Transcendental Hodge algebra

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