The $p$-curvature conjecture for the non-abelian Gauss-Manin connection

TL;DR

This paper extends the $p$-curvature conjecture to the non-abelian setting using Simpson's non-abelian Hodge theory, introducing a non-abelian Gauss-Manin connection and proving a key technical result.

Contribution

It introduces the non-abelian Gauss-Manin connection on the stack of vector bundles with integrable connection and proves an analogue of Katz's main theorem in this context.

Findings

Defined the non-abelian Gauss-Manin connection and its $p$-curvature.

Proved the analogue of Katz's main technical result for the non-abelian connection.

Generalized the $p$-curvature conjecture to the non-abelian setting.

Abstract

Originally conjectured unpublished by Grothendieck, then formulated precisely by Katz, the $p$-curvature conjecture is a local-global principle for algebraic differential equations. It is at present open, though various cases are known. Katz subsequently proved this conjecture in a wide range of cases, for differential equations corresponding to the Gauss-Manin connection on algebraic de Rham cohomology. This dissertation addresses the non-abelian analogue of Katz' theorem, in the sense of Simpson's non-abelian Hodge theory, surveyed by Simpson and later developed in characteristic $p$ by Ogus and Vologodsky. Specifically, there is a canonical non-abelian Gauss-Manin connection on $M_{dR}$, the stack of vector bundles with integrable connection, which is the appropriate definition of non-abelian de Rham cohomology. In this dissertation, I introduce this connection and its $p$-curvature; this requires the generalization of the $p$-curvature conjecture due to Bost, Ekedahl and Shepherd-Barron. Then, I prove that the analogue of the main technical result of Katz' theorem holds for this connection.