On Okounkov's conjecture connecting Hilbert schemes of points and multiple q-zeta values

TL;DR

This paper verifies Okounkov's conjecture relating Hilbert schemes of points on surfaces to multiple q-zeta values, providing explicit calculations and proofs especially for abelian surfaces, and determines universal constants for tangent bundle Chern classes.

Contribution

It confirms Okounkov's conjecture modulo lower weight terms and fully for abelian surfaces, and computes universal constants for tangent bundle Chern classes.

Findings

Verification of Okounkov's conjecture modulo lower weight terms

Complete proof of the conjecture for abelian surfaces

Determination of universal constants for tangent bundle Chern classes

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].