Nagoya Mathematical Journal

A Noether-Lefschetz theorem for varieties of r-planes in complete intersections

Zhi Jiang

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We prove a Noether-Lefschetz type theorem for varieties of r-planes in complete intersections. We then use it to study the Abel-Jacobi map of planes on a smooth cubic fivefold.

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Nagoya Math. J., Volume 206 (2012), 39-66.

First available in Project Euclid: 22 May 2012

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Zentralblatt MATH identifier

Primary: 14C22: Picard groups 14D07: Variation of Hodge structures [See also 32G20]
Secondary: 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.)


Jiang, Zhi. A Noether-Lefschetz theorem for varieties of $r$ -planes in complete intersections. Nagoya Math. J. 206 (2012), 39--66. doi:10.1215/00277630-1548484.

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  • [AK] A. B. Altman and S. L. Kleiman, Foundations of the theory of Fano schemes, Compos. Math. 34 (1977), 3–47.
  • [BVV] W. Barth and A. Van de Ven, Fano varieties of lines on hypersurfaces, Arch. Math. (Basel) 31 (1978), 96–104.
  • [BD] A. Beauville and R. Donagi, La variété des droites d’une hypersurface cubique de dimension 4, C. R. Math. Acad. Sci. Paris 301 (1985), 703–706.
  • [BM] S. Bloch and J. P. Murre, On the Chow groups of certain types of Fano threefolds, Compos. Math. 39 (1979), 47–105.
  • [BV] L. Bonavero and C. Voisin, Fano schemes and Moishezon manifolds (in French), C. R. Math. Acad. Sci. Paris 323 (1996), 1019–1024.
  • [B1] C. Borcea, Deforming varieties of k-planes of projective complete intersections, Pacific J. Math. 143 (1990), 25–36.
  • [B2] C. Borcea, “Homogeneous vector bundles and families of Calabi-Yau threefolds, II” in Several Complex Variables and Complex Geometry (Santa Cruz, 1989), Proc. Sympos. Pure Math. 52, Part 2, Amer. Math. Soc., Providence, 1991, 83–91.
  • [Bot] R. Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1975), 203–248.
  • [CG] C. H. Clemens and P. A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356.
  • [DM] O. Debarre and L. Manivel, Sur la variété des espaces linéaires contenus dans une intersection complète, Math. Ann. 312 (1998), 549–574.
  • [DR] I. Dimitrov and M. Roth, Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one, preprint, arXiv:0909.2280v1 [math.AG]
  • [ELV] H. Esnault, M. Levine, and E. Viehweg, Chow groups of projective varieties of very small degree, Duke Math. J. 87 (1997), 29–58.
  • [IM] A. Iliev and L. Manivel, Cubic hypersurfaces and integrable systems, Amer. J. Math. 130 (2008), 1445–1475.
  • [J] Z. Jiang, On the restriction of holomorphic forms, Manuscripta Math. 124 (2007), 173–182.
  • [LM] J. M. Landsberg and L. Manivel, On the projective geometry of rational homogeneous varieties, Comment. Math. Helv. 78 (2003), 65–100.
  • [PS] C. Peters and J. Steenbrink, Mixed Hodge Structures, Ergeb. Math. Grenzgeb. (3) 52, Springer, Berlin, 2008.
  • [P] G. P. Pirola, Base number theorem for abelian varieties: An infinitesimal approach, Math. Ann. 282 (1988), 361–368.
  • [Re] M. Reid, The complete intersection of two or more quadrics, Ph.D. dissertation, University of Cambridge, Cambridge, England, 1972.
  • [Ro] X. Roulleau, Elliptic curve configurations on Fano surfaces, Manuscripta Math. 129 (2009), 381–399.
  • [S] J. Spandaw, Noether-Lefschetz Problems for Degeneracy Loci, Mem. Amer. Math. Soc. 161 (2003), no. 764.
  • [W] J. Weyman, Cohomology of Vector Bundles and Syzygies, Cambridge Tracts in Math. 149, Cambridge University Press, Cambridge, 2003.
  • [V1] C. Voisin, Hodge Theory and Complex Algebraic Geometry, II, Cambridge Stud. Adv. Math. 77, Cambridge University Press, Cambridge, 2003.
  • [V2] C. Voisin, Coniveau 2 complete intersections and effective cones, Geom. Funct. Anal. 19 (2010), 1494–1513.