Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Arithmetic Torelli maps for cubic surfaces and threefolds


Author: Jeffrey D. Achter
Journal: Trans. Amer. Math. Soc. 366 (2014), 5749-5769
MSC (2010): Primary 14J10; Secondary 11G18, 14H40, 14K30
DOI: https://doi.org/10.1090/S0002-9947-2014-05978-3
Published electronically: June 16, 2014
MathSciNet review: 3256183
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base, and in particular identifies the moduli space of cubic surfaces with an open substack of a certain moduli space of abelian varieties. This answers, away from the prime $ 2$, an old question of Deligne and a recent question of Kudla and Rapoport.


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

  • [1] Jeffrey D. Achter, On the abelian fivefolds attached to cubic surfaces, Math. Res. Lett. 20 (2013), no. 5, 805-824. DOI 10.4310/MRL.2013.v20.n5.a1.
  • [2] Daniel Allcock, The moduli space of cubic threefolds, J. Algebraic Geom. 12 (2003), no. 2, 201-223. MR 1949641 (2003k:14043), https://doi.org/10.1090/S1056-3911-02-00313-2
  • [3] Daniel Allcock, James A. Carlson, and Domingo Toledo, A complex hyperbolic structure for moduli of cubic surfaces, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 1, 49-54 (English, with English and French summaries). MR 1649469 (2001h:14042), https://doi.org/10.1016/S0764-4442(97)82711-5
  • [4] Daniel Allcock, James A. Carlson, and Domingo Toledo, The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Algebraic Geom. 11 (2002), no. 4, 659-724. MR 1910264 (2003m:32011), https://doi.org/10.1090/S1056-3911-02-00314-4
  • [5] Daniel Allcock, James A. Carlson, and Domingo Toledo, The moduli space of cubic threefolds as a ball quotient, Mem. Amer. Math. Soc. 209 (2011), no. 985, xii+70. MR 2789835 (2012b:32021), https://doi.org/10.1090/S0065-9266-10-00591-0
  • [6] Allen B. Altman and Steven L. Kleiman, Foundations of the theory of Fano schemes, Compositio Math. 34 (1977), no. 1, 3-47. MR 0569043 (58 #27967)
  • [7] Arnaud Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 3, 309-391 (French). MR 0472843 (57 #12532)
  • [8] Arnaud Beauville, Les singularités du diviseur $ \Theta $ de la jacobienne intermédiaire de l'hypersurface cubique dans $ {\mathbf {P}}^{4}$, Algebraic threefolds (Varenna, 1981), Lecture Notes in Math., vol. 947, Springer, Berlin, 1982, pp. 190-208. MR 672617 (84c:14030)
  • [9] Arnaud Beauville, Moduli of cubic surfaces and Hodge theory (after Allcock, Carlson, Toledo), Géométries à courbure négative ou nulle, groupes discrets et rigidités, Sémin. Congr., vol. 18, Soc. Math. France, Paris, 2009, pp. 445-466. MR 2655320 (2011g:32022)
  • [10] 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 (2005c:14001)
  • [11] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822 (91i:14034)
  • [12] James A. Carlson and Domingo Toledo, Cubic surfaces with special periods, Proc. Amer. Math. Soc. 141 (2013), no. 6, 1947-1962. MR 3034422, https://doi.org/10.1090/S0002-9939-2013-11484-X
  • [13] Sebastian Casalaina-Martin and Robert Friedman, Cubic threefolds and abelian varieties of dimension five, J. Algebraic Geom. 14 (2005), no. 2, 295-326. MR 2123232 (2006g:14071), https://doi.org/10.1090/S1056-3911-04-00379-0
  • [14] Sebastian Casalaina-Martin and Radu Laza, The moduli space of cubic threefolds via degenerations of the intermediate Jacobian, J. Reine Angew. Math. 633 (2009), 29-65. MR 2561195 (2011a:14071), https://doi.org/10.1515/CRELLE.2009.059
  • [15] C. Herbert Clemens and Phillip A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281-356. MR 0302652 (46 #1796)
  • [16] Pierre Deligne, Les intersections complètes de niveau de Hodge un, Invent. Math. 15 (1972), 237-250 (French). MR 0301029 (46 #189)
  • [17] P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5-89. MR 849651 (88a:22023a)
  • [18] Michel Demazure and Alexander Grothendieck, Schémas en groupes, séminaire de géométrie algébrique du bois marie 1962/64 (sga 3), Lecture Notes in Mathematics, no. 151, 152, 153, Springer-Verlag, Berlin, 1962/1964. MR 0274460 (43:223c)
  • [19] I. Dolgachev, B. van Geemen, and S. Kondō, A complex ball uniformization of the moduli space of cubic surfaces via periods of $ K3$ surfaces, J. Reine Angew. Math. 588 (2005), 99-148. MR 2196731 (2006h:14051), https://doi.org/10.1515/crll.2005.2005.588.99
