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Let $X$ be a complex minimal surface of general type, id est $K_X$ is big and nef. It is well-known that $\displaystyle\int_X3c_2(X)-c_1(X)^2\geq0$, and the equality holds if and only if $X$ is uniformized by $\mathbb{B}\subset\mathbb{C}^2$ (the open ball). Ever in this case: $X$ does not contain neither rational smooth curves nor smooth curves of genus $1$.

In the opposite case, id est assuming that $\displaystyle\int_X3c_2(X)-c_1(X)^2>0$, there are examples of such $X$'s which do not contain any smooth rational curves.

Questions: let $X$ be a complex minimal surface of general type such that $\displaystyle\int_X3c_2(X)-c_1(X)^2>0$: does $X$ contain any smooth curves of genus $1$? Does the previous answer change if one assumes $\Omega^1_X$ nef but not ample and $K_X$ ample?

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    $\begingroup$ The Fano surface of lines $F(X)$ of a cubic threefold $X$ is a surface of general type. It contains an elliptic curve when $X$ has an Eckardt point. This happens exactly when $\Omega_X^1$ is not ample. $\endgroup$
    – AG learner
    Commented Dec 21, 2023 at 4:47

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Take a product $X= C_1\times C_2$, where $C_i$ are smooth curves of genus greater than $1$. $X$ has general type, and is uniformized by a product of two disks. Also $X$ won't contain an elliptic curve because it maps trivially to each $C_i$.

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  • $\begingroup$ Ok: $X$ is of general type with $K_X$ ample and $\Omega^1_X$ nef. However, why $\Omega^1_X$ is not ample? $\endgroup$ Commented Sep 1, 2023 at 12:50
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    $\begingroup$ Restrict $\Omega_X^1$ to $C_1\times pt$, then one factor is a trivial line bundle, so it can't be ample. $\endgroup$ Commented Sep 1, 2023 at 13:19

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