By Aldo Andreotti, Wilhelm Stoll

ISBN-10: 354005670X

ISBN-13: 9783540056706

ISBN-10: 3540369953

ISBN-13: 9783540369950

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Open neighborhood Define V = @-l(w) ~ U p . d. 22. (Open mapping theorem of Remmert Let X be a complex space. space of pure dimension n. pure rank n. Proof. [12], Satz 28). Let Y be a locally irreducible, complex Let @: X -*Y be a holomorphlc map of Then @ is open. Let U ~ ~ be open in X. with b = @(a). Take b ~ @(U). 21). d. An open holomorphlc map @: X - ~ Y into a locally irreducible complex space Y exists such that @ does not have pure rank. 6). 2~. @: X ~ Y Let X and Y be complex spaces. be a holomorphlc map of rank n < ~.

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. 13 shows that ~IU s has pure rank n. This proves 3. and 4. " Take c e U s- 5. analytic subset of U s. Then S = s-l({c} × U~) is an Again siS: S ~ {c} × U~ is proper, open, surJective and holomorphlc.

6. 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.

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

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