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Characteristic cycles and the microlocal geometry of the Gauss map, II

  • Thomas Krämer EMAIL logo


We show that any Weyl group orbit of weights for the Tannakian group of semisimple holonomic 𝒟 -modules on an abelian variety is realized by a Lagrangian cycle on the cotangent bundle. As applications we discuss a weak solution to the Schottky problem in genus five, an obstruction for the existence of summands of subvarieties on abelian varieties, and a criterion for the simplicity of the arising Lie algebras.


I would like to thank Giulio Codogni, Michael Dettweiler, Gavril Farkas, Javier Fresán, Bert van Geemen, Bruno Klingler, Mihnea Popa and Stefan Schreieder for stimulating discussions and comments, and the anonymous referee for helpful remarks.


[1] A. Andreotti and A. L. Mayer, On period relations for abelian integrals on algebraic curves, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3) 21 (1967), 189–238. Search in Google Scholar

[2] M. F. Atiyah and D. O. Tall, Group representations, λ-rings and the J-homomorphism, Topology 8 (1969), 253–297. 10.1016/0040-9383(69)90015-9Search in Google Scholar

[3] A. A. Beĭlinson, J. Bernstein and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy 1981), Astérisque 100, Soc. Math. France, Paris (1982), 5–171. Search in Google Scholar

[4] C. Birkenhake and H. Lange, Complex abelian varieties, 2nd ed., Grundlehren Math. Wiss. 302, Springer, Berlin 2004. 10.1007/978-3-662-06307-1Search in Google Scholar

[5] T. Bröcker and T. tom Dieck, Representations of compact Lie groups, Grad. Texts in Math. 98, Springer, New York 1985. 10.1007/978-3-662-12918-0Search in Google Scholar

[6] T. Bühler, Exact categories, Expo. Math. 28 (2010), no. 1, 1–69. 10.1016/j.exmath.2009.04.004Search in Google Scholar

[7] S. E. Cappell, L. Maxim, J. Schürmann, J. L. Shaneson and S. Yokura, Characteristic classes of symmetric products of complex quasi-projective varieties, J. reine angew. Math. 728 (2017), 35–63. 10.1515/crelle-2014-0114Search in Google Scholar

[8] S. Casalaina-Martin, M. Popa and S. Schreieder, Generic vanishing and minimal cohomology classes on abelian fivefolds, J. Algebraic Geom. 27 (2018), no. 3, 553–581. 10.1090/jag/691Search in Google Scholar

[9] G. Codogni, S. Grushevsky and E. Sernesi, The degree of the Gauss map of the theta divisor, Algebra Number Theory 11 (2017), 983–1001. 10.2140/ant.2017.11.983Search in Google Scholar

[10] M. A. A. de Cataldo and L. Migliorini, The decomposition theorem, perverse sheaves and the topology of algebraic maps, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 4, 535–633. 10.1090/S0273-0979-09-01260-9Search in Google Scholar

[11] O. Debarre, Annulation de thêtaconstantes sur les variétés abéliennes de dimension quatre, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 20, 885–888. Search in Google Scholar

[12] O. Debarre, Complex tori and abelian varieties, SMF/AMS Texts Monogr. 11, American Mathematical Society, Providence 2005. Search in Google Scholar

[13] P. Deligne, Catégories tensorielles, Moscow Math. J. 2 (2002), 227–248. 10.17323/1609-4514-2002-2-2-227-248Search in Google Scholar

[14] P. Deligne and J. S. Milne, Tannakian categories, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math. 900, Springer, Berlin (1982), 101–228. 10.1007/978-3-540-38955-2_4Search in Google Scholar

[15] L. Ein and R. Lazarsfeld, Singularities of theta divisors and the birational geometry of irregular varieties, J. Amer. Math. Soc. 10 (1997), no. 1, 243–258. 10.1090/S0894-0347-97-00223-3Search in Google Scholar

[16] H. Flenner, Join varieties and intersection theory, Recent progress in intersection theory (Bologna 1997), Trends Math., Birkhäuser, Boston (2000), 129–197. 10.1007/978-1-4612-1316-1_5Search in Google Scholar

[17] H. Flenner, L. O’Carroll and W. Vogel, Joins and intersections, Springer Monogr. Math., Springer, Berlin 1999. 10.1007/978-3-662-03817-8Search in Google Scholar

[18] J. Franecki and M. Kapranov, The Gauss map and a noncompact Riemann–Roch formula for constructible sheaves on semiabelian varieties, Duke Math. J. 104 (2000), no. 1, 171–180. 10.1215/S0012-7094-00-10417-6Search in Google Scholar

[19] W. Fulton, Intersection theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin 1998. 10.1007/978-1-4612-1700-8Search in Google Scholar

[20] R. Goodman and N. R. Wallach, Symmetry, representations, and invariants, Grad. Texts in Math. 255, Springer, New York 2009. 10.1007/978-0-387-79852-3Search in Google Scholar

[21] F. Heinloth, A note on functional equations for zeta functions with values in Chow motives, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 6, 1927–1945. 10.5802/aif.2318Search in Google Scholar

[22] R. Hotta, K. Takeuchi and T. Tanisaki, D-modules, perverse sheaves, and representation theory, Progr. Math. 236, Birkhäuser, Boston 2008. 10.1007/978-0-8176-4523-6Search in Google Scholar

[23] R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc. 8, Bar-Ilan University, Ramat Gan (1995), 1–182. Search in Google Scholar

