
TL;DR
This paper introduces an algebraic structure on the sum of transcendental Hodge lattices of projective manifolds, explicitly computes it for hyperkähler manifolds, and applies this to derive a lower bound on the dimension of tori symplectically embedded in such manifolds.
Contribution
It establishes a natural algebraic structure on transcendental Hodge lattices and computes it explicitly for hyperkähler manifolds, leading to new geometric bounds.
Findings
The sum of transcendental Hodge lattices forms a natural algebraic structure.
Explicit computation of this algebraic structure for hyperkähler manifolds.
A lower bound on the dimension of tori embedded in hyperkähler manifolds.
Abstract
The transcendental Hodge lattice of a projective manifold is the smallest Hodge substructure in -th cohomology which contains all holomorphic -forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkahler manifold. As an application, we obtain a theorem about dimension of a compact torus admitting a symplectic embedding to a hyperkahler manifold . If is generic in a -dimensional family of deformations, then .
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