Darboux coordinates on projective spaces

Question

I am trying to perform some computations in local coordinates on $\Bbb P ^n \Bbb C$ seen as a symplectic manifold, in order to get a better feeling of some facts. While I do know the coefficients of the symplectic form in the usual charts (where one projective coordinate is $1$), they do not lend themselves easily to the types of calculations that I need to do, which would be far simpler in Darboux coordinates. Consequently, does anyone know whether one can explicitly exhibit Darboux coordinates on $\Bbb P ^n \Bbb C$?

Answer 1

According to "Hamiltonian manifolds and moment maps" by Nicole Berline and Michèle Vergne (chapter 2.3, page 19), or to the lectures notes by Yael Karshon (page 1), Darboux coordinate patches can be obtained as follows:

• let $U = \{ [z_0 : \dots : z_n] \in \Bbb P ^n \Bbb C \mid z_0 \ne 0 \}$, as usual

• let $B = \{ (x_1, \dots, x_n, y_1, \dots, y_n) \in \Bbb R ^{2n} \mid x_1^2 + \dots x_n^2 + y_1^2 + \dots + y_n^2 < 1 \}$ (the open ball of radius $1$ and center $0$)

• define the parametrization $\varphi : B \to U$ by

$$\varphi (x_1, \dots, x_n, y_1, \dots, y_n) = \left[\sqrt{1 - x_1^2 - \dots - x_n^2 - y_1^2 - \dots - y_n^2} : x_1 + y_1 {\rm i} : \dots : x_n + y_n {\rm i} \right]$$

Similar formulae, but more annoying to write, can be produced for the other charts given by $z_i \ne 0$, for every $i = 1, \dots, n$.