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Is the subspace of $\mathbb C^2$ of those vectors with real coordinates a complex submanifold?

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Perhaps part of the issue here is that, as a topological space, your subspace is homeomorphic to $\mathbb{C}$ and can thus certainly be endowed with the structure of a complex manifold. But being a complex *sub*manifold of $\mathbb{C}^2$ is a stronger condition, as Mariano's response addresses. –  Pete L. Clark Jan 12 '11 at 18:41
    
Thanks Pete. I see know how my thinking was wrong. –  jamie Jan 13 '11 at 14:49

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Call that subspace $X$. Then the composition of the inclusion $X\to \mathbb C^2$ with the first projection $\mathbb C^2\to\mathbb C$ is an holomorphic map defined on a $1$-dimensional complex manifold which takes only real values. Such a function is necessarily constant. Now do the same for the second projection. What can you conclude?

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It makes for a fun exercise to determine the $\mathbb R$-lineal subspaces of dimension $2$ in $\mathbb C^2$ which are complex submanifolds. –  Mariano Suárez-Alvarez Jan 12 '11 at 18:35
    
Ah, thanks! I see now. –  jamie Jan 13 '11 at 14:48

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