On Okounkov’s Conjecture Connecting Hilbert Schemes of Points and Multiple $q$-Zeta Values

Abstract

We compute the generating series for the intersection pairings between the total Chern classes of the tangent bundles of the Hilbert schemes of points on a smooth projective surface and the Chern characters of tautological bundles over these Hilbert schemes. Modulo the lower weight term, we verify Okounkov's conjecture [Oko] connecting these Hilbert schemes and multiple $q$-zeta values. In addition, this conjecture is completely proved when the surface is abelian. We also determine some universal constants in the sense of Boissi\' ere and Nieper-Wisskirchen [Boi, BN] regarding the total Chern classes of the tangent bundles of these Hilbert schemes. The main approach of this paper is to use the set-up of Carlsson and Okounkov outlined in [Car, CO] and the structure of the Chern character operators proved in [LQW2].