Difference between revisions of "Privileged compact set"
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− | in | + | 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 | ||
− | + | $$ \tag{1 } | |
+ | 0 \rightarrow {\mathcal L} _ {n} \rightarrow \dots \rightarrow \ | ||
+ | {\mathcal L} _ {1} \rightarrow {\mathcal L} _ {0} \mathop \rightarrow \limits ^ \phi {\mathcal F} \rightarrow 0, | ||
+ | $$ | ||
− | + | 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|Exact sequence]]; [[Split sequence|Split sequence]]). Here | is exact and split (cf. [[Exact sequence|Exact sequence]]; [[Split sequence|Split sequence]]). Here | ||
− | + | $$ | |
+ | B ( K, {\mathcal L} _ {i} ) = \ | ||
+ | B ( K, {\mathcal O} ) ^ {r _ {i} } , | ||
+ | $$ | ||
− | and | + | 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 | + | 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 | + | 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 | + | 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 | + | 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> |
Latest revision as of 01:23, 19 January 2022
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 |
Privileged compact set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Privileged_compact_set&oldid=13339