In algebraic geometry, a formal holomorphic function along a subvariety ''V'' of an algebraic variety ''W'' is an algebraic analog of a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
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The theory of formal holomorphic functions has largely been replaced by the theory of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 scheme
In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off ...
s 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.
References
*
*{{citation, mr=0041487
, last=Zariski, first= Oscar
, title=Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields
, series=Mem. Amer. Math. Soc., volume= 5 , year=1951
Algebraic geometry