Let X be a nonsingular complex algebraic curve and let Y be a nonsingular rational complex algebraic surface. Given a compact subset K of X, every holomorphic map from a neighborhood of K in X into Y can be approximated by rational maps from X into Y having no poles in K. If Y is a nonsingular projective complex surface with the first Betti number nonzero, then such an approximation is impossible.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap80-0-5,
author = {J. Bochnak and W. Kucharz},
title = {Approximation of holomorphic maps by algebraic morphisms},
journal = {Annales Polonici Mathematici},
volume = {81},
year = {2003},
pages = {85-92},
zbl = {1028.32007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap80-0-5}
}
J. Bochnak; W. Kucharz. Approximation of holomorphic maps by algebraic morphisms. Annales Polonici Mathematici, Tome 81 (2003) pp. 85-92. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap80-0-5/