# The relation between the weak Lefschetz theorem and the strong Lefschetz theorem

The Weak Lefschetz Theorem states that for a compact Kahler manifold, $Pic(X) \rightarrow H^{1,1}(X, \mathbb{Z})$ is surjective.

The Hard Lefschetz Theorem states that for a compact Kahler manifold with Lefschetz operator $L$, $$L^{n-k}: H^k(X,\mathbb{R}) \cong H^{2n-k}(X,\mathbb{R})$$ and for any $k$ $$H^k(X, \mathbb{R}) = \bigoplus_{i \geq 0}L^iH^{k-2i}(X, \mathbb{R})_p$$ The subscript $p$ denotes the primitive cohomology groups.

The decomposition above respects the $p,q$ form decomposition of Dolbeault cohomology.

My question is: what is the relation between these two Lefschetz theorems? Does one imply the other? Any good references?

-
To avoid confusion: I think what you call the Weak Lefschetz Theorem is usually called the Lefschetz (1,1) Theorem, and the Weak Lefschetz Theorem is the theorem also called the Lefschetz Hyperplane Theorem. (At least according to Iskovskikh: encyclopediaofmath.org/index.php/Lefschetz_theorem) – user5117 Apr 23 '13 at 15:30