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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.

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    $\begingroup$ If $\omega _{\mathrm{FS}}$ is what I think, namely the standard Kähler form on $\mathbb{P}^n$, there is something wrong with the degree in your equality — unless you assume $m=2$. $\endgroup$
    – abx
    Commented Oct 3, 2018 at 8:57
  • $\begingroup$ Yes you are right! I have been thinking about the surface case mainly, I am pretty sure the equality should in higher dimensions. $\endgroup$
    – ben
    Commented Oct 3, 2018 at 9:00

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