The tangent bundle of complex projective space and its Pontryagin classes

By [F1] the space ${\mathbb{C}\mathbb{P}}^n$ is a compact complex manifold of real dimension $2n$, hence a closed smooth manifold with the complex orientation, and it is a finite CW complex. By [F2] its integral cohomology is $H^{\ast }({\mathbb{C}\mathbb{P}}^n;\mathbb{Z})=\mathbb{Z}[u]\mathrm{/}(u^{n+1})$ with $u=c_1(\gamma )=e({\gamma }_{\mathbb{R}})$ generating each even degree, and by [F4] applied to the dual line $x=c_1({\gamma }^{\ast })=-u$; therefore $x$ also generates $\mathbb{Z}[u]$ as a ring, $x^{n+1}=0$, and $x^k\mathrm{\ne }0$ for $0\le k\le n$. The vanishing $x^{n+1}=0$ is also the monic relation of the projective bundle theorem applied to the trivial rank-$(n+1)$ bundle over a point [F2].

Euler sequence. At a line $\mathrm{\ell }\subset {\mathbb{C}}^{n+1}$ choose a linear complement $H$, so that nearby lines are graphs of linear maps $a:\mathrm{\ell }\to H$, and identify ${\mathbb{C}}^{n+1}\mathrm{/}\mathrm{\ell }$ with $H$ by the quotient map. Differentiating the graph chart at $a=0$ identifies $T_{[\mathrm{\ell }]}{\mathbb{C}\mathbb{P}}^n$ with $\mathrm{Hom}(\mathrm{\ell },H)\cong \mathrm{Hom}(\mathrm{\ell },{\mathbb{C}}^{n+1}\mathrm{/}\mathrm{\ell })$; this identification is independent of the complement, because for a smooth family of nonzero representatives $v(t)$ of the moving line the derivative defines $v(0)\mapsto v^{\mathrm{\prime }}(0)<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>\mathrm{\ell }$, and rescaling $v$ by a nonzero scalar multiplies both the input and the derivative modulo $\mathrm{\ell }$ by that scalar. The graph charts show that the identification and its inverse vary smoothly, so it is an isomorphism of complex vector bundles $T{\mathbb{C}\mathbb{P}}^n\cong \mathrm{Hom}(\gamma ,{\underset{‾}{\mathbb{C}}}^{n+1})\mathrm{/}\mathrm{Hom}(\gamma ,\gamma )$. Explicitly, composition with the fiberwise quotient map ${\mathbb{C}}^{n+1}\to {\mathbb{C}}^{n+1}\mathrm{/}\gamma$ defines a complex bundle map $q:\mathrm{Hom}(\gamma ,{\underset{‾}{\mathbb{C}}}^{n+1})\to T{\mathbb{C}\mathbb{P}}^n$; fiberwise it is surjective, with a linear lift supplied by any complement, and its kernel is the line $\mathrm{Hom}(\gamma ,\gamma )$ of scalar multiples of the inclusion, so $0\to \underset{‾}{\mathbb{C}}\to \mathrm{Hom}(\gamma ,{\underset{‾}{\mathbb{C}}}^{n+1})\stackrel{q}{\to }T{\mathbb{C}\mathbb{P}}^n\to 0$ is short exact as bundles, with the local lifts of the graph charts exhibiting the local subbundle structure rather than a dimension count. The evaluation of the $n+1$ coordinate functionals of the fixed space ${\mathbb{C}}^{n+1}$ gives a canonical identification $\mathrm{Hom}(\gamma ,{\underset{‾}{\mathbb{C}}}^{n+1})\cong ({\gamma }^{\ast }{)}^{\oplus (n+1)}$ [F9]. Give ${\gamma }^{\ast }$ the metric dual to the standard metric on $\gamma$ and the direct sum its product metric; then [F9] splits the sequence over the base, giving a complex bundle isomorphism $\underset{‾}{\mathbb{C}}\oplus T{\mathbb{C}\mathbb{P}}^n\cong ({\gamma }^{\ast }{)}^{\oplus (n+1)}$ that need not be canonical.

