Extension theorems (in analytic geometry)
Statements on the continuation (extension) of functions, sections of analytic sheaves, analytic sheaves, analytic subsets, holomorphic and meromorphic mappings, from the complement in an analytic space
of a set
(as a rule, also analytic) to the whole space
. Two theorems of B. Riemann form the classical results concerning continuation of functions.
Riemann's first theorem states that every analytic function on , where
is a normal complex space and
an analytic subspace of codimension
, can be continued to an analytic function on
. Riemann's second theorem states that every analytic function
on
that is locally bounded on
, where
is a nowhere-dense analytic subset in a normal complex space
, can be continued to an analytic function on
. There are generalizations of these theorems to arbitrary complex spaces
, as well as to sections of coherent analytic sheaves (cf. 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 -dimensional complex-analytic subset in
, where
is a complex-analytic space and
a closed subset having zero
-dimensional Hausdorff measure, can be extended to a pure
-dimensional complex-analytic subset in
. Bishop's theorem states that every pure
-dimensional complex-analytic subset
in
, where
is a complex-analytic space and
is a complex-analytic subset, can be extended to a pure
-dimensional complex-analytic subset
in
if
has locally finite volume in some neighbourhood
of
in
.
There are criteria for extendability of analytic mappings, generalizing the classical Picard theorem. E.g., every analytic mapping , where
is a complex manifold,
is an analytic nowhere-dense set and
is a hyperbolic compact complex manifold, can be extended to an analytic mapping
. Every analytic mapping
that is not everywhere-degenerate, where
is a complex manifold,
is an analytic subset and
is a compact complex manifold with negative first Chern class, can be extended to a meromorphic mapping
.
References
[1] | P.A. Griffiths, J. King, "Nevanlinna theory and holomorphic mappings between algebraic varieties" Acta. Math. , 130 (1973) pp. 145–220 |
[2] | S. Kobayashi, "Hyperbolic manifolds and holomorphic mappings" , M. Dekker (1970) |
[3] | R. Harvey, "Holomorphic chains and their boundaries" , Proc. Symp. Pure Math. , 30 , Amer. Math. Soc. (1977) pp. 309–382 |
Comments
Bishop's theorem has been generalized in several directions. Let be an open subset of
and
a complex-analytic subset of
. First, Skoda's theorem states that if
is a positive closed current of bi-degree
on
which has locally finite mass in a neighbourhood of
, then
extends to a positive closed current on
. (A current on
is a continuous linear functional on the space of all complex differential forms of class
on
, with compact support, in the strong topology, cf. [a1] and Differential form.) Next, H. El Mir showed that one may take
to be a closed complete pluripolar set, which is more general than a closed analytic set, and then
as above will still extend. (A pluripolar set
in
is a set such there exists a plurisubharmonic function
defined in some neighbourhood of
such that
, the
set of
. It is a complete pluripolar set if there is such a
with
equal to the
set of
.) N. Sibony generalized these results even further: If
is a pluripositive current of bi-degree
on
which has locally finite mass in a neighbourhood of
, then
extends to a pluripositive current on
.
One recovers Bishop's theorem from Skoda's using the fact that to every pure -dimensional analytic subset
of
is associated a current
, the current of integration over the regular points of
. This is a positive closed current of bi-degree
. One can return from currents to analytic sets using Siu's theorem (cf. [a4]) on analyticity of sets associated to positive Lelong numbers. (The Lelong number at a point
of a pure
-dimensional analytic set
in
is the number
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The limit exists (cf., e.g., [a1]); in this formula, , the volume of the unit ball in
, and
(i.e. the part of
contained in the ball with centre
and radius
), cf. also [a1].)
References
[a1] | E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian) |
[a2] | N. Sibony, "Quelques problèmes de prolongement de courants en analyse complexe" Duke Math. J. , 52 (1985) pp. 157–197 |
[a3] | Y.T. Siu, "Techniques of extension of analytic objects" , M. Dekker (1974) |
[a4] | Y.T. Siu, "Analyticity of sets associated to Lelong numbers and the extension of closed positive currents" Inv. Math. , 27 (1974) pp. 53–156 |
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