Work in progress with Michael Groechenig.
If $X$ is a smooth variety over a perfect field $k$ of characteristic $p>0$ which lifts to $W_2(k)$, its de Rham complex up to a certain level splits. It is the spectacular theorem of Deligne-Illusie from 1987, which led to manifold developments. It enabled Ogus-Vologodsky to develop a version of the Simpson correspondence, and later to Lan-Sheng-Zuo to define when $X$ is projective Higgs-de Rham flows, which over finite fields for flat connections with vanishing Chern classes, are preperiodic.
On the other hand, over the field of complex numbers, Brunebarbe-Klingler-Totaro proved, in answer to a question I had posed, that if $X$ smooth projective has no degree $\geq 1$ symmetric differential forms, then all flat connections are motivic, in fact they have finite monodromy. Their proof is transcendental.
We investigate a version of this theorem over a finite field. Under Deligne-Illusie $W_2(\mathbb{F}_q)$ assumption, we suspect that precisely the same vanishing condition on global symmetric differential forms forces flat connections with vanishing Chern classes to have finite monodromy. As of today, we can prove it in rank $\leq 3$.
De Kęstutis Česnavičius
De Takeshi Saito
De Miaofen Chen
De Ana-Maria Castravet
De Ana-Maria Castravet
De Ana-Maria Castravet