[24] G. Kennedy, MacPherson’s Chern classes of singular algebraic varieties, Comm. Algebra 18 (1990), no. 9, 2821–2839. 10.1080/00927879008824054Search in Google Scholar

[25] S. L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. Search in Google Scholar

[26] D. Knutson, λ-rings and the representation theory of the symmetric group, Lecture Notes in Math. 308, Springer, Berlin 1973. 10.1007/BFb0069217Search in Google Scholar

[27] T. Krämer, Cubic threefolds, Fano surfaces and the monodromy of the Gauss map, Manuscripta Math. 149 (2016), no. 3–4, 303–314. 10.1007/s00229-015-0785-zSearch in Google Scholar

[28] T. Krämer, Characteristic cycles and the microlocal geometry of the Gauss map I, preprint (2018), 10.1515/crelle-2020-0048Search in Google Scholar

[29] T. Krämer, Summands of theta divisors on Jacobians, Compos. Math. 156 (2020), no. 7, 1457–1475. 10.1112/S0010437X20007204Search in Google Scholar

[30] T. Krämer and R. Weissauer, On the Tannaka group attached to the theta divisor of a generic principally polarized abelian variety, Math. Z. 281 (2015), no. 3–4, 723–745. 10.1007/s00209-015-1505-9Search in Google Scholar

[31] T. Krämer and R. Weissauer, Semisimple super Tannakian categories with a small tensor generator, Pacific J. Math. 276 (2015), no. 1, 229–248. 10.2140/pjm.2015.276.229Search in Google Scholar

[32] T. Krämer and R. Weissauer, Vanishing theorems for constructible sheaves on abelian varieties, J. Algebraic Geom. 24 (2015), no. 3, 531–568. 10.1090/jag/645Search in Google Scholar

[33] J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs 1984), Springer, New York (1986), 103–150. 10.1007/978-1-4613-8655-1_5Search in Google Scholar

[34] P. Nang and K. Takeuchi, Characteristic cycles of perverse sheaves and Milnor fibers, Math. Z. 249 (2005), no. 3, 493–511. 10.1007/s00209-004-0712-6Search in Google Scholar

[35] G. Pareschi and M. Popa, Generic vanishing and minimal cohomology classes on abelian varieties, Math. Ann. 340 (2008), no. 1, 209–222. 10.1007/s00208-007-0146-7Search in Google Scholar

[36] D. Quillen, Higher algebraic K-theory. I, Algebraic K-theory, I: Higher K-theories (Seattle 1972), Lecture Notes in Math. 341, Springer, Berlin (1973), 85–147. 10.1007/BFb0067053Search in Google Scholar

[37] Z. Ran, On subvarieties of abelian varieties, Invent. Math. 62 (1981), no. 3, 459–479. 10.1007/BF01394255Search in Google Scholar

[38] C. Sabbah, Quelques remarques sur la géométrie des espaces conormaux, Differential systems and singularities (Luminy 1983), Astérisque 130, Société Mathématique de France, Paris (1985), 161–192. Search in Google Scholar

[39] C. Schnell, Holonomic D-modules on abelian varieties, Publ. Math. Inst. Hautes Études Sci. 121 (2015), 1–55. 10.1007/s10240-014-0061-xSearch in Google Scholar

[40] S. Schreieder, Theta divisors with curve summands and the Schottky problem, Math. Ann. 365 (2016), no. 3–4, 1017–1039. 10.1007/s00208-015-1287-8Search in Google Scholar

[41] S. Schreieder, Decomposable theta divisors and generic vanishing, Int. Math. Res. Not. IMRN 2017 (2017), no. 16, 4984–5009. 10.1093/imrn/rnw160Search in Google Scholar

[42] J. Schürmann and M. Tibăr, Index formula for MacPherson cycles of affine algebraic varieties, Tohoku Math. J. (2) 62 (2010), no. 1, 29–44. 10.2748/tmj/1270041025Search in Google Scholar

[43] R. Smith and R. Varley, Components of the locus of singular theta divisors of genus 5, Algebraic geometry – Sitges (Barcelona 1983), Lecture Notes in Math. 1124, Springer, Berlin (1985), 338–416. 10.1007/BFb0075005Search in Google Scholar

[44] J. R. Stembridge, Multiplicity-free products and restrictions of Weyl characters, Represent. Theory 7 (2003), 404–439. 10.1090/S1088-4165-03-00150-XSearch in Google Scholar

[45] R. Varley, Weddle’s surfaces, Humbert’s curves, and a certain 4-dimensional abelian variety, Amer. J. Math. 108 (1986), no. 4, 931–951. 10.2307/2374519Search in Google Scholar

[46] W. Vogel, Lectures on results on Bezout’s theorem, Tata Inst. Fund. Res. Lect. Math. Phys. 74, Springer, Berlin 1984. 10.1007/978-3-662-00493-7Search in Google Scholar

[47] R. Weissauer, Brill–Noether sheaves, preprint (2007), Search in Google Scholar

[48] R. Weissauer, Degenerate perverse sheaves on abelian varieties, preprint (2015), Search in Google Scholar

[49] R. Weissauer, On subvarieties of abelian varieties with degenerate Gauss mapping, preprint (2015), Search in Google Scholar

Received: 2019-11-11
Revised: 2020-08-19
Published Online: 2021-01-09
Published in Print: 2021-05-01

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