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

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9

Families and Deformations

This chapter and the next one are devoted to variations of the Hodge structure, which will be one of the main objects of study of the second volume. Here, we content ourselves with establishing their essential properties. The preceding chapters allowed us to show the existence of a Hodge structure on the cohomology of a Kähler manifold, depending only on its complex structure. Now, we wish to describe how this Hodge structure varies with the complex structure.

In this chapter, we will establish various results from the theory of deformations of a compact complex manifold, which will enable us in the following chapter to formalise the notion of a period map (or a variation of Hodge structure), and to study its infinitesimal ...

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