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

By Aldo Andreotti, Wilhelm Stoll

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The set of elements f ¢ ~(U) fiB ~ 0 for each branch B of U. ring of ~(U). ~(U) Let O~(U) be the total quotient If U is irreducible, then V ( U ) = ~(U) - {0} and is a field called the quotient field of ~(U). and if V and U are open, the restriction map r~: maps ~(U) into Then on X. %= such that If ~ + V ~ U ~(U) ~ ~(V) ~(V) and induces a restriction map r~: ~ ( U ) - ~ V ) . { %(U),r~} and ~ = {~(U),r~} are presheaves of rings The associated sheaf ~ to ~ is the sheaf ~ = ~ X of ~erms of holomorphlc functions on X, also called the structure sheaf of X.

Analytic ~ p - 1}. N dlmaE ~ m - l, since E is thin in X. c E. = m - dim B. all simple points E. Hence = dlmaE - dim B _< m - 1 - dim B. Now, B is also a branch of ~-l(~(a)) for all x ¢ E. Then D is w i t h a ~ B and dim B = dima~-l(~(a)) and p ~ r a n ~ map. Then rank ~IE ~ p - i. Let a be a simple point of E. of @-l(~(a)) A E Now, be a holomorphic _-< p}. Define D = (x ~ Elrankx~IE in E. Suppose Define E = {x ~ Xlrankx@ Suppose space. of E belong by Lemma Take a simple Then dim B = dimx~-l(@(a)) Hence ranka~IE to D.

Hence ^ E ~ = ~(~ . Let~be the set of branches of X. Each branch A B ~ ~ is open in the normal space X. for every x ~ B. Hence rankx~o~ = rankx~o~IB Therefore = {x ~ B l r a n ~ o ~ l B ~ p} = ~ N B . 23. = E~ is almost thin of If B + E B, then E B is a thin analytic subset of the pure dimensional complex space B. 20, rank @o~IE B ~ p - 1. By Proposition is almost thin of dimension p - 1. 23, ~(~(EB)) = almost count- able BE Be is almost thin of dimension p. Take a ¢ E. Then ranka@IE -< p has to be proved.

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