[Paper Review] Non-abelian Hodge theory for algebraic curves in characteristic p

Tsao-Hsien Chen, Xinwen Zhu|arXiv (Cornell University)|Jun 3, 2013

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper establishes a non-abelian Hodge correspondence for algebraic curves in positive characteristic p, showing that the moduli space of flat G-bundles (de Rham local systems) is equivalent to the moduli space of G-Higgs bundles on the Frobenius twist C′ of the curve C. The correspondence is realized via a system of equations involving p-curvature and horizontality, generalizing Simpson's correspondence to positive characteristic using Cartier descent and Frobenius pullback.

ABSTRACT

Let G be a reductive group over an algebraically closed field of positive characteristic. Let C be a smooth projective curve over k. We give a description of the moduli space of flat G-bundles in terms of the moduli space of G-Higgs bundles over the Frobenius twist C' of C. This description can be regarded as the non-abelian Hodge theory for curves in positive characteristic.

Motivation & Objective

• To extend Simpson's non-abelian Hodge correspondence to algebraic curves over fields of positive characteristic p.
• To construct a correspondence between de Rham G-local systems on a curve C and G-Higgs bundles on its Frobenius twist C′.
• To resolve the lack of a full non-abelian Hodge theory in positive characteristic by introducing a characteristic p analogue of Hitchin's equations.
• To establish this correspondence using p-curvature and Cartier descent, leveraging the geometry of Frobenius morphisms and gerbes.
• To show that solutions of the characteristic p Hitchin-type equations yield a bijection between stable Higgs bundles and irreducible local systems, compatible with moduli space structures.

Proposed method

• Propose a characteristic p analogue of Hitchin's equations involving the p-curvature Ψ(∇) and horizontality of Ψ(∇) under a modified connection ∇−θ.
• Use the canonical connection ∇^can on the Frobenius pullback bundle to define the second equation in the system.
• Apply Cartier descent to show that a flat connection ∇ with vanishing p-curvature arises as a pullback from a bundle on C′.
• Show that the horizontality condition implies the Higgs field Ψ(∇) is a pullback of a Higgs field φ′ on C′, thus constructing a G-Higgs bundle on C′.
• Use the Azumaya property of crystalline differential operators to prove existence of local solutions for GL_n, and extend to reductive groups.
• Construct a pseudo-Bun_{G'}-torsor via the p-curvature map h_p, and identify it as the stack of splittings of a G′-gerbe associated to the Higgs field.

Experimental results

Research questions

• RQ1 Can a full non-abelian Hodge correspondence be established in positive characteristic p, analogous to Simpson's correspondence over C?
• RQ2 How can the Hitchin equations be reformulated in characteristic p to relate flat connections to Higgs bundles?
• RQ3 What role does the Frobenius twist C′ play in realizing the correspondence between de Rham local systems on C and Higgs bundles on C′?
• RQ4 How does the p-curvature map interact with the geometry of connections and Higgs fields in positive characteristic?
• RQ5 Can the correspondence be realized via a moduli-theoretic construction using gerbes and torsors in positive characteristic?

Key findings

• The moduli space of flat G-bundles on a smooth projective curve C over a field of characteristic p is canonically isomorphic to the moduli space of G-Higgs bundles on the Frobenius twist C′ of C.
• Solutions to the characteristic p Hitchin-type equations (Ψ(∇−θ)=0 and ((∇−θ)⊗∇^can)(Ψ(∇))=0) exist locally étale-locally on C for G=GL_n, via the Azumaya property of crystalline differential operators.
• The p-curvature map h_p: LocSys_G → Γ(C′, Lie G′ ⊗ Ω_{C′}) is smooth and surjective at the level of points, with its differential equal to −dh_p.
• The tangent space of the moduli of local systems at the trivial connection is isomorphic to the hypercohomology of the de Rham complex, and the map c: T_x → H^0(C′, Lie G′ ⊗ Ω_{C′}) is identified with −dh_p.
• The pseudo-Bun_{G'}-torsor arising from the p-curvature map is equivalent to the stack of splittings of a G′-gerbe associated to the Higgs field, providing a geometric interpretation of the correspondence.
• The correspondence is compatible with the Cartier isomorphism and generalizes the classical Simpson correspondence to positive characteristic, using Frobenius pullback and descent.