By Ciro Ciliberto, E. Laura Livorni, Andrew J. Sommese

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C„) ^ ( 0 , . . , 0) of natural numbers, there are no non-constant meromorphic functions on M = WIG. (c) Surgery We explain another method to construct complex manifolds. Let M be a complex manifold, and 5 c M a compact submanifold of M. We construct a new complex manifold M = {M — S)u S by replacing S by another compact complex manifold S as follows: Take domains W and W^ such that 5 c: Wi c [ \ ^ J c: \y c: M, where [ W] is assumed to be compact. Let 5 be a compact complex submanifold of a complex manifold W such that there are a domain W] with 5 c: W, c [ W'j] c: W, and a bihojomorphic map O of W - 5 onto W - 5 such that ^{W^-S) = W^-S.

By local coordinates with centre q we mean a biholomorphic map Zq\ p-^ Zq{p) of a domain U{q) containing q onto a domain of C" containing ( 0 , . . , 0) such that Zq{q) = 0. Let Zq'. , z''q{p)) bc givcn local coordinates with centre q. 1. (0) = {z|z = ( z ^ . . , z J , | z ^ | < r ^ . . , | z " | < r " } , such that [(7,(0)] c:z^((7(^)), where r = {r\ ... r"") and r ' > O f o r all /. 1. Let M"" he a complex manifold. Suppose that for each point qe M, local coordinates z^ with centre q and a coordinate polydisk UjK^q^iq) are given.

I Corollary. Let f{z) he a holomorphic function in a domain of C", and S the analytic hypersurface defined by the equation f{z) = 0. We assume OeS. Supppose fo{z) = 0 is a minimal equation of S at 0. Then if e>0 is small enough, fiz) = 0 is a minimal equation ofS at cfor every ce S with \c\ < s. 14. Let U he a. 17, if fiz) = 0 and g{z) = 0 are both minimal equations of S in U, u{z) = f(z)/g{z) is a non-vanishing holomorphic function in U. 2. Holomorphic Map In this section we consider a map O: z^w = <^(z) of a domain D c C " into C".