On the Picard Group of Low-codimension Subvarieties

Arrondo Esteban, Enrique; Caravantes, Jorge

On the Picard Group of Low-codimension Subvarieties

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Official URL

http://arxiv.org/pdf/math/0511267.pdf

Publication date

2009

Authors

Arrondo Esteban, Enrique

Caravantes, Jorge

Publisher

Indiana University

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URI

https://hdl.handle.net/20.500.14352/42091

Citation

[1] E. Arrondo, Subcanonicity of codimension two subvarieties, Rev. Mat. Compl. 18 (2005), 69–80. [2] E, Arrondo, M.L. Fania, Evidence to subcanonicity of codimension two subvarieties of G(1, 4), to appear on Int. J. Math. [3] W. Barth, Transplanting cohomology classes in complex projective space, Amer. J. Math. 92 (1970), 951–967. [4] W. Barth, M.E. Larsen, On the homotopy-groups of complex projective manifolds, Math Scand. 30 (1972), 88-94. [5] W. Barth and A. Van de Ven, On the geometry in codimension 2 of Grassmann manifolds, Lecture Notes in Math. 412, Springer Verlag (1974), 1–35. [6] O. Debarre, Th´eor`emes de connexit´e pour les produits d’espaces projectifs et les grassmanniennes, Amer. J. Math. 118 (1996), no. 6, 1347–1367. [7] R. Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017–1032. [8] S.L. Kleiman, D. Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061-1082. [9] M.E. Larsen, On the topology of projective manifolds, Invent. Math. 19 (1973), 251–260. [10] A.J. Sommese, Complex subspaces of homogeneous complex manifolds. II. Homotopy results, Nagoya Math. J. 86, 101–129.

Abstract

We introduce a method to determine if n-dimensional smooth subvarieties of an ambient space of dimension at most 2n - 2 inherits the Picard group from the ambient space (as it happens when the ambient space is a projective space, according to results of Barth and Larsen). As an application, we give an affirmative answer (LIP to some mild natural numerical conditions) when the ambient space is a Grassmannian of lines (thus improving results of Barth, Van de Ven and Sommese) or a product of two projective spaces of the same dimension.