Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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 submanifold 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
up vote 3 down vote accepted

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?

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.