Duistermaat-Heckman Formula and Bott’s Original Idea
1. Duistermaat-Heckman Formula
In this section, we consider the case of that $\left(M^{2l},\omega\right)$ is a symplectic manifold. Let $(M,\omega)$ be a symplectic manifold with $\omega$ is a symplectic structure. It means
- $\omega$ is a non-singular 2-form. i.e. If for any $Y\in \Gamma(TM)$ there always have $\omega(X,Y)=0$, then $X=0$.
- $\rd \omega=0$.
Assume $\omega$ is $S^1$-invariant and the $S^1$-action on $(M,\omega)$ is Hamiltonian. It means there exists a smooth function $\mu\in C^{\infty}(M)$ such that $\rd \mu=i_K\omega$. the $\mu$ is called with momentum map. We still assume that the zero set of $K$ is discrete.
定理 1. \[\int_{M}\exp\left(\sqrt{-1}\mu\right)\frac{\omega}{(2\pi)^ll!}
=\left(\sqrt{-1}\right)^l\sum_{p\in\mathrm{zero}(K)}\frac{\exp\left(\sqrt{-1}\mu(p)\right)}{\lambda(p)}.\]
=\left(\sqrt{-1}\right)^l\sum_{p\in\mathrm{zero}(K)}\frac{\exp\left(\sqrt{-1}\mu(p)\right)}{\lambda(p)}.\]
证明 . Since
\begin{align*}
\rd_{K}(\omega-\mu)=&(\rd+i_K)(\omega-\mu)\\
=&\rd\omega+i_k\omega-\rd\mu-i_K\mu\\
=&\rd\omega+i_K\omega-i_K\omega=\rd\omega=0,
\end{align*}
one sees that $\exp\left(\sqrt{-1}\mu-\sqrt{-1}\omega\right)$ is also $\rd_K$-closed. Using Berline-Vergne localization formula, one has
\[
\int_{M}\exp\left(\sqrt{-1}\mu-\sqrt{-1}\omega\right)=
(2\pi)^l\sum_{p\in\mathrm{zero}(K)}\frac{\exp\left(\sqrt{-1}\mu(p)\right)}{\lambda(p)}
\]i.e.,
\begin{align*}
\int_M\exp\left(\sqrt{-1}\mu\right)\left(-\sqrt{-1}\right)^l\frac{\omega^l}{l!}
=&(2\pi)^l\sum_{p\in\mathrm{zero}(K)}\frac{\exp\left(\sqrt{-1}\mu(p)\right)}{\lambda(p)}\\
\int_M\exp\left(\sqrt{-1}\mu\right)\frac{\omega^l}{(2\pi)^ll!}
=&\left(-\sqrt{-1}\right)^{-l}\sum_{p\in\mathrm{zero}(K)}\frac{\exp\left(\sqrt{-1}\mu(p)\right)}{\lambda(p)}\\
=&\left(\sqrt{-1}\right)^{l}\sum_{p\in\mathrm{zero}(K)}\frac{\exp\left(\sqrt{-1}\mu(p)\right)}{\lambda(p)}
\end{align*}
\begin{align*}
\rd_{K}(\omega-\mu)=&(\rd+i_K)(\omega-\mu)\\
=&\rd\omega+i_k\omega-\rd\mu-i_K\mu\\
=&\rd\omega+i_K\omega-i_K\omega=\rd\omega=0,
\end{align*}
one sees that $\exp\left(\sqrt{-1}\mu-\sqrt{-1}\omega\right)$ is also $\rd_K$-closed. Using Berline-Vergne localization formula, one has
\[
\int_{M}\exp\left(\sqrt{-1}\mu-\sqrt{-1}\omega\right)=
(2\pi)^l\sum_{p\in\mathrm{zero}(K)}\frac{\exp\left(\sqrt{-1}\mu(p)\right)}{\lambda(p)}
\]i.e.,
\begin{align*}
\int_M\exp\left(\sqrt{-1}\mu\right)\left(-\sqrt{-1}\right)^l\frac{\omega^l}{l!}
=&(2\pi)^l\sum_{p\in\mathrm{zero}(K)}\frac{\exp\left(\sqrt{-1}\mu(p)\right)}{\lambda(p)}\\
\int_M\exp\left(\sqrt{-1}\mu\right)\frac{\omega^l}{(2\pi)^ll!}
=&\left(-\sqrt{-1}\right)^{-l}\sum_{p\in\mathrm{zero}(K)}\frac{\exp\left(\sqrt{-1}\mu(p)\right)}{\lambda(p)}\\
=&\left(\sqrt{-1}\right)^{l}\sum_{p\in\mathrm{zero}(K)}\frac{\exp\left(\sqrt{-1}\mu(p)\right)}{\lambda(p)}
\end{align*}
About the generalization of above theorem in the case where the zero set of $K$ may not be discrete you can see[DH]: “J.J.Duistermaat and G.Heckman, Onthevaroation inthe cohomology of the symplectic from of the reduce phase space”.
2. Bott’s Original Idea From Bismut’s lemma, If $\rd_K\Omega =0$, one has
\[
\int_M\omega=\int_M\omega\exp\left(-T\rd_K\theta\right), \quad for~~any~~T>0.
\]when $\mathrm{zero(K)}=\emptyset $,
\[
\int_M\omega=\lim_{T\rightarrow+\infty}\int_M\omega\exp\left(-T\rd_K\theta\right)=0.
\]Bott’s original proof is different from above. Now we give a description of Bott’s idea.
Let $\omega$ be a $\rd_K$-closed form on $M$ with $K$ has no zeros on $M$. Since $\rd_K(\theta)=|K|^2+\rd\theta=|K|^2\left(1+\frac{\rd\theta}{|K|^2}\right)$, by $\frac{1}{1+x}=\sum_{i=0}^{\infty}(-1)^ix^i$, we can define
\[
\left(\rd_K\theta\right)^{-1}=
\frac{1}{|K|^2}\left[\sum_{i=0}^{\infty}(-1)^i\left(\frac{\rd\theta}{|K|^2}\right)^i\right].
\]From $K$ is $S^1$-invariant, the $\theta$ is also $S^1$-invariant. It shows $\mathcal{L}_K\theta=0$, i.e., $\rd_K^2\theta =0$. one can verify that
\begin{align*}
\rd_K\left[\rd_K^{-1}\wedge\theta\wedge\omega\right]=&
\rd_K\left(\rd_K\theta\right)^{-1}\wedge\theta\wedge\omega+
\left(\rd_K\right)^{-1}\theta \wedge\rd_K\theta\wedge\omega-\left(\rd_k\theta\right)^{-1} \wedge\theta\wedge\rd\omega\\
=&\left(\rd_K\theta\right)^{-2}\wedge \left(\rd_K\theta\right)^2+\omega-0\\
=&\omega.
\end{align*}
From this formula and the Stokes formula, one can directly calculate
\[
\int_M\omega=0.
\]
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