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 be a compact Stein set in (cf. Stein manifold) and let be the restriction to of the sheaf of germs of holomorphic functions in . Then is called privileged with respect to a coherent analytic sheaf on (cf. Coherent analytic sheaf) if there is an exact sequence of mappings of -sheaves

(1)

in which for some , , such that the induced sequence of continuous operators

(2)

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

and is the Banach space of continuous functions on that are holomorphic in the interior of , endowed with the max-norm. Here, the sequence (2) is said to be split if the kernel and the image of the differential have, for every term, a direct closed complement. This condition for being split is equivalent to: There is a linear continuous operator in (2) mapping into such that (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 lies in the interior of . Then there is a morphism of the complex (2) into the fibre of the complex (1) over , mapping an element of , i.e. a function on with values in , into its germ at . This implies that the sequence

(3)

is semi-exact. The compact set is called an -privileged neighbourhood of if it is an -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 every point of its domain of definition has a fundamental system of -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 -privileged, relating the sheaf with the boundary of (cf. [1]).

One also considers privileged compact sets in relation to a sheaf given on an arbitrary complex space ; here one has in mind compact sets that are privileged with respect to sheaves , where is a chart on .

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