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Singular Lagrangians in the Hitchin moduli space and conformal limits

Szehong Kwong

TL;DR

The paper develops a local analytic model for singular points in the SL(r,C) Hitchin moduli space by combining the Kuranishi framework with a graded deformation theory arising from systems of Hodge bundles. It defines a central locus S, realized as an affine GIT quotient of a BB-slice, and proves that S is complex Lagrangian on its intersection with the stable locus; under Assumptions I (Φ0=0) or II (abelian automorphism group), S contains stable Higgs bundles. It then establishes the existence of conformal limits for stable Higgs bundles lying on the central locus, first on the open stratum and then for polystable SHBs with distinct summands (abelian stabilizer), by constructing harmonic metrics adapted to the C*-flow. Collectively, these results generalize conformal-limit phenomena and provide a central, Lagrangian substructure inside the upward flow through singular fixed points, with implications for mirror symmetry and the geometry of non-stable strata in the Hitchin system.

Abstract

In the moduli space of semistable $\text{SL}(r, \mathbb{C})$-Higgs bundles, we show that there exists a sublocus of the upward flow through a polystable $\mathbb{C}^{*}$-fixed point, which is Lagrangian on its intersection with the stable locus. This intesesction is always non-empty in the case when the Higgs field of the fixed point vanishes, or when the automorphism group of its polystable representative is abelian. Under the same assumptions, we show that the conformal limit of a stable Higgs bundle lying on this locus exists.

Singular Lagrangians in the Hitchin moduli space and conformal limits

TL;DR

The paper develops a local analytic model for singular points in the SL(r,C) Hitchin moduli space by combining the Kuranishi framework with a graded deformation theory arising from systems of Hodge bundles. It defines a central locus S, realized as an affine GIT quotient of a BB-slice, and proves that S is complex Lagrangian on its intersection with the stable locus; under Assumptions I (Φ0=0) or II (abelian automorphism group), S contains stable Higgs bundles. It then establishes the existence of conformal limits for stable Higgs bundles lying on the central locus, first on the open stratum and then for polystable SHBs with distinct summands (abelian stabilizer), by constructing harmonic metrics adapted to the C*-flow. Collectively, these results generalize conformal-limit phenomena and provide a central, Lagrangian substructure inside the upward flow through singular fixed points, with implications for mirror symmetry and the geometry of non-stable strata in the Hitchin system.

Abstract

In the moduli space of semistable -Higgs bundles, we show that there exists a sublocus of the upward flow through a polystable -fixed point, which is Lagrangian on its intersection with the stable locus. This intesesction is always non-empty in the case when the Higgs field of the fixed point vanishes, or when the automorphism group of its polystable representative is abelian. Under the same assumptions, we show that the conformal limit of a stable Higgs bundle lying on this locus exists.

Paper Structure

This paper contains 12 sections, 28 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Kuranishi local model around a polystable Higgs bundle
  3. The moduli space of semistable Higgs bundles
  4. The local Kuranishi model around a polystable Higgs bundle
  5. Stability in linear representations
  6. The central locus
  7. $\mathbb{C}^{*}$-fixed points in $\mathcal{M}_{H}$ and system of Hodge bundles (SHB)
  8. The deformation complex at a polystable SHB
  9. Model for the central locus
  10. Conformal limits
  11. On the open stratum
  12. On polystable SHBs with pairwise distinct stable summands

Key Result

Theorem 1.1

Along the upward flow through $[(\overline{\partial}_0,\Phi_0)]$, there exists a sub-locus parametrized by an affine GIT quotient, which is complex Lagrangian along the points which represent stable Higgs bundles. When Assumption assumption_I or Assumption assumption_II holds, $S$ contains some stable Higgs bundle.

Theorems & Definitions (48)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 2.1: Fan22, Section 3
  • Theorem 2.2: KN79, Kempf-Ness
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • ...and 38 more