分层动机不变量与庞加莱多项式的二元变形
Stratified motivic invariants and bivariate deformations of Poincaré polynomials
AI总结：
【一句话总结】针对分层簇\(X\)和动机不变量\(\mathbb{H}\)，考虑插值不变量的分层不变量，引入分层簇的阶梯塔概念，证明相关显式归纳公式，表明文献中特定庞加莱多项式的神秘二元变形是分层虚拟庞加莱多项式。
AI中文摘要：
对于分层簇\(X\)和动机不变量\(\mathbb{H}\)，我们考虑插值\(X\)及其内部\(X^{\circ}\)不变量的分层不变量。基于模理论的观察，引入分层簇的阶梯塔概念，并证明了亏格为\(0\)的稳定曲线的模空间\(\overline{M}_{0,n}\)和任意光滑射影簇\(Y\)的富尔顿 - 麦克弗森簇\(Y[n]\)的分层不变量的显式归纳公式。利用这些，表明文献\(\cite{BercziKiem2026}\)中\(\overline{M}_{0,n + 1}\)和\(\mathbb{P}^1[n]\)的偶次庞加莱多项式的神秘二元变形正是分层虚拟庞加莱多项式。
英文摘要：
For a stratified variety $X$ and a motivic invariant $\mathbb{H}$, we consider the stratified invariant which interpolates the invariant of $X$ and that of its interior $X^{\circ}$. Based on observations in moduli theory, we introduce the notion of an echelon tower of stratified varieties and then prove explicit inductive formulae for the stratified invariants of the moduli spaces $\overline{M}_{0,n}$ of stable curves of genus $0$ and the Fulton-MacPherson varieties $Y[n]$ for any smooth projective variety $Y$. Using these, we show that the mysterious bivariate deformations of the even degree Poincaré polynomials of $\overline{M}_{0,n+1}$ and $\mathbb{P}^1[n]$ in \cite{BercziKiem2026} are nothing but stratified virtual Poincaré polynomials.