Chern classes. Since $c(\underset{‾}{\mathbb{C}})=1$ and Chern classes of isomorphic bundles agree, the splitting of step 1.2 and the Whitney formula [F3] give $c(T{\mathbb{C}\mathbb{P}}^n)=c(({\gamma }^{\ast }{)}^{\oplus (n+1)})=c({\gamma }^{\ast }{)}^{n+1}=(1+x{)}^{n+1}$ in $H^{\ast }({\mathbb{C}\mathbb{P}}^n;\mathbb{Z})=\mathbb{Z}[x]\mathrm{/}(x^{n+1})$, where the line normalization enters only through the definition of $x=c_1({\gamma }^{\ast })$; equivalently $c_i(T{\mathbb{C}\mathbb{P}}^n)=\left(\genfrac{}{}{0px}{}{n+1}{i}\right)x^i$ for $0\le i\le n$ and $c_i=0$ for $i>n$ by the rank convention.

Pontryagin classes. Let $V$ be a complex bundle and write $V_{\mathbb{C}}=(V_{\mathbb{R}}){\otimes }_{\mathbb{R}}\mathbb{C}$. If $v\mapsto \bar{v}$ denotes the canonical antilinear map $V\to \bar{V}$, the complex-linear isomorphism is $v\otimes z\mapsto (zv,z\bar{v})$. Its inverse sends $(a,\bar{b})$ to $\frac{a+b}{2}\otimes 1+\frac{a-b}{2i}\otimes i$; here $a,b$ are both vectors of $V$ and multiplication by $z$ on $\bar{V}$ is its conjugate complex structure. The formulas agree in local frames and on overlaps [F9]. Applying this to $V=T{\mathbb{C}\mathbb{P}}^n$ and conjugating the sequence of step 1.2 gives an exact sequence with middle term $(\overline{{\gamma }^{\ast }}{)}^{\oplus (n+1)}$ and a conjugate splitting, so ${\underset{‾}{\mathbb{C}}}^2\oplus T{\mathbb{C}\mathbb{P}}^n\oplus \overline{T{\mathbb{C}\mathbb{P}}^n}\cong ({\gamma }^{\ast }{)}^{\oplus (n+1)}\oplus (\overline{{\gamma }^{\ast }}{)}^{\oplus (n+1)}$ and, since the trivial summands have Chern class $1$, by [F3] and the conjugate-line formula [F4] with $c_1(\overline{{\gamma }^{\ast }})=-x$, $c((T{\mathbb{C}\mathbb{P}}^n{)}_{\mathbb{C}})=(1+x{)}^{n+1}(1-x{)}^{n+1}=(1-x^2{)}^{n+1}\mathrm{.}$ Its $2k$th Chern class is $(-1{)}^k\left(\genfrac{}{}{0px}{}{n+1}{k}\right)x^{2k}$, so the definition [F5] gives $p_k(T{\mathbb{C}\mathbb{P}}^n)=(-1{)}^k{c}_{2k}((T{\mathbb{C}\mathbb{P}}^n{)}_{\mathbb{C}})=\left(\genfrac{}{}{0px}{}{n+1}{k}\right)x^{2k}$, i.e. $p(T{\mathbb{C}\mathbb{P}}^n)={\sum }_k\left(\genfrac{}{}{0px}{}{n+1}{k}\right)x^{2k}=(1+x^2{)}^{n+1}$ in the truncated ring, with $p_k=0$ already for $2k>n$ because $x^{2k}$ exceeds the top cohomology degree. Note that $\overline{T{\mathbb{C}\mathbb{P}}^n}$ here is the conjugate complex bundle, obtained by conjugating transition matrices; no claim about anti-holomorphic tangency is made.

