Tautological and non-tautological cycles on the moduli space of Abelian varieties

• 成果类型: Article
• 署名作者: Canning, Samir; Oprea, Dragos; Pandharipande, Rahul
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich; University of California System; University of California San Diego
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01367-4
• 发表日期: 2025
• 页码: 659-723
• 关键词: chern classes; locus; ring; COHOMOLOGY

摘要

The tautological Chow ring of the moduli space Ag\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}_{g}$\end{document} of principally polarized abelian varieties of dimension g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g$\end{document} was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes from Ag\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}_{g}$\end{document} to the moduli space Mgct\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{M}_{g}<^>{\operatorname{ct}}$\end{document} of genus g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g$\end{document} of curves of compact type, we prove that the product class [A1xA5]is an element of CH5(A6)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[\mathcal{A}_{1}\times \mathcal{A}_{5}]\in \mathsf{CH}<^>{5}( \mathcal{A}_{6})$\end{document} is non-tautological, the first construction of an interesting non-tautological algebraic class on the moduli spaces of abelian varieties. For our proof, we use the complete description of the tautological ring R & lowast;(M6ct)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf{R}<^>{*}(\mathcal{M}_{6}<^>{\operatorname{ct}})$\end{document} in genus 6 conjectured by Pixton and recently proven by Canning-Larson-Schmitt. The tautological ring R & lowast;(M6ct)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf{R}<^>{*}(\mathcal{M}_{6}<^>{\operatorname{ct}})$\end{document} has a 1-dimensional Gorenstein kernel, which is geometrically explained by the Torelli pullback of [A1xA5]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[\mathcal{A}_{1}\times \mathcal{A}_{5}]$\end{document}. More generally, the Torelli pullback of the difference between [A1xAg-1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[\mathcal{A}_{1}\times \mathcal{A}_{g-1}]$\end{document} and its tautological projection always lies in the Gorenstein kernel of R & lowast;(Mgct)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf{R}<^>{*}(\mathcal{M}_{g}<^>{\operatorname{ct}})$\end{document}. The product map A1xAg-1 -> Ag\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}_{1}\times \mathcal{A}_{g-1}\rightarrow \mathcal{A}_{g}$\end{document} is a Noether-Lefschetz locus with general Neron-Severi rank 2. A natural extension of van der Geer's tautological ring is obtained by including more general Noether-Lefschetz loci. Results and conjectures related to cycle classes of Noether-Lefschetz loci for all g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g$\end{document} are presented.