Hilbert schemes of points and quasi-modularity (Q2658475)

Let \(X\) be a smooth complex algebraic surface. The Hilbert scheme \(X^{[n]}\) of \(n\) points on \(X\) is a smooth variety of dimension \(2n\). \textit{E. Carlsson} studied the generating series for the intersection pairings between the total Chern class of the tangent bundle and the Chern classes of tautological bundles on \((\mathbb C^2)^{[n]}\), proving that the reduced series \(\langle \text{ch}_{k_1} \dots \text{ch}_{k_N} \rangle^\prime\) is a quasi-modular form [Adv. Math. 229, 2888--2907 (2012; Zbl 1255.14005)]. \textit{A. Okounkov} conjectured that these reduced series are multiple \(q\)-zeta values [Funct. Anal. Appl. 48, 138--144 (2014; Zbl 1327.14026)]. \textit{Z. Qin} and \textit{F. Yu} [Int. Math. Res. Not. 2018, 321--361 (2018; Zbl 1435.14007)] proved the conjecture modulo lower weight terms via the reduced series \[ \overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q) = (q;q)_\infty^{\chi (X)} \cdot \sum_n q^n \int_{X^{[n]}} (\Pi_{i=1}^N G_{k_i} (\alpha_i,n)) c(T_{X^{[n]}}) \] where \(0 \leq k_i \in \mathbb Z\), \(\alpha_i \in H^* (X), (q;q)_\infty = \Pi_{n=1}^\infty (1-q^n)\) and \(G_{k_i}(\alpha_i, n) \in H^* (X^{[n]})\) are classes which play a role in the study of the geometry of \(X^{[n]}\) (see work of \textit{Z. Qin} [Hilbert schemes of points and infinite dimensional Lie algebras. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1403.14003)]). In the paper under review, the authors further study the series \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\). Defining functions \(\Theta_k^\alpha (q)\) depending on \(\alpha \in H^* (X)\) and \(k \geq 0\), they fix \(0 \leq k_1, \dots, k_N \in \mathbb Z\) and \(\alpha_1, \dots, \alpha_N \in H^* (X, \mathbb Q)\) and prove the following: (1) If \(\langle K_X^2,\alpha_i \rangle =0\) and \(2|k_i\) for each \(i\), then the leading term \(\Pi_{i=1}^N \Theta_k^\alpha (q)\) of \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) is either \(0\) or a quasi-modular form of weight \(\sum (k_i+2)\). (2) Suppose \(|\alpha_i|=4\) for each \(i\). If \(2|k_i\) for each \(i\), then \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) is a quasi-modular form of weight \(\sum (k_i+2)\). if \(2 \not |k_i\) for some \(i\), then \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)=0\). These results are proved by relating the leading term of \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) for \(X\) to the leading term of \(\langle \text{ch}_{k_1} \dots \text{ch}_{k_N} \rangle^\prime\) for \(\mathbb C^2\) studied by Carlsson [loc. cit.].