I'm currently taking a course on characteristic classes. In proving the uniqueness of the Chern classes we used the following.
Let $h_n \in H^2_{dR}(\mathbb{C}P^n)$ be the Poincaré dual of $\mathbb{C}P^{n-1}$ as a submanifold of $\mathbb{C}P^n$. Let $i$ denote the standard injection of $\mathbb{C}P^1$ into $\mathbb{C}P^n$. Then $\int_{\mathbb{C}P^1}i^*h_n=1$.
In this case I'm not quite sure how I'd go about relating the Poincare dual to it's integral under the pullback. I know that this might follow from the axioms of the Chern classes but I was wondering if there is a more direct argument that I'm missing.
This is a direct computation. The Kähler form for $\Bbb CP^n$ (which generates the integral cohomology) can be easily integrated over a linear $\Bbb CP^1$ to get $1$. On the other hand, Poincaré duality tells you that the Poincaré dual of a hyperplane (i.e., $\Bbb CP^{n-1}$) must integrate over a $\Bbb CP^1$ to give the intersection number of that $\Bbb CP^1$ with the hyperplane; this intersection number is patently $1$. Thus, the Poincaré dual is in fact the Kähler form.