View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e0370701.png" /> in an [[Analytic space|analytic space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e0370702.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e0370703.png" /> (as a rule, also analytic) to the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e0370704.png" />. Two theorems of B. Riemann form the classical results concerning continuation of functions.

Riemann's first theorem states that every analytic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e0370705.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e0370706.png" /> is a normal [[Complex space|complex space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e0370707.png" /> an analytic subspace of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e0370708.png" />, can be continued to an analytic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e0370709.png" />. Riemann's second theorem states that every analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707011.png" /> that is locally bounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707013.png" /> is a nowhere-dense analytic subset in a normal complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707014.png" />, can be continued to an analytic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707015.png" />. There are generalizations of these theorems to arbitrary complex spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707016.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707017.png" />-dimensional complex-analytic subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707019.png" /> is a complex-analytic space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707020.png" /> a closed subset having zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707021.png" />-dimensional [[Hausdorff measure|Hausdorff measure]], can be extended to a pure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707022.png" />-dimensional complex-analytic subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707023.png" />. Bishop's theorem states that every pure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707024.png" />-dimensional complex-analytic subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707027.png" /> is a complex-analytic space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707028.png" /> is a complex-analytic subset, can be extended to a pure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707029.png" />-dimensional complex-analytic subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707031.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707032.png" /> has locally finite volume in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707034.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707035.png" />.

There are criteria for extendability of analytic mappings, generalizing the classical [[Picard theorem|Picard theorem]]. E.g., every analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707037.png" /> is a complex manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707038.png" /> is an analytic nowhere-dense set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707039.png" /> is a hyperbolic compact complex manifold, can be extended to an analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707040.png" />. Every analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707041.png" /> that is not everywhere-degenerate, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707042.png" /> is a complex manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707043.png" /> is an analytic subset and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707044.png" /> is a compact complex manifold with negative first [[Chern class|Chern class]], can be extended to a meromorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707045.png" />.

====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707046.png" /> be an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707048.png" /> a complex-analytic subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707049.png" />. First, Skoda's theorem states that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707050.png" /> is a positive closed current of bi-degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707051.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707052.png" /> which has locally finite mass in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707053.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707054.png" /> extends to a positive closed current on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707055.png" />. (A current on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707056.png" /> is a continuous linear functional on the space of all complex differential forms of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707057.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707058.png" />, with compact support, in the strong topology, cf. [[#References|[a1]]] and [[Differential form|Differential form]].) Next, H. El Mir showed that one may take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707059.png" /> to be a closed complete pluripolar set, which is more general than a closed analytic set, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707060.png" /> as above will still extend. (A pluripolar set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707061.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707062.png" /> is a set such there exists a [[Plurisubharmonic function|plurisubharmonic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707063.png" /> defined in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707064.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707065.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707066.png" /> set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707067.png" />. It is a complete pluripolar set if there is such a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707068.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707069.png" /> equal to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707070.png" /> set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707071.png" />.) N. Sibony generalized these results even further: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707072.png" /> is a pluripositive current of bi-degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707073.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707074.png" /> which has locally finite mass in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707076.png" /> extends to a pluripositive current on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707077.png" />.

One recovers Bishop's theorem from Skoda's using the fact that to every pure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707078.png" />-dimensional analytic subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707079.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707080.png" /> is associated a current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707081.png" />, the current of integration over the regular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707082.png" />. This is a positive closed current of bi-degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707083.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707084.png" /> of a pure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707085.png" />-dimensional [[Analytic set|analytic set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707086.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707087.png" /> is the number

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707088.png" /></td> </tr></table>

The limit exists (cf., e.g., [[#References|[a1]]]); in this formula, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707089.png" />, the volume of the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707090.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707091.png" /> (i.e. the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707092.png" /> contained in the ball with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707093.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037070/e03707094.png" />), 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>