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Hodge Theory and Complex Algebraic Geometry I by Claire Voisin, Leila Schneps

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10

Variations of Hodge Structure

In this chapter, we introduce the period domain and the period map for a family of Kähler manifolds. The period domain parametrises the Hodge filtrations with fixed Hodge numbers on a fixed vector space V. It is an open set in a flag space, which can also be considered as a submanifold of a product of Grassmannians, via the map which associates the sequences of spaces Fi V to a filtration F·V on V. We thus devote a section to the construction of Grassmannians as complex manifolds, and the description of their tangent space.

Proposition 10.1 The tangent space of the Grassmannian of the subspaces of V at the point WV is canonically isomorphic to Hom (W, V/W ).

We then proceed to the study of the local period ...

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