Hyperk\"ahler bases for six rational bordism theories

TL;DR

This paper constructs explicit bases for various rational bordism theories using geometric tools like tori and Hilbert schemes of K3 surfaces, leveraging a theorem on the Milnor genus.

Contribution

It introduces explicit bases for rational symplectic and Spin bordism theories using geometric constructions and a recent theorem on Hilbert schemes of K3 surfaces.

Findings

Explicit bases for rational bordism theories are constructed.

The work connects geometric objects with algebraic bordism invariants.

Utilizes a theorem on the Milnor genus of Hilbert schemes of K3s.

Abstract

We use tori and Hilbert schemes of K3 surfaces to construct explicit bases for the real, complex, and quaternionic versions of rational symplectic and rational Spin bordism. The key input to our work is a theorem of Oberdieck, Song, and Voisin on the Milnor genus of Hilbert schemes of K3s.