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The cohomology of rank two stable bundle moduli: mod two nilpotency & skew Schur polynomials

Christopher Scaduto, Matthew Stoffregen

Abstract

We compute cup product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping class group action.

The cohomology of rank two stable bundle moduli: mod two nilpotency & skew Schur polynomials

Abstract

We compute cup product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping class group action.

Paper Structure

This paper contains 15 sections, 24 theorems, 121 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background & Generators
  3. Integral generators for the cohomology of $M_g$
  4. The redundancy of some generators over $\mathbb{Z}_p$
  5. Integral generators for the cohomology of $N_g$
  6. Twisting by a line bundle to define $z_i$ and $\xi_i$
  7. Generators for the invariant subring of $N_g$
  8. The intersection pairings for Newstead classes
  9. The computation of integral intersection pairings
  10. Extracting pairings via specializations
  11. Chern numbers for the tangent bundle
  12. Skew Schur functions
  13. Mod two nilpotency
  14. Computations
  15. Background on symmetric functions

Key Result

Theorem 1.1

The nilpotency degree of $\alpha$ as viewed in $H^2(N_g;\mathbb{Z}_2)$ is equal to $g$: On the other hand, the nilpotency degree of $a_1$ as viewed in $H^2(M_g;\mathbb{Z}_2)$ is equal to $2g$:

Figures (2)

  • Figure 1: With $n=6$, the skew tableaux appearing in the skew Schur functions and $1/Q(T)$ and $1/E(T)$.
  • Figure 2: Here we list the SSYT of shape $\lambda_2/\mu_2$ with type $\nu$ for each partition $\nu$. Using equation (\ref{['eq:ssyt']}) we conclude $s_{\lambda_2/\mu_2} = m_{(2,2,2)} + 2m_{(2,2,1,1)} + 5m_{(2,1,1,1,1)} + 14m_{(1,1,1,1,1,1)}$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 2.1: ab Thm 9.11
  • Corollary 2.2
  • Corollary 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 26 more