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Symplectic Geometry of Higgs Moduli and the Hilbert Scheme of Points

Zelin Jia

TL;DR

This work establishes a global holomorphic symplectomorphism between the moduli space $\mathcal{M}_H^s(n,0)$ of stable marked Higgs bundles on an elliptic curve $C$ and the Hilbert scheme of $n$ points on the cotangent bundle $\mathrm{Hilb}^n(T^*C)$. It combines spectral data via the BNR correspondence, the Fourier–Mukai transform, and the Biswas–Mukherjee method to compare holomorphic symplectic forms, showing $f^*(\mathrm{d}\Omega)=\phi^*(\mathrm{d}\Omega_{(n,0)})$ and that the Hitchin-base contribution vanishes, hence a genuine symplectomorphism. The generic case is analyzed through a natural twisting that reduces to $(T^*C)^n_{\star}$, and this is extended to the whole spaces, with the Hilbert–Chow morphism providing a holomorphic symplectic resolution. The results illuminate a deep link between Hitchin integrable systems on elliptic curves and the geometry of Hilbert schemes, framed by spectral data, parabolic/marked structures, and derived equivalences.

Abstract

We show that the isomorphism between the moduli space of certain parabolic Higgs bundles over an elliptic curve and the Hilbert scheme of n points of the cotangent bundle of the elliptic curve is a symplectomorphism with respect to their natural symplectic structures.

Symplectic Geometry of Higgs Moduli and the Hilbert Scheme of Points

TL;DR

This work establishes a global holomorphic symplectomorphism between the moduli space of stable marked Higgs bundles on an elliptic curve and the Hilbert scheme of points on the cotangent bundle . It combines spectral data via the BNR correspondence, the Fourier–Mukai transform, and the Biswas–Mukherjee method to compare holomorphic symplectic forms, showing and that the Hitchin-base contribution vanishes, hence a genuine symplectomorphism. The generic case is analyzed through a natural twisting that reduces to , and this is extended to the whole spaces, with the Hilbert–Chow morphism providing a holomorphic symplectic resolution. The results illuminate a deep link between Hitchin integrable systems on elliptic curves and the geometry of Hilbert schemes, framed by spectral data, parabolic/marked structures, and derived equivalences.

Abstract

We show that the isomorphism between the moduli space of certain parabolic Higgs bundles over an elliptic curve and the Hilbert scheme of n points of the cotangent bundle of the elliptic curve is a symplectomorphism with respect to their natural symplectic structures.
Paper Structure (6 sections, 9 theorems, 56 equations)

This paper contains 6 sections, 9 theorems, 56 equations.

Table of Contents

  1. Introduction & Preliminary
  2. Marked Higgs bundles and Hilbert scheme of $n$ points
  3. Symplectic structures on marked Higgs bundles and Hilbert scheme of $n$ points
  4. Symplectic structure and the projective bundle
  5. Identification of symplectic forms
  6. Symplectic structures in the generic case

Key Result

Theorem 1.3

HiNiSim The moduli space $\mathcal{M}_H^{ss}(n,d)$ whose closed points parametrize semistable Higgs bundles is a quasi-projective algebraic variety. The moduli space $\mathcal{M}_H(n,d)$ whose closed points parametrize stable Higgs bundles is open and dense in $\mathcal{M}_H^{ss}(n,d)$, moreover, $\

Theorems & Definitions (24)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 2.1
  • Proposition 2.2: BNR
  • proof
  • Theorem 2.3: BNR
  • ...and 14 more