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?



Existence of elliptic curves on surfaces of general type
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 $C\_i$.


