7 - Hodge Structures and Polarisations  pp. 156-183

Hodge Structures and Polarisations

By Claire Voisin and Leila Schneps

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In this chapter, we give a synthesis of all the results proved up to now, and using it, we prove that the rational cohomology of a complex compact polarised manifold admits a decomposition as a direct sum of polarised Hodge structures.

To begin with, we define the integral and rational Hodge structures. These are the structures which lie naturally on the integral or rational cohomology of a compact Kähler manifold; they are given by the Hodge decomposition of the cohomology with complex coefficients. We study the case of the Hodge structure of weight 1; giving such a structure is equivalent to giving a complex torus. In the last section, we will study morphisms of Hodge structures, and the functoriality properties under direct or inverse image of the Hodge structure on the cohomology of Kähler manifolds relative to the holomorphic maps between two such manifolds.We also prove a very simple result on morphisms of Hodge structures, whose generalisation to mixed Hodge structures (which will be explained in the second volume of this work) has numerous applications.

Lemma 7.1The morphisms of Hodge structures are strict for the Hodge filtration.

Polarisation is the major notion introduced in this chapter. The Lefschetz decomposition and the Hodge index theorem allow us to write the cohomology of a compactKähler manifold as a direct sum of primitive components, compatible with the Hodge decomposition, on which the Hermitian intersection form given by the Lefschetz operator has signs defined on each component of type (p, q).

This Lefschetz decomposition is not a decomposition as a direct sum of rational sub-Hodge structures, except when the operator L preserves the rational cohomology.