Schur-Weyl Duality and Higher Abel-Jacobi Invariants for Tautological Cycles in $\mathcal{M}_{g,n}$
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866912509485645824 |
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| author | Rahmati, Mohammad Reza |
| author_facet | Rahmati, Mohammad Reza |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_05326 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Schur-Weyl Duality and Higher Abel-Jacobi Invariants for Tautological Cycles in $\mathcal{M}_{g,n}$ Rahmati, Mohammad Reza Algebraic Geometry Representation Theory 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. |
| title | Schur-Weyl Duality and Higher Abel-Jacobi Invariants for Tautological Cycles in $\mathcal{M}_{g,n}$ |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2011.05326 |