## Obstruction for real subvariety to be embedded as complex subvariety

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Let $X$ be a nonsingular complex projective variety. Suppose $X$ is embedded as a nonsingular real subvariety of complex projective space ${\mathbb{CP}}^n$.

When can we embed ${\mathbb{CP}}^n$ in some larger complex projective space ${\mathbb{CP}}^N$ such that the image of $X$ is now a nonsingular complex subvariety of this larger ${\mathbb{CP}}^n$?

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Colin, do you mean to embed $\mathbb{CP}^n$ as a real submanifold?  – Oleg Eroshkin Oct 2 2010 at 15:29
The fact that $\mathbb{CP}^n$ has a complex structure seems to be irrelevant for Colin's question. The question only uses the structure of real variety of $\mathbb{CP}^n$. – André Henriques Oct 2 2010 at 18:24
I mean embed both $X$ and ${\mathbb{CP}}^n$ as complex submanifolds of ${\mathbb{CP}}^N$. – Colin Tan Oct 3 2010 at 3:09

## 2 Answers

I am assuming from your comment that you are demanding that the embedding $X \to \mathbb{CP}^N$ preserves the complex structure that $X$ originally had (although it is still not completely clear). In this case, the real embedding $X \to \mathbb{CP}^n$ has to be a complex embedding. Otherwise, the actions of multiplication by $i$ on the tangent spaces of $X$ and $\mathbb{CP}^N$ will be incompatible.

For example, you can take $X = \mathbb{CP}^1$, with complex conjugation as the real embedding into $\mathbb{CP}^1$. This is a real-algebraic isomorphism, but orientation-reversing. Then, you will never have a compatible way to embed both varieties as complex subvarieties in $\mathbb{CP}^N$ by complex algebraic maps.

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thanks scott. you answered my question (in this form well).   this question still does not get at the situation i'm really considering. when i can think of how better to phrase my question I will ask it again on MO.  – Colin Tan Oct 3 2010 at 6:38

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Actually answer is no even when you allow real embedding of $\mathbb{CP}^n$. Consider any null-homotopic embedding of $X$ into $\mathbb{CP}^n$. Such embedding cannot be a pullback of a complex submanifold of $\mathbb{CP}^N$.

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