Non-Abelian p-Curvature and a Non-Abelian Katz's Formula

1. Problem Statement

The paper addresses the generalization of Katz's formula, a fundamental result in arithmetic geometry relating the Kodaira–Spencer map of a smooth proper morphism $f: X \to S$ to the $p$-curvature of the Gauss–Manin connection in characteristic $p$.

• Context: In classical (abelian) Hodge theory, Katz's formula describes how the Gauss–Manin connection acts on the associated graded pieces of the Hodge and conjugate filtrations.
• The Gap: Recent work by Lam and Litt [12] established a non-abelian variant of this formula, relating non-abelian Higgs fields to non-abelian $p$-curvature. However, their proof relied on specific constructions (syntomification) that were somewhat ad hoc.
• Goal: The author aims to provide a conceptual, stack-theoretic proof of the non-abelian Katz's formula, following a suggestion by Lam and Litt to utilize the theory of sheared de Rham stacks. The goal is to simplify the construction of the objects involved and clarify the underlying geometric mechanisms.

2. Methodology

The paper employs the language of derived algebraic geometry and stacks, specifically utilizing the framework of sheared de Rham stacks developed by Bhatt, Kanaev, Mathew, Vologodsky, Zhang, and Drinfeld.

• Sheared de Rham Stacks ($(S/k)^c_{dR}$): Instead of working with vector bundles with flat connections directly, the author works with vector bundles on the sheared de Rham stack. This stack geometrizes the category of $D$-modules (vector bundles with flat connections) in characteristic $p$.
  • Unlike the classical de Rham stack (which handles connections with nilpotent $p$-curvature), the sheared version handles general flat connections by quotienting by a groupoid of nilpotent divided powers rather than just nilpotent elements.
• Filtrations via Stacks: The paper constructs the Hodge filtration and the conjugate filtration not as filtrations on a vector bundle, but as geometric families of stacks over $\mathbb{A}^1/\mathbb{G}_m$:
  • Hodge-filtered stack: $(S/k)^c_{dR,+}$, which geometrizes $\lambda$-connections.
  • Conjugate-filtered stack: $(S/k)^c_{dR,c}$, which geometrizes connections equipped with a horizontal map related to $p$-curvature.
• Non-Abelian Connections: The author defines non-abelian connections as Cartesian diagrams involving mapping stacks into the classifying stack $B SL_n$. This allows the treatment of non-linear differential equations (moduli of bundles with connections) within the stacky framework.
• Griffiths Transversality: The paper introduces a non-abelian analogue of Griffiths transversality, allowing the extraction of "Higgs fields" and "$p$-curvature" as morphisms between tangent stacks of the associated graded moduli spaces.

3. Key Contributions and Constructions

A. The Sheared de Rham Stack and $p$-Curvature

The author rigorously defines the sheared de Rham stack $(S/k)^c_{dR}$ and explains its relationship to the Frobenius twist $S'$.

• Key Insight: The map $(S/k)^c_{dR} \to S'$ is a torsor for $B V(F^* \Omega^1_{S'/k})$.
• $p$-Curvature as a Pullback: The $p$-curvature of a connection is defined geometrically as the pullback of the connection along the map from the "twisted" stack $S^{\psi_p}$ to $(S/k)^c_{dR}$. This recovers the classical definition of $p$-curvature (an endomorphism $\psi_p: E \to F^* \Omega^1_S \otimes E$) in a purely stack-theoretic manner.

B. Non-Abelian Higgs Fields and $p$-Curvature

The paper constructs the non-abelian analogues of the Higgs field and $p$-curvature:

1. Non-Abelian Higgs Field ($\Theta_{X/S}$): Derived from the Hodge filtration on the moduli stack of flat connections $M_{dR}(X/S, n)$. By base-changing to the Hodge stack $(S/k)_{Hodge}$, the author constructs a morphism:
   $$\Theta_{X/S}: \pi^* T_{S/k}(-1) \to T M_{Dol}$$
   where $M_{Dol}$ is the Dolbeault moduli stack.
2. Non-Abelian $p$-Curvature ($\psi_{X/S}|_{\lambda=0}$): Derived from the conjugate filtration. By base-changing to the conjugate Hodge stack, a morphism is constructed:
   $$\psi_{X/S}|_{\lambda=0}: \pi_c^* F^*_{S/k} T_{S'/k}(+1) \to T M_{Dol,c}$$

C. The Main Theorem (Non-Abelian Katz's Formula)

The central result establishes a precise relationship between the Higgs field and the $p$-curvature via the Frobenius twist.

Theorem 5.12: There exists a Cartesian diagram of stacks relating the conjugate and Hodge moduli spaces:
$$\begin{array}{ccc} M_{Dol,c} & \xrightarrow{\phi} & M_{Dol} \\ \downarrow & & \downarrow \\ S \times B\mathbb{G}_m & \xrightarrow{(F_{abs}, [-1])} & S \times B\mathbb{G}_m \end{array}$$
The main formula states:
$$\phi^* \Theta_{X/S} = \psi_{X/S}|_{\lambda=0}$$
where:

• $\phi$ is the morphism induced by the Frobenius twist and a sign change $[-1]$ on the $\mathbb{G}_m$-action.
• $\Theta_{X/S}$ is the non-abelian Higgs field.
• $\psi_{X/S}|_{\lambda=0}$ is the non-abelian $p$-curvature.

4. Results and Technical Details

• Sign Convention: The paper explicitly addresses the sign $[-1]$ in the formula. In the abelian case, Katz's formula involves a sign of $-1$ (often attributed to a historical algebraic error corrected by André). The stacky approach forces this sign naturally because the twist in the conjugate filtration ($+1$) must be reconciled with the twist in the Higgs field ($-1$) via the Frobenius map.
• Comparison to Lam–Litt: While Lam and Litt [12] proved a similar result, their construction required an intermediate object $\Psi_{X/S}$ and a localization argument to divide by a parameter $\lambda$. The stacky approach constructs the object $\psi_{X/S}$ directly without needing to invert $\lambda$, offering a more global and conceptual proof.
• Geometric Interpretation: The proof relies on the observation that the Hodge and conjugate filtrations can be viewed as living on the same underlying stack structure, related by the Frobenius twist and a shift in the $\mathbb{G}_m$-weight. The "gluing" of these filtrations (ignoring the Frobenius twist) is visualized as gluing two copies of $\mathbb{A}^1/\mathbb{G}_m$ along their closed points.

5. Significance

• Conceptual Clarity: The paper successfully translates a concrete arithmetic phenomenon (Katz's formula) into the language of sheared de Rham stacks. This provides a unified geometric framework for understanding both the Hodge and conjugate filtrations in the non-abelian setting.
• Simplification: It simplifies the construction of the non-abelian $p$-curvature and Higgs field, removing the need for the specific syntomic constructions used by Lam and Litt.
• Foundation for Future Work: By establishing this formula using the stacky approach, the paper paves the way for applying techniques from $p$-adic Hodge theory (specifically syntomification and the theory of de Rham stacks) to the Ekedahl–Shepherd-Barron–Taylor conjecture (the non-abelian $p$-curvature conjecture).
• Generalizability: The constructions of non-abelian Griffiths-filtered connections and the extraction of invariants (like Higgs fields) from split gerbes are presented as general tools that may be useful in other contexts of non-linear Hodge theory.

In summary, Michael Barz's paper provides a rigorous, stack-theoretic derivation of the non-abelian Katz's formula, demonstrating that the relationship between the Kodaira–Spencer class and $p$-curvature is a natural consequence of the geometry of sheared de Rham stacks and the Frobenius twist.