By Aldo Andreotti, Wilhelm Stoll

ISBN-10: 354005670X

ISBN-13: 9783540056706

ISBN-10: 3540369953

ISBN-13: 9783540369950

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Additional resources for Analytic and Algebraic Dependence of Meromorphic Functions

Example text

7. on W - (SwL9 @I(Sv)). and thin because @i then @l I is holomorphlc and This proves 4, 5 and 6. If s = i, then N2; S' and S are empty by 1 and 6. Also by 6, @ is InJective. Hence, If in addition Y is normal, @ is topological and locally biholo- morphic. Hence, ~ is blholomorphlc. 2. Let b e Y - S' be a simple point of Y. be an open, connected, simply connected neighborhood consisting of simple points only. ~-l(b). 5 implies that @ is topological. Define V = @-I(u). s [gVk, k=l By 5, ~: V -~U is an s-sheeted cover- one number Pk with 1 _-< PX -< r.

Let ~: X ~ Y be a light, holo- An open neighborhood U of a is said to be distinguished if and only if U is compact and {a} = ~ - l ( ~ ( a ) ) N ~ The distinguished neighborhoods of a form a neighborhood base of a. 1) ~(a) = llm sup # ~-l(~(z)) C ~ U Z -@a is finite, integral, independent of U and called the order of ^ at a. Let ~: X ~ X be the normalization of X. ,b r be the different points of ~-l(a). The map ~ = @o~ is light. 2) v (a) = kZ__l~(bl). If X and Y are pure m-dimensional and if Y is locally irreducible, then v = ~.

Hence ~ ( A ~ M ) = A' A Let A be open in M, then Now, ~(A) = A' is open in U~. L is open in L. By induction ~-l(0) {-]M = ~-l(0) has pure codimenslon 1 The map VIM: M ~ L = @-l(b) has pure dimension (m-l) - (n-l) = M - n = q Let X and Y be complex spaces. map. is open. d. Let @: X -*Y be a holomorphlc For y ~ Y, let T@(y) be the number of branches of @-l(y). - ~ ~. Lemma 2. 7 . Let X and Y be complex spaces of pure dimension m and n respectively with q = m - n ~ 0. q-flberlng, Proof. holomorphic map.

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


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