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

Abstract

We prove that the complex cobordism class of any hyper-Kähler 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.

On fibrations and measures of irrationality of hyper-Kähler manifolds

“…In fact we prove in Theorem 4.5 that the dimension of the locus X k where f has rank k is expected to be at least 2k and the dimension of its image in B is expected to be at least k, unless some unexpected cohomological vanishing (29), (which is stronger than (3)), holds. As a consequence of our arguments, we get a positivity result for the Chern classes which provides some evidence for the questions asked in [35], see also [36].…”