On the Euler characteristics of certain moduli spaces of \(1\)-dimensional closed subschemes (Q2025313)

Let \(X\) be a smooth complex projective variety of dimension \(r\). For \(\beta \in H_2 (X, \mathbb Z)\) and \(n \in \mathbb Z\), let \(\mathcal J_n (X, \beta)\) be the moduli space of one dimensional closed subschemes \(Z \subset X\) with \(\chi ({\mathcal O}_Z)=n\) and \([Z]=\beta\). Letting \(X^{[n]}\) denote the Hilbert scheme of \(n\) points on \(X\), \textit{Li and Qin} conjectured that the reduced partition function of Euler characteristics \[ \frac{\sum_{n \in \mathbb Z} \chi(\mathcal J_n (X, \beta)) q^n}{\sum_{n \geq 0} \chi(X^{[n]}) q^n} \] is a rational function of \(q\) and invariant under \(q \mapsto 1/q\) when \(K_X =0\) [\textit{W.-P. Li} and \textit{Z. Qin}, Commun. Anal. Geom. 14, 387--410 (2006; Zbl 1111.14002)]. Assuming \(X\) admits a Zariski-locally trivial vibration \(\mu: X \to S\) with \(S\) smooth projective and the fibers of \(\mu\) smooth connected genus \(g\) curves, they verified their conjecture if \(2 \leq \dim X \leq 3\) or \(K_X =0\). Expanding on their ideas, the author assumes a fibration \(\mu:X \to S\) as above and computes the partition function \(\displaystyle \sum_{n \geq 0} \chi (\mathfrak M_{2,n}) q^n\), where \(\mathfrak M_{d,n} = \mathcal J_{d(1-g)+n} (X, d \beta)\). The answer is given in terms of \(\chi (S)\), \(\sum_{n \geq 0} \chi (X^{[n]}) q^n\), and the numbers of certain \(r\)-dimensional partitions. The proof is achieved by decomposition \(\mathfrak M_{2,n}\) into two parts, depending on whether \(Z\) is the support of a component of a fiber of \(\mu\) or the union of the supports of two such fibers and using virtual Hodge polynomials to reduce the calculation to a an appropriate local model. The calculation confirms the conjecture of Li and Qin in some special cases.