Signature and first Pontryagin number of the complex projective plane

Given: AC, the tautological complex line $\gamma \to {\mathbb{C}\mathbb{P}}^2$, the class $y=c_1({\gamma }^{\ast })$, and the complex orientation of ${\mathbb{C}\mathbb{P}}^2$.

The in-run supplier gives $H^{\ast }({\mathbb{C}\mathbb{P}}^n;\mathbb{Z})=\mathbb{Z}[x]\mathrm{/}(x^{n+1})$ with $x=c_1({\gamma }^{\ast })$, $⟨x^n,[{\mathbb{C}\mathbb{P}}^n]⟩=1$, and $p(T{\mathbb{C}\mathbb{P}}^n)=(1+x^2{)}^{n+1}$ in the truncated ring (The tangent bundle of complex projective space and its Pontryagin classes).

The middle-dimensional intersection form is $Q_M(x,y)=⟨x⌣y,[M]⟩$ on $H^{2k}(M;\mathbb{R})$, and the signature is the difference $p-q$ of the positive and negative inertia indices of the nondegenerate symmetric form $Q_M$ (The middle-dimensional intersection form of a closed oriented 4k-manifold, The signature of a closed oriented manifold of dimension divisible by four).

The first Pontryagin number is $p_1[M]=⟨p_1(TM),[M]⟩\in \mathbb{Z}$, the Kronecker evaluation of the Pontryagin class on the fundamental class (Pontryagin numbers of a closed oriented manifold, Kronecker evaluation pairing).

For every closed oriented $4$-manifold $M$, $\sigma (M)=p_1[M]\mathrm{/}3$ (The four-dimensional signature formula), and on projective spaces the signature and the L-genus agree: $\sigma ({\mathbb{C}\mathbb{P}}^{2k})=1=L[{\mathbb{C}\mathbb{P}}^{2k}]$ for every $k\ge 0$ (The signature and the L-genus agree on complex projective space).