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In https://mathoverflow.net/questions/29964/which-spheres-are-complex-manifolds , I find that $S^2=\mathbb{C}P^1$ is a complex manifold and $S^6$ is an almost complex manifold.

Are there references about:

What is the smallest integer $N$ such that there exists an embedding of $S^2$ into $\mathbb{C}^N$ such that the image of $S^2$ is a complex submanifold?

What is the smallest integer $N$ such that there exists an embedding of $S^6$ into $\mathbb{C}^N$ such that the image of $S^6$ is an almost complex submanifold?

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There is no such $N$ since $\mathbb C^N$ has no compact complex submanifolds (this follows from the maximum principle: the coordinate functions $\Re z_i$ and $\Im z_i$ are pluriharmonic and therefore cannot have local max or min on any complex submanifold).

Also note that every "almost complex submanifold" (i.e. tangent spaces are complex subspaces) of a complex manifold is in fact a genuine (analytic) complex submanifold. So you can't hope to embed $S^6$ into any complex manifold unless you can equip $S^6$ with an integrable almost complex structure (a well-known open problem).

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