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A notion that is often used in the theory of complex spaces, in particular in the moduli theory of complex spaces. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p0748701.png" /> be a compact Stein set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p0748702.png" /> (cf. [[Stein manifold|Stein manifold]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p0748703.png" /> be the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p0748704.png" /> of the sheaf of germs of holomorphic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p0748705.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p0748706.png" /> is called privileged with respect to a coherent analytic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p0748707.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p0748708.png" /> (cf. [[Coherent analytic sheaf|Coherent analytic sheaf]]) if there is an exact sequence of mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p0748709.png" />-sheaves
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<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/p/p074/p074870/p07487010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487011.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487013.png" />, such that the induced sequence of continuous operators
+
A notion that is often used in the theory of complex spaces, in particular in the moduli theory of complex spaces. Let  $  K $
 +
be a compact Stein set in  $  \mathbf C  ^ {n} $(
 +
cf. [[Stein manifold|Stein manifold]]) and let  $  {\mathcal O} _ {K} $
 +
be the restriction to  $  K $
 +
of the sheaf of germs of holomorphic functions in  $  \mathbf C  ^ {n} $.  
 +
Then  $  K $
 +
is called privileged with respect to a coherent analytic sheaf  $  {\mathcal F} $
 +
on  $  K $(
 +
cf. [[Coherent analytic sheaf|Coherent analytic sheaf]]) if there is an exact sequence of mappings of  $  {\mathcal O} _ {K} $-
 +
sheaves
  
<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/p/p074/p074870/p07487014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
0 \rightarrow  {\mathcal L} _ {n}  \rightarrow \dots \rightarrow \
 +
{\mathcal L} _ {1}  \rightarrow  {\mathcal L} _ {0}  \mathop \rightarrow \limits ^  \phi    {\mathcal F}  \rightarrow  0,
 +
$$
  
<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/p/p074/p074870/p07487015.png" /></td> </tr></table>
+
in which  $  {\mathcal L} _ {i} = {\mathcal O} _ {K} ^ {r _ {i} } $
 +
for some  $  r _ {i} \geq  0 $,
 +
$  i = 0 \dots n $,
 +
such that the induced sequence of continuous operators
 +
 
 +
$$ \tag{2 }
 +
0  \rightarrow  B ( K, {\mathcal L} _ {n} )  \rightarrow ^ { d }  \
 +
B ( K, {\mathcal L} _ {n - 1 }  )  \rightarrow \dots
 +
$$
 +
 
 +
$$
 +
\dots \rightarrow  B ( K, {\mathcal L} _ {1} )  \rightarrow ^ { d }  B ( K, {\mathcal L} _ {0} )
 +
$$
  
 
is exact and split (cf. [[Exact sequence|Exact sequence]]; [[Split sequence|Split sequence]]). Here
 
is exact and split (cf. [[Exact sequence|Exact sequence]]; [[Split sequence|Split sequence]]). Here
  
<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/p/p074/p074870/p07487016.png" /></td> </tr></table>
+
$$
 +
B ( K, {\mathcal L} _ {i} )  = \
 +
B ( K, {\mathcal O} ) ^ {r _ {i} } ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487017.png" /> is the Banach space of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487018.png" /> that are holomorphic in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487019.png" />, endowed with the max-norm. Here, the sequence (2) is said to be split if the kernel and the image of the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487020.png" /> have, for every term, a direct closed complement. This condition for being split is equivalent to: There is a linear continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487021.png" /> in (2) mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487022.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487024.png" /> (a homotopy operator). The properties of the sequence (2) being exact and split do not depend on the choice of (1).
+
and $  B ( K, {\mathcal O} ) $
 +
is the Banach space of continuous functions on $  K $
 +
that are holomorphic in the interior of $  K $,  
 +
endowed with the max-norm. Here, the sequence (2) is said to be split if the kernel and the image of the differential $  d $
 +
have, for every term, a direct closed complement. This condition for being split is equivalent to: There is a linear continuous operator $  h $
 +
in (2) mapping $  B ( K, {\mathcal L} _ {i} ) $
 +
into $  B ( K, {\mathcal L} _ {i + 1 }  ) $
 +
such that $  dhd = d $(
 +
a homotopy operator). The properties of the sequence (2) being exact and split do not depend on the choice of (1).
  
