Read e-book online Analytic and Algebraic Dependence of Meromorphic Functions PDF

By Aldo Andreotti, Wilhelm Stoll

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Let C be a branch Let b e B be a simple point of F ~ E . of F with b ¢ C and dim C = dimbF. Let B be a branch Then dim C = dimbF = dimbX - rankb~ -~ dimbX - p. Let S F be the set of non-simple points of F. s = Min{dimzXlZ Define ¢ C} D = {z • CldimzX > s}. J ~ ) . subset of C. Take Then rankz@ = dlmzX - dlmzF = s - dlmzC ~ dlmbX - dim C ~ p. Therefore C - (DU SF) ~ E. Because D and S F ~ C are thin in C, 30 and because E is analytic, irreducible. Hence C ~ E. C ~ F NE, where C c B 1 where B 1 ~s a branch of F N E .

Define z = 6(y) and define F , F ' , ~ Let B be a branch of ~-l(y) N Us. then ~(s(x)) = B(@(y)) If x ¢ ~-l(y)(% Us ' = z and x ¢ s-l(~-l(x)) = F. Hence 55 ~-l(y) N U s ~ F. A branch C of F contain8 B. Because ~IC is constant and ~(B) = {y}, the branch C is contained in ~-l(y) N U s . Hence B = C. Therefore, each branch of ~-l(y) A U s is a branch of F, which has at most s branches. at most s branches slonal. Hence, of F. Hence ~-l(y) N ue is an union of Especially ~-l(y) A U s is pure q-dimen- ~IU s is q-fiberlng with q = m - n.

Y be complex (0penmapplng theorem of Remmert). spaces of pure dimension m and n respectively. pose that Y is locally irreducible. Then the following three statements are equivalent. The map ~ is q-flberlng with m - n = q. b) The map ~ has pure rank n. c) The map ~ is open. 13, a) and b) are equivalent. 22, b) implies c). By Theorem Now, it shall be shown that c) implies a). If n = 0, the statement if true. ,n - i. Sup- Let ~: X ~ Y be holomorphic. a) Proof. Let X and Suppose that it is true for Then it shall be proved for n.

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