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Transcendental Hodge algebra

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Abstract

The transcendental Hodge lattice of a projective manifold M is the smallest Hodge substructure in pth cohomology which contains all holomorphic p-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkähler manifold. As an application, we obtain a theorem about dimension of a compact torus T admitting a holomorphic symplectic embedding to a hyperkähler manifold M. If M is generic in a d-dimensional family of deformations, then \(\dim T\ge 2^{[(d+1)/2]}\).

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Authors and Affiliations

  1. Laboratory of Algebraic Geometry, Department of Mathematics, National Research University HSE, 7 Vavilova Str., Moscow, Russia

    Misha Verbitsky

  2. Université Libre de Bruxelles, CP 218, Bd du Triomphe, 1050, Brussels, Belgium

    Misha Verbitsky

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  1. Misha Verbitsky
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Correspondence to Misha Verbitsky.

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Partially supported by RSCF, grant number 14-21-00053 within AG laboratory, NRU-HSE.

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Verbitsky, M. Transcendental Hodge algebra. Sel. Math. New Ser. 23, 2203–2218 (2017). https://doi.org/10.1007/s00029-017-0307-9

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