Chern class of tautological and hyperplane lines on complex projective space

By [F3], $c_1(\gamma )=t$, and by [F1], $x=-t$. Since [F2] supplies a generator, rather than merely a nonzero class, $x$ is a generator as well. Restriction carries this identity to the standard ${\mathbb{C}\mathbb{P}}^1$. It remains to check the asserted sign on that complex-oriented sphere.

Write $M={\mathbb{C}\mathbb{P}}^1$ and $L={\gamma }^{\ast }{\mathrm{\mid }}_M$. The linear functional $(z_0,z_1)\mapsto z_1$ restricts on each line to a section $s$ of $L$. This section is continuous in bundle charts and vanishes precisely at $p=[1:0]$. On the affine chart $w=z_1\mathrm{/}{z}_0$, the tautological frame is $(1,w)$; its dual frame expresses $s$ as the scalar $w$. Both this base coordinate and the dual fiber coordinate have their complex orientations.

Let $E$ be the total space of $L$ and $E^{\times }$ the complement of its zero section. Since $L_{\mathbb{R}}$ is an integrally oriented numerable rank-two bundle over the CW complex $M$, [F4] supplies its unique normalized class $u_L\in H^2(D(L),S(L);\mathbb{Z})$. Inclusion $(D(L),S(L))\to (E,E^{\times })$ induces an isomorphism by the pair sequences, since $D(L)\to E$ and $S(L)\to E^{\times }$ are homotopy equivalences by radial contraction and radial normalization; let $U\in H^2(E,E^{\times };\mathbb{Z})$ be the unique class restricting to $u_L$. The section gives $v=s^{\ast }U\in H^2(M,M\setminus \{p\};\mathbb{Z})$. Its absolute image is $e(L_{\mathbb{R}})=c_1(L)$, because the section $s$ and the zero section are homotopic by fiberwise multiplication by a parameter in $[0,1]$, and relative-to-absolute maps commute with pullback.

Choose a coordinate disk $V$ about $p$, centered at $w=0$. Excision identifies $H^2(M,M\setminus \{p\};\mathbb{Z})$ with $H^2(V,V\setminus \{0\};\mathbb{Z})$: remove $M\setminus V$, a closed subset of the open relative subspace. In the trivialization $E{\mathrm{\mid }}_V=V\times \mathbb{C}$, the Thom class is the pullback of the positive generator of $H^2(\mathbb{C},\mathbb{C}\setminus \{0\};\mathbb{Z})$ by the fiber projection. Indeed contraction of $V$ to its center gives an equivalence of pairs with the central fiber, where [F4] fixes that generator. The section is $w\mapsto (w,w)$, so its pullback is the positive local orientation cohomology class: the composite with the fiber projection is the identity coordinate $w\mapsto w$. Thus the local class $v$ evaluates to $+1$ on the complex-oriented local homology generator.

The fundamental class $[M]$ maps to that positive local generator by [F6]. Evaluation therefore gives $⟨c_1(L),[M]⟩=1$: the absolute image of $v$ is $c_1(L)$ by step 2.1, and evaluation commutes with the relative quotient on chains. Explicitly, a relative cocycle is a cochain vanishing on chains in $M\setminus \{p\}$, so evaluating its absolute image on a cycle equals evaluating the relative cocycle on that cycle's relative image. This also shows that excision and restriction preserve this pairing. Hence $x$ is positive in the stated convention, and $c_1(\gamma )=-x$ is negative.

Since ${\gamma }^{\ast }$ is a line, [F3] gives $c({\gamma }^{\ast })=1+c_1({\gamma }^{\ast })=1+x$. For $n=1$ the restriction used above is the identity. The empty base does not occur; $n=0$ is excluded from the generator claim because $H^2({\mathbb{C}\mathbb{P}}^0;\mathbb{Z})=0$. All bundles used are over finite CW complexes and their complex orientations supply the integral Thom normalization. AC is inherited as stated in [A1].