pure Hodge structure

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Mixed Hodge structures

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special values of L functions

Computation in Hodge cohomology using the plectic conjecture, for abelian characters of CM fields and Hilbert modular forms. Yields a definition of a category of mixed Hodge structures: Expected to exist for arithmetic cohomology of a Shimura variety, using toroidal compactification See Tannakian categories.
Kahler

pure Hodge structure and Hodge theory
K3 Surfaces

See Hodge structure