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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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V.31 The Riemann-Roch Theorem

A RIEMANN SURFACE [III.79] is a MANIFOLD [I.3 §6.9] that “looks locally like Image,” in the usual sense of this sort of phrase. In other words, every point has a neighborhood that can be mapped bijectively to an open subset of Image, and where two such neighborhoods overlap, the “transition functions” are HOLOMORPHIC [I.3 §5.6]. One can think of a Riemann surface as the most general sort of set on which the notion of a holomorphic function (that is, a complex-differentiable function) of one complex variable makes sense.

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