Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 4

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

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n4.a13

Authors

Georg Oberdieck (Mathemathisches Institut, Universität Bonn, Germany)

Jieao Song (CNRS, Institut de Mathématiques de Jussieu, Paris, France)

Claire Voisin (CNRS, Institut de Mathématiques de Jussieu, Paris, France)

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.

Keywords

Chern numbers, hyper-Kähler manifolds

2010 Mathematics Subject Classification

Primary 14J99. Secondary 14C17.

Full Text (PDF format)

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