Qin关于点的Hilbert概形的拟模性猜想
Qin's quasimodularity conjecture for Hilbert schemes of points
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中文总结 AI 辅助
本文借助GPT-5.6 Sol证明了Qin关于点的Hilbert概形上扭结丛交点数的约化生成级数为拟模形式的猜想，并给出了显式算法。
中文摘要 AI 辅助
设$X$是一个具有数值平凡典范类的光滑射影复曲面。借助GPT-5.6 Sol，我们证明了Qin的猜想：即$X^{[n]}$的全Chern类与扭结丛的Chern特征的交点数的约化生成级数是一个具有预测的混合权界标的拟模形式。主要成分是一个Wick定理型公式，用于计算$\bigoplus_n H^*(X^{[n]})$上Nakajima算子的正规有序乘积的迹。遵循Li-Qin-Wang的论证，约化生成级数的计算被归结为计算某些算子值流与Carlsson-Okounkov算子乘积的超迹的常数项。我们的迹公式表明，该超迹可以用两个拟椭圆函数$\widehat Z$、$P$和一个拟模函数$T$来表示，根据Goujard-Moller定理，它们的常数项是拟模形式。最后，我们给出了计算一般约化生成级数的显式算法。
英文摘要
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.
