AI 中文总结
本研究证明环面纽结奇点上点Quot概型的模空间可由仿射胞腔铺成，确定了其格罗滕迪克动机，提出了相关的Rogers–Ramanujan型恒等式、同调与几何对象的对应猜想及反常滤过猜想并部分验证。
AI 中文摘要
对于满足gcd(a,b)=1的a,b，我们证明了Bbbk[[T]]^n中余维数为m的Bbbk[[T^a,T^b]]-子模的模空间可由仿射胞腔铺成，证明思路是：一个与之密切相关的模空间在自然G_m作用下的每个Białynicki-Birula stratum（比亚维尼茨基-比鲁拉 stratum）都是不动点轨迹上的仿射丛，且不动点轨迹是迭代Grassmannian（格拉斯曼）丛。作为应用，我们用显式双变量级数N_{a,b;n}(q,t)确定了该模空间在格罗滕迪克簇环中的 motive（动机），并利用它显式计算了F_q[[T^a,T^b]]上有限模范畴的广群体积。\n级数N_{a,b;n}承载了我们随后提出的猜想。当n=∞时，我们通过对t变量取特殊化，提出了一族双无限的Rogers–Ramanujan（罗杰斯-拉马努金）型恒等式；我们将其乘积侧与W-代数极小模型W_a(a,a+b)上模的归一化特征等同起来，并观察到它与colored Jones tails（着色琼斯尾）的联系。当n<∞时，我们猜想N_{a,b;n}由环面纽结T(a,b)的三分次S^n-着色HOMFLY同调的底部α行计算得到，且同一底部行还可计算Bbbk[[T^a,T^b]]^n中有限余维Bbbk[[T^a,T^b]]-子模的Quot概型，以及非约化曲线(Y^a-X^b)^n=0的点Hilbert概型；这三个量是三分次在三个点处的特殊值，当n=1时，它们可同时恢复Oblomkov–Rasmussen–Shende猜想与Kivinen–Trinh猜想。\n最后我们猜想，这三个点未观测到的三分次的一个方向，是模空间自身上的perverse filtration（反常滤过），我们通过计算GL_2谱曲线族的分解定理，验证了它对n=2时光滑芽的预测。
英文摘要
For $\gcd(a,b)=1$, we show that the moduli space of $m$-codimensional $\Bbbk[\![T^a,T^b]\!]$-submodules of $\Bbbk[\![T]\!]^n$ is paved by affine cells, by proving that each Białynicki-Birula stratum of a closed related moduli space with respect to the natural $\mathbb{G}_m$-action is an affine bundle over the fixed point locus and that the fixed point locus is an iterated Grassmannian bundle. As an application, we determine the motive of this moduli space in the Grothendieck ring of varieties in terms of an explicit two-variable series $N_{a,b;n}(q,t)$, and use it to explicit compute the groupoid volume of the category of finite modules over $\mathbb{F}_q[\![T^a,T^b]\!]$. The series $N_{a,b;n}$ carries the conjectures we then formulate. At $n=\infty$ we conjecture a bi-infinite family of Rogers--Ramanujan type identities by specializing the $t$-variable; we identify their product side with the normalized character of a module over the $\mathcal{W}$-algebra minimal model $\mathcal{W}_a(a,a+b)$, and observe a connetion to colored Jones tails. At $n<\infty$ we conjecture that $N_{a,b;n}$ is computed by the bottom $α$-row of the trigraded $S^n$-colored HOMFLY homology of the torus knot $T(a,b)$, and that this same bottom row also computes the Quot schemes of finite codimensional $\Bbbk[\![T^a,T^b]\!]$-submoudles of $\Bbbk[\![T^a,T^b]\!]^n$ and the punctual Hilbert schemes of the non-reduced curve $(Y^a-X^b)^n=0$; the three quantities are special values at three points of the trigrading, and when $n=1$ they recover both the conjectures of Oblomkov--Rasmussen--Shende and of Kivinen--Trinh. Finally we conjecture that the one direction of the trigrading these three points do not see is a perverse filtration on the moduli spaces themselves, and we verify its prediction for a smooth germ at $n=2$ by computing the decomposition theorem for the $\mathrm{GL}_2$ spectral-curve family.
Comments60 pages
