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 $\alpha \in \Omega^{m-3}(M)$?


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