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Harmonic Bundles with Symplectic Structures

Takashi Ono

Abstract

We study harmonic bundles with an additional structure called symplectic structure. We study them for the case of the base manifold is compact and non-compact. For the compact case, we show that a harmonic bundle with a symplectic structure is equivalent to principle $Sp(2n,C)$-bundle with a reductive flat connection. For the non-compact case, we show that a polystable good filtered Higgs bundle with a perfect skew-symmetric pairing is equivalent to a good wild harmonic bundle with a symplectic structure.

Harmonic Bundles with Symplectic Structures

Abstract

We study harmonic bundles with an additional structure called symplectic structure. We study them for the case of the base manifold is compact and non-compact. For the compact case, we show that a harmonic bundle with a symplectic structure is equivalent to principle -bundle with a reductive flat connection. For the non-compact case, we show that a polystable good filtered Higgs bundle with a perfect skew-symmetric pairing is equivalent to a good wild harmonic bundle with a symplectic structure.
Paper Structure (46 sections, 33 theorems, 62 equations)

This paper contains 46 sections, 33 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Harmonic bundles
  3. Harmonic Bundles with Symplectic Structures
  4. Results
  5. Compact case
  6. Non-Compact case
  7. Harmonic bundles with symplectic structure
  8. Skew-symmetric pairings of vector spaces
  9. Harmonic bundles with symplectic structure
  10. Harmonic metrics on Principal $G$-bundles
  11. Harmonic bundles and Principal $\mathrm{Sp}(2n,\mathbb{C})$-Bundles
  12. Good filtered Higgs bundles and Good Wild Harmonic bunldes
  13. Filtered sheaves
  14. Filtered sheaves
  15. Filtered Higgs sheaves
  16. ...and 31 more sections

Key Result

Theorem 1.1

Suppose $X$ is a compact Kähler manifold. $(E,\overline\partial_E,\theta)$ admits a harmonic metric if and only $(E,\overline\partial_E, \theta)$ is a polystable Higgs bundle and $c_1(E)=c_2(E)=0$. If $h_1$ and $h_2$ are harmonic metrics, then there exists a decomposition $(E,\overline\partial_E, \t

Theorems & Definitions (67)

  • Theorem 1.1: HS1
  • Theorem 1.2: BBM1M2S1S2
  • Theorem 1.3: Theorem \ref{['KH']}
  • Theorem 1.4: Theorem \ref{['H-P']}
  • Proposition 1.1: Proposition \ref{['w-f']}
  • Proposition 1.2: Proposition \ref{['B4']} and \ref{['B5']}
  • Theorem 1.5: qm1
  • Definition 2.1
  • Lemma 2.1
  • proof
  • ...and 57 more