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I am reading the paper "Two Theorems on Extensions of Holomorphic Mappings" by PHILLIP A. GRIFFITHS. In Example 2 of the paper, there is a proposition saying that: Let $N$ be a complex manifold, $S\subset N$ a proper analytic subset, M a projective algebraic variety, and $f: N-S\to M$ a holomorphic map. Then $S$ is a removable singularity for $f$ (in the sense that the closure of the the graph is an analytic subvariety of $N\times M$) if and only if the pull-backs $f^{*}(\varphi)$ of all rational functions $\varphi$ on M extend to meromorphic functions on N.

I do not know why this proposition holds, even if the paper tells me it is obvious.

Any hint is welcomed. Many thanks!

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  • $\begingroup$ You need to assume that $\phi$ is defined somewhere on the image of $f$ to define $f^* \phi$, so as stated this doesn't quite work. $\endgroup$
    – Ben McKay
    Dec 9, 2018 at 10:39

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It is just the fact that affine coordinates of any projective embedding of $M$ are rational functions, so the coordinate functions compose with $f$ to extend to meromorphic functions, giving a meromorphic map $f \colon N-\!\! \rightarrow M$.

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