Curvature form, tangent bundle and structural group.

Question

Let $T\mathbb{C}P^n$ the tangent bundle over $\mathbb{C}P^n$. We have that the Chern classes are the coefficients of the characteristic polynomial of curvature form $\Omega$ of $T\mathbb{C}P^n$: $$ det \left (\frac{it\Omega}{2\pi} + Id \right) = \sum_{i}^n c_k(V)t^k $$ where $\Omega:=d\omega + \frac{1}{2}[\omega, \omega]$ with $\omega$ the connection form. I have two questions: $$ $$ 1) How can I calculate explicitly $\Omega$ for $T\mathbb{C}P^2$? What is the reason of this definition? $$ $$ 2) How can find explicitly the transition functions $g_{\alpha \beta}: U \rightarrow GL_2(\mathbb{C})$? ($U$ is a trivial open in $\mathbb{C}P^2$) How can I reduce $GL_2(\mathbb{C})$ in $SU(2)$? And in what sense Chern classes represent the obstruction to reduce $U(n)$ in $SU(n)$?

Answer 1

One approach is to learn about moving frames. You naturally work with $\mathbb CP^n$ by lifting to a unitary frame on $\mathbb C^{n+1}$. (This is not unrelated to how Chern first developed Chern classes.) A bit more abstractly, you can think of $\mathbb CP^n$ as a symmetric space and get the connection and curvature form directly from the Lie algebraic structure. Unfortunately, this is not stuff that is clearly worked out in textbooks, to my knowledge.