  We investigate a method proposed by E. Arrondo and J. Caravantes to study the
Picard group of a smooth low-codimension subvariety X in a variety Y when Y is
homogeneous. We prove that this method is strongly related to the signature
\sigma_Y of the Poincare pairing on the middle cohomology of Y. We give under
some topological assumptions a bound on the rank of Picard group Pic(X) in
terms of \sigma_Y and remove these assumptions for grassmannians to generalise
the main result of E. Arrondo and J. Caravantes.
