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:

 1. $\Omega_X$ is neither globally generated nor ample;
 2. $\operatorname{Sym}^n \Omega_X$ 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][1], 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?

[1]: https://mathoverflow.net/questions/412306/projective-variety-of-general-type-such-that-sm-omega-x1-is-globally-genera/