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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?

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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}, }

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  • $\begingroup$ I am late to the party, sorry. It seems to me that the Matsushima calls "ample" what nowadays is called "globally generated". In fact, saying that $T^*M$ is ample usually means, instead, that the sheaf $\mathcal{O}_{\mathbb{P}(T^*M)}(1)$ is ample, and this is a different notion. $\endgroup$ Jan 18, 2022 at 14:35
  • $\begingroup$ Also, where is in Matsushima's paper the proof of injectivity of the Albanese map? I can find only the proof of injectivity of its differential (that is equivalent to $T^*M$ being globally generated). On the other hand, the answer is accepted, so I am probably missing something... $\endgroup$ Jan 18, 2022 at 15:09

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