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Nonlinear Harmonic Bundles

Mao Sheng

TL;DR

The paper introduces nonlinear harmonic bundles as a nonlinear analogue of harmonic bundles in nonabelian Hodge theory by replacing linear fibers with complex manifolds and equipping differentiable bundles with two integrable complex structures connected by a beta-automorphism. It develops a generalized Simpson mechanism, Chern connection theory, and curvature-based harmonicity conditions, anchored in almost connections and almost Higgs fields. It then connects these ideas to relative nonabelian Hodge moduli spaces, proving a rank-one case that ties Gauss-Manin data to Kodaira-Spencer Higgs fields and suggesting a nonlinear Hodge structure on the base. Finally, it establishes a Torelli-type result for abelian varieties, showing that nonlinear Higgs data on moduli spaces captures the isomorphism class of polarized families under strong monodromy, thereby enriching the classical VHS/Torelli framework with nonlinear Higgs dynamics.

Abstract

We generalize the notion of harmonic bundles in nonabelian Hodge theory to the nonlinear setting.

Nonlinear Harmonic Bundles

TL;DR

The paper introduces nonlinear harmonic bundles as a nonlinear analogue of harmonic bundles in nonabelian Hodge theory by replacing linear fibers with complex manifolds and equipping differentiable bundles with two integrable complex structures connected by a beta-automorphism. It develops a generalized Simpson mechanism, Chern connection theory, and curvature-based harmonicity conditions, anchored in almost connections and almost Higgs fields. It then connects these ideas to relative nonabelian Hodge moduli spaces, proving a rank-one case that ties Gauss-Manin data to Kodaira-Spencer Higgs fields and suggesting a nonlinear Hodge structure on the base. Finally, it establishes a Torelli-type result for abelian varieties, showing that nonlinear Higgs data on moduli spaces captures the isomorphism class of polarized families under strong monodromy, thereby enriching the classical VHS/Torelli framework with nonlinear Higgs dynamics.

Abstract

We generalize the notion of harmonic bundles in nonabelian Hodge theory to the nonlinear setting.

Paper Structure

This paper contains 7 sections, 7 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Almost connections and almost Higgs fields
  3. Chern connections and complex conjugations
  4. Simpson mechanism
  5. Nonlinear harmonic bundles
  6. Relative nonabelian Hodge moduli spaces
  7. A Torelli theorem

Key Result

Proposition 2.3

Let $(\alpha, T_{rel})$ be a complex fiber bundle over $S$. Then a complex structure on $(\alpha, T_{rel})$ gives rise to a canonical $\bar{\partial}$-operator on $(\alpha, T_{rel})$. Conversely, a $\bar{\partial}$-operator on $(\alpha, T_{rel})$ gives a canonical almost complex structure on $(\alph

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Example 2.6
  • Example 2.7
  • Definition 2.8
  • ...and 29 more