[Paper Review] On the Hitchin fibration for algebraic surfaces

Tsao-Hsien Chen, Ngô Bảo Châu|arXiv (Cornell University)|Nov 7, 2017

Mathematics and Applications3 citations

TL;DR

This paper extends Hitchin's integrable system for Higgs bundles on Riemann surfaces to algebraic surfaces in characteristic zero, constructing a Hitchin fibration via spectral data and proving that the image of the spectral map lies in a canonical subvariety $ \mathscr{A}_C$ associated to a curve $C$ inside the surface. The key result establishes that the generic fibers of the Hitchin map are abelian varieties, generalizing Hitchin's original result to higher dimensions using degeneration techniques and nilpotent actions on Higgs bundles.

ABSTRACT

In this paper, we explore the structure of the Hitchin map for higher dimensional varieties with emphasis on the case of algebraic surfaces.

Motivation & Objective

• To generalize Hitchin's integrable system for Higgs bundles on Riemann surfaces to higher-dimensional algebraic surfaces.
• To define and study the Hitchin fibration for Higgs bundles on smooth projective surfaces over an algebraically closed field of characteristic zero.
• To construct a spectral data map from the moduli space of Higgs bundles to a parameter space $\mathscr{A}_C$ associated with a curve $C$ embedded in the surface.
• To prove that the image of the spectral data map is precisely $\mathscr{A}_C$, and that the generic fibers of the Hitchin map are abelian varieties.

Proposed method

• Uses the spectral data construction via the stack $[\mathfrak{C}_G^d / (G \times \mathrm{GL}_d)]$ to interpret Higgs bundles as maps to a quotient stack.
• Applies the Chevalley restriction theorem and its generalization to $d$-dimensional commuting tuples in the Lie algebra, conjectured to be isomorphic to $k[\mathfrak{t}^d]^W$.
• Constructs a $\mathrm{GL}_d$-equivariant lifting of the spectral map to $\mathrm{Spec}(k[\mathfrak{t}^d]^W)$, enabling the definition of the Hitchin map.
• Analyzes the action of the tangent sheaf $\mathcal{T}_{X^0/C}$ on the restriction of a Higgs bundle to the open locus $X^0$, proving nilpotence via trace arguments.
• Uses the fact that $\pi: X \to C$ has reduced fibers to show that sections of symmetric powers of $\Omega^1_X$ extend from the open locus to the whole surface.
• Applies the factorization of the Hitchin map through $\mathscr{B}_C$ to prove that the image of the spectral data map lies in $\mathscr{A}_C$.

Experimental results

Research questions

• RQ1 How can the Hitchin fibration for Higgs bundles be generalized from curves to algebraic surfaces?
• RQ2 What is the structure of the spectral data map for Higgs bundles on surfaces, and where does it land in the parameter space?
• RQ3 Under what conditions does the image of the spectral data map coincide with a canonical subvariety $\mathscr{A}_C$ associated to a curve $C$ in the surface?
• RQ4 Are the generic fibers of the Hitchin map on surfaces still abelian varieties, as in the classical case?
• RQ5 How does the nilpotent action of the relative tangent sheaf $\mathcal{T}_{X^0/C}$ on Higgs bundles constrain the spectral data?

Key findings

• The image of the spectral data map $\mathrm{sd}_X$ is exactly the subvariety $\mathscr{A}_C$, which is isomorphic to the moduli space of spectral data over the curve $C$.
• The generic fibers of the Hitchin map $h_X$ are abelian varieties, generalizing Hitchin's original result from curves to surfaces.
• The spectral data map factors through $\mathscr{A}_C$, and $\mathscr{A}_C$ is irreducible and closed in $\mathscr{A}_X$, so $\overline{\mathscr{A}_X^{\heartsuit}} = \mathscr{A}_C$.
• The nilpotence of the action of $\mathcal{T}_{X^0/C}$ on the restriction of a Higgs bundle to $X^0$ is proven via trace arguments, showing eigenvalues must vanish.
• Sections $a_i$ of $\mathrm{S}^i\Omega^1_X$ extend from the open locus $X^0$ to the whole surface because the complement has codimension $\geq 2$, implying $a_i \in \mathrm{H}^0(C, \mathrm{S}^i\Omega^1_C)$.
• The factorization of the Hitchin map through $\mathscr{B}_C$ implies that $h_X(E,\theta) \in \mathscr{B}_C(F)$, which confirms the image lies in $\mathscr{A}_C$.