A noncompact variety with a proper fixed component

Each listed chart is isomorphic to ${\mathbb{C}}^2$, is $T$-invariant, and the two cover $X$. The action fixes $([x:y],z)$ exactly when $z=0$, so $F={\mathbb{P}}^1\times \{0\}\cong {\mathbb{P}}^1$ is smooth and proper. Its normal bundle is the trivial complex line over ${\mathbb{P}}^1$ with character $\chi$; hence $e_T({\nu }_F)=u$ by [F3].

Let $i:p\hookrightarrow F$ be the point inclusion and choose a complex coordinate disk $D$ about $p$. The complex orientation of its normal line gives the relative Thom class $\tau \in H^2(D,D\setminus \{p\};\mathbb{Q})$ whose restriction evaluates as $+1$ on the oriented disk. By [F4], $h=i_{!}1$ is the image of this positive Thom class under excision and the relative-to-absolute map. At the $m$th finite Borel approximation, the trivial $T$-action gives $F_m={\mathbb{C}\mathbb{P}}^m\times F$ and the neighborhood ${\mathbb{C}\mathbb{P}}^m\times D$; the Thom class there is $1\times \tau$. Let $h_m\in H^2(F_m;\mathbb{Q})$ be its image, first by excision and then by the relative-to-absolute map for $(F_m,F_m\setminus ({\mathbb{C}\mathbb{P}}^m\times \{p\}))$, and let $i_m:{\mathbb{C}\mathbb{P}}^m\times \{p\}\hookrightarrow F_m$ be the induced inclusion. The local cap-Thom calculation sends $(1\times \tau )\cap [{\mathbb{C}\mathbb{P}}^m\times D]$ to $[{\mathbb{C}\mathbb{P}}^m\times \{p\}]$ with coefficient $+1$: the positive generator of $(D,D\setminus \{p\})$ caps the oriented disk to the positive point class, and the product orientation contributes no sign. Excision therefore gives $h_m\cap [F_m]=i_{m\ast }[{\mathbb{C}\mathbb{P}}^m]$ in Borel–Moore degree $2m$. Passing to the stable normalized Borel–Moore groups yields $h\cap [F{]}_T=i_{\ast }[p{]}_T$. Since the action on $F$ is trivial, $\alpha {\mathrm{\mid }}_F=h$.

Both maps $i:p\to F$ and $p_F:F\to \mathrm{p}\mathrm{t}$ are proper. By functoriality of Borel–Moore pushforward and step 1.2, $p_{F\ast }(h\cap [F{]}_T)=p_{F\ast }{i}_{\ast }[p{]}_T=(p_F\circ i{)}_{\ast }[p{]}_T=1$, because $p_F\circ i$ is the identity of the point and its complex-oriented degree-zero class is $1$. This also verifies the stated normalization ${\int }_{{\mathbb{P}}^1}h=1$ through the cap-class identification, rather than inferring a Borel–Moore trace directly from cohomological functoriality.

The localization theorem [F2] gives $p_{\ast }^{\mathrm{l}\mathrm{o}\mathrm{c}}(\alpha \cap [X{]}_T)=p_{F\ast }((h\mathrm{/}u)\cap [{\mathbb{P}}^1{]}_T)=1\mathrm{/}u$ by step 2.1. The target is $(S_T^{-1}{H}_{\ast }^{\mathrm{B}\mathrm{M},T}(\mathrm{p}\mathrm{t};\mathbb{Q}){)}_2=(S_T^{-1}{R}_T{)}^{-2}$, containing $1\mathrm{/}u$, although the unlocalized $H_2$ of the point is zero. The output has Borel–Moore degree $2$, matching $2{\dim }_{\mathbb{C}}X-\mathrm{deg}\alpha =4-2$.