Hilbert schemes on blowing ups and the free Boson

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.

arXiv commentThe Grothendieck group of coherent sheaves is also included. More typos are fixed