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By Aldo Andreotti, Wilhelm Stoll
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Additional info for Analytic and Algebraic Dependence of Meromorphic Functions
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. Pick b ¢ ~(X). Let ~: U~ -~U~ be a chart of Y centered at b 57 with ~(b) = 0 e ~n.
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.
If p is a chart at b (respectively centered at b), then ((%,p) is a product representation of ~ over b (respectively centered over b). The product representation is said to be smooth if and only if (% and p are biholomorphic. A smooth product representation of at a exists if and only if @ is regular at a. If ((%,p) is a product representation of @, then U(% is pure m-dimensional, U p is pure n-dimenslonal with q = m - n. If q = O, T! then U a'' is a point and U(% × U~ can be identified with U ~ such that becomes the identity.