Chern classes tangent bundle.

I studied Chern classes but I don't find explicit examples of calculation. So I'd like to consider the tangent bundle over $\mathbb{C}P^n$ ($T\mathbb{C}P^n$) and develop the theory beginning from this example. I have a preliminar question: how can I imagine the first Chern classes $c_1 \in H^*(\mathbb{C}P^1, \mathbb{Z})$? I read that it is an hyperplane, but what hyperplane? Why is it the fundamental homology class? Now we have tha Chern classes are the coefficient of characteristic polynomial of the curvature form $(\Omega)$ of $T\mathbb{C}P^n$, where $\Omega:= d\omega + \frac{1}{2}[\omega,\omega]$ with $\omega$ the connection form. How can I calculate $\Omega$ for $\mathbb{C}P^n$ and why this coefficients are cohomology classes? And how can I calculate $c(T\mathbb{C}P^n)$? Futhermore, how can I imagine Chern classes as the obstruction to reduce the structure group of $T\mathbb{C}P^n$ $(U(n))$ to $SU(n)$? Can you explain me and explicit calculus?

• That is a large number of questions. Here are some pointers. First, you should read about de Rham cohomology to learn why differential forms give cohomology classes. I think the classic reference for that is the text of Bott--Tu. Second, it's also useful to know about different constructions of Chern classes. For example, there is a construction using the classifying map of a vector bundle that does not use any machinery of differential geometry. (Personally, I prefer this approach.) You can read about that approach in Milnor--Stasheff, "Characteristic Classes". – user64687 May 2 '13 at 22:11
• (In particular, the latter approach makes the calculation of the chern classes of projective space very straightforward.) – user64687 May 2 '13 at 22:12
• @AsalBeagDubh Beag Dubh I know De Rham cohomology... I don't understan why these coefficients are differential forms. I calculated chern calasses of projective spaces. But I'd like to calculate Chern classes of $T\mathbb{C}P^n$. – ArthurStuart May 2 '13 at 22:16
• Sorry if I misread your questions. I don't understand your distinction between Chern classes of projective space and Chern classes of $T(CP^n)$: in my experience when people talk about the Chern classes of a manifold, they mean exactly the Chern classes of its tangent bundle. (Also, to answer your first question briefly: all hyperplanes in projective space are in the same cohomology class.) – user64687 May 2 '13 at 22:18
• @AsalBeagDubh I'm sorry... I think so. But how can I calculate $\Omega$ for $T\mathbb{C}P^n$? – ArthurStuart May 2 '13 at 22:21