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 $ X \setminus A $ in an 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 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).

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, 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. 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, can be extended to a meromorphic mapping $ X \rightarrow Y $.

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 $ 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. [a1] and 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 $ \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. [a4]) on analyticity of sets associated to positive Lelong numbers. (The Lelong number at a point $ a $ of a pure $ p $- dimensional 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., [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 [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