Let $M$ be projective complex manifold. The Lefschetz (1,1) theorem syas that the cycle map 
$$
\text{cl}:Pic(M) \to \text{Hod}^1(M)
$$
is surjective. 

Quesion.
Is there an interesting example of (1,1)-form $\omega \in H^1(M,\Omega_M^1)$ which isn`t spanned by $Pic(M)_{\mathbb{C}}$?