Transcendental Hodge algebra

Abstract

The transcendental Hodge lattice of a projective manifold M is the smallest Hodge substructure in p-th 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 2 [(d+1)/2] .

Footnotes to papers of O’Grady and Markman

“…The Betti numbers of hyper-Kähler manifolds have been studied in [6,15] which establishes very precise bounds in dimension 4 and in [16] which claims similar bounds in dimension 6 (but the proof seems to be incomplete). The paper [7] gives very precise conjectural bounds (for example bounds on b 2 depending only on the dimension), depending on a conjecture on the Looijenga-Lunts representation [9,19]. The subject remains however wide open.…”

Footnotes to papers of O'Grady and Markman

“…The Betti numbers of hyper-Kähler manifolds have been studied in [17], [8] which establishes very precise bounds in dimension 4 and in [18] which claims similar bounds in dimension 6 (but the proof seems to be incomplete). The paper [9] gives very precise conjectural bounds (for example bounds on b 2 depending only on the dimension), depending on a conjecture on the Looijenga-Lunts representation [11], [19]. The subject remains however wide open.…”