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Abelian varieties with no power isogenous to a Jacobian
Published 12 Jan 2024 in math.AG, math.CV, and math.NT | (2401.06577v2)
Abstract: For a curve of genus at least four which is either very general or very general hyperelliptic, we classify all ways in which a power of its Jacobian can be isogenous to a product of Jacobians of curves. As an application, we show that, for a very general principally polarized abelian variety of dimension at least four, or the intermediate Jacobian of a very general cubic threefold, no power is isogenous to a product of Jacobians of curves. This confirms some cases of the Coleman-Oort conjecture. We further deduce from our results some progress on the question whether the integral Hodge conjecture fails for such abelian varieties.
- Valery Alexeev “Compactified Jacobians and Torelli map” In Kyoto University. Research Institute for Mathematical Sciences. Publications 40.4, 2004, pp. 1241–1265
- Thorsten Beckmann and Olivier Gaay Fortman “Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties” In Compositio Mathematica 159.6, 2023, pp. 1188–1213
- “Complex abelian varieties” 302, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Springer-Verlag, Berlin, 2004, pp. xii+635
- Fabio Bardelli and Gian Pietro Pirola “Curves of genus g𝑔gitalic_g lying on a g𝑔gitalic_g-dimensional Jacobian variety” In Inventiones Mathematicae 95.2, 1989, pp. 263–276
- James Carlson “Extensions of mixed Hodge structures” In Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979 Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 107–127
- “The intermediate Jacobian of the cubic threefold” In Annals of Mathematics. Second Series 95, 1972, pp. 281–356
- Ching-Li Chai “Compactification of Siegel moduli schemes” 107, London Mathematical Society Lecture Note Series Cambridge University Press, Cambridge, 1985, pp. xvi+326
- “A numerical criterion for simultaneous normalization” In Duke Mathematical Journal 133.2, 2006, pp. 347–390
- “Abelian varieties isogenous to a Jacobian” In Annals of Mathematics. Second Series, 2012, pp. 589––635
- Alberto Collino “The fundamental group of the Fano surface. I” In Algebraic threefolds (Varenna, 1981) 947, Lecture Notes in Math. Springer, Berlin-New York, 1982, pp. 209–218, 219–220
- Olivier Debarre “Polarisations sur les variétés abéliennes produits” In C. R. Acad. Sci. Paris Sér. I Math. 323.6, 1996, pp. 631–635
- “Degeneration of abelian varieties” 22, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) Springer-Verlag, Berlin, 1990, pp. xii+316
- Allen Hatcher “Algebraic Topology” Cambridge University Press, Cambridge, 2002, pp. xii+544
- James Humphreys “Reflection groups and Coxeter groups” 29, Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge, 1990, pp. xii+204
- Aise Johan Jong “Smoothness, semi-stability and alterations” In Institut des Hautes Études Scientifiques. Publications Mathématiques, 1996, pp. 51–93
- Martin Kneser “Klassenzahlen definiter quadratischer Formen” In Archiv der Mathematik 8, 1957, pp. 241–250
- János Kollár “Shafarevich maps and plurigenera of algebraic varieties” In Inventiones Mathematicae 113.1, 1993, pp. 177–215
- “Measures of association between algebraic varieties, II: Self-correspondences” In Épijournal de Géométrie Algébrique 10, 2023
- “Curves of genus g𝑔gitalic_g on an abelian variety of dimension g𝑔gitalic_g” In Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae. New Series 13.4, 2002, pp. 523–535
- “The Oort conjecture on Shimura curves in the Torelli locus of curves” In Journal de Mathématiques Pures et Appliquées. Neuvième Série 123, 2019, pp. 41–77
- Valeria Ornella Marcucci “On the genus of curves in a Jacobian variety” In Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 12.3, 2013, pp. 735–754
- “Symmetric bilinear forms”, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73 Springer-Verlag, New York-Heidelberg, 1973, pp. viii+147
- James Milne “Abelian Varieties” In Arithmetic Geometry Springer New York, 1986, pp. 103–150
- Valeria Marcucci, Juan Carlos Naranjo and Gian Pietro Pirola “Isogenies of Jacobians” In Algebraic Geometry 3.4, 2016, pp. 424–440
- “The Torelli locus and special subvarieties” In Handbook of Moduli. Vol. II 25, Adv. Lect. Math. (ALM) Int. Press, Somerville, MA, 2013, pp. 549–594
- David Morrison “The Clemens-Schmid exact sequence and applications” In Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) 106, Ann. of Math. Stud. Princeton Univ. Press, Princeton, NJ, 1984, pp. 101–119
- “Abelian varieties isogenous to no Jacobian” In Annals of Mathematics. Second Series 191.2, 2020, pp. 635–674
- Juan Carlos Naranjo and Gian Pietro Pirola “Hyperelliptic Jacobians and isogenies” In Advances in Mathematics 335, 2018, pp. 896–909
- Juan Carlos Naranjo and Gian Pietro Pirola “On the genus of curves in the generic Prym variety” In Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae. New Series 5.1, 1994, pp. 101–105
- “Mixed Hodge Structures” 52, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics Springer-Verlag, Berlin, 2008, pp. xiv+470
- Chad Schoen “Bounds for rational points on twists of constant hyperelliptic curves” In Journal für die Reine und Angewandte Mathematik. [Crelle’s Journal] 411, 1990, pp. 196–204
- Jacob Tsimerman “The existence of an abelian variety over ℚ¯¯ℚ\overline{\mathbb{Q}}over¯ start_ARG blackboard_Q end_ARG isogenous to no Jacobian” In Annals of Mathematics. Second Series 176.1, 2012, pp. 637–650
- Claire Voisin “Hodge Theory and Complex Algebraic Geometry. II” 77, Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge, 2007, pp. x+351
- Claire Voisin “On the universal CH0Csubscript𝐻0{\mathrm{C}}H_{0}roman_C italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT group of cubic hypersurfaces” In Journal of the European Mathematical Society (JEMS) 19.6, 2017, pp. 1619–1653
- Claire Voisin “Cycle classes on abelian varieties and the geometry of the Abel-Jacobi map” To appear in Pure and Applied Mathematics Quarterly, 2022 arXiv:2212.03046 [math.AG]
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