I would like some help finding references for a analytic treatment of rational maps between compact complex manifolds (that is holomorphic maps defined away from a codimension at least 2 subvariety). In particular maps $f:X^m \dashrightarrow \mathbb{P}^n \quad$.

A specific question which may be obvious but whose answer eludes me is the following. Suppose that $\omega$ is a K\"ahler form on $X^m$ and $\omega_{\text{FS}}$ is the Fubini-Study form on $\mathbb{P}^n$, then when is $$\int_X f^* \omega_{\text{FS}} \wedge \omega =\int_{X \setminus V} f^* \omega_{\text{FS}} \wedge \omega < \infty?$$

I think this should be true if $f \in W_{loc}^{1,2}(X,\mathbb{P}^n)$ since in this case $$\omega^{m-1} \wedge f^* \omega_{\text{FS}} = c(n,m) |\nabla f|^2 \omega^m,$$ holds almost everywhere. But I do not know whether one can expect such regularity of rational maps in general.