Is the subspace of $\mathbb C^2$ of those vectors with real coordinates a complex submanifold?
2026-03-28 04:32:55.1774672375
question about a complex manifold
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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?