Let $\alpha : \mathbb{C} \rightarrow M$ be an immersion and $M$ a $n$ dimensional complex manifold with complex structure $I$. Does then follow that $\alpha (\mathbb{C})$ is a one dimensional submanifold of $M$ ? If, yes, does the induced complex structure on $\alpha (\mathbb{C})$ by the immersion coincide with $I$ ?
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If $\alpha$ is only required to be a smooth embedding, then we can set $M = \mathbb{C}^2$, and embed $\mathbb{C}$ as a totally real subspace, e.g., $x + iy \mapsto (x,y)$. This is not a complex submanifold in the usual sense of the word, and the complex structure induced by $\alpha$ on its image is not compatible with $I$. If $\alpha$ is required to be an immersion of complex analytic spaces but not an embedding, then the image may have singularities. This is even true if we require $\alpha$ to be a homeomorphism to its image, since the normalization map of the cuspidal cubic curve is a homeomorphism whose image is a topological manifold that is not a complex manifold. If $\alpha$ is an analytic cover of its image, then the image is a complex submanifold of the ambient space, and the induced complex structure coincides with $I$. |
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