Sheaf cohomology with bounds and bounded holomorphic functions

Abstract

Suppose U is the unit disc in C. For O r }. A subvariety V of pure codimension 1 in UN is called a Rudin subvariety if for some r VnQN= 0. A Rudin subvariety is called a special Rudin subvariety if there is >0 such that, for 1 a. If a holomorphic function f generates the ideal-sheaf of its zero-set E, then we write Z(f) = E. The Banach space of all bounded holomorphic functions on a reduced complex space X under the sup norm is denoted by H0(X) and the norm of f H??(X) is denoted by |Iflix. The following two theorems were proved by W. Rudin [2] and H. Alexander [I] respectively.