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

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7

Hodge Structures and Polarisations

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 ...

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