this is a rough outline of some things that i'm thinking about right now. feel free to drop me an e-mail about any of this:
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comparing complex and \(p\)-adic period maps like in lawrence-venkatesh and betts-stix. in particular, one can construct a period map for varieties over \(\mathbb{Q}_p\) that encodes the ''variation of hodge structures on the unipotent fundamental group''. here are some questions:
- is there a \(p\)-adic analogue of pulte's theorem?
- how can the motivic nature of the Hodge structure on \(\pi_1\) be used to study period maps?
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behaviour of de rham local systems under morphisms of fundamental groups: the pullback of a local system along a geometric morphism preserves de rhamness. what happens if only a morphism on fundamental groups is given?
i'll probably fail to update this page regularly, so chances are that what is written here is wrong, trivial or perhaps actually turned out to be interesting.