I was working on the proof of Duistermaat-Heckman theorem in Introduction to Symplectic Topology by Dusa McDuff. He used a lemma called localization. It can be found on page 194. You can find the lemma and the proof here.
and here is the definition of the differential
I have some questions about this proof.
1.What dose it mean to "average a form over certain group action"?
2.Why is $\phi^{*} \tau-\tau$ exact?
3.Why dose the first part of the proof work for $\phi^{*} \tau$? I cant seem to find the needed one-form here in the first part.
Forgive me for these stupid questions. Any comment or answer is appreciated.
Averaging a form $\alpha$ (or anything really) on a manifold $M$ over a compact Lie group $G$ acting on it means to take $\int_G \phi_g^*\alpha dg$ (where $\phi_g$ is the multiplication by $g$ map, and $dg$ is the invariant measure on $G$).
$\phi^*\tau-\tau$ is exact because $\phi$ is isotopic to the identity, so it must induce the identity map on cohomology.
The point is that $\phi^*\tau$ vanishes in a neighborhood of the fixed point set, so you can find $\alpha$ such that $\alpha(X)=1$ away from such a neighborhood, then the same argument should work.