9 - Families and Deformations  pp. 219-238

Families and Deformations

By Claire Voisin and Leila Schneps

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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 properties. Starting from the notion of a family of compact complex manifolds, we show that by Ehresmann's theorem, such a family can be considered locally as a family of complex structures on a fixed differentiable manifold. In particular, the cohomology groups of the fibres Xt of this family can be considered locally as constant spaces by these trivialisations, and this will allow us to locally define the period map in the following chapter: indeed, the Hodge structure on the cohomology of the fibre Xt can be considered as a variable Hodge structure on a constant lattice.

The notion of a family of complex manifolds will give rise to the notion of a holomorphic deformation of the complex structure.We will concentrate here on the study of these families to first order, or on the functor of infinitesimal deformation of the complex structure.