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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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Mumford-Tate groups and Hodge structures
Indeed. Look at Hodge structures as finite dimensional vector spaces $V$ over $\mathbb Q$ together with an action of the Deligne torus $\mathbb S$ on $V \otimes \mathbb C$, i.e. a morphism of algebrai …