Download Analytic and Algebraic Dependence of Meromorphic Functions by Aldo Andreotti, Wilhelm Stoll PDF

By Aldo Andreotti, Wilhelm Stoll

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Observe that AX is an analytlc subset of X~. ~v: X~ ~ X xv. ,p. is holomorphic. ,Xp) Moreover 6@ = If y e y, then 6~l(y) = ~-l(y) X ... × ~-l(y) = (~-l(y))q. 7) %l(x ) = (-l(y))V-i × & } × (-l(y))p-v. 28. Let X and Y be complex spaces. be a light holomorphic map. Let p > 1 be an integer. branch of the diagonal &X in X~. is irreducible, Proof. If X A X is a branch of XE. ,b) of ~ with a e B. 6X: A X -~X is biholomorphic, dimbX => ranka~ 1 Let B be a Then B is a branch of X~. Let C be a branch of X p with B _ C C.

Spaces. Define thin of d i m e n s i o n Let ~: X ~ Y be D = {x ~ X I r a n k x @ < n). n - 2. A Proof. Let ~: X -~X be the n o r m a l i z a t i o n of X. A Define A D = (x ~ X l r a n k x ~ o ~ with ~ (8) = D. Take a e B. such that < n). Let ~ b e By Lemma rank ~ o v I B By T h e o r e m D and D are a n a l y t i c of X. 14, for all x Take B e ~ . U of a in B exists E U. A point x ¢ U ^ exists such that ~(x) ra nk x ~0~ = rank branch of X. a simple rank is a simple (X ) ~ =< n. By Lemma point Then ~ B exists rankv(x) ~ = n.

Sists of at most s points. Therefore, The map ~: S ~ U ~ Let K be a compact subset of U~. a compact subset of S. (c,~(~(x))). ~: S ~ U ~ ~-l(y) N U s. (c,~(y)) then = ~(x). ~-l(~(x)) A S con- Then K' = s-l({c} × ~(K)) is Therefore, Take y e ~(Us). of at most is light. Take x e ~-l(K) A S . Hence, x ~ K'. is proper. S consists If z e ~-l(~(x)) A S , = (c,~(~(z))) light, Then s(x) = ~-l(K) A S ~ K'. The map Let B be a branch of Because s(B) = U s" × {~(y) }, a point x e B with s(x) = exists. Hence x e B N s .

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