On the Hodge structures of compact hyperkähler manifolds

The purpose of this note is to give an account of a well-known folklore result: the Hodge structure on the second cohomology of a compact hyperk\"ahler manifold uniquely determines Hodge structures on all higher cohomology groups. We discuss the precise statement and its proof, which are somewhat difficult to locate in the literature.
Comment: 8 pages. Includes a more precise version of Theorem 3.6 and other minor corrections. To appear in Math. Res. Lett