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The study of harmonic differential forms on complex projective varieties, their invariantly defined filtrations, their integrals over topological cycles, especially over subvarieties, the deformations of these integrals and filtrations in families, and a multitude of generalizations.
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When can Hodge filtrations (decompositions?) be described explicitly in terms of periods?
Building on Donu Arapura's answer (although it's again not clear you can do it in an hour), I would suggest the computing the Hodge filtration in $H^2$ of a product $E_1\times E_2$ of two elliptic cur …