach

Camilla Felisetti, Olga Trapeznikova

Abstract

We present explicit formulas for the intersection pairing in the intersection cohomology of the moduli space $M_0(r)$ of rank-$r$, degree-$0$ semistable bundles on a Riemann surface. The key idea is to realize this intersection cohomology as a canonical subspace of the cohomology of a smooth moduli space of parabolic bundles, where the pairing can be computed via the Hecke correspondence and the Jeffrey-Kirwan iterated residue formulas. This approach provides a simpler alternative to the blow-up construction of Jeffrey-Kirwan-Kiem-Woolf, yielding formulas for the intersection pairing on $M_0(r)$, for arbitrary $r$, with a clear geometric interpretation.

Intersection theory on singular moduli spaces of vector bundles: a parabolic approach

Abstract

We present explicit formulas for the intersection pairing in the intersection cohomology of the moduli space of rank-, degree- semistable bundles on a Riemann surface. The key idea is to realize this intersection cohomology as a canonical subspace of the cohomology of a smooth moduli space of parabolic bundles, where the pairing can be computed via the Hecke correspondence and the Jeffrey-Kirwan iterated residue formulas. This approach provides a simpler alternative to the blow-up construction of Jeffrey-Kirwan-Kiem-Woolf, yielding formulas for the intersection pairing on , for arbitrary , with a clear geometric interpretation.
Paper Structure (24 sections, 23 theorems, 109 equations)

This paper contains 24 sections, 23 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Remarks
  3. Contents of the paper
  4. Intersection cohomology and the Decomposition Theorem
  5. Intersection cohomology
  6. The Decomposition Theorem
  7. Perverse cohomology groups
  8. A canonical embedding of $IH^*(\mathcal{M}_0(r))$ into $H^*(\mathcal{P}_0(r))$
  9. Moduli spaces of parabolic bundles
  10. Application of the Decomposition theorem to $\pi\colon \mathcal{P}_0(r)\to \mathcal{M}_0(r)$
  11. A canonical embedding of $IH^*(\mathcal{M}_0(r))$ into $H^*(\mathcal{P}_0(r))$
  12. The Poincaré--Verdier pairing on $IH^*(\mathcal{M}_0(r))$
  13. Jeffrey--Kirwan formulas for the intersection pairing in $H^*(\mathcal{M}_1(r))$
  14. Generators of the cohomology ring $H^*(\mathcal{M}_1(r))$
  15. Notation: Hamiltonian bases and $\mathrm{iBer}$
  16. ...and 9 more sections

Key Result

Theorem 2.2

Let $f:X\rightarrow Y$ be a proper algebraic map from a nonsingular projective variety $X$ of dimension $N$ to a projective variety $Y$. Denote by $r(f) := \dim(X\times_YX)-\dim(X)$ the defect of semismallness of $f$. Here $\mathcal{L}^i_\alpha$ are (shifted) local systems on $Y_\alpha$, defined by

Theorems & Definitions (48)

  • Definition 2.1
  • Theorem 2.2: Decomposition Theorem
  • Corollary 2.3
  • Theorem 2.4: Relative Hard Lefschetz
  • Theorem 2.5: Poincaré-Verdier duality
  • Remark 2.6
  • Proposition 3.1
  • Remark 3.2
  • Proposition 3.3
  • Definition 3.4
  • ...and 38 more