View source for Extension theorems (in analytic geometry)

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Statements on the continuation (extension) of functions, sections of analytic sheaves, analytic sheaves, analytic subsets, holomorphic and meromorphic mappings, from the complement  $  X \setminus  A $
in an [[Analytic space|analytic space]]  $  X $
of a set  $  A $(
as a rule, also analytic) to the whole space  $  X $. 
Two theorems of B. Riemann form the classical results concerning continuation of functions.

Riemann's first theorem states that every analytic function on  $  X \setminus  A $, 
where  $  X $
is a normal [[Complex space|complex space]] and  $  A $
an analytic subspace of codimension  $  \geq  2 $, 
can be continued to an analytic function on  $  X $. 
Riemann's second theorem states that every analytic function  $  f $
on  $  X \setminus  A $
that is locally bounded on  $  X $, 
where  $  A $
is a nowhere-dense analytic subset in a normal complex space  $  X $, 
can be continued to an analytic function on  $  X $. 
There are generalizations of these theorems to arbitrary complex spaces  $  X $, 
as well as to sections of coherent analytic sheaves (cf. [[Local cohomology|Local cohomology]]).

Important results concerning extension of analytic subsets are the theorems of Remmert–Stein–Shiffman and Bishop. The Remmert–Stein–Shiffman theorem states that every pure  $  p $-
dimensional complex-analytic subset in  $  X \setminus  A $, 
where  $  X $
is a complex-analytic space and  $  A $
a closed subset having zero  $  ( 2p - 1) $-
dimensional [[Hausdorff measure|Hausdorff measure]], can be extended to a pure  $  p $-
dimensional complex-analytic subset in  $  X $. 
Bishop's theorem states that every pure  $  p $-
dimensional complex-analytic subset  $  V $
in  $  X \setminus  A $, 
where  $  X $
is a complex-analytic space and  $  A $
is a complex-analytic subset, can be extended to a pure  $  p $-
dimensional complex-analytic subset  $  \overline{V}\; $
in  $  X $
if  $  V $
has locally finite volume in some neighbourhood  $  U $
of  $  A $
in  $  X $.

There are criteria for extendability of analytic mappings, generalizing the classical [[Picard theorem|Picard theorem]]. E.g., every analytic mapping  $  X \setminus  A \rightarrow Y $, 
where  $  X $
is a complex manifold,  $  A $
is an analytic nowhere-dense set and  $  Y $
is a hyperbolic compact complex manifold, can be extended to an analytic mapping  $  X \rightarrow Y $. 
Every analytic mapping  $  X \setminus  A \rightarrow Y $
that is not everywhere-degenerate, where  $  X $
is a complex manifold,  $  A $
is an analytic subset and  $  Y $
is a compact complex manifold with negative first [[Chern class|Chern class]], can be extended to a meromorphic mapping  $  X \rightarrow Y $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.A. Griffiths,   J. King,   "Nevanlinna theory and holomorphic mappings between algebraic varieties"  ''Acta. Math.'' , '''130'''  (1973)  pp. 145–220</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Kobayashi,   "Hyperbolic manifolds and holomorphic mappings" , M. Dekker  (1970)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Harvey,   "Holomorphic chains and their boundaries" , ''Proc. Symp. Pure Math.'' , '''30''' , Amer. Math. Soc.  (1977)  pp. 309–382</TD></TR></table>

====Comments====
Bishop's theorem has been generalized in several directions. Let  $  X $
be an open subset of  $  \mathbf C  ^ {n} $
and  $  A $
a complex-analytic subset of  $  X $. 
First, Skoda's theorem states that if  $  T $
is a positive closed current of bi-degree  $  ( p, p) $
on  $  X \setminus  A $
which has locally finite mass in a neighbourhood of  $  A $, 
then  $  T $
extends to a positive closed current on  $  X $. 
(A current on  $  X $
is a continuous linear functional on the space of all complex differential forms of class  $  C  ^  \infty  $
on  $  X $, 
with compact support, in the strong topology, cf. [[#References|[a1]]] and [[Differential form|Differential form]].) Next, H. El Mir showed that one may take  $  A $
to be a closed complete pluripolar set, which is more general than a closed analytic set, and then  $  T $
as above will still extend. (A pluripolar set  $  A $
in  $  \mathbf C  ^ {n} $
is a set such there exists a [[Plurisubharmonic function|plurisubharmonic function]]  $  \phi $
defined in some neighbourhood of  $  A $
such that  $  A \subset  \{ {z } : {\phi ( z) = - \infty } \} $, 
the  $  - \infty $
set of  $  \phi $. 
It is a complete pluripolar set if there is such a  $  \phi $
with  $  A $
equal to the  $  - \infty $
set of  $  \phi $.) 
N. Sibony generalized these results even further: If  $  T $
is a pluripositive current of bi-degree  $  ( p, p) $
on  $  X \setminus  A $
which has locally finite mass in a neighbourhood of  $  A $, 
then  $  T $
extends to a pluripositive current on  $  X $.

One recovers Bishop's theorem from Skoda's using the fact that to every pure  $  p $-
dimensional analytic subset  $  V $
of  $  X $
is associated a current  $  [ V] $, 
the current of integration over the regular points of  $  V $. 
This is a positive closed current of bi-degree  $  ( p, p) $. 
One can return from currents to analytic sets using Siu's theorem (cf. [[#References|[a4]]]) on analyticity of sets associated to positive Lelong numbers. (The Lelong number at a point  $  a $
of a pure  $  p $-
dimensional [[Analytic set|analytic set]]  $  A $
in  $  \mathbf C  ^ {n} $
is the number

$$ 
n ( A , a )  =  \lim\limits _
{r \rightarrow 0 }  
\frac{ \mathop{\rm vol} _ {2p}  A _ {r} }{c ( p) r  ^ {2p} }
 .
$$

The limit exists (cf., e.g., [[#References|[a1]]]); in this formula,  $  c ( p) = \pi  ^ {p} / p ! $, 
the volume of the unit ball in  $  \mathbf C  ^ {n} $, 
and  $  A _ {r} = \{ {z \in A } : {| z - a | < r } \} $(
i.e. the part of  $  A $
contained in the ball with centre  $  a $
and radius  $  r $), 
cf. also [[#References|[a1]]].)

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.M. Chirka,   "Complex analytic sets" , Kluwer  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Sibony,   "Quelques problèmes de prolongement de courants en analyse complexe"  ''Duke Math. J.'' , '''52'''  (1985)  pp. 157–197</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Y.T. Siu,   "Techniques of extension of analytic objects" , M. Dekker  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Y.T. Siu,   "Analyticity of sets associated to Lelong numbers and the extension of closed positive currents"  ''Inv. Math.'' , '''27'''  (1974)  pp. 53–156</TD></TR></table>