Hilbert schemes of K3 surfaces, generalized Kummer, and cobordism classes of hyper-Kähler manifolds

We prove that the complex cobordism class of any hyper-K\{a}hler manifold dimension $2n$ is a unique combination with rational coefficients classes products punctual Hilbert schemes $K3$ surfaces. also similar result using generalized Kummer varieties instead schemes. As key step, we establish closed formula for top Chern character their tangent bundles.