Coordinates on Flag Manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:50:29Z http://mathoverflow.net/feeds/question/29507 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29507/coordinates-on-flag-manifolds Coordinates on Flag Manifolds Charlie Frohman 2010-06-25T13:00:57Z 2010-06-25T13:26:19Z <p>Suppose you want to work with complete flags $\mathbb{F}_3$ on $\mathbb{C}^3$. Given a flag</p> <p>$${0}\leq V_1\leq V_2 \leq \mathbb{C}^3$$</p> <p>you can think of $V_1$ as the span of a vector $\vec{u}$, and then you can choose a vector $\vec{v}$ that is Hermitian orthogonal to $\vec{u}$ so that $V_2=&lt;\vec{u},\vec{v}>$. Finally you can choose $\vec{w}$ so that it is Hermitian orthogonal to $&lt;\vec{u},\vec{v}>$. This gives an embedding</p> <p>$$\mathbb{F}_3\rightarrow \mathbb{C}P(2)^3 .$$</p> <p>Since the Hermitian inner product involves complex conjugates, this embedding cannot possibly be holomorphic. For instance if the first line in homogeneous coordinates is $[a,b,c]$ and the second is $[d,e,f]$ then they satisfy $a\overline{d}=b\overline{e}+c\overline{f}=0$ where the overline indicates complex conjugate. Is there some way of playing around with complex structures to fix this? Is there a similar map, that is better behaved?</p> http://mathoverflow.net/questions/29507/coordinates-on-flag-manifolds/29514#29514 Answer by Diego Matessi for Coordinates on Flag Manifolds Diego Matessi 2010-06-25T13:26:19Z 2010-06-25T13:26:19Z <p>I think you can use wedge products. Choose $v \in V_1$, then $u \in V_2$, which is linearly independent. Map the flag to $([v], [v \wedge u]) \in (CP^{2})^2$. This should be well defined and holomorphic.</p>