Absolute gap-sheaves and extensions of coherent analytic sheaves

Abstract

Thimm introduced the concept of gap-sheaves for analytic subsheaves of finite direct sums of structure-sheaves on domains of complex number spaces (Definition 9, [13]) and proved that these gap-sheaves are coherent if the subsheaves themselves are coherent (Satz 3, [13]). This concept of gap-sheaves can be readily generalized to analytic subsheaves of arbitrary analytic sheaves on general complex spaces (Definition 1, [12]). All the gap-sheaves of coherent analytic subsheaves of arbitrary coherent analytic sheaves on general complex spaces are coherent (Theorem 3, [12]). The gap-sheaves of a given analytic subsheaf depend not only on the subsheaf itself but also on the analytic sheaf in which the given subsheaf is embedded as a subsheaf. In this paper we introduce a new notion of gap-sheaves which we call absolute gap-sheaves (Definition 3 below). These gap-sheaves arise naturally from the problem of removing singularities of local sections of a coherent analytic sheaf. They depend only on a given analytic sheaf and neither require nor depend upon an embedding of the given sheaf as a subsheaf in another analytic sheaf. We give here a necessary and sufficient condition for the coherence of absolute gap-sheaves of coherent sheaves (Theorem 1 below). This yields some results concerning removing singularities of local sections of coherent sheaves (see Remark following Corollary 2 to Theorem 1). Then we use absolute gap-sheaves to derive a theorem (Theorem 2 below) which generalizes Serre's Theorem on the extension of torsion-free coherent analytic sheaves (Theorem 1, [11]). Finally a result on extensions of global sections of coherent analytic sheaves is derived (Theorem 4 below). Unless specified otherwise, complex spaces are in the sense of Grauert (?1, [5]). If Y is an analytic subsheaf of an analytic sheaf Y on a complex space (X, X), then Y: S denotes the ideal-sheaf f defined by fX ={3 E I lsST c Yx} for x E X. E(9, -) denotes {x E X I x =& }x Supp 3denotes the support of 3C If t E r(X, J), then Supp t denotes the support of t. For x E X, tx denotes the germ of t at x. By the annihilator-ideal-sheaf Q1of Y we mean the ideal-sheaf v defined by /x = {s E Xx I s$x = O} for x E X. If 0: (X, -(X', *') is a holomorphic map (i.e. a morphism of ringed spaces) from (X, to another complex space (X', k'), then R06(Y) denotes the zeroth direct image of T under O. If fE I(X, X) and x E X, we say thatf vanished at x iffx is not a unit in Xx.