# On extensions of holomorphic mappings with image in a projective algebraic variety

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!

• 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. – Ben McKay Dec 9 '18 at 10:39

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 $$f \colon N-\!\! \rightarrow M$$.