ABBV on the Projective Line with the Standard Circle Action

A point $[z_0:z_1]$ is fixed by every $\lambda$ precisely when it lies on one coordinate axis, so the fixed set is $\{p_0,p_{\mathrm{\infty }}\}$. In the coordinate $w=z_1\mathrm{/}{z}_0$ near $p_0$, the action is $w\mapsto \lambda w$, giving tangent weight $+1$; in $v=z_0\mathrm{/}{z}_1$ near $p_{\mathrm{\infty }}$, it is $v\mapsto {\lambda }^{-1}v$, giving weight $-1$. The complex orientations agree with $M$, so the Euler denominators are $u$ and $-u$.

The tautological fiber at $p_0$ is the trivial coordinate line, and at $p_{\mathrm{\infty }}$ it is the weight-$+1$ coordinate line. Dualizing gives the fixed-fiber weights $0$ and $-1$ for $\mathcal{O}(1)$, hence $x{\mathrm{\mid }}_{p_0}=0$ and $x{\mathrm{\mid }}_{p_{\mathrm{\infty }}}=-u$.

Substituting these restrictions and tangent Euler classes in [F2] gives ${\int }_{M}x=0\mathrm{/}u+(-u)\mathrm{/}(-u)=1$.

The linear functional $\mathrm{\ell }(z_0,z_1)=z_0$ defines a section of $\mathcal{O}(1)$ that vanishes only at $p_{\mathrm{\infty }}$. In the local coordinate $v=z_0\mathrm{/}{z}_1$ and the dual tautological frame there, its coefficient is $v$, so its unique zero is transverse with positive local degree $+1$. By [F3], the normalized Euler pairing counts this transverse zero with its positive local degree. Hence the ordinary Euler number is $1$. Restricting the Borel pushforward to the basepoint of connected $BT$ gives this ordinary Euler number, and $H^0(BT;\mathbb{R})=\mathbb{R}$; therefore ${\int }_{{\mathbb{C}\mathbb{P}}^1}x=1$, independently confirming the localized sum.