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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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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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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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