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?



Which complex manifolds embed into tori?
See:


@article {MR0350057,
 AUTHOR = {Matsushima, Yozo},
 TITLE = {Holomorphic immersions of a compact {K}\"ahler manifold into
 complex tori},
 JOURNAL = {J. Differential Geometry},
 FJOURNAL = {Journal of Differential Geometry},
 VOLUME = {9},
 YEAR = {1974},
 PAGES = {309--328},
 ISSN = {0022-040X},
 MRCLASS = {32C10 (32J99)},
 MRNUMBER = {0350057 (50 #2550)},
MRREVIEWER = {B. Smyth},
}