Suppose that a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487025.png" /> lies in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487026.png" />. Then there is a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487027.png" /> of the complex (2) into the fibre of the complex (1) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487028.png" />, mapping an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487029.png" />, i.e. a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487030.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487031.png" />, into its germ at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487032.png" />. This implies that the sequence
+
Suppose that a point $  z $
 +
lies in the interior of $  K $.  
 +
Then there is a morphism $  \pi $
 +
of the complex (2) into the fibre of the complex (1) over $  z $,  
 +
mapping an element of $  B ( K, {\mathcal L} _ {i} ) $,  
 +
i.e. a function on $  K $
 +
with values in $  \mathbf C ^ {r _ {i} } $,  
 +
into its germ at $  z $.  
 +
This implies that the sequence
  
<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/p/p074/p074870/p07487033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
B ( K, {\mathcal L} _ {1} )  \rightarrow ^ { d }  \
 +
B ( K, {\mathcal L} _ {0} )   \mathop \rightarrow \limits ^ { {\pi \phi }}  \
 +
{\mathcal F} _ {z}  $$
  
is semi-exact. The compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487034.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487036.png" />-privileged neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487037.png" /> if it is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487038.png" />-privileged set and if (3) is an exact sequence. This property, too, does not depend on the choice of (1).
+
is semi-exact. The compact set $  K $
 +
is called an $  {\mathcal F} $-
 +
privileged neighbourhood of $  z $
 +
if it is an $  {\mathcal F} $-
 +
privileged set and if (3) is an exact sequence. This property, too, does not depend on the choice of (1).
  
For an arbitrary coherent analytic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487039.png" /> every point of its domain of definition has a fundamental system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487040.png" />-privileged neighbourhoods. One can choose as such neighbourhoods semi-discs with certain, inequality-type, relations between the radii. There is a sufficient condition for a polycylinder to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487041.png" />-privileged, relating the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487042.png" /> with the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487043.png" /> (cf. [[#References|[1]]]).
+
For an arbitrary coherent analytic sheaf $  {\mathcal F} $
 +
every point of its domain of definition has a fundamental system of $  {\mathcal F} $-
 +
privileged neighbourhoods. One can choose as such neighbourhoods semi-discs with certain, inequality-type, relations between the radii. There is a sufficient condition for a polycylinder to be $  {\mathcal F} $-
 +
privileged, relating the sheaf $  {\mathcal F} $
 +
with the boundary of $  K $(
 +
cf. [[#References|[1]]]).
  
One also considers privileged compact sets in relation to a sheaf given on an arbitrary complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487044.png" />; here one has in mind compact sets that are privileged with respect to sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487046.png" /> is a chart on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074870/p07487047.png" />.
+
One also considers privileged compact sets in relation to a sheaf given on an arbitrary complex space $  X $;  
 +
here one has in mind compact sets that are privileged with respect to sheaves $  f _ {*} ( {\mathcal F} ) $,  
 +
where $  f $
 +
is a chart on $  X $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Douady,  "Le problème des modules pour les sous-espaces analytiques compacts d'un espace analytique donné"  ''Ann. Inst. Fourier'' , '''16'''  (1966)  pp. 1–95</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Douady,  "Le problème des modules pour les sous-espaces analytiques compacts d'un espace analytique donné"  ''Ann. Inst. Fourier'' , '''16'''  (1966)  pp. 1–95</TD></TR></table>

