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Non-abelian Hodge correspondence and moduli spaces of flat bundles on Sasakian manifolds with fixed basic structures

Hisashi Kasuya

Abstract

We show that the moduli space of simple flat bundles over a compact Sasakian manifold is a finite disjoint union of moduli spaces of simple flat bundles with fixed basic structures. This gives a detailed description of the non-abelian Hodge correspondence on a compact Sasakian manifold at the level of moduli spaces. As an application, we give an analogue of Hitchin's properness of maps defined by the coefficients of the characteristic polynomial of Higgs fields.

Non-abelian Hodge correspondence and moduli spaces of flat bundles on Sasakian manifolds with fixed basic structures

Abstract

We show that the moduli space of simple flat bundles over a compact Sasakian manifold is a finite disjoint union of moduli spaces of simple flat bundles with fixed basic structures. This gives a detailed description of the non-abelian Hodge correspondence on a compact Sasakian manifold at the level of moduli spaces. As an application, we give an analogue of Hitchin's properness of maps defined by the coefficients of the characteristic polynomial of Higgs fields.

Paper Structure

This paper contains 12 sections, 18 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Non-abelian Hodge correspondence on compact Sasakian manifolds
  3. Sasakian manifolds
  4. Hodge theory
  5. Basic bundles
  6. Non-abelian Hodge correspondence
  7. Moduli spaces
  8. Moduli spaces of flat connections
  9. Moduli spaces of flat connections with fixed basic structures
  10. Moduli spaces of harmonic bundles and basic (pluri)harmonic bundles
  11. Moduli spaces of stable basic Higgs bundles
  12. Compactness and properness

Key Result

Theorem 1.1

${\mathcal{M}}^{s}_{Bflat }(E, D_{\xi})$ is an open and closed set in ${\mathcal{M}}^{s}_{flat }(SL_r)$ and ${\mathcal{M}}^{s}_{flat }(SL_r)$ is a finite disjoint union where the union runs over the set of isomorphism classes of basic vector bundles $(E, D_{\xi})$ coming from simple flat complex vector bundles $(E, D)$ of rank $r$ with the trivial determinants.

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 2.1: BK
  • Definition 2.2
  • Theorem 2.3: BKBK2
  • Remark 3.1
  • Theorem 3.2
  • ...and 19 more