# Analytic Aspects of Rational Maps

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.

• 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 $m=2$. – abx Oct 3 '18 at 8:57
• Yes you are right! I have been thinking about the surface case mainly, I am pretty sure the equality should in higher dimensions. – ben Oct 3 '18 at 9:00