MathDB

Qin's quasi-modularity conjecture for Hilbert schemes of points

Let XX be a smooth projective complex surface, and let X[n]X^{[n]} be the Hilbert scheme of nn points on XX. For line bundles L1,…,LNL_1,\ldots,L_N on XX, let chk(Li[n]){\rm ch}_k(L_i^{[n]}) denote the kk-th Chern character of the tautological bundle, and define the reduced generating series

⟨chk1L1⋯chkNLN⟩′=(q;q)∞χ(X)∑n≥0qn∫X[n]∏i=1Nchki(Li[n])⋅c(TX[n]),\big \langle {\rm ch}_{k_1}^{L_1}\cdots {\rm ch}_{k_N}^{L_N}\big \rangle'=(q;q)_\infty^{\chi(X)}\sum_{n\geq 0}q^n\int_{X^{[n]}}\prod_{i=1}^N{\rm ch}_{k_i}(L_i^{[n]})\cdot c(T_{X^{[n]}}),

where (q;q)∞=∏m≥1(1−qm)(q;q)_\infty=\prod_{m\geq 1}(1-q^m). Qin's conjecture. If the canonical divisor of XX is numerically trivial, then ⟨chk1L1⋯chkNLN⟩′\big \langle {\rm ch}_{k_1}^{L_1}\cdots {\rm ch}_{k_N}^{L_N}\big \rangle' is a quasi-modular form of weight at most

∑i=1N(ki+2).\sum_{i=1}^N(k_i+2).

This conjecture predicts strong modularity properties for intersection-theoretic generating series on Hilbert schemes of points. The paper verifies it for the case ⟨ch1L1ch1L2⟩′\langle {\rm ch}_1^{L_1}{\rm ch}_1^{L_2}\rangle', while the general statement remains open.

Sources & referencesView supporting material

Primary source

Mazen M Alhwaimel, “Toward Qin's Conjecture on Hilbert schemes of points and quasi-modular forms”, arXiv:2407.01554 (2024).

Additional references

2 papers in this index state this conjecture (2023–2024). The statement above is taken from the most recent of them; the others are arXiv:2310.10812.

algebraic-geometryhilbert-schemesquasi-modular-forms

Progress summary

Refreshed
Partially solved

A 2024 paper verifies one two-factor case of Qin’s conjecture, while the general statement remains open and no later complete proof or counterexample was found.

Qin’s conjecture predicts that certain intersection generating series for Hilbert schemes of points are quasi-modular when the surface has numerically trivial canonical class, with an explicit weight bound. The full assertion is not yet proved.

Known results

  • Qin and Yu (2018): verified the conjecture modulo lower-weight terms.
  • Carlsson (2012): proved quasi-modularity for the corresponding series on (C2)[n](\mathbb{C}^2)^{[n]}.
  • Okounkov (2014): conjectured that the reduced series are multiple qq-zeta values.

2024–2025 partial developments

The 2024 paper proves the conjecture for ⟨ch1L1ch1L2⟩′\langle \mathrm{ch}_{1}^{L_1}\mathrm{ch}_{1}^{L_2}\rangle', giving the predicted weight bound 66. A May 2025 paper proves a related single equivariant series and obtains further restricted partial results, but does not establish Qin’s general statement.

Current status (as of August 2026): The case ⟨ch1L1ch1L2⟩′\langle \mathrm{ch}_{1}^{L_1}\mathrm{ch}_{1}^{L_2}\rangle' and several related special cases are settled, but Qin’s conjecture for arbitrary NN and kik_i remains open.

Sources

Solutions 0

No solutions have been posted yet.