Definition For Complex Manifolds
More precisely, 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 is then the set of all meromorphic functions on the variety. 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). In any case, the meromorphic functions form a field, the function field.
Read more about this topic: Function Field Of An Algebraic Variety
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