By Aldo Andreotti, Wilhelm Stoll

ISBN-10: 354005670X

ISBN-13: 9783540056706

ISBN-10: 3540369953

ISBN-13: 9783540369950

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**Extra info for Analytic and Algebraic Dependence of Meromorphic Functions**

**Sample text**

P P is proper, an open neighborhood V of b w i t h V c ~ U - p ~-l(v) ~ Ig=lUal exists. be a proper, Then x@ is locally bounded. Hence x@(y) = 0 if y ~ Y - ~(X). be considered. Let ~: X ~ Y i=l ~i of ~ centered Because and 58 Let s i be the sheet number of ~i" Define s -- s I + ... + Sp. Then s ~ x~(y) for all y ~ V is claimed. the set of branches of ~-l(y). ~-l(y) ~ U~i. Pick y ¢ V. Let ~ b e Let ~ i be the set of branches of By definition x~(y) = # ~ . 5, s i -> # ~ i " If H • ~i' one and only one ki(H ) ¢ ~ map ki: ~ i ~ ~Is defined.

Is a Stein space. Suppose that Y Let @: X -~ Y be a holomorphic map. Let V be an open subset of Y such that V is compact and U ffi @-l(v) ~ ~. f ¢ ~(Y). and h l V e Then g ¢ ~(Y) ~(V) Proof. and h ¢ ~(Y) exist such that h-f = g and ho@IU ~ ~(U). Let ~m be the set of branches of U. @-l(pf) f-] B is thin on B. ~(XB) g V - Pf. Take x B c B - @-l(pf). Hence, D = {YBIB e ~ } subset of V - Pf. 4. such that h l V e ~ 0 if B e ~ . d. Let X and Y be complex spaces. an holomorphlc map. f = g ~ ~(Y) for each B ~ ~ , Take -~ ~(X) ~ ~(X) Let @: X -~Y be extends to one and satisfying the following condition: i.

BlJectlve Here, @(V) and holomorphic, sets of non-simple points neighborhood that @IV is not inJective Then an open neighborhood if InJective. = #~-l(y) v (x) = ~@(x) V of x in U exists of x for every open ~ 2, which is wrong. such that @: V ~ Y = W is open and @l = @: V ~ W is open, hence topological. Let S V and S W be the of V and W respectively. The map 48 @ii: W ~ V is continuous and holomorphlc Here @I(Sv) is analytic, is topological. because @i is proper, Hence, if Y is normal, @i is biholomorphic.

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

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