Meromorphic mapping

of complex spaces

A generalization of the notion of a meromorphic function. Let $ X $ and $ Y $ be complex spaces (cf. Complex space), let $ A $ be an open subset of $ X $ such that $ X \setminus A $ is a nowhere-dense analytic subset (cf. Analytic set) and suppose that an analytic mapping $ f: A \rightarrow Y $ has been given. Then $ f $ is called a meromorphic mapping of $ X $ into $ Y $ if the closure $ \Gamma _ {f} $ of the graph $ A ^ \star $ of $ f $ in $ X \times Y $ is an analytic subset of $ X \times Y $ and if the projection $ \pi : \Gamma _ {f} \rightarrow X $ is a proper mapping (cf. also Proper morphism). The set $ \Gamma _ {f} $ is called the graph of the meromorphic mapping $ f $. The mapping $ \pi : \Gamma _ {f} \rightarrow X $ is surjective and defines a bijective mapping of the set of irreducible components. If $ A _ {0} ^ {f} \subset X $ denotes the largest open subset to which $ f $ can be extended as an analytic mapping, then $ I _ {f} = X \setminus A _ {0} ^ {f} $ is a nowhere-dense analytic subset of $ X $, called the set of indeterminacy of $ f $. The set $ \pi ^ {-} 1 ( A _ {0} ^ {f} ) = A _ {0} ^ {f \star } $ is open and dense in $ \Gamma _ {f} $; also, $ A ^ \star \subseteq A _ {0} ^ {f \star } $ and $ \Gamma _ {f} \setminus A _ {0} ^ {f \star } $ is analytic and nowhere dense in $ \Gamma _ {f} $. The restriction $ \pi : A _ {0} ^ {f\star } \rightarrow A _ {0} ^ {f} $ is an isomorphism of analytic spaces. If $ X $ is a normal complex space (cf. Normal analytic space), then $ \mathop{\rm codim} I _ {f} \geq 2 $ and $ \mathop{\rm dim} _ {z} \pi ^ {-} 1 ( x) > 0 $ if and only if $ z \in \pi ^ {-} 1 ( x) $ and $ x \in I _ {f} $. If $ X $ is not normal, $ \pi ^ {-} 1 ( x) $ may consist of a finite number of points, even if $ x \in I _ {f} $. In the case $ Y = \mathbf C P ^ {1} $ the notion of a meromorphic mapping reduces to that of a meromorphic function.

Let $ f: X \rightarrow Y $, $ g: Y \rightarrow Z $, $ k: X \rightarrow Z $ be meromorphic mappings of complex spaces. One says that the composite $ g \circ f $ of the mappings $ f $ and $ g $ is defined and equals $ k $ if there is an open dense subset $ U $ of $ X $ such that $ U \subseteq A _ {0} ^ {f} $, $ f( U) \subset A _ {0} ^ {g} $, $ U \subset A _ {0} ^ {k} $, and $ k | _ {U} = g \circ f | _ {U} $. A meromorphic mapping $ f: X \rightarrow Y $ is called bimeromorphic if there is a meromorphic mapping $ g : Y \rightarrow X $ such that $ f \circ g = 1 \mid _ {Y} $ and $ g \circ f = 1 \mid _ {X} $. Composition of two bimeromorphic mappings $ X \rightarrow Y $ and $ Y \rightarrow Z $ 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

References

[a1]	H. Whitney, "Complex analytic varieties" , Addison-Wesley (1972) pp. Sect. 6.3