Qin's quasimodularity conjecture for Hilbert schemes of points

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 ⨁_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 Ẑ, 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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BibTeX RIS

BibTeX

@misc{alekseev2026qins,
  title = {{Qin's quasimodularity conjecture for Hilbert schemes of points}},
  author = {Victor Alekseev and Avik Chakravarty and Daebeom Choi and Shengjing Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33884},
  eprint = {2609.33884},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33884},
}

RIS

TY  - UNPB
TI  - Qin's quasimodularity conjecture for Hilbert schemes of points
AU  - Victor Alekseev
AU  - Avik Chakravarty
AU  - Daebeom Choi
AU  - Shengjing Xu
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33884
UR  - https://arxiv.org/abs/2609.33884
AB  - 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 ⨁_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 Ẑ, 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.
ER  -

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