Let $X$ be a projective surface of general type over $\mathbb{C}$, and assume that $\Omega_X$ is globally generated. Then the Albanese map $a \colon X \to \operatorname{Alb}(X)$ is a local embedding, namely, a finite, étale cover onto its image $a(X) \subset \operatorname{Alb}(X)$. It is also well known that $a(X)$ generate $\operatorname{Alb}(X)$.

> **Question.** Is the normal bundle of $a(X)$ an ample vector bundle?

By a result of Hartshorne (see [1]), in order to have a positive answer, it suffices to check that every curve in $a(X)$ generates $\operatorname{Alb}(X)$, see also MO187149 https://mathoverflow.net/questions/187149/on-a-proposition-in-hartshornes-paper-ample-vector-bundles-on-curves. In particular, this shows that the answer to the previous question is "yes" when $\operatorname{Alb}(X)$ is *simple*. But what can happen in general?   

**References.**

[1]. <cite authors="Hartshorne, R.">_Hartshorne, R._, [**Ample vector bundles on curves**](http://dx.doi.org/10.1017/S0027763000014379), Nagoya Math. J. 43, 73-89 (1971). [ZBL0218.14018](https://zbmath.org/?q=an:0218.14018).</cite>  

[1]: https://mathoverflow.net/questions/187149/on-a-proposition-in-hartshornes-paper-ample-vector-bundles-on-curves