(i) In 1960 Serre proved a famous analogue of the Weil conjectures for Kähler manifolds. This poses an obvious question: Does there exist an analogue of a Kähler structure for (non-singular) projective varieties over a finite field. That is, do there exist things like almost complex structures, Lefschetz operators, Kähler identities, etc?
(ii) Moreover, the study of generalised Hodge structures is a well-studied field. Does there exist a subfield of generalised Kähler structures?