The harmonic cohomology of a Donaldson symplectic submanifold and of an
Auroux symplectic submanifold are compared with that of its ambient space. We
also study symplectic manifolds satisfying a weakly Lefschetz property, that
is, the s-Lefschetz propery. In particular, we consider the symplectic
blow-ups of the complex projective space along weakly Lefschetz symplectic
submanifolds. As an application we construct, for each even integer s≥2,
compact symplectic manifolds which are s-Lefschetz but not (s+1)-Lefschetz.Comment: 22 pages; many improvements from previous versio