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Connections, metrics and Higgs fields on complex fiber bundles

Nianzi Li, Mao Sheng

Abstract

We give a representation of the extension class associated to a holomorphic fibration by curvature, generalizing the work of Atiyah on holomorphic principal bundles in a natural way. As an application, we obtain a nonlinear analogue of the classical result of Weil on characterizing the existence of flat connections on holomorphic vector bundles over compact Riemann surfaces. We further establish a faithful functor from the category of nonlinear flat bundles reductive of Kähler type to the category of nonlinear Higgs bundles over the same base, which is assumed to be a compact complex manifold of Kähler type. Finally, we establish a notion of nonlinear harmonic bundle and prove that the variation of nonabelian Hodge structure is a nonlinear harmonic bundle in the rank one case and in the semisimple case.

Connections, metrics and Higgs fields on complex fiber bundles

Abstract

We give a representation of the extension class associated to a holomorphic fibration by curvature, generalizing the work of Atiyah on holomorphic principal bundles in a natural way. As an application, we obtain a nonlinear analogue of the classical result of Weil on characterizing the existence of flat connections on holomorphic vector bundles over compact Riemann surfaces. We further establish a faithful functor from the category of nonlinear flat bundles reductive of Kähler type to the category of nonlinear Higgs bundles over the same base, which is assumed to be a compact complex manifold of Kähler type. Finally, we establish a notion of nonlinear harmonic bundle and prove that the variation of nonabelian Hodge structure is a nonlinear harmonic bundle in the rank one case and in the semisimple case.
Paper Structure (24 sections, 93 theorems, 292 equations)

This paper contains 24 sections, 93 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\bar{\partial}$-operators and almost complex structures
  4. Connections
  5. Curvature classes
  6. Decomposition of $\bar{\partial}_X$
  7. Associated bundles
  8. Holomorphic connections
  9. Extension classes
  10. Nonlinear Riemann-Hilbert correspondence
  11. Relatively Kähler fibrations
  12. Fiberwise Kähler metrics
  13. Relatively Kähler fibrations
  14. Associated relatively Kähler bundles
  15. Nonlinear harmonic metrics
  16. ...and 9 more sections

Key Result

Theorem 1.3

There is a canonically associated $\bar{\partial}_X$-closed tensor $\mathcal{R}\in A^{0,1}(X,f^*\Omega_S\otimes T_{X/S})$ to a pure complex connection $\nabla^{\mathbb{C}}=\nabla^{1,0}+ \nabla^{0,1}$, whose class $[\mathcal{R}]\in H^1(X,f^*\Omega_S\otimes T_{X/S})$ equals $A(X)$. When $\nabla^{1,0}$

Theorems & Definitions (224)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5: Corollary \ref{['cor: Weil']}
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8: Theorem \ref{['thm:HolomorphicRH']}
  • Definition 1.10
  • Theorem 1.11: Proposition \ref{['prop:faithful_functor']}
  • ...and 214 more