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Formal holomorphic function
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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 Oscar Zariski (1949, 1951).
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The theory of formal holomorphic functions has largely been replaced by the theory of formal schemes which generalizes it: a formal holomorphic function on a variety is essentially just a section of the structure sheaf of a related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.
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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
- 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 0041488
- Zariski, Oscar (1951), Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields, Mem. Amer. Math. Soc., vol. 5, MR 0041487
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