# 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