On the symplectic forms of Groechenig's Higgs moduli over an elliptic curve

Technical Summary: On the Symplectic Forms of Groechenig's Higgs Moduli over an Elliptic Curve

Problem Statement
This paper addresses the geometric relationship between two distinct moduli spaces arising in the context of an elliptic curve $E$ and a finite cyclic group $\Gamma \subset \text{Aut}(E, o)$ with order $|\Gamma| \in \{1, 2, 3, 4, 6\}$.

1. The Hilbert Scheme Side: The space $S[m]_\Gamma$, defined as the Hilbert scheme of $m$ points on the surface $S_\Gamma = \Gamma\text{-Hilb}(T^*E)$. For $|\Gamma|=1$, $S_\Gamma = T^*E$. For $|\Gamma|>1$, $S_\Gamma$ is the unique crepant resolution of the quotient surface $X_\Gamma = T^*E/\Gamma$, which possesses rational double points.
2. The Parabolic Higgs Side: The moduli space $M_{\Gamma,m}$ of stable, strongly parabolic Higgs bundles on a coarse curve $C_\Gamma$ (either $E$ or $\mathbb{P}^1$ depending on $\Gamma$), with specific parabolic data determined by the affine Dynkin types $\tilde{A}_0, \tilde{D}_4, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$.

Previous work by Gorsky, Nekrasov, and Rubtsov (for $|\Gamma|=1$) and Groechenig (for $|\Gamma|>1$) established that these spaces are isomorphic as algebraic varieties via relative Fourier–Mukai transforms. However, the question of whether these isomorphisms preserve the natural holomorphic symplectic structures on both sides remained open. The paper aims to prove that the isomorphism $\Phi_{\Gamma,m}: S[m]_\Gamma \to M_{\Gamma,m}$ is a holomorphic symplectomorphism.

Methodology
The proof proceeds by explicit calculation of the symplectic forms on a dense open locus and extending the result globally. The methodology involves three main components:

1. Deformation Theory and Symplectic Forms:
   The paper utilizes the deformation complex of parabolic Higgs bundles (following Biswas and Ramanan) to construct the holomorphic symplectic form $\Omega_{M_{\Gamma,m}}$ on the moduli space $M_{\Gamma,m}$. This involves identifying the tangent space with the first hypercohomology of a two-term complex involving parabolic endomorphisms and strongly parabolic Higgs fields. The symplectic form is derived from a trace pairing on this complex.

2. Local Analysis on the Reduced Locus:
   The author restricts attention to the open dense locus $U_m \subset S[m]_\Gamma$ consisting of reduced subschemes (distinct points).

   • Case $|\Gamma|=1$: The isomorphism is analyzed using the relative Fourier–Mukai transform. The paper demonstrates that on the reduced locus, the parabolic Higgs moduli space corresponds to a direct sum of line bundles with distinct spectral data. By comparing the deformation complexes of the parabolic and ordinary (semistable) Higgs bundles, the author shows that the symplectic form on the Higgs side matches the natural symplectic form on the Hilbert scheme of points on $T^*E$.
   • Case $|\Gamma|>1$: The author employs the "orbifold–parabolic correspondence." They relate the parabolic Higgs bundles on the coarse curve $C_\Gamma$ to $\Gamma$-equivariant (orbifold) Higgs bundles on $E$. Using the Cartan–Leray spectral sequence, they establish an isomorphism between the hypercohomology of the parabolic deformation complex on $C_\Gamma$ and the $\Gamma$-invariant part of the hypercohomology of the orbifold complex on $E$. This allows the symplectic pairing on the parabolic side to be computed via the pairing on the elliptic curve, scaled by $1/|\Gamma|$.

3. Global Extension and Hitchin Compatibility:
   Since both symplectic forms are regular (holomorphic) and the spaces are smooth and irreducible, equality on the dense open set $U_m$ implies global equality. Additionally, the paper verifies that the isomorphism is compatible with the Hitchin maps (spectral data maps) on both sides, ensuring the integrable system structures align.

Key Contributions and Results

• Main Theorem (Theorem 1.1): The paper proves that for all five cases ($|\Gamma| \in \{1, 2, 3, 4, 6\}$), the isomorphism $\Phi_{\Gamma,m}$ constructed by Groechenig is a holomorphic symplectomorphism. Specifically, $\Phi_{\Gamma,m}^* \Omega_{M_{\Gamma,m}} = \omega_{S[m]_\Gamma}$, where $\omega_{S[m]_\Gamma}$ is the symplectic form induced by the cotangent bundle structure on $S_\Gamma$.
• Explicit Parabolic Data: The paper provides a detailed classification of the parabolic flags, weights, and stability conditions required to define the moduli spaces $M_{\Gamma,m}$ for each affine Dynkin type. This includes the specific construction of the "marked" line in the $\tilde{A}_0$ case and the multi-step flags for the $\tilde{D}_4, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$ cases.
• Orbifold-Parabolic Correspondence: The paper rigorously establishes the compatibility of the symplectic forms under the correspondence between parabolic bundles on the coarse quotient and equivariant bundles on the cover, clarifying the scaling factor $1/|\Gamma|$ in the Serre duality pairing.
• Integrable Systems: As a consequence, the paper constructs an algebraic integrable system (a Hitchin system) on the Hilbert scheme of points $S[m]_\Gamma$, confirming that the spectral data maps on the Hilbert scheme side correspond to the characteristic polynomial maps on the Higgs side.

Significance
The paper completes the geometric picture initiated by Gorsky, Nekrasov, Rubtsov, and Groechenig. While the isomorphism between these moduli spaces was previously known, the preservation of the symplectic structure is a crucial property for applications in mathematical physics, particularly in the context of the geometric Langlands program and the study of hyperkähler manifolds. By proving the symplectomorphism, the paper confirms that the Hilbert scheme of points on the crepant resolution of $T^*E/\Gamma$ carries the same symplectic geometry as the moduli space of parabolic Higgs bundles. This unifies the study of these spaces across the trivial and non-trivial cyclic group cases, providing a consistent framework for the five affine Dynkin types associated with elliptic curves.