  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.
