If $X$ is a compact Kahler manifold then it's well-known that $X$ can be embedded into a projective space if and only if it admits an ample line bundle. Suppose now that we look for other things to embed $X$ into, like complex tori, and ask for necessary and sufficient conditions for this to happen.
If $X$ is a submanifold of a torus $A$, then the short exact sequence $$ 0 \longrightarrow N_{X/A}^* \longrightarrow \Omega_{A|X}^1 \longrightarrow \Omega_X^1 \longrightarrow 0 $$ over $X$ shows that the cotangent bundle $\Omega^1_X$ is globally generated, because $A$ is a torus and so $\Omega^1_A$ is trivial. It's not hard to see that the converse has some truth to it, that is if $\Omega_X^1$ is globally generated then the Albanese morphism $\alpha : X \to \operatorname{Alb} X$ is a local immersion (by looking at its differential), which seems to me like it should imply that the image of $\alpha$ in $\operatorname{Alb} X$ is smooth (locally the image looks like $X$) and that $\alpha : X \to \alpha(X)$ is a finite morphism (by compactness of everything).
Is there a simple condition on $X$ that ensures that the Albanese morphism is actually injective and thus an embedding when $\Omega_X^1$ is globally generated?
