For every smooth projective surface S with numerically trivial canonical class, and for all tautological insertions (in particular, line bundles L_1,…,L_N and indices k_1,…,k_N), the reduced generating series of tautological integrals on the Hilbert schemes of points S^[n], including series of the form langle ch_k_1^L_1… ch_k_N^L_Nrangle', is a quasi-modular form in q.

Qin’s quasi-modularity conjecture

For every smooth projective surface $S$ with numerically trivial canonical class, and for all tautological insertions (in particular, line bundles $L_1,\ldots,L_N$ and indices $k_1,\ldots,k_N$ ), the reduced generating series of tautological integrals on the Hilbert schemes of points $S^{[n]}$ , including series of the form $\left\langle \mathrm{ch}_{k_1}^{L_1}\cdots \mathrm{ch}_{k_N}^{L_N}\right\rangle'$ , is a quasi-modular form in $q$ .

References

Primary source

arXiv

Additional references

Quasi-modularity of q-traces and integrals over Hilbert schemes — arXiv — Killian Hong-Minh, Sergey Mozgovoy

Posted 5 days ago · edited 5 days ago

Progress summary

Refreshed 5 days ago

Claimed solved

A new preprint claims to prove Qin’s conjecture by establishing quasi-modularity for a broad class of generating series, but the claim has not been independently assessed.

Qin’s conjecture predicts quasi-modularity for reduced generating series of tautological integrals on smooth projective surfaces with numerically trivial canonical divisor. The intended general statement includes series such as $\left\langle \mathrm{ch}_{k_1}^{L_1}\cdots\mathrm{ch}_{k_N}^{L_N}\right\rangle'$ ; the conjecture was directly formulated and partially verified in 2024.

Known results

$\left\langle \mathrm{ch}_1^{L_1}\mathrm{ch}_1^{L_2}\right\rangle'$ was proved quasi-modular by Alhwaimel (2024).

October 2026 claimed proof

Hong-Minh and Mozgovoy’s preprint establishes quasi-modularity for a broad class of normalized $q$ -traces and applies this to the Hilbert-scheme tautological integrals covered by Qin’s conjecture, claiming the general result. The evidence is currently only the primary preprint, so this remains unverified.

Current status (as of October 2026): The conjecture has a claimed general proof in a new preprint, while independent verification is not yet recorded.

Sources

No solutions have been posted yet.