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arXiv · 2609.33884

Qin's quasimodularity conjecture for Hilbert schemes of points

Victor Alekseev·Avik Chakravarty·Daebeom Choi·Shengjing Xu
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Abstract

Let $X$ be a smooth projective complex surface with numerically trivial canonical class. With the help of GPT-5.6 Sol, we prove Qin's conjecture that the reduced generating series of intersection numbers of Chern characters of tautological bundles against the total Chern class of $X^{[n]}$ is a quasimodular form with the predicted mixed-weight bound. The main ingredient is a Wick's theorem-type formula for computing traces of normally ordered products of Nakajima operators on $\bigoplus_n H^*(X^{[n]})$. Following the argument of Li-Qin-Wang, the computation of the reduced generating series is reduced to computing the constant term of the supertrace of a product of certain operator-valued currents and the Carlsson-Okounkov operator. Our trace formula shows that this supertrace can be expressed in terms of two quasi-elliptic functions $\widehat Z$, $P$ and a quasimodular function $T$, whose constant terms are quasimodular forms by the theorem of Goujard-Moller. Finally, we give an explicit algorithm for computing the general reduced generating series.

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Victor Alekseev, Avik Chakravarty, Daebeom Choi, Shengjing Xu. 2026-09-27. Qin's quasimodularity conjecture for Hilbert schemes of points. https://arxiv.org/abs/2609.33884

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