Weakly Lefschetz symplectic manifolds

For a symplectic manifold, the harmonic cohomology of symplectic divisors (introduced by Donaldson, 1996) and of the more general symplectic zero loci (introduced by Auroux, 1997) are compared with that of its ambient space. We also study symplectic manifolds satisfying a weakly Lefschetz property, that is, the $s$-Lefschetz property. In particular, we consider the symplectic blow-ups $\widetilde{CP}{}^m$ of the complex projective space ${CP}^m$ along weakly Lefschetz symplectic submanifolds $M\subset{CP}^m$. As an application we construct, for each even integer $s\geq 2$, compact symplectic manifolds which are $s$-Lefschetz but not $(s+1)$-Lefschetz.