Meromorphic mapping

of complex spaces

A generalization of the notion of a meromorphic function. Let and be complex spaces (cf. Complex space), let be an open subset of such that is a nowhere-dense analytic subset (cf. Analytic set) and suppose that an analytic mapping has been given. Then is called a meromorphic mapping of into if the closure of the graph of in is an analytic subset of and if the projection is a proper mapping (cf. also Proper morphism). The set is called the graph of the meromorphic mapping . The mapping is surjective and defines a bijective mapping of the set of irreducible components. If denotes the largest open subset to which can be extended as an analytic mapping, then is a nowhere-dense analytic subset of , called the set of indeterminacy of . The set is open and dense in ; also, and is analytic and nowhere dense in . The restriction is an isomorphism of analytic spaces. If is a normal complex space (cf. Normal analytic space), then and if and only if and . If is not normal, may consist of a finite number of points, even if . In the case the notion of a meromorphic mapping reduces to that of a meromorphic function.

Let , , be meromorphic mappings of complex spaces. One says that the composite of the mappings and is defined and equals if there is an open dense subset of such that , , , and . A meromorphic mapping is called bimeromorphic if there is a meromorphic mapping such that and . Composition of two bimeromorphic mappings and is always defined.

References

 [1] A. Andreotti, W. Stoll, "Analytic and algebraic dependence of meromorphic functions" , Springer (1971) [2] R. Remmert, "Holomorphe und meromorphe Abbildungen komplexer Räume" Math. Ann. , 133 : 3 (1957) pp. 328–370