Transcendental Hodge algebra

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 $M$ 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 hyperkahler manifold. As an application, we obtain a theorem about dimension of a compact torus $T$ admitting a symplectic embedding to a hyperkahler manifold $M$. If $M$ is generic in a $d$-dimensional family of deformations, then $\dim T\geq 2^{[(d+1)/2]}$.