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I am sure this is well known but I am having trouble finding the appropriate literature.

I am in interested in the following question:

Given a rational map $f:M --\rightarrow \mathbb C P^n$ what can we say about the pullback of the Fubini-Study form $f^* \omega_{\text{FS}}$ which is a smooth positive $(1,1)$ form on $M \setminus V$ for some subvariety $V \subset M$ of codimension at least 2.

For example I am interested in the case of $(M^n,\omega)$ a Kahler manifold and when the above form has finite mass, i.e. when $$\int_{M} f^*\omega_{\text{FS}} \wedge \omega^{n-1} < \infty$$

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