Obstruction for real subvariety to be embedded as complex subvariety - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:47:24Z http://mathoverflow.net/feeds/question/40828 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40828/obstruction-for-real-subvariety-to-be-embedded-as-complex-subvariety Obstruction for real subvariety to be embedded as complex subvariety Colin Tan 2010-10-02T12:02:55Z 2010-10-04T16:20:38Z <p>Let $X$ be a nonsingular complex projective variety. Suppose $X$ is embedded as a nonsingular real subvariety of complex projective space ${\mathbb{CP}}^n$. </p> <p>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 <i>complex</i> subvariety of this larger ${\mathbb{CP}}^n$?</p> http://mathoverflow.net/questions/40828/obstruction-for-real-subvariety-to-be-embedded-as-complex-subvariety/40909#40909 Answer by S. Carnahan for Obstruction for real subvariety to be embedded as complex subvariety S. Carnahan 2010-10-03T04:53:27Z 2010-10-03T04:53:27Z <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/40828/obstruction-for-real-subvariety-to-be-embedded-as-complex-subvariety/41037#41037 Answer by Oleg Eroshkin for Obstruction for real subvariety to be embedded as complex subvariety Oleg Eroshkin 2010-10-04T16:20:38Z 2010-10-04T16:20:38Z <p>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 $\mathbb{CP}^N$.</p>