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''of complex spaces''

A generalization of the notion of a [[Meromorphic function|meromorphic function]]. Let  $  X $
and  $  Y $
be complex spaces (cf. [[Complex space|Complex space]]), let  $  A $
be an open subset of  $  X $
such that  $  X \setminus  A $
is a nowhere-dense analytic subset (cf. [[Analytic set|Analytic set]]) and suppose that an [[Analytic mapping|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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Andreotti,   W. Stoll,   "Analytic and algebraic dependence of meromorphic functions" , Springer  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Remmert,   "Holomorphe und meromorphe Abbildungen komplexer Räume"  ''Math. Ann.'' , '''133''' :  3  (1957)  pp. 328–370</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Whitney,   "Complex analytic varieties" , Addison-Wesley  (1972)  pp. Sect. 6.3</TD></TR></table>