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 On a proposition in Hartshorne's 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. Hartshorne, R., Ample vector bundles on curves, Nagoya Math. J. 43, 73-89 (1971). ZBL0218.14018.