  • [20] Igor V. Dolgachev and Shigeyuki Kondō, Moduli of $ K3$ surfaces and complex ball quotients, Arithmetic and geometry around hypergeometric functions, Progr. Math., vol. 260, Birkhäuser, Basel, 2007, pp. 43-100. MR 2306149 (2008e:14053), https://doi.org/10.1007/978-3-7643-8284-1_3
  • [21] Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendix by David Mumford. MR 1083353 (92d:14036)
  • [22] Alexander Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ. Math. (1964-1967), no. 20, 24, 28, 32. MR 0238860 (39:220)
  • [23] Steven L. Kleiman, The Picard scheme, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 235-321. MR 2223410
  • [24] Shigeyuki Kondō, The moduli space of curves of genus 4 and Deligne-Mostow's complex reflection groups, Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, Tokyo, 2002, pp. 383-400. MR 1971521 (2004h:14033)
  • [25] Stephen Kudla and Michael Rapoport, Special cycles on unitary Shimura varieties II: global theory, December 2009, arXiv:0912.3758.
  • [26] -, On occult period maps, Pacific J. Math. 260 (2012), no. 2, 565-581. MR 3001805
  • [27] Eduard Looijenga and Rogier Swierstra, On period maps that are open embeddings, J. Reine Angew. Math. 617 (2008), 169-192. MR 2400994 (2010a:32030), https://doi.org/10.1515/CRELLE.2008.029
  • [28] Keiji Matsumoto and Tomohide Terasoma, Theta constants associated to cubic threefolds, J. Algebraic Geom. 12 (2003), no. 4, 741-775. MR 1993763 (2004h:14041), https://doi.org/10.1090/S1056-3911-03-00348-5
  • [29] Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461 (90i:13001)
  • [30] James S. Milne, Algebraic geometry (v5.20), 2009, Available at www.jmilne.org/math/, pp. 239+vi.
  • [31] Ben Moonen, Models of Shimura varieties in mixed characteristics, Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser., vol. 254, Cambridge Univ. Press, Cambridge, 1998, pp. 267-350. MR 1696489 (2000e:11077), https://doi.org/10.1017/CBO9780511662010.008
  • [32] 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 (52 #415)
  • [33] David Mumford, Stability of projective varieties, Enseignement Math. (2) 23 (1977), no. 1-2, 39-110. MR 0450272 (56 #8568)
  • [34] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906 (95m:14012)
  • [35] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906 (95m:14012)
  • [36] J. P. Murre, Algebraic equivalence modulo rational equivalence on a cubic threefold, Compositio Math. 25 (1972), 161-206. MR 0352088 (50 #4576a)
  • [37] J. P. Murre, Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of Mumford, Compositio Math. 27 (1973), 63-82. MR 0352089 (50 #4576b)
  • [38] J. P. Murre, Some results on cubic threefolds, Classification of algebraic varieties and compact complex manifolds, Springer, Berlin, 1974, pp. 140-160. Lecture Notes in Math., Vol. 412. MR 0374145 (51 #10345)
  • [39] Isao Naruki, Cross ratio variety as a moduli space of cubic surfaces, Proc. London Math. Soc. (3) 45 (1982), no. 1, 1-30. With an appendix by Eduard Looijenga. MR 662660 (84d:14020), https://doi.org/10.1112/plms/s3-45.1.1
  • [40] Arthur Ogus, $ F$-isocrystals and de Rham cohomology. II. Convergent isocrystals, Duke Math. J. 51 (1984), no. 4, 765-850. MR 771383 (86j:14012), https://doi.org/10.1215/S0012-7094-84-05136-6
  • [41] M. Rapoport, Complément à l'article de P. Deligne ``La conjecture de Weil pour les surfaces $ K3$'', Invent. Math. 15 (1972), 227-236 (French). MR 0309943 (46 #9046)
  • [42] C. S. Seshadri, Quotient spaces modulo reductive algebraic groups, Ann. of Math. (2) 95 (1972), 511-556; errata, ibid. (2) 96 (1972), 599. MR 0309940 (46 #9044)
  • [43] Tetsuji Shioda, The Hodge conjecture for Fermat varieties, Math. Ann. 245 (1979), no. 2, 175-184. MR 552586 (80k:14035), https://doi.org/10.1007/BF01428804
  • [44] A. Silverberg, Fields of definition for homomorphisms of abelian varieties, J. Pure Appl. Algebra 77 (1992), no. 3, 253-262. MR 1154704 (93f:14022), https://doi.org/10.1016/0022-4049(92)90141-2

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14J10, 11G18, 14H40, 14K30

Retrieve articles in all journals with MSC (2010): 14J10, 11G18, 14H40, 14K30


Additional Information

Jeffrey D. Achter
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
Email: j.achter@colostate.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-05978-3
Received by editor(s): February 16, 2012
Received by editor(s) in revised form: September 11, 2012
Published electronically: June 16, 2014
Additional Notes: This work was partially supported by a grant from the Simons Foundation (204164).
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society