I am working on the complex numbers field $\mathbb{C}$, for simplicity. However you can relax this assumption if you want.

Let $X$ be an elliptic surface, id est there is a proper morphism $\pi\colon X\to C$ of projective varieties, where $C$ is a smooth curve, such that the genus of all smooth fibres is $1$.

Example. $X=E\times C$ and $\pi$ is the canonical projection on the second factor. $X$ is an elliptic surface. If $g(C)\geq1$ then $\Omega^1_X$ is nef.

Question. Are there other examples of elliptic surfaces $X$ such that $\Omega^1_X$ is nef? If not, how does one prove the non-existence of these surfaces?