Hilbert schemes on blowing ups and the free Boson

TL;DR

This paper explores the cohomology of moduli spaces of rank 1 perverse coherent sheaves on surface blow-ups, revealing their structure as tensor products involving Hilbert schemes and Fock spaces, with algebra actions given geometrically.

Contribution

It identifies the cohomology of these moduli spaces as tensor products with Fock spaces and describes algebra actions via geometric correspondences, extending previous understanding.

Findings

Cohomology of moduli spaces is expressed as tensor products with Fock spaces.

Actions of Clifford and Heisenberg algebras are realized geometrically.

Cohomology of Hilbert schemes on blow-ups is linked to that on original surfaces.

Abstract

For different cohomology theories (including the Hochschild homology, Hodge cohomology, Chow groups, and Grothendieck groups of coherent sheaves), we identify the cohomology of moduli space of rank 1 perverse coherent sheaves on the blow-up of a surface by Nakajima-Yoshioka as the tensor product of cohomology of Hilbert schemes with a Fermionic Fock space. As the stable limit, we identify the cohomology of Hilbert schemes of blow-up of a surface as the tensor product of cohomology of Hilbert schemes on the surface with a Bosonic Fock space. The actions of the infinite-dimensional Clifford algebra and Heisenberg algebra are all given by geometric correspondences.