I am studying Kahler Manifolds for applications in non-Hermitian Quantum Mechanics. I am struggling to get an intuitive understanding of the almost complex structure $J$.
I would like to formulate $J$ for specific Kahler Manifolds in particular coordinate systems. For example:
$\mathbb{C}P^1$ is a Kahler manifold and parametrised by $z$. The Kahler form for this manifold is $K=\tfrac{i}{2}\frac{dz\wedge{d}\bar{z}}{(1+z\bar{z})^2}$ and metric is defined via $ds^2=\frac{Re(dz\otimes{d}\bar{z})}{(1+z\bar{z})^2}$. In this example, what is $J$ such that $\omega(X,Y)=g(JX,Y)$?
As $\mathbb{C}P^1\cong{S^2}$, we can parametrise $S^2$ with coordinates $x,y,z$. Can we express a symplectic form, metric tensor and almost complex structure in terms of $x,y,z$ on $S^2$? Would this now be a Poisson Manifold and would the techniques have to be adjusted?
Thank you in advance for your help,
Wasim
The simplest Kahler manifold is $\mathbb{C}^n=\mathbb{R}^{2n}$. Write the $i$th complex coordinate on $\mathbb{C}^n$ as $z_i=x_i+\sqrt{-1}y_i$. The natural complex structure on $\mathbb{C}^n$ is $$ J\frac{\partial}{\partial x_i}=\frac{\partial}{\partial y_i},~~J\frac{\partial}{\partial y_i}=-\frac{\partial}{\partial x_i} $$ The standard symplectic form $\omega_0$ on $\mathbb{R}^{2n}$ is then a Kahler form for $(\mathbb{C}^n,J)$. Explicitly, $\omega_0$ is $$ \omega_0=\sum_i dx_i\wedge dy_i $$ One can then check that $\omega_0(J\cdot,J\cdot)=\omega_0(\cdot,\cdot)$ and $$ \omega_0(u,Ju)>0 $$ for all nonzero $u\in T\mathbb{R}^{2n}$.