I have shown that the FS form on an affine chart of $CP^1$ is $1/4$ times the usual area form (written in $\theta$,h coordinates).
So the area of $CP^1$ wrt FS form should be $1/4$ of usual area right?
But when I integrate the FS form that is $(dx\wedge dy)/(1+x^2+y^2)^2$ over $S^2$ (since it is diffeomorphic to $CP^1$) I am getting $\pi /2$. Area of $S^2$ is $4\pi$ right? So it doesn't add up.
Where am I making a mistake?
We use polar coordinates $dx\wedge dy=r dr d\theta$, then set $t=r^2$
$\int \int {dx\wedge dy\over (1+x^2+y^2)^2}= \int_0^\infty {r dr\over (1+r^2)^2}\int_0^{2\pi}d\theta =$ $ (2\pi)\times \int _0^\infty {1\over 2} {dt\over (1+t)^2 }= 2\pi \times {1\over 2}= \pi$