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

Zhi Jiang

paper

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

arXiv:1010.5210

Abstract

Let X be a very general complete intersection in complex projective space and we denote by $F_{r} (X)$ the variety of r-planes in X, for $r \geq 1$ . We show that the Picard number of $F_{r} (X)$ is 1, as soon as $dim F_{r} (X) \geq 2$ , except when X is a quadric of dimension 2r or 2r+2, or X is a complete intersection of two quadrics of dimension 2r+2. We also apply this result to determine the cohomology class of the variety of planes of a cubic fivefold contained (by the Abel-Jacobi map) in the intermediate Jacobian.

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