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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. Let $E$ be an elliptic curve, $X=E\times C$ 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?

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  • $\begingroup$ There are also examples where you form the quotient by an action of a finite group $G$ that acts freely on both $X$ and $C$ such that projection is equivariant. $\endgroup$ Commented Dec 30, 2023 at 14:29
  • $\begingroup$ @JasonStarr Can you explain some of these examples? Or can you give me a reference? Thank you. $\endgroup$ Commented Jan 2 at 16:30

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