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Specialized Simpson's main estimates for cyclic harmonic $G$-bundles

Takuro Mochizuki

Abstract

We study a generalization of specialized Simpson's main estimate in the context of cyclic harmonic $G$-bundles induced by split automorphisms. We apply it to the classification of Toda type $G$-harmonic bundles.

Specialized Simpson's main estimates for cyclic harmonic $G$-bundles

Abstract

We study a generalization of specialized Simpson's main estimate in the context of cyclic harmonic -bundles induced by split automorphisms. We apply it to the classification of Toda type -harmonic bundles.
Paper Structure (124 sections, 150 theorems, 127 equations)

This paper contains 124 sections, 150 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Harmonic bundles on Riemann surfaces
  3. Simpson's main estimate
  4. Specialized Simpson's main estimate in the case of symmetric Higgs bundles
  5. Similar estimates for Toda equations
  6. Goal of this paper
  7. Split automorphisms for simple complex algebraic groups
  8. Split automorphism of finite order
  9. Canonical maximal compact groups
  10. Example 1
  11. Example 2
  12. Specialized Simpson's main estimate for cyclic harmonic $G$-bundles
  13. Harmonic $G$-bundles
  14. Compatibility with an automorphism $\sigma$
  15. Specialized Simpson's main estimates
  16. ...and 109 more sections

Key Result

Theorem 1.1

If $X$ is compact, the above correspondences induce equivalences of harmonic bundles, polystable Higgs bundles of degree $0$, and semisimple flat bundles.

Theorems & Definitions (166)

  • Theorem 1.1: Corlette-Donaldson-Hitchin-Simpson
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 1.7
  • Proposition 1.8: Corollary \ref{['cor;25.6.21.10']} and Corollary \ref{['cor;25.9.27.2']}
  • Proposition 1.9: Proposition \ref{['prop;25.9.27.3']} and Proposition \ref{['prop;25.9.27.4']}
  • Remark 1.10
  • ...and 156 more