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From Hitchin Systems to Rational Elliptic Surfaces with C*-actions via Orbifold Hilbert Schemes

Yonghong Huang

TL;DR

The work develops a comprehensive framework to compactify two-dimensional Hitchin systems for affine tilde Dynkin types via orbifold Hilbert schemes, producing four rational elliptic surfaces with a $\mathbb{C}^*$-action and extending Hitchin fibrations. It establishes a general theory for moduli of sheaves on orbifold surfaces, proving smoothness, connectedness, and Poisson structures, and shows Hilbert schemes $\mathrm{Hilb}^{\alpha}(\mathcal{X})$ provide Poisson-respecting resolutions of symmetric products through the Hilbert–Chow morphism. These tools are then applied to compactify the Hitchin systems as $\mathrm{Hilb}^1(\mathbb{P}(T^{\vee}\mathcal{X}_{i}\oplus \mathcal{O}_{\mathcal{X}_{i}}))$, yielding explicit rational elliptic surfaces $\widetilde{X}_{i}$ with boundary and singular fibers, together with compatible Poisson structures and minimal resolutions of the corresponding GIT quotients. The results give concrete geometric realizations of these Hitchin systems, illuminate wall-crossing and polarization phenomena in the orbifold setting, and provide a robust path to understanding Calabi–Yau-like orbifold compactifications in dimension two.

Abstract

Using orbifold Hilbert schemes, we compactify all two-dimensional Hitchin systems corresponding to types A0-tilde, D4-tilde, E6-tilde, E7-tilde, and E8-tilde, thereby obtaining four rational elliptic surfaces with C*-actions. Their singular fibers and relative minimal models are listed in the main table. To this end, we prove that Hilbert schemes of orbifold surfaces are connected smooth projective schemes under suitable conditions, and we use the Hilbert-Chow morphism to construct the minimal resolutions of the coarse moduli spaces.

From Hitchin Systems to Rational Elliptic Surfaces with C*-actions via Orbifold Hilbert Schemes

TL;DR

The work develops a comprehensive framework to compactify two-dimensional Hitchin systems for affine tilde Dynkin types via orbifold Hilbert schemes, producing four rational elliptic surfaces with a -action and extending Hitchin fibrations. It establishes a general theory for moduli of sheaves on orbifold surfaces, proving smoothness, connectedness, and Poisson structures, and shows Hilbert schemes provide Poisson-respecting resolutions of symmetric products through the Hilbert–Chow morphism. These tools are then applied to compactify the Hitchin systems as , yielding explicit rational elliptic surfaces with boundary and singular fibers, together with compatible Poisson structures and minimal resolutions of the corresponding GIT quotients. The results give concrete geometric realizations of these Hitchin systems, illuminate wall-crossing and polarization phenomena in the orbifold setting, and provide a robust path to understanding Calabi–Yau-like orbifold compactifications in dimension two.

Abstract

Using orbifold Hilbert schemes, we compactify all two-dimensional Hitchin systems corresponding to types A0-tilde, D4-tilde, E6-tilde, E7-tilde, and E8-tilde, thereby obtaining four rational elliptic surfaces with C*-actions. Their singular fibers and relative minimal models are listed in the main table. To this end, we prove that Hilbert schemes of orbifold surfaces are connected smooth projective schemes under suitable conditions, and we use the Hilbert-Chow morphism to construct the minimal resolutions of the coarse moduli spaces.

Paper Structure

This paper contains 26 sections, 70 theorems, 106 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Notations and conventions
  4. Néron-Severi group and change of polarization
  5. Néron-Severi group of $\mathcal{X}$
  6. Change of Polarization
  7. The Poisson structure on the moduli spaces of stable sheaves
  8. The Atiyah class
  9. The Smoothness of the moduli space of sheaves
  10. The Kodaira-Spencer map
  11. The Poisson structure on $\mathcal{M}_\upsilon$
  12. The Connectedness of the moduli spaces of sheaves
  13. Orbifold Hilbert schemes
  14. Hilbert schemes
  15. Hilbert-Chow morphism
  16. ...and 11 more sections

Key Result

Theorem 1.1

Figures (13)

  • Figure 1: Configuration of curves on $\widetilde{X}_2$
  • Figure 2: Configuration before blowing down $D_\infty$
  • Figure 3: Fiber of type IV after blowing down $D_\infty$
  • Figure 4: Configuration of curves on $\widetilde{X}_3$
  • Figure 5: Configuration of the fiber $\pi_4^{-1}(\infty)$ before contractions.
  • ...and 8 more figures

Theorems & Definitions (134)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Theorem 2.2: Brochard
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 124 more