$\u001e{mathcal{M}}_{g,n}$中 tautological 闭链的 Schur-Weyl 对偶与高阶 Abel-Jacobi 不变量
Schur-Weyl Duality and Higher Abel-Jacobi Invariants for Tautological Cycles in $\mathcal{M}_{g,n}$
AI总结：
本文研究带标记点的亏格g曲线模空间的Hodge理论，构建表示论框架建立辛李代数的Schur-Weyl对偶与高阶Abel-Jacobi不变量的联系，引入高阶Faber-Pandharipande闭链并证明其非挠性与滤过位置，首次搭建两类不变量的系统关联。
摘要：
本文研究具有n个标记点的亏格g曲线模空间的Hodge理论，建立了$\u001e{mathfrak{sp}}_{g}$的Schur-Weyl对偶与高阶Abel-Jacobi不变量之间的新联系。我们构建了一个表示论框架，利用与$\u001e{mathfrak{sp}}_{2g}$作用相容的Leray滤过和动机分解，按辛李代数表示分解$C_{g}^{n}$中tautological闭链的高阶Abel-Jacobi不变量。本工作的核心是引入**高阶Faber-Pandharipande闭链**$FP_n = π_1^{\u00d7 2}(Δ_{12}^n \u00b7 ψ_1)$（属于$CH^{n+1}(C_g^2)$），这是一类推广经典构造的新tautological闭链族。我们证明在最优亏格约束下这些闭链是非挠的：对于以(n-1)维空间为基的族，当$g \u2265 3n+1$时，$FP_n$不有理等价于零。此外，我们确定了$FP_n$在$C_g^2 \to M_g$的Leray滤过中的精确位置，证明它位于深度n+1且不再更深，在$H^{n+1}(M_g, R^{n+1}f_*\u001e{mathbb{Q}})$的$V_{(n+1,1)}$同型分量上有明确的非零性。这一结果首次建立了Schur-Weyl对偶与高阶超越不变量之间的系统性联系，揭示了高阶对角线编码了标准tautological类无法观测到的几何现象。
英文摘要：
This article investigates the Hodge theory of the moduli space of genus $g$ curves with $n$ marked points, establishing new connections between Schur-Weyl duality for $\mathfrak{sp}_{g}$ and higher Abel-Jacobi invariants. We develop a represe\\ ntation-theoretic framework that decomposes higher Abel-Jacobi invariants of tautological cycles in $C_{g}^{n}$ according to symplectic Lie algebra representations, leveraging the Leray filtration and motivic decompositions compatible with $\mathfrak{sp}_{2g}$-actions. Central to this work is the introduction of \textbf{higher Faber-Pandharipande cycles} $FP_n = π_1^{\times 2}(Δ_{12}^n \cdot ψ_1)$ in $CH^{n+1}(C_g^2)$, a new family of tautological cycles generalizing classical constructions. We prove these cycles are non-torsion under optimal genus constraints: for families over $(n-1)$-dimensional bases, $FP_n$ is not rationally equivalent to zero when $g \geq 3n+1$. Furthermore, we determine the precise position of $FP_n$ in the Leray filtration of $C_g^2 \to M_g$, showing it lies in depth $n+1$ but no deeper, with explicit non-vanishing in $H^{n+1}(M_g, R^{n+1}f_*\mathbb{Q})$ on the $V_{(n+1,1)}$-isotypic component. This yields the first systematic link between Schur-Weyl duality and higher transcendental invariants, revealing that higher diagonals encode geometric phenomena invisible to standard tautological classes.
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