Symplectic leaves of meromorphic Hitchin systems

Technical Summary: Symplectic Leaves of Meromorphic Hitchin Systems

Problem Statement
The paper addresses the geometric structure of the moduli space of stable meromorphic Higgs bundles, denoted $\text{Higgss}(C, D)$, where $C$ is a smooth projective curve and $D$ is a reduced effective divisor. While this space carries a holomorphic Poisson structure (established by Bottacin and Markman), it lacks a global holomorphic symplectic structure. A fundamental problem in Poisson geometry is the identification and understanding of its symplectic leaves.

In the context of meromorphic Higgs bundles, the residue of the Higgs field at the marked points $D$ provides the local model. The authors investigate the loci $\text{Higgs}_{\mathcal{O},s}(C, D)$ defined by fixing the residue of the Higgs field to lie in a specific product of adjoint orbits $\mathcal{O} = \prod \mathcal{O}_i$. The central questions are:

1. Are these loci $\text{Higgs}_{\mathcal{O},s}(C, D)$ indeed symplectic leaves of the ambient Poisson variety?
2. Can the Hitchin map, when restricted to these leaves, be partially compactified to form an algebraically completely integrable system (ACIS), particularly when the orbits are not of maximal rank (leading to singular spectral curves) or when the leaves are not closed?

Methodology
The authors employ a combination of deformation theory, algebraic geometry, and the spectral correspondence to resolve these questions.

1. Infinitesimal Analysis: The authors utilize the deformation theory of meromorphic Higgs bundles with fixed residue conditions. They construct the tangent complex $C^\bullet_{\mathcal{O},E}$ and its Serre dual to explicitly identify the tangent space of $\text{Higgs}_{\mathcal{O},s}(C, D)$ with the characteristic distribution of the Bottacin–Markman Poisson structure. This confirms that the locus is tangent to the Poisson structure.
2. Parabolic Resolution: To address the singularities arising from orbit closures and the lack of properness in the restricted Hitchin map, the authors introduce $\vec{\xi}$-parabolic Higgs bundles. These are triples $(E, E^\bullet_D, \Phi)$ where the Higgs field $\Phi$ has residues in $\mathcal{O}$, but the bundle $E$ is equipped with a flag structure $E^\bullet_D$ at the marked points compatible with the residue eigenvalues $\vec{\xi}$.
3. Spectral Correspondence: Building on previous work ([LL24], [LL25]), the authors utilize a spectral correspondence for $\vec{\xi}$-parabolic Higgs bundles. This relates the moduli space to a moduli space of pure one-dimensional sheaves on a specific surface $Z_{\vec{\xi}}$ (constructed via iterated blow-ups of a ruled surface). This correspondence allows them to describe the fibers of the Hitchin map as compactified Jacobians of spectral curves.
4. Forgetful Morphism: A key technical tool is the construction of a forgetful map $F: \text{Higgs}^{\vec{\xi}\text{-par}}_{\vec{\alpha}}(C, D, \vec{m}, \vec{\alpha}) \to \text{Higgs}_{\mathcal{O},ss}(C, D)$, which discards the parabolic flags. The authors analyze the stability conditions (using sufficiently small parabolic weights $\vec{\alpha}$) to ensure this map is well-defined, projective, and an isomorphism over the stable locus.

Key Contributions and Results

• Identification of Symplectic Leaves:
  The paper proves that for a product of conjugacy classes $\mathcal{O}$ satisfying the trace condition, the locus $\text{Higgs}_{\mathcal{O},s}(C, D)$ is an irreducible symplectic leaf of $\text{Higgss}(C, D)$. Its closure in the moduli space is $\text{Higgs}_{\mathcal{O},s}(C, D)$ (Theorem 1.3, Theorem 3.12, Theorem 3.13). This resolves the connectedness issue and confirms the symplectic nature of these strata.

• Partial Compactification to ACIS:
  The authors demonstrate that the moduli space of $\vec{\xi}$-parabolic Higgs bundles, $\text{Higgs}^{\vec{\xi}\text{-par}}(C, D, \vec{m}, \vec{\alpha})$, provides a partial compactification of the restricted Hitchin map.

  • The restricted Hitchin map $h^{(\vec{m})}_{\vec{\xi}}: \text{Higgs}^{\vec{\xi}\text{-par}} \to \mathcal{B}^{(\vec{m})}_{\vec{\xi}}$ is shown to be a proper, flat, Lagrangian fibration with generic fibers being abelian varieties (Theorem 1.7, Theorem 5.4).
  • This establishes that the restricted system is an algebraically completely integrable system, even when the underlying spectral curves are singular or the original leaf is not closed.

• Symplectic Resolution:
  The forgetful map $F$ is shown to be a symplectic resolution of the normalization of the closure $\text{Higgs}_{\mathcal{O},ss}(C, D)$ (Theorem 1.9, Theorem 5.11). Specifically, the normalization of the closure of the meromorphic leaf has symplectic singularities, and the smooth moduli space of $\vec{\xi}$-parabolic Higgs bundles resolves these singularities while preserving the symplectic structure.

• Connectedness of Betti Moduli Spaces:
  Via Simpson's tame non-abelian Hodge correspondence, the connectedness results for the Dolbeault side (parabolic Higgs bundles) are transferred to the Betti side.

  • The paper proves that the stable moduli space of filtered local systems with prescribed residue diagrams is connected (Theorem 1.10, Theorem 6.6).
  • A significant application is to character varieties $\chi(C^\circ, \mathcal{C})$ with prescribed local monodromy classes $\mathcal{C}$. Under the assumption that the eigenvalues of the monodromy matrices share a common argument (a condition ensuring compatibility with scalar graded residues), the irreducible locus $\chi(C^\circ, \mathcal{C})^{\text{irr}}$ is shown to be connected whenever it is non-empty (Theorem 1.11, Theorem 6.7). This extends previous connectedness results to non-generic monodromy classes.

Significance
The paper claims significance in providing a complete geometric picture of the symplectic leaves of meromorphic Hitchin systems, which were previously understood primarily in the maximal rank (closed leaf) case. By introducing $\vec{\xi}$-parabolic Higgs bundles, the authors:

1. Resolve the singularities of orbit closures in the moduli space of meromorphic Higgs bundles.
2. Extend the theory of algebraically completely integrable systems to arbitrary rank symplectic leaves, addressing the "Lax project" for non-maximal rank cases.
3. Establish new connectedness results for character varieties with non-generic local monodromy, bridging the gap between the Dolbeault and Betti sides of non-abelian Hodge theory for singular strata.

The work relies on the independent contributions of Bottacin and Markman regarding the Poisson structure and builds upon the authors' previous spectral correspondence results for parabolic Higgs bundles. It does not propose new experimental applications but rather deepens the theoretical understanding of the moduli spaces of Higgs bundles and their relation to representation theory.