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 |
Comments
References
[a1] | H. Whitney, "Complex analytic varieties" , Addison-Wesley (1972) pp. Sect. 6.3 |
Meromorphic mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Meromorphic_mapping&oldid=47823