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Summands of theta divisors on Jacobians

Part of: Abelian varieties and schemes (Co)homology theory Categories with structure Linear algebraic groups and related topics

Published online by Cambridge University Press:  08 July 2020

Thomas Krämer
Thomas Krämer*
Affiliation:
Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099Berlin, Germany email thomas.kraemer@math.hu-berlin.de
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Abstract

We show that the only summands of the theta divisor on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the powers of the curve and the Fano surface of lines on the threefold. The proof only uses the decomposition theorem for perverse sheaves, some representation theory and the notion of characteristic cycles.

Keywords

Jacobian varietytheta divisorGauss mapcharacteristic cycleperverse sheafTannakian category

MSC classification

Primary: 14K12: Subvarieties
Secondary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories 20G05: Representation theory
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 7 , July 2020 , pp. 1457 - 1475
Copyright
© The Author 2020

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References

Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers, Astérisque, vol. 100 (Société Mathématique de France, 1982).Google Scholar
Bourbaki, N., Groupes et algèbres de Lie — Chapitres 4 à 6 (Springer, 2009), reprint of the first edition from 1968.Google Scholar
Bröcker, T. and tom Dieck, T., Representations of compact lie groups, Graduate Texts in Mathematics, vol. 98 (Springer, 1985).CrossRefGoogle Scholar
Casalaina-Martin, S., Popa, M. and Schreieder, S., Generic vanishing and minimal cohomology classes on abelian fivefolds, J. Algebra Geom. 27 (2018), 553–581.CrossRefGoogle Scholar
Clemens, C. H. and Griffiths, P. A., The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356.CrossRefGoogle Scholar
de Cataldo, M. and Migliorini, L., The decomposition theorem, perverse sheaves and the topology of algebraic maps, Bull. Amer. Math. Soc. (N.S.) 46 (2009), 535–633.CrossRefGoogle Scholar
Debarre, O., Minimal cohomology classes and Jacobians, J. Algebra Geom. 4 (1995), 321–335.Google Scholar
Debarre, O., Complex tori and abelian varieties, SMF/AMS Texts and Monographs, vol. 11 (American Mathematical Society, Société Mathématique de France, 2005).Google Scholar
Deligne, P. and Milne, J. S., Tannakian categories, Hodge Cycles, Motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900 (Springer, 1982), 101–228.Google Scholar
Gabber, O. and Loeser, F., Faisceaux pervers ℓ-adiques sur un tore, Duke Math. J. 83 (1996), 501–606.CrossRefGoogle Scholar
Goodman, R. and Wallach, N. R., Symmetry, representations and invariants (Springer, 2009).CrossRefGoogle Scholar
Heinloth, F., A note on functional equations for zeta functions with values in Chow motives, Ann. Inst. Fourier (Grenoble) 57 (2007), 1927–1945.CrossRefGoogle Scholar
Krämer, T., Perverse sheaves on semiabelian varieties, Rend. Semin. Mat. Univ. Padova 132 (2014), 83–102.10.4171/RSMUP/132-7CrossRefGoogle Scholar
Krämer, T., Cubic threefolds, Fano surfaces and the monodromy of the Gauss map, Manuscripta Math. 149 (2016), 303–314.CrossRefGoogle Scholar
Krämer, T., Characteristic cycles and the microlocal geometry of the Gauss map I, Preprint (2016), arXiv:1604.02389.Google Scholar
Krämer, T., Characteristic cycles and the microlocal geometry of the Gauss map II, Preprint (2018), arXiv:1807.01929.Google Scholar
Krämer, T. and Weissauer, R., Semisimple super Tannakian categories with a small tensor generator, Pacific J. Math. 276 (2015), 229–248.CrossRefGoogle Scholar
Krämer, T. and Weissauer, R., Vanishing theorems for constructible sheaves on abelian varieties, J. Algebra Geom. 24 (2015), 531–568.CrossRefGoogle Scholar
Martens, H. H., On the variety of special divisors on a curve, J. Reine Angew. Math. 227 (1967), 111–120.Google Scholar
Ran, Z., On subvarieties of abelian varieties, Invent. Math. 62 (1980), 459–480.CrossRefGoogle Scholar
Schnell, C., Holonomic 𝒟-modules on abelian varieties, Publ. Math. Inst. Hautes Études Sci. 121 (2015), 1–55.CrossRefGoogle Scholar
Schreieder, S., Theta divisors with curve summands and the Schottky problem, Math. Ann. 365 (2016), 1017–1039.Google Scholar
Schreieder, S., Decomposable theta divisors and generic vanishing, Int. Math. Res. Not. IMRN 2017 (2017), 4984–5009.Google Scholar
Weissauer, R., Brill-Noether sheaves, Preprint (2006), arXiv:math/0610923.Google Scholar
Weissauer, R., Vanishing theorems for constructible sheaves on abelian varieties over finite fields, Math. Ann. 365 (2016), 559–578.Google Scholar