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The theory of formal holomorphic functions has largely been replaced by the theory of formal schemes which generalizes it: a formal holomorphic function on a variety is essentially just a section of the structure sheaf of a related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.
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.