Hyperkähler bases for six rational bordism theories

TL;DR

This paper constructs explicit, geometrically meaningful bases for rational Sp and Spin bordism theories in real, complex, and quaternionic flavors by leveraging hyperkähler geometry (notably Hilbert schemes of K3 surfaces) and torus factors. The key input is the Milnor genus computation for Hilbert schemes of K3s due to Oberdieck–Song–Voisin, which, together with carefully defined Sp^x and Spin^x structures, yields rational isomorphisms between the corresponding bordism theories and provides concrete generating sets B_*^x formed from K3^{[n]}-type manifolds and torus products. The authors demonstrate both the generation and linear independence of these sets over $Q$, using OSV22 and characteristic-number pairings to control independence, and they compute explicit ranks in each degree. They also justify the Sp^x constructions via rational homotopy equivalences of Thom spectra and classifying spaces, establishing a solid foundation for rational bordism bases with hyperkähler geometry, while noting a near miss for the Sp^c and Sp^h cases due to non-hyperkähler factors in certain generators. The resulting framework provides explicit, high-signal generators for four rational bordism rings and clarifies the interplay between hyperkähler geometry, torus factors, and spin/Sp-structures in bordism theory.

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.

Figures (1)

• Figure 1: Characteristic numbers of $B^c_{4n+2\delta}$

Theorems & Definitions (53)

• Theorem 1.6
• Remark 1.9
• Remark 1.10
• Remark 2.7
• Lemma 2.16
• proof
• Proposition 2.19
• proof
• Lemma 2.24
• proof