Privileged compact set

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