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

By Aldo Andreotti, Wilhelm Stoll

ISBN-10: 354005670X

ISBN-13: 9783540056706

ISBN-10: 3540369953

ISBN-13: 9783540369950

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Example text

Therefore ~(Us) ~ U~ and on U s. d. Let 9: X ~ Y be a holomorphic map. An analytic subset S of an open subset U of X is said to be a section of ~ over the open subset V of Y if and only if ~(S) ~ V and if 9: S ~ V and light. ~-l(y) A is proper The section S is said to be central in U if and only if S ~ ~ for every y e ~(U). The section S is said to be 53 strlctly central in U if and only if each branch of ~-l(y) A U intersects S if y ¢ ~(U). ~: S ~ V The section S covers V if and only if is also open and surJective.

18. (and only one|). Together, Hence Define rankx~ Because t = s u p { r ] C C X r) is defined. only in f i n i t e l y many T h e n C 1 = Xt+ 1 • a ¢ C with dimaX= t ~ d i m x X ~ s. then d l m x X -~ r for all x analytic s. subset Then subset of X. Since the C is a n a l y t i c = s = t, h e n c e x each p o i n t supremum of the ~ r} is of X is con- and thin in C. If x Therefore r. is f i n i t e and s = t and C - C 1 ~ C - D. c C - C 1. E X; C ~ X r for some n u m b e r T h e n a ¢ X t and s ~ t.

2, open, @-l(s') v By T h e o r e m S' is thin, because = ~@. 1, @0 = ~: X - is surJective, open, By Proposition S' = ~(N2) dim Y = m > dim N 2. = S is a thin analytic subset of X. S-~Y- light and proper. 1, N 2 is analytic. Y - S' is irreducible. V~(x0;Y ) = v@(x0) and exists. > #~-l(y) c X - N 2 and v (x) = 1 if x ~ @-l(y) m Hence s = Take x e X - S. with U ~ X - S. neighborhood Hence, z v (x;y) xeX ~ Let U be a distinguished Suppose V of x in U. blJectlve Here, @(V) and holomorphic, sets of non-simple points neighborhood that @IV is not inJective Then an open neighborhood if InJective.

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


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