On the image of Hitchin morphism for some classical groups on algebraic suefaces

Abstract: In this article, we study the image of the Hitchin morphism for some classical groups over an algebraic surface. The Hitchin morphism is a map from the moduli stack of $G$-Higgs bundles $\mathscr{M}{X,G}$ to the Hitchin base $\mathscr{A}{X,G}$, where $X$ is a smooth projective variety. In general, this morphism is not surjective when the dimension of $X$ is greater than one. Chen and Ng{ô} showed that the Hitchin morphism factors through a closed subscheme $\mathscr{B}{X,G}$ of the Hitchin base, which is called the spectral base. They conjectured that the image of the Hitchin morphism is exactly the spectral base. When $X$ is a smooth projective surface, we prove that this conjecture holds for the special linear algebraic group of odd rank. We also confirm this conjecture for the classical groups ${\rm SL}_n$ and ${\rm Sp}{2n}$ when $X$ is a product of smooth curves.

• The paper demonstrates that the image of the Hitchin morphism precisely equals the spectral base for classical groups under specific geometric conditions.
• It employs spectral cover analysis and determinantal formulas to rigorously control the geometry and cohomology of G-Higgs bundles on algebraic surfaces.
• The results advance non-abelian Hodge theory and provide a template for extending these methods to broader settings in the geometric Langlands program.

The Image of the Hitchin Morphism for Classical Groups on Algebraic Surfaces

Introduction and Context

This article rigorously investigates the image of the Hitchin morphism—originally defined in the context of Higgs bundles over algebraic curves—for certain classical groups when extended to algebraic surfaces. Specifically, the work targets the identification of the geometric image of the Hitchin morphism for principal $G$-Higgs bundles, with $G$ being classical groups such as $\mathrm{SL}_n$ (special linear), and $\mathrm{Sp}_{2n}$ (symplectic), defined over smooth projective surfaces. Chen and Ngô previously introduced the notion of a spectral base—a closed subscheme of the Hitchin base capturing the image of the Hitchin morphism—and conjectured that the Hitchin morphism onto this base is surjective in higher dimensions. This article establishes this conjecture for several important cases, extending known results for $\mathrm{GL}_n$ in dimension two and $\mathrm{GL}_2$ in arbitrary dimension.

Mathematical Foundations and Framework

Notational and Structural Setting

Let $X$ denote a smooth projective surface over an algebraically closed field of characteristic zero, and let $G$ be a connected reductive algebraic group. A $G$-Higgs bundle on $X$ is specified by a principal $G$0-bundle $G$1 and a Higgs field $G$2 satisfying the usual integrability constraint. The moduli space of semistable $G$3-Higgs bundles, $G$4, is thus equipped with the Hitchin morphism $G$5 mapping onto the Hitchin base $G$6, constructed from invariant polynomials on the Lie algebra $G$7 of $G$8, following Hitchin's and Simpson’s foundational work.

When $G$9 is a curve, $\mathrm{SL}_n$0 is dominant (and surjective for $\mathrm{SL}_n$1), while for $\mathrm{SL}_n$2, the morphism fails to be surjective in general. Instead, Chen and Ngô’s spectral base $\mathrm{SL}_n$3 conjecturally supports the actual locus realizable by Higgs bundles.

Main Results and Technical Approach

Surjectivity for Spectral Base

The article rigorously proves that the image of the Hitchin morphism is precisely the spectral base $\mathrm{SL}_n$4 in new settings:

• $\mathrm{SL}_n$5 for odd $\mathrm{SL}_n$6 and arbitrary smooth projective surface $\mathrm{SL}_n$7.
• $\mathrm{SL}_n$8 (arbitrary $\mathrm{SL}_n$9) and $\mathrm{Sp}_{2n}$0 when $\mathrm{Sp}_{2n}$1 is a product of smooth projective curves.

The proofs combine a spectral cover analysis, determinantal formulas for pushforwards of reflexive and Cohen–Macaulay modules, and an explicit construction of Higgs bundles with prescribed spectral data.

Key Construction Principles

The main strategy leverages faithful representations $\mathrm{Sp}_{2n}$2 for the classical groups, embedding the spectral base $\mathrm{Sp}_{2n}$3 into $\mathrm{Sp}_{2n}$4 and analyzing the pull-back and twist operations to realize required determinant and symmetry properties:

• For $\mathrm{Sp}_{2n}$5, the surjectivity rests on constructing $\mathrm{Sp}_{2n}$6-rank Higgs bundles on $\mathrm{Sp}_{2n}$7 with trivial determinant and trace-free Higgs field, through systematic manipulation of the spectral data and associated line bundles.
• When $\mathrm{Sp}_{2n}$8, with $\mathrm{Sp}_{2n}$9 smooth curves, the surface’s spectral base admits a decomposition corresponding to base spectral data on each factor, with the normalization of the surface spectral cover reducible to products of normalized curve spectral covers. This is critical for tracking the required geometric and cohomological properties.
• For $\mathrm{GL}_n$0, the spectral cover admits an involution corresponding to the symplectic form’s symmetries. The construction provides Higgs bundles equipped with perfect alternating forms and anti-self-adjoint Higgs fields, matching the spectral datum.

A cornerstone is an explicit determinantal formula for pushforward sheaves under finite flat morphisms between normal surfaces, employed to control the determinant of the underlying vector bundles.

Surjectivity claims are proven in all cases considered, and for $\mathrm{GL}_n$1 (odd $\mathrm{GL}_n$2), the conjecture of Chen–Ngô is thus confirmed for algebraic surfaces.

Technical Implications

This work clarifies the precise shape of the "non-abelian Hodge correspondence" in higher dimension, especially in the context of the geometric Langlands program and the algebro-geometric analysis of moduli of $\mathrm{GL}_n$3-Higgs bundles. The proof techniques are robust and generalizable—especially the decomposition of spectral data and the application of normalization and base change arguments. The explicit realization of various determinant and symmetry constraints on these bundles—especially under group embeddings—provides a template for potential extensions to further classical or exceptional groups.

The methods also establish cohomological and geometric control of Cohen–Macaulay and reflexive sheaves on singular spectral varieties, which is relevant in context of singular support phenomena and representation-theoretic questions for moduli of local systems.

Outcomes and Future Directions

The surjectivity results presented resolve outstanding cases of Chen–Ngô’s conjecture among classical groups, suggesting that, for classical groups over projective surfaces, the spectral base is the optimal surrogate for the Hitchin base. From a theoretical perspective, this facilitates the study of the fibers of the Hitchin map, integrability structures, and supports for category-theoretic correspondences in the geometric Langlands program.

Further investigations may focus on:

• Generalizations to other classical or exceptional groups beyond the cases handled.
• Analysis of the geometry and singularities of the spectral base in the wild ramification or non-reductive setting.
• Studying the interaction with the cohomological Hall algebra structures and the construction of global Springer theories on these moduli spaces.
• Explicit description of the connected components and their Hodge-theoretic or motivic invariants.

Conclusion

This article significantly advances the understanding of the image of the Hitchin morphism for classical groups on algebraic surfaces, confirming the spectral base surjectivity conjecture in several new cases. The technical framework blends deep aspects of spectral cover theory, normalization, and the algebraic geometry of Cohen–Macaulay sheaves, positioning these methods as foundational for higher-dimensional Non-Abelian Hodge theory and related areas. The explicit constructions and thorough proofs reinforce the spectral base as the sharp algebro-geometric target for the Hitchin morphism in higher dimensions.