Atiyah–Bott–Berline–Vergne Integration Formula

Localize the restriction map $i^{\ast }:H_T^{\ast }(M;\mathbb{R})\to H_T^{\ast }(F;\mathbb{R})$ at $S$. By [L2] it is an isomorphism in this compact smooth finite-type case. Let $\beta =i^{\ast }\alpha =({\beta }_j{)}_j$ under $H_T^{\ast }(F)={⨁}_j{H}_T^{\ast }(F_j)$.

By [F3], each $e_T({\nu }_j)$ is invertible after localization. Form $\gamma ={\sum }_j{i}_{j\ast }^T({\beta }_j\mathrm{/}{e}_T({\nu }_j))\in S^{-1}{H}_T^{\ast }(M;\mathbb{R})$; the sum is finite because $F$ has finitely many components.

All maps and identities in [F2] are $\mathcal{R}$-linear and extend $S^{-1}\mathcal{R}$-linearly to the localized modules. For a component $F_{\mathrm{\ell }}$, pull back $\gamma$. The $j=\mathrm{\ell }$ term is $e_T({\nu }_{\mathrm{\ell }}){\beta }_{\mathrm{\ell }}\mathrm{/}{e}_T({\nu }_{\mathrm{\ell }})={\beta }_{\mathrm{\ell }}$ by the localized self-intersection identity; each $j\mathrm{\ne }\mathrm{\ell }$ term restricts to zero by the localized disjoint-support identity. Hence $i^{\ast }\gamma =\beta =i^{\ast }\alpha$. Since the localized restriction map in step 1.1 is injective, $\gamma =\alpha$.

The $\mathcal{R}$-linear pushforwards in [F1] and the identity ${\int }_M{i}_{j\ast }^T(\delta )={\int }_{F_j}\delta$ in [F2] extend $S^{-1}\mathcal{R}$-linearly, so they apply to $\delta ={\beta }_j\mathrm{/}{e}_T({\nu }_j)$. Applying ${\int }_M$ to step 3.1 gives ${\int }_M\alpha ={\sum }_j{\int }_M{i}_{j\ast }^T({\beta }_j\mathrm{/}{e}_T({\nu }_j))={\sum }_j{\int }_{F_j}{\beta }_j\mathrm{/}{e}_T({\nu }_j)$, which is the fixed-component formula in the localized coefficient ring. If $F=\mathrm{\varnothing }$, [L2] says $S^{-1}{H}_T^{\ast }(M)=0$, so ${\int }_M\alpha$ localizes to zero; because $\mathcal{R}$ is a domain and every element of $S$ is nonzero, $\mathcal{R}\hookrightarrow S^{-1}\mathcal{R}$ is injective and the empty sum is indeed zero. If a normal rank is zero, its Euler denominator is $1$ by [F3], so that term is the ordinary integral over that fixed component.

If $F$ is nonempty and each fixed component is a point $p$, then ${\nu }_p=T_{p}M$ has no trivial weight and hence $n=2m$. With the stated ambient normal and positive point orientations, its complex character-line decomposition has Euler class $e_T(T_{p}M)={ϵ}_p{\prod }_{a=1}^m{u}_{{\chi }_{p,a}}$ by [F1]. Substituting into step 4.1 and using ${ϵ}_p^{-1}={ϵ}_p$ yields the isolated-fixed-point formula. If $n=0$, the product is empty and the same calculation reads as the sum of the oriented point evaluations. For empty $F$, step 4.1 already gives zero in every dimension, without introducing $m$.