Connections and Symmetric Differential Forms

Abstract: (work in progress with Michael Groechenig) If $X$ is smooth complex projective and does not have any non-trivial symmetric differential forms, then all its complex local systems have finite monodromy (Brunebarbe-Klingler-Totaro ’13, in answer to a question I had posed). The proof relies on positivity theory stemming from Hodge Theory.

The aim is to understand a suitable formulation in characteristic $p>0$.

If $X$ is smooth projective over the algebraic closure $k$ of finite field, and does not have non-trivial differential forms, one may ask whether all convergent isocrystals have finite monodromy. This is true if $X$ lifts to $W(k)$. If $X$ lifts to $W_2(k)$, one can show that stable rank $2$ connections of degree $0$ have finite monodromy (i.e. are trivializable by a finite étale cover).

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

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