Revision as of 08:07, 6 June 2020


A notion that is often used in the theory of complex spaces, in particular in the moduli theory of complex spaces. Let $ K $ be a compact Stein set in $ \mathbf C ^ {n} $( cf. Stein manifold) and let $ {\mathcal O} _ {K} $ be the restriction to $ K $ of the sheaf of germs of holomorphic functions in $ \mathbf C ^ {n} $. Then $ K $ is called privileged with respect to a coherent analytic sheaf $ {\mathcal F} $ on $ K $( cf. Coherent analytic sheaf) if there is an exact sequence of mappings of $ {\mathcal O} _ {K} $- sheaves

$$ \tag{1 } 0 \rightarrow {\mathcal L} _ {n} \rightarrow \dots \rightarrow \ {\mathcal L} _ {1} \rightarrow {\mathcal L} _ {0} \mathop \rightarrow \limits ^ \phi {\mathcal F} \rightarrow 0, $$

in which $ {\mathcal L} _ {i} = {\mathcal O} _ {K} ^ {r _ {i} } $ for some $ r _ {i} \geq 0 $, $ i = 0 \dots n $, such that the induced sequence of continuous operators

$$ \tag{2 } 0 \rightarrow B ( K, {\mathcal L} _ {n} ) \rightarrow ^ { d } \ B ( K, {\mathcal L} _ {n - 1 } ) \rightarrow \dots $$

$$ \dots \rightarrow B ( K, {\mathcal L} _ {1} ) \rightarrow ^ { d } B ( K, {\mathcal L} _ {0} ) $$

is exact and split (cf. Exact sequence; Split sequence). Here

$$ B ( K, {\mathcal L} _ {i} ) = \ B ( K, {\mathcal O} ) ^ {r _ {i} } , $$

and $ B ( K, {\mathcal O} ) $ is the Banach space of continuous functions on $ K $ that are holomorphic in the interior of $ K $, endowed with the max-norm. Here, the sequence (2) is said to be split if the kernel and the image of the differential $ d $ have, for every term, a direct closed complement. This condition for being split is equivalent to: There is a linear continuous operator $ h $ in (2) mapping $ B ( K, {\mathcal L} _ {i} ) $ into $ B ( K, {\mathcal L} _ {i + 1 } ) $ such that $ dhd = d $( a homotopy operator). The properties of the sequence (2) being exact and split do not depend on the choice of (1).

Suppose that a point $ z $ lies in the interior of $ K $. Then there is a morphism $ \pi $ of the complex (2) into the fibre of the complex (1) over $ z $, mapping an element of $ B ( K, {\mathcal L} _ {i} ) $, i.e. a function on $ K $ with values in $ \mathbf C ^ {r _ {i} } $, into its germ at $ z $. This implies that the sequence

$$ \tag{3 } B ( K, {\mathcal L} _ {1} ) \rightarrow ^ { d } \ B ( K, {\mathcal L} _ {0} ) \mathop \rightarrow \limits ^ { {\pi \phi }} \ {\mathcal F} _ {z} $$

is semi-exact. The compact set $ K $ is called an $ {\mathcal F} $- privileged neighbourhood of $ z $ if it is an $ {\mathcal F} $- privileged set and if (3) is an exact sequence. This property, too, does not depend on the choice of (1).

For an arbitrary coherent analytic sheaf $ {\mathcal F} $ every point of its domain of definition has a fundamental system of $ {\mathcal F} $- privileged neighbourhoods. One can choose as such neighbourhoods semi-discs with certain, inequality-type, relations between the radii. There is a sufficient condition for a polycylinder to be $ {\mathcal F} $- privileged, relating the sheaf $ {\mathcal F} $ with the boundary of $ K $( cf. [1]).

One also considers privileged compact sets in relation to a sheaf given on an arbitrary complex space $ X $; here one has in mind compact sets that are privileged with respect to sheaves $ f _ {*} ( {\mathcal F} ) $, where $ f $ is a chart on $ X $.

References

[1] A. Douady, "Le problème des modules pour les sous-espaces analytiques compacts d'un espace analytique donné" Ann. Inst. Fourier , 16 (1966) pp. 1–95
How to Cite This Entry:
Privileged compact set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Privileged_compact_set&oldid=48297
This article was adapted from an original article by V.P. Palamodov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article