# Existence of elliptic curves on surfaces of general type

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?

• 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. Commented Dec 21, 2023 at 4:47

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