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.

We study characteristic classes on hyperkähler manifolds with a view towards the Verbitsky component. The case of the second Chern class leads to a conditional upper bound on the second Betti number in terms of the Riemann-Roch polynomial, which is also valid for singular examples. We discuss the general structure of characteristic classes and the Riemann-Roch polynomial on hyperkähler manifolds using among other things Rozansky-Witten theory.

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.

The moduli space for polarized hyperkähler manifolds of ${\textrm {K3}^{[m]}}$-type or ${\textrm {Kum}}_m$-type with a given polarization type is not necessarily connected, which is a phenomenon that only happens for $m$ large. The period map restricted to each connected component gives an open embedding into the period domain, and the complement of the image is a finite union of Heegner divisors. We give a simplified formula for the number of connected components, as well as a simplified criterion to enumerate the Heegner divisors in the complement. In particular, we show that the image of the period map may be different when restricted to different components of the moduli space.