Hilbert schemes of K3 surfaces, generalized Kummer, and cobordism classes of hyper-K\"ahler manifolds

TL;DR

This paper demonstrates that hyper-K"ahler manifolds' cobordism classes can be expressed as rational combinations of products of Hilbert schemes of K3 surfaces and generalized Kummer varieties, with a new formula for their top Chern character.

Contribution

It establishes a unique cobordism class decomposition for hyper-K"ahler manifolds using Hilbert schemes and Kummer varieties, and derives a closed formula for their tangent bundle's top Chern character.

Findings

Cobordism classes of hyper-K"ahler manifolds are generated by Hilbert schemes and Kummer varieties.

Derived a closed formula for the top Chern character of tangent bundles.

Unified the cobordism classification of hyper-K"ahler manifolds.

Abstract

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.