Pure and Applied Mathematics Quarterly
Volume 18 (2022)
Special issue celebrating the work of Herb Clemens
Guest Editor: Ron Donagi
Hilbert schemes of K3 surfaces, generalized Kummer, and cobordism classes of hyper-Kähler manifolds
Pages: 1723 – 1748
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.
Chern numbers, hyper-Kähler manifolds
2010 Mathematics Subject Classification
Primary 14J99. Secondary 14C17.
G.O. is funded by the Deutsche Forschungsgemeinschaft (DFG) - OB 512/1-1.
C. Voisin is supported by the ERC Synergy Grant HyperK (Grant agreement No. 854361).
Received 11 October 2021
Received revised 28 March 2022
Accepted 5 April 2022
Published 25 October 2022