# Formal holomorphic function

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In [algebraic geometry](/source/Algebraic_geometry), a **formal holomorphic function** along a subvariety *V* of an [algebraic variety](/source/Algebraic_variety) *W* is an algebraic analog of a [holomorphic function](/source/Holomorphic_function) defined in a neighborhood of *V*. They are sometimes just called holomorphic functions when no confusion can arise. They were introduced by [Oscar Zariski](/source/Oscar_Zariski) ([1949](#CITEREFZariski1949), [1951](#CITEREFZariski1951)).

The theory of formal holomorphic functions has largely been replaced by the theory of [formal schemes](/source/Formal_scheme) which generalizes it: a formal holomorphic function on a variety is essentially just a section of the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](/source/Completion_(algebra)) 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

- Zariski, Oscar (1949), "A fundamental lemma from the theory of holomorphic functions on an algebraic variety", *Ann. Mat. Pura Appl. (4)*, **29**: 187–198, [MR](/source/MR_(identifier)) [0041488](https://mathscinet.ams.org/mathscinet-getitem?mr=0041488)

- Zariski, Oscar (1951), *Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields*, Mem. Amer. Math. Soc., vol. 5, [MR](/source/MR_(identifier)) [0041487](https://mathscinet.ams.org/mathscinet-getitem?mr=0041487)

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