Assume $ds^2$ is Fubini-Study metric, and $\omega = -\frac{1}{2}Im \{ds^2\}$ is its associated form.
Let $V_n = \int_{\mathbb{C}P^n} \omega^n / n!$ the volume of $\mathbb{C}P^n$.
Let $V \subset \mathbb{C}P^n$ be a complex closed submanifold, with $\deg V = d , \ \dim_{\mathbb{C}} V = m$.
I want to show that the form $d\cdot \frac{v_m m!}{v_n n!} \omega^{n-m}=W$, satisfies:
$\int_M \psi \wedge W = \int_V \psi$, for $\psi$ which is a closed form of degree $2m$.
Any references or answers for this question?
Thanks in advance.