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Hodge structures on cohomology algebras and geometry

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Abstract

We study restrictions on cohomology algebras of compact Kähler manifolds, imposed by the presence of a polarized Hodge structure on cohomology groups, compatible with the cup-product, but not depending on the h p,q numbers or the symplectic structure. To illustrate the effectiveness of these restrictions, we give a number of examples of compact symplectic manifolds satisfying the formality condition, the Lefschetz property and having commutative or trivial π 1, but not having the cohomology algebra of a compact Kaehler manifold. We also prove a stability theorem for these restrictions : if a compact Kähler manifold is homeomorphic to a product X × Y, with one summand satisfying b 1 = 0, then the cohomology algebra of each summand carries a polarized Hodge structure.

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  1. Institut de mathématiques de Jussieu, CNRS, UMR, 7586, Paris, France

    Claire Voisin

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  1. Claire Voisin
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Voisin, C. Hodge structures on cohomology algebras and geometry. Math. Ann. 341, 39–69 (2008). https://doi.org/10.1007/s00208-007-0181-4

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  • DOI: https://doi.org/10.1007/s00208-007-0181-4

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