Formal holomorphic function
In algebraic geometry, a formal holomorphic function along a subvariety "V" of an algebraic variety "W" is an algebraic analog of a holomorphic function defined in a neighborhood of "V". They are sometimes just called holomorphic functions when no confusion can arise. They were introduced by . The theory of formal holomorphic functions has largely been replaced by the theory of formal schemes which generalizes it: a formal holomorphic function on a variety is essentially just a section of the structure sheaf of a related formal scheme. Definition. If "V" is an affine subvariety of the affine variety  W defined by an ideal "I" of the coordinate ring "R" of "W", then a formal holomorphic function along "V" is just an element of the completion of "R" at the ideal "I". In general holomorphic functions along a subvariety "V" of "W" are defined by gluing together holomorphic functions on affine subvarieties.