Let $P_n\mathbb{C}$ denote the $n$-dimensional complex projective space. We define Chern classes via the Chern-Weil theory, and then I already proved that the first Chern class $-c_1(L)$ for the canonical line bundle
$$L=\{(\xi, z)\in P_1\mathbb{C}\times \mathbb{C}^2:z\in \xi\}$$
of $P_1\mathbb{C}$ is the generator of $H^2(P_1\mathbb{C};\mathbb{Z})=\mathbb{Z}$ by direct calculation
$$\int_{P_1\mathbb{C}}c_1(L)=-1.$$
In general, does the first Chern class $\alpha=-c_1(L)$ for the line bundle
$$L=\{(\xi, z)\in P_n\mathbb{C}\times \mathbb{C}^{n+1}:z\in \xi\}$$
of $P_n\mathbb{C}$ generate $H^2(P_n\mathbb{C};\mathbb{Z})=\mathbb{Z}$? Furthermore, for $1\leq i\leq n$, does
$$\alpha^i=\alpha\smallsmile\cdots\smallsmile\alpha\in H^{2i}(P_n\mathbb{C};\mathbb{Z})$$
generate $H^{2i}(P_n\mathbb{C};\mathbb{Z})=\mathbb{Z}$? (In fact, I try showing that the cohomology algebra is the free abelian group
$$H^*(P_n\mathbb{C};\mathbb{Z})=\bigoplus_{i=0}^{\infty}H^{i}(P_n\mathbb{C};\mathbb{Z})
=\bigoplus_{i=0}^{n}H^{2i}(P_n\mathbb{C};\mathbb{Z})
=\mathbb{Z}\oplus \mathbb{Z}\alpha \oplus \mathbb{Z}\alpha^2 \oplus \cdots \oplus \mathbb{Z}\alpha^n
=\mathbb{Z}[\alpha]/(\alpha^{n+1})
$$
generated by $1, \alpha=-c_1(L), \alpha^2, \cdots,\alpha^n$.)
I think it might be difficult by the conventional direct calculation. How do I prove it? Any help is welcome!
Best Answer
The proof is somewhat easier if you use the abstract properties of the Chern Class. Here are a few facts that will be helpful in producing the proof:
Let $L_0$ denote the total space of $L$ with the zero section removed, i.e. $L_0=\{l\in L| l\neq 0\}$ with $\pi: L_0\to P_n\mathbb{C}$ the restriction of the projection $L\to P_n\mathbb{C}$. We can think of $L$ as an oriented real vector bundle by considering each fiber as a real vector space and by giving it the orientation on each fiber which corresponds $[u, iu]$ for a nonzero vector $u$. Under this definition $c_1(L)=e(L)$ the euler class of $L$. From the Euler class of a vector bundle we yield the following long exact sequence in cohomology with coefficients in $\mathbb{Z}$ (the Gysin sequence). $$ \cdots\to H^i(P_n\mathbb{C})\stackrel{\wedge e}{\to} H^{i+2}(P_n\mathbb{C})\stackrel{\pi^*}{\to} H^{i+n}(L_0)\to H^{i+1}(P_n\mathbb{C})\to\cdots$$ Is there some familiar space that $L_0$ is diffeomorphic/homotopic to? Since we define $L$ as a subset of $P_n\mathbb{C}\times \mathbb{C}^{n+1}$ we induce a map $L\to \mathbb{C}^{n+1}$ by restriction of the projection onto the second factor. When we restrict once again to $L_0$ we yield a map $L_0\to \mathbb{C}^{n+1}\backslash\{0\}$. What can be said about this map?