Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

On the bicanonical map of irregular varieties


Authors: Miguel Angel Barja, Martí Lahoz, Juan Carlos Naranjo and Giuseppe Pareschi
Journal: J. Algebraic Geom. 21 (2012), 445-471
DOI: https://doi.org/10.1090/S1056-3911-2011-00565-1
Published electronically: July 27, 2011
MathSciNet review: 2914800
Full-text PDF

Abstract | References | Additional Information

Abstract: From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional generalization to this context of curves of genus $ 2$: varieties birationally equivalent to the theta-divisor of an indecomposable principally polarized abelian variety. The proof is based on the (generalized) Fourier-Mukai transform.


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

  • [BCP] I. Bauer, F. Catanese, R. Pignatelli, Complex surfaces of general type: some recent progress, In: Global aspects of complex geometry, 1-58, Springer, Berlin (2006). MR 2264106 (2007i:14039)
  • [BL] C. Birkenhake and H. Lange, Complex abelian varieties, 2nd edition, Springer 2004. MR 2062673 (2005c:14001)
  • [Ca] F. Catanese, Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations, Invent. Math. 104 (1991), 389-407. MR 1098610 (92f:32049)
  • [CCM] F. Catanese, C. Ciliberto, M. Mendes Lopes, On the classification of irregular surfaces of general type with non-birational bicanonical map, Trans. Amer. Math. Soc. 350 (1998), 275-308. MR 1422597 (98h:14043)
  • [CH] J. A. Chen, C. Hacon, Linear series on irregular varieties, In Algebraic geometry in East Asia (Kyoto, 2001), 143-153, World Sci. Publ., River Edge, NJ (2002). MR 2030451 (2005a:14010)
  • [Ci] C. Ciliberto, The bicanonical map for surfaces of general type, In: Algebraic Geometry - Santa Cruz 1995, Proc. Symphos. Pure Math., 62, Part I, 57-84, Amer. Mat. Soc.. Providence (1997). MR 1492518 (98m:14040)
  • [CM] C. Ciliberto, M. Mendes Lopes, On surfaces with $ p_g=q=2$ and non-birational bicanonical map, Adv. Geom 2 (2002), 281-300. MR 1924760 (2004d:14053)
  • [EL] L. Ein and R. Lazarsfeld, Singularities of theta divisors and the birational geometry of irregular varieties, J. Amer. Math. Soc. 10 (1997), 243-258. MR 1396893 (97d:14063)
  • [E] D. Eisenbud, Commutative algebra with a view towards algebraic geometry, Springer-Verlag GTM 150, New York, 1995. MR 1322960 (97a:13001)
  • [GL1] M. Green and R. Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), 389-407. MR 910207 (89b:32025)
  • [GL2] -, Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc. 1 (1991), no. 4, 87-103. MR 1076513 (92i:32021)
  • [Ha] C. Hacon, A derived category approach to generic vanishing, J. Reine Angew. Math. 575 (2004), 173-187. MR 2097552 (2005m:14026)
  • [HP] C. Hacon and R. Pardini, On the birational geometry of varieties of maximal Albanese dimension, J. Reine Angew. Math. 546 (2002), 177-199. MR 1900998 (2003d:14016)
  • [Hu] D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry, Oxford University Press, 2006. MR 2244106 (2007f:14013)
  • [Ke] G. Kempf, Complex abelian varieties and theta functions, Springer 1991. MR 1109495 (92h:14028)
  • [Ko1] -, Higher direct images of dualizing sheaves II, Ann. of Math. 124 (1986), 171-202. MR 847955 (87k:14014)
  • [Ko2] -, Singularities of pairs, Algebraic geometry-Santa Cruz, 1995, 221-287, Proc. Sympos. Pure Math., 62, Part 1, Amer. Math. Soc., Providence, RI, 1997. MR 847955 (87k:14014)
  • [LP] R. Lazarsfeld and M. Popa, Derivative complex, BGG correspondence, and numerical inequalities for compact Kähler manifolds, Invent. Math. 182 (2010), 605-633. MR 2737707
  • [Mu1] S. Mukai, Semi-homogeneous vector bundles on an abelian variety, J. Math. Kyoto Univ., 18 (1978), 239-272. MR 0498572 (58:16667)
  • [Mu2] -, Duality between $ D(X)$ and $ D(\widehat X)$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153-175. MR 607081 (82f:14036)
  • [M] D. Mumford, Abelian varieties, Second edition, Oxford University Press, 1970. MR 0282985 (44:219)
  • [PP1] G. Pareschi and M. Popa, Regularity on abelian varieties, I, J. Amer. Math. Soc., 16 (2003), 285-302. MR 1949161 (2004c:14086)
  • [PP2] -, Regularity on abelian varieties, II. Basic results on linear series and defining equations, J. Algebraic Geom. 13 (2004), 167-193. MR 1949161 (2004c:14086)
  • [PP3] -, $ M$-regularity and the Fourier-Mukai transform, Pure and Applied Math. Quarterly, F. Bogomolov issue II, 4 no. 3 (2008). MR 2008719 (2005a:14059)
  • [PP4] -, $ GV$-sheaves, Fourier-Mukai transform, and Generic Vanishing, Amer. J. Math. 133 (2011), 235-271. MR 2752940
  • [PP5] -, Regularity on abelian varieties III: relationship with Generic Vanishing and applications in Grassmannians, Moduli Spaces and Vector Bundles (D.A. Ellwood and E. Previate, eds.), 141-168, Clay Mathematics Proceedings 14, Amer. Math. Soc., Providence, RI, 2011, 141-168. MR 2008719 (2005a:14059)
  • [PP6] -, Strong generic vanishing and a higher dimensional Castelnuovo-de Franchis inequality, Duke Math. J. 150 (2009), 269-285. MR 2569614 (2011b:14044)


Additional Information

Miguel Angel Barja
Affiliation: Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, ETSEIB Avda. Diagonal 647, 08028 Barcelona, Spain
Email: miguel.angel.barja@upc.edu

Martí Lahoz
Affiliation: Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain
Email: marti.lahoz@ub.edu

Juan Carlos Naranjo
Affiliation: Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain
Email: jcnaranjo@ub.edu

Giuseppe Pareschi
Affiliation: Dipartimento di Matematica, Università di Roma, Tor Vergata, V. le della Ricerca Scientifica, I-00133 Roma, Italy
Email: pareschi@mat.uniroma2.it

DOI: https://doi.org/10.1090/S1056-3911-2011-00565-1
Received by editor(s): July 29, 2009
Received by editor(s) in revised form: February 12, 2010
Published electronically: July 27, 2011
Additional Notes: The first, second and third authors were partially supported by the Proyecto de Investigación MTM2006-14234. The first and second authors were also partially supported by 2005SGR-557 and the third author was partially supported by 2005SGR-787.

American Mathematical Society