Peng Zhou

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====== 2023-06-17 ======
  * equivariant cohomology and localization


===== equivariant =====
Let $G$ be a compact connected Lie group, acting on a smooth manifold $M$. Suppose we have another $G$-manifold $N$, and a map $\pi: M \to N$, then I want to integrate along the fiber
$$ \pi_*: H^*(M) \to H^{*-d}(N). $$
This should be a module map of $H^*(N)$, where we view $H^*(M)$ as a module over $H^*(N)$. 

Now, given that we have $G$-action, we have the universal family of $M$ and $N$ over $BG$, and we have $H^*_G(N)$-mod hom $H_G^*(M) \to H^{*-d}_G(N)$. 

Say, we have a $G/B = U/T$ a flag variety. And, we have left $T$-action, then we have a bunch of fixed points. Suppose we want to do some integration, of the type $e^W dvol$, I mean, we can do some fiberwise integral, then restrict to the base, and do an integral over there. But that does not simplify anything.

The idea is to introduce some small monodromy along the $T$-orbit, so that integration along them will cancel out. I know only one example, namely, how $S^1$ acts on $\C$, and with $e^{-|z|^2}$ as weight. What if, we have some other $f(r^2)$ function as weight? Then, we would have 
$$ \omega = f(r^2) d(r^2) d\theta + u \phi(r^2), \quad \phi'(r^2) d r^2 = f(r^2) d(r^2) $$
But, I don't know whether we can add some constant to $\phi$. 

Oh, I see, I better set the value of $\phi$ at $\infty$ to be zero, so that the contribution of $\phi(\infty)$ would be zero.

wait, but there is no small monodromy around the $S^1$-orbit.