Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



The primitive cohomology of the theta divisor of an abelian fivefold

Authors: E. Izadi, Cs. Tamás and J. Wang
Journal: J. Algebraic Geom. 26 (2017), 107-175
Published electronically: June 28, 2016
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Abstract | References | Additional Information

Abstract: The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $ g$ contains a Hodge structure of level $ g-3$ which we call the primal cohomology. The Hodge conjecture predicts that this is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. In this paper we use the Prym map to show that this version of the Hodge conjecture is true for the theta divisor of a general abelian fivefold.

References [Enhancements On Off] (What's this?)

  • [ACGH] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932 (86h:14019)
  • [B1] Arnaud Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), no. 2, 149-196. MR 0572974 (58 #27995)
  • [B2] Arnaud Beauville, Sous-variétés spéciales des variétés de Prym, Compositio Math. 45 (1982), no. 3, 357-383 (French). MR 656611 (83f:14025)
  • [BC] Charles Barton and C. H. Clemens, A result on the integral Chow ring of a generic principally polarized complex Abelian variety of dimension four, Compositio Math. 34 (1977), no. 1, 49-67. MR 0447237 (56 #5552)
  • [C] C. H. Clemens, Degeneration of Kähler manifolds, Duke Math. J. 44 (1977), no. 2, 215-290. MR 0444662 (56 #3012)
  • [De] Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5-77 (French). MR 0498552 (58 #16653b)
  • [Do] Ron Donagi, The fibers of the Prym map, Curves, Jacobians, and abelian varieties (Amherst, MA, 1990) Contemp. Math., vol. 136, Amer. Math. Soc., Providence, RI, 1992, pp. 55-125. MR 1188194 (94e:14037),
  • [DS] Ron Donagi and Roy Campbell Smith, The structure of the Prym map, Acta Math. 146 (1981), no. 1-2, 25-102. MR 594627 (82k:14030b),
  • [Du] Alan H. Durfee, Mixed Hodge structures on punctured neighborhoods, Duke Math. J. 50 (1983), no. 4, 1017-1040. MR 726316 (85m:14012),
  • [H] Fumio Hazama, The generalized Hodge conjecture for stably nondegenerate abelian varieties, Compositio Math. 93 (1994), no. 2, 129-137. MR 1287693 (95d:14011)
  • [I] Elham Izadi, Some remarks on the Hodge conjecture for abelian varieties, Ann. Mat. Pura Appl. (4) 189 (2010), no. 3, 487-495. MR 2657421 (2011f:14073),
  • [ILS] E. Izadi, H. Lange, and V. Strehl, Correspondences with split polynomial equations, J. Reine Angew. Math. 627 (2009), 183-212. MR 2494932 (2010g:14038),
  • [IvS] E. Izadi and D. van Straten, The intermediate Jacobians of the theta divisors of four-dimensional principally polarized abelian varieties, J. Algebraic Geom. 4 (1995), no. 3, 557-590. MR 1325792 (96e:14053)
  • [KS] George R. Kempf and Frank-Olaf Schreyer, A Torelli theorem for osculating cones to the theta divisor, Compositio Math. 67 (1988), no. 3, 343-353. MR 959216 (89g:14020)
  • [Ma] I. G. Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962), 319-343. MR 0151460 (27 #1445)
  • [Mo] David R. Morrison, The Clemens-Schmid exact sequence and applications, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 101-119. MR 756848
  • [Sc] Wilfried Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211-319. MR 0382272 (52 #3157)
  • [So] A. Soucaris, The ampleness of the theta divisor on the compactified Jacobian of a proper and integral curve, Compositio Math. 93 (1994), no. 3, 231-242. MR 1300762 (95m:14017)
  • [St] Joseph Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1975/76), no. 3, 229-257. MR 0429885 (55 #2894)
  • [T] Montserrat Teixidor i Bigas, For which Jacobi varieties is $ {\rm Sing}\,\Theta $ reducible?, J. Reine Angew. Math. 354 (1984), 141-149. MR 767576 (86c:14025),

Additional Information

E. Izadi
Affiliation: Department of Mathematics, University of California San Diego, 9500 Gilman Drive #0112, La Jolla, California 92093-0112

Cs. Tamás
Affiliation: Department of Mathematics and Statistics, Langara College, 100 West 49th Avenue, Vancouver, BC, Canada V5Y 2Z6

J. Wang
Affiliation: Department of Mathematics, University of California San Diego, 9500 Gilman Drive #0112, La Jolla, California 92093-0112

Received by editor(s): December 24, 2013
Received by editor(s) in revised form: November 1, 2014
Published electronically: June 28, 2016
Additional Notes: The authors are indebted to the referees for a careful reading of this manuscript and many helpful comments and suggestions. The first author was partially supported by the National Science Foundation and the National Security Agency. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the NSF or NSA
Dedicated: Dedicated to Herb Clemens
Article copyright: © Copyright 2016 University Press, Inc.

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