On the nilpotent residue non-abelian Hodge correspondence for higher-dimensional quasiprojective varieties

Technical Summary: On the Nilpotent Residue Non-Abelian Hodge Correspondence for Higher-Dimensional Quasiprojective Varieties

Problem Statement
The paper addresses the topological nature of the non-Abelian Hodge correspondence for higher-dimensional quasiprojective varieties. Specifically, it considers a projective log smooth complex variety $(\bar{X}, D)$, where $\bar{X}$ is a smooth projective variety and $D$ is a simple normal crossing divisor. The authors focus on two moduli spaces:

1. $M^{\text{nilp}}_{\text{Dol}}(\bar{X}, D, r)$: The moduli space of rank $r$ polystable logarithmic Higgs bundles with nilpotent residues and vanishing Chern classes ($c_i(E)=0$).
2. $M^{\text{nilp}}_{\text{DR}}(\bar{X}, D, r)$: The moduli space of rank $r$ polystable logarithmic connections with nilpotent residues.

In the work of [BBT24], a continuous bijection $S^{\text{nilp}}_{(\bar{X}, D)}: M^{\text{nilp}}_{\text{Dol}}(\bar{X}, D, r)(\mathbb{C}) \to M^{\text{nilp}}_{\text{DR}}(\bar{X}, D, r)(\mathbb{C})$ was established. While this map was proven to be a homeomorphism in the Euclidean topology when $\bar{X}$ is a curve, the higher-dimensional case remained open regarding the continuity of the inverse map. The central problem is to determine whether this continuous bijection is, in fact, a homeomorphism for higher-dimensional varieties.

Methodology
The paper employs a reduction strategy using "Lefschetz curves" to transfer topological properties from the curve case to the higher-dimensional case. The methodology proceeds through the following steps:

1. Lefschetz Curves: The authors utilize the concept of a Lefschetz curve $\iota: \bar{C} \hookrightarrow \bar{X}$, defined as a smooth complete intersection curve transverse to $D$ such that the induced map on fundamental groups $\pi_1(\bar{C} \setminus D) \twoheadrightarrow \pi_1(\bar{X} \setminus D)$ is surjective.
2. Restriction Maps: Such a curve induces restriction maps $i^*_{\text{Dol}}$ and $i^*_{\text{DR}}$ between the moduli spaces of $\bar{X}$ and $\bar{C}$. Since the correspondence is already known to be a homeomorphism for curves (by [BBT24]), the map $S^{\text{nilp}}_{(\bar{C}, D \cap \bar{C})}$ is a homeomorphism.
3. Commutative Diagrams: The authors construct a commutative diagram involving the Hitchin maps ($h$) for both the variety and the curve, and the restriction maps between the Hitchin bases.
4. Properness and Injectivity: To prove the main result, the authors establish two critical properties:
   • Properness of $i^*_{\text{Dol}}$: They prove the existence of a Lefschetz curve such that the restriction map on the Higgs moduli space is proper. This is achieved using the valuative criterion for properness, relying on the properness of the Hitchin map $h_{(\bar{X}, D)}$ (cited from [Lan14]) and the injectivity of the restriction map on the Hitchin base.
   • Injectivity of the Hitchin Base Restriction: They demonstrate that there exists a Lefschetz curve $\bar{C}$ such that the restriction map $\text{res}$ from the space of global sections of symmetric powers of logarithmic differentials on $\bar{X}$ to those on $\bar{C}$ is injective. This is proven by constructing a finite set of points and tangent directions where sections of the symmetric powers do not vanish, and then invoking [ACV19] to find a curve passing through these points with prescribed tangents.

Key Contributions and Results

• Theorem 1: The paper proves the existence of a Lefschetz curve $\bar{C} \subset \bar{X}$ such that the restriction map $i^*_{\text{Dol}}: M^{\text{nilp}}_{\text{Dol}}(\bar{X}, D, r) \to M^{\text{nilp}}_{\text{Dol}}(\bar{C}, D \cap \bar{C}, r)$ is proper.
• Corollary 2: As a direct consequence of Theorem 1 and the known curve case, the comparison map $S^{\text{nilp}}_{(\bar{X}, D)}$ is a homeomorphism in the Euclidean topology for higher-dimensional projective log smooth varieties.
• Technical Lemma (Proposition 3): The paper clarifies that the Hitchin map for logarithmic Higgs bundles with nilpotent residues and vanishing Chern classes is proper, noting that the condition of vanishing Chern classes defines a closed subscheme within the moduli space of torsion-free sheaves.
• Technical Lemma (Proposition 4): The paper establishes the existence of a Lefschetz curve that induces an injective restriction on the Hitchin base, a crucial step for ensuring the properness of the moduli space restriction.

Significance and Claims
The paper claims to resolve the topological question of the non-Abelian Hodge correspondence in the higher-dimensional setting for the specific case of nilpotent residues. By proving that the continuous bijection established in [BBT24] is a homeomorphism, the authors confirm that the moduli spaces of logarithmic Higgs bundles and logarithmic connections are topologically equivalent in the Euclidean topology, extending the known curve results to arbitrary dimensions.

The work relies heavily on the interplay between the geometry of Lefschetz curves and the algebraic properties of the Hitchin fibration. The significance lies in the rigorous extension of the homeomorphism result, utilizing the properness of the Hitchin map and the specific geometric construction of curves that preserve the necessary cohomological and fundamental group data. The paper does not propose new applications or future directions but focuses strictly on establishing the topological equivalence of these moduli spaces.