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Generators for the moduli space of parabolic bundle

Lisa Jeffrey, Yukai Zhang

Abstract

The purpose of this note is to find explicit representatives in deRham cohomology for the generators of the cohomology of the moduli space of parabolic bundles, analogous to the results of \cite{groupcoho} for the moduli space of vector bundles. Further we use the explicit generators to compute the intersection pairing of its cohomology.

Generators for the moduli space of parabolic bundle

Abstract

The purpose of this note is to find explicit representatives in deRham cohomology for the generators of the cohomology of the moduli space of parabolic bundles, analogous to the results of \cite{groupcoho} for the moduli space of vector bundles. Further we use the explicit generators to compute the intersection pairing of its cohomology.
Paper Structure (9 sections, 13 theorems, 53 equations)

This paper contains 9 sections, 13 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. The moduli space of parabolic bundles
  3. Notation of Chern-Weil homomorphism
  4. Leray–Hirsch theorem and fibre bundle $M/T \to M/G$
  5. Using Chern-Weil homomorphism to express the generators
  6. Application to moduli space of parabolic bundles
  7. Intersection Pairing
  8. Fibre integration and push-forward map
  9. Intersection pairing of $G/T$

Key Result

Theorem 3.1

For an invariant polynomial of degree $k$ and a 2-form $\Omega$ which is the curvature associated to the connection $\theta$, we have: Furthermore, the class $[\Lambda] \in H^{2k}(M)$ is independent of the connection $\theta$.

Theorems & Definitions (24)

  • Theorem 3.1: Chern-Weil homomorphism
  • Remark
  • Theorem 4.1: Leray–Hirsch theorem
  • Theorem 4.2
  • Corollary 4.2.1
  • Remark
  • Remark
  • Corollary 4.2.2
  • proof
  • Remark
  • ...and 14 more