Generic vanishing and minimal cohomology classes on abelian fivefolds

Casalaina-Martin, Sebastian; Popa, Mihnea; Schreieder, Stefan

doi:10.1090/jag/691

Authors: Sebastian Casalaina-Martin, Mihnea Popa and Stefan Schreieder
Journal: J. Algebraic Geom. 27 (2018), 553-581
DOI: https://doi.org/10.1090/jag/691
Published electronically: December 7, 2017
MathSciNet review: 3803607
Full-text PDF

Abstract | References | Additional Information

Abstract: We classify $GV$-subschemes of five-dimensional ppavs, proving the main conjecture in a work by Pareschi and the second author in this case. This result is implied by a more general statement about subvarieties of minimal cohomology class whose sum is a theta divisor.

References

Arnaud Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), no. 2, 149–196. MR 572974, DOI https://doi.org/10.1007/BF01418373
Arnaud Beauville, Surfaces algébriques complexes, Société Mathématique de France, Paris, 1978 (French). Avec une sommaire en anglais; Astérisque, No. 54. MR 0485887
O. Debarre, Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. France 110 (1982), no. 3, 319–346 (French, with English summary). With an appendix by A. Beauville. MR 688038
Christina Birkenhake and Herbert Lange, Complex abelian varieties, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 2004. MR 2062673
Sebastian Casalaina-Martin, Cubic threefolds and abelian varieties of dimension five. II, Math. Z. 260 (2008), no. 1, 115–125. MR 2413346, DOI https://doi.org/10.1007/s00209-007-0264-7
Sebastian Casalaina-Martin, Singularities of the Prym theta divisor, Ann. of Math. (2) 170 (2009), no. 1, 162–204. MR 2521114, DOI https://doi.org/10.4007/annals.2009.170.163
C. Herbert Clemens and Phillip A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356. MR 302652, DOI https://doi.org/10.2307/1970801
Olivier Debarre, Minimal cohomology classes and Jacobians, J. Algebraic Geom. 4 (1995), no. 2, 321–335. MR 1311353
Olivier Debarre, Complex tori and abelian varieties, SMF/AMS Texts and Monographs, vol. 11, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2005. Translated from the 1999 French edition by Philippe Mazaud. MR 2158864
Lawrence Ein and Robert Lazarsfeld, Singularities of theta divisors and the birational geometry of irregular varieties, J. Amer. Math. Soc. 10 (1997), no. 1, 243–258. MR 1396893, DOI https://doi.org/10.1090/S0894-0347-97-00223-3
William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323
Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558
Andreas Höring, $M$-regularity of the Fano surface, C. R. Math. Acad. Sci. Paris 344 (2007), no. 11, 691–696 (English, with English and French summaries). MR 2334677, DOI https://doi.org/10.1016/j.crma.2007.04.008
Andreas Höring, Minimal classes on the intermediate Jacobian of a generic cubic threefold, Commun. Contemp. Math. 12 (2010), no. 1, 55–70. MR 2649227, DOI https://doi.org/10.1142/S0219199710003737
Luigi Lombardi and Sofia Tirabassi, $GV$-subschemes and their embeddings in principally polarized abelian varieties, Math. Nachr. 288 (2015), no. 11-12, 1405–1412. MR 3377125, DOI https://doi.org/10.1002/mana.201400238
David Mumford, Prym varieties. I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 325–350. MR 0379510
Giuseppe Pareschi and Mihnea Popa, Regularity on abelian varieties. I, J. Amer. Math. Soc. 16 (2003), no. 2, 285–302. MR 1949161, DOI https://doi.org/10.1090/S0894-0347-02-00414-9
Giuseppe Pareschi and Mihnea Popa, Castelnuovo theory and the geometric Schottky problem, J. Reine Angew. Math. 615 (2008), 25–44. MR 2384330, DOI https://doi.org/10.1515/CRELLE.2008.008
Giuseppe Pareschi and Mihnea Popa, Generic vanishing and minimal cohomology classes on abelian varieties, Math. Ann. 340 (2008), no. 1, 209–222. MR 2349774, DOI https://doi.org/10.1007/s00208-007-0146-7
Giuseppe Pareschi and Mihnea Popa, GV-sheaves, Fourier-Mukai transform, and generic vanishing, Amer. J. Math. 133 (2011), no. 1, 235–271. MR 2752940, DOI https://doi.org/10.1353/ajm.2011.0000
Ziv Ran, On subvarieties of abelian varieties, Invent. Math. 62 (1981), no. 3, 459–479. MR 604839, DOI https://doi.org/10.1007/BF01394255
Ziv Ran, A characterization of five-dimensional Jacobian varieties, Invent. Math. 67 (1982), no. 3, 395–422. MR 664113, DOI https://doi.org/10.1007/BF01398929
Stefan Schreieder, Theta divisors with curve summands and the Schottky problem, Math. Ann. 365 (2016), no. 3-4, 1017–1039. MR 3521080, DOI https://doi.org/10.1007/s00208-015-1287-8
Stefan Schreieder, Decomposable theta divisors and generic vanishing, Int. Math. Res. Not. IMRN 16 (2017), 4984–5009. MR 3687123, DOI https://doi.org/10.1093/imrn/rnw160

Additional Information

Sebastian Casalaina-Martin
Affiliation: Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309-0395
MR Author ID: 754836
Email: casa@math.colorado.edu

Mihnea Popa
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
MR Author ID: 653676
Email: mpopa@math.northwestern.edu

Stefan Schreieder
Affiliation: Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Address at time of publication: Mathematisches Institut, LMU München, Theresienstr. 39, 80333 München, Germany
MR Author ID: 982064
Email: schreieder@math.lmu.de

Received by editor(s): March 1, 2016
Received by editor(s) in revised form: July 3, 2016
Published electronically: December 7, 2017
Additional Notes: The first author was partially supported by Simons Foundation Collaboration Grant for Mathematicians (317572). The second author was partially supported by the NSF grant DMS-1405516 and by a Simons Fellowship.
Article copyright: © Copyright 2017 University Press, Inc.