Small codimension smooth subvarieties in even-dimensional homogeneous spaces with Picard group Z

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.