Let $X$ be a minimal surface of general type. Recall that a vector bundle $\mathscr{E}$ on $X$ is called *globally generated* if the evaluation map of global sections $$e \colon H^0(X, \, \mathscr{E}) \otimes \mathcal{O}_X \to \mathscr{E}$$ is surjective. Instead, $\mathscr{E}$ is called *ample* if the line bundle $\mathcal{O}_{\mathbb{P}(\mathscr{E})}(1)$ is ample on $\mathbb{P}(\mathscr{E})$. By the cohomological characterization of ampleness, if $\mathscr{E}$ is ample then the symmetric power $\operatorname{Sym}^n \mathscr{E}$ is globally generated (and ample) for $n \gg 0$.

Now let us take $\mathscr{E}=\Omega_X$, the cotangent bundle. I'm looking for examples of $X$ such that:

- $\Omega_X$ is neither globally generated nor ample;
- $\operatorname{Sym}^n \Omega_X$ is globally generated for $n \geq n_0$, where $n_0 \geq 2$ is an explicit constant (I would be already happy with $n_0=2$).

Note that these conditions provide several restrictions. For instance, 1. tells us that the Albanese map $a_X \colon X \to \operatorname{Alb}(X)$ is not a local immersion, since the non-surjectivity of the evaluation map for $\Omega_X$ at a point $x \in X$ is equivalent to the non-injectivity of the differential $da_X(x)$. Moreover, 2. implies that $K_X$ is ample: indeed, by my previous question MO412306, if $\operatorname{Sym}^n \mathscr{E}$ is globally generated then $X$ contains no rational curves at all; as a consequence, the failure of $a_X$ to be locally immersive is not related to the contraction of rational curves.

I have looked for such examples, so far without success. Probably, one of the reasons is that I do not know a geometrical characterization of the global generation of $\operatorname{Sym}^n \Omega_X$ in terms of the Albanese map. So let me ask the following

Question.What are examples of minimal surfaces of general type that satisfy 1. and 2. above? More generally, how can I check in general whether $\operatorname{Sym}^n \Omega_X$ is globally generated?

**Edit.** Will Sawin's nice answers provide examples $X$ which are étale quotients $Y \to X$, where $Y$ is a suitable hypersurface in a reducible abelian threefold. Then $X$ has zero second Segre class, namely, $s_2(X)=c_1(X)^2-c_2(X)=0$, because $Y$ has the same property and this is preserved by étale maps.

Question 2.Can we obtain examples with positive $s_2$?

My guess is that this should be possible, by generalizing Will's construction to étale quotients of suitable complete intersections in reducible Abelian fourfolds.

amplein place of $ample$. $\endgroup$