  In this paper we study the behaviour of the Lefschetz property under the
blow-up construction. We show that it is possible to reduce the dimension of
the kernel of the Lefschetz map if we blow up along a suitable submanifold
satisfying the Lefschetz property. We use that, together with results about
Massey products, to construct nonformal (simply connected) symplectic manifolds
satisfying the Lefschetz property.
