Hilbert schemes of K3 surfaces, generalized Kummer, and cobordism classes of hyper-K\"ahler manifolds

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