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I encountered the following problem in Complex Geometry-an introduction of Dianel Huybrechts:

Show that any submanifold of a complex torus has nonnegative Kodaira dimension.

Since $\operatorname{kod}(T^n)=0$, I think it's sufficient to show that if $f: Y\to X$ is an embedding, then $f^*:Q(X)\to Q(Y)$ is a field embedding, where $Q(X)$ is the canonical ring of $X$. I wonder how to prove it. Thanks in advance.

eulershi
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    I think this might be simpler than that. The tangent and cotangent bundles of a torus are trivial, therefore globally generated. It follows that the cotangent bundle of a submanifold of a torus is globally generated. Therefore the canonical bundle has a nonzero section. – Gunnar Þór Magnússon Apr 11 '23 at 11:32

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