# Pull backs along rational maps

Let $$M^m$$ be a compact complex $$m$$-dimensional manifold and $$f: M \dashrightarrow C\mathbb{P}^n$$ a rational map (i.e. holomorphic map defined away from a subvariety, $$V$$, of codimension at least 2). Let $$\omega_{\text{FS}}$$ denote the Fubini-Study form on $$C\mathbb{P}^n$$.

I would like to know the right way to interpret the pull-back form $$f^* \omega_{\text{FS}}$$. This is a closed differential form on $$M \setminus V$$ and hence represents a class in $$H^2(M\setminus V,\mathbb{R})$$ (or maybe even $$H^2(M\setminus V,\mathbb{Q})$$). Is there an appropriate class in $$H^2(M,\mathbb{R})$$ one should associate to this form (maybe this is asking whether there is some notion in which the form could be extended over $$V$$?

A related question is whether $$\int_M f^*\omega_{\text{FS}} \wedge d \alpha =0$$ for all $$\alpha \in \Omega^{m-3}(M)$$?