Top pairing. On $({\gamma }^{\ast }{)}^{\oplus n}$ take the section $s$ whose $i$-th component is the restriction to each line of the $i$-th coordinate functional of ${\mathbb{C}}^{n+1}$, for $1\le i\le n$. Its zero set is the single line ${\mathrm{\ell }}_0=[0:\cdots :0:1]$: a line where all first $n$ coordinate functionals vanish is spanned by the last standard basis vector. In the chart $w=(w_1,\dots ,w_n)\mapsto [w_1:\cdots :w_n:1]$ at ${\mathrm{\ell }}_0$ the tautological frame is $(w_1,\dots ,w_n,1)$ and the dual frame restricts each coordinate functional to the scalar $w_i$, so $s$ is exactly $w\mapsto w$; its derivative at $0$ is the complex identity, of positive real determinant, and the zero is transverse and isolated. By [F6] the Euler class of $({\gamma }^{\ast }{)}^{\oplus n}$ is $e=c_n(({\gamma }^{\ast }{)}^{\oplus n})=x^n$. For the sign, let $E$ be the total space of $({\gamma }^{\ast }{)}^{\oplus n}$, let $U\in H^{2n}(E,E^{\times };\mathbb{Z})$ be the transfer of the normalized Thom class, which exists and is unique by [F7] and is identified across the radial homotopy equivalences of pairs by the pair sequences of [F7], and let $\beta =s^{\ast }U\in H^{2n}({\mathbb{C}\mathbb{P}}^n,{\mathbb{C}\mathbb{P}}^n\setminus \{{\mathrm{\ell }}_0\};\mathbb{Z})$. Its absolute image is $e$, because the section $s$ is homotopic to the zero section by the fiberwise interpolation $v\mapsto tv$, $t\in [0,1]$, and relative-to-absolute maps commute with pullback [F7]. Choose a closed coordinate ball $B$ about ${\mathrm{\ell }}_0$ small enough that $s{\mathrm{\mid }}_B$ lies in a bundle chart of $({\gamma }^{\ast }{)}^{\oplus n}$ and identify that chart with $B\times {\mathbb{C}}^n$; excising $Z={\mathbb{C}\mathbb{P}}^n\setminus B$, whose closure is contained in the open relative subspace ${\mathbb{C}\mathbb{P}}^n\setminus \{{\mathrm{\ell }}_0\}$, identifies $\beta$ with the class of $s{\mathrm{\mid }}_B^{\ast }U$ in $H^{2n}(B,B\setminus \{{\mathrm{\ell }}_0\})$ [F7]. In the chart the section is the identity $w\mapsto w$, so $s{\mathrm{\mid }}_B^{\ast }U$ is the pullback of the fiber-normalized generator of $H^{2n}({\mathbb{C}}^n,{\mathbb{C}}^n\setminus \{0\};\mathbb{Z})$, that is, the positive local orientation cohomology class at ${\mathrm{\ell }}_0$. By [F8] the fundamental class $[{\mathbb{C}\mathbb{P}}^n]$ restricts to the positive local generator at ${\mathrm{\ell }}_0$, and evaluating the absolute image of $\beta$ on it evaluates the relative cocycle on the relative image of the fundamental class: a relative cocycle vanishes on chains in the omitted subspace, and the relative fundamental class is the positive local generator, so the value is $1$. Hence $⟨x^n,[{\mathbb{C}\mathbb{P}}^n]⟩=1$, which fixes the positive sign that the ring presentation alone does not fix.

Conclusion. By steps 2.1 and 3.1 together with [F9], for $k\ge 0$ the Pontryagin number of ${\mathbb{C}\mathbb{P}}^{2k}$ is $p_k[{\mathbb{C}\mathbb{P}}^{2k}]=⟨p_k(T{\mathbb{C}\mathbb{P}}^{2k}),[{\mathbb{C}\mathbb{P}}^{2k}]⟩=\left(\genfrac{}{}{0px}{}{2k+1}{k}\right)⟨x^{2k},[{\mathbb{C}\mathbb{P}}^{2k}]⟩=\left(\genfrac{}{}{0px}{}{2k+1}{k}\right)$, giving $p_1[{\mathbb{C}\mathbb{P}}^2]=\left(\genfrac{}{}{0px}{}{3}{1}\right)=3$ and $p_2[{\mathbb{C}\mathbb{P}}^4]=\left(\genfrac{}{}{0px}{}{5}{2}\right)=10$. All cohomology classes are read in the truncated ring $\mathbb{Z}[x]\mathrm{/}(x^{n+1})$, so monomials beyond the manifold dimension are zero as stated. For $n=0$ the space is a point, $\gamma$ is the trivial line over it, $x\in H^2(\mathrm{p}\mathrm{t})=0$, the sequence is $0\to \mathbb{C}\to \mathbb{C}\to 0\to 0$, the ring presentation is $\mathbb{Z}=\mathbb{Z}[x]\mathrm{/}(x)$ with the empty product $x^0=1$, and $p_0[{\mathbb{C}\mathbb{P}}^0]=⟨1,[\mathrm{p}\mathrm{t}]⟩=1$ with the positive point class, so the rank-zero convention is covered. AC is used exactly as declared: through the projective bundle and Chern class construction [F2, F3, F4], the Thom isomorphism and uniqueness [F7], and the metric splitting [F9]; no further selection is made.