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

By Aldo Andreotti, Wilhelm Stoll

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Extra info for Analytic and Algebraic Dependence of Meromorphic Functions

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S [gVk, k=l By 5, ~: V -~U is an s-sheeted cover- one number Pk with 1 _-< PX -< r. implies p = Pk" Because U is simply where a k g V k and V k ~ ] v ~ that @: V k -*U is blholomorphic. Bp ~ ~-l(a) + ~. ,a s be the points of lug map, which is locally biholomorphic. connected, V = Let U = ~ if k + ~ such Then ~ _ c Bpl for one and only If Bp is a branch of X, then Hence a k e Bp for some k with 1 -< k -< s, which Therefore r =< s; Let X be a complex space. d. A proper, light, open and holo- morphic map s: U s ~ U' of an open subset U s into an open, connected s subset U~ of @m is called a chart of X.

5) n~(G;b) = x (a) = b If G ~ X, define Z v~(x;b) eG as the valence of ~ for b in G. 6) i f ~(a) Obviously # ~-l(b) n S ~ n~(S,b) ~ ~. If G is compact, n~(G;b) < ~. If X and Y are pure m-dlmenslon- al complex spaces, if Y is locally irreducible and irreducible, and if ~: X ~ Y is light, proper and holomorphic, then n~(X;y) is a positive, finite, constant function of y e Y, whose constant function value is called the sheet number of ~. 2. Let X and Y be pure m-dimensional complex spaces. Suppose that Y is irreducible and locally irreducible.

However, the natural map ~ ( U ) InJectlve and defines ~ ( U ) a subpresheaf of F ( ~ ) . as a subrlng of F(U,~0. A section I n , o v e r is Hence ~ i s the open set U ~ ~ is 60 called a meromorphlc function on U and ~ ( U ) of meromorphic functions on U. ~(U) is a field. Clearly, ~(U) ~ ( U ) (Observe, that f c ~ ~x. ). Moreover, ~(U)= is the ring If and only if U is irreducible, be the set of units in ~ ( U ) . x E U}. = F(U,~) Then ~(U). Let ~ * ( U ) ~*(U) = {f ~ ( U ) I f x + 0 if x with f + 0 may not be an unit in ~(U)N ~*(U).

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