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

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