Hilbert schemes of points on surfaces and multiple q-zeta values

For a line bundle $L$ on a smooth projective surface $X$ and nonnegative integers $k_1,\dots ,k_N$, Okounkov \cite{Oko} introduced the reduced generating series $⟨{\mathrm{c}\mathrm{h}}_{k_1}^L\cdots {\mathrm{c}\mathrm{h}}_{k_N}^L{⟩}^{\mathrm{\prime }}$ for the intersection numbers among the Chern characters of the tautological bundles over the Hilbert schemes of points on $X$ and the total Chern classes of the tangent bundles of these Hilbert schemes, and conjectured that they are multiple $q$-zeta values of weight at most ${\sum }_{i=1}^N(k_i+2)$. The second-named author further conjectured in \cite{Qin2} that these reduced generating series are quasi-modular forms if the canonical divisor of $X$ is numerically trivial. In this paper, we verify these two conjectures for $⟨{\mathrm{c}\mathrm{h}}_2^L{⟩}^{\mathrm{\prime }}$. The main approaches are to apply the procedure laid out in \cite{QY} and to establish various identities for multiple $q$-zeta values and quasi-modular forms.