  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.
