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function field of an algebraic variety
In algebraic geometry, the function field of an algebraic variety ''V'' consists of objects which are interpreted as rational functions on ''V''. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.
==Definition for complex manifolds==
In complex algebraic geometry the objects of study are complex analytic varieties, on which we have a local notion of complex analysis, through which we may define meromorphic functions. The function field of a variety is then the set of all meromorphic functions on the variety. (Like all meromorphic functions, these take their values in Cu.) Together with the operations of addition and multiplication of functions, this is a field in the sense of algebra.
For the Riemann sphere, which is the variety P1 over the complex numbers, the global meromorphic functions are exactly the rational functions (that is, the ratios of complex polynomial functions).
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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