Absolute gap-sheaves and extensions of coherent analytic sheaves

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 Sf is an analytic subsheaf of an analytic sheaf Jona complex space {X, ¿f), then ¿f : 2T denotes the ideal-sheaf J defined by Jx = {s e3fx \ s^x^£Q for xeX.E(ïf, 3T) denotes {xeX\ Sfx^yx}.Supp ST denotes the support of ST.If t e F{X, !7~), 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 ¿/of J~ we mean the ideal-sheaf sé defined by K = {s e -K | s9~x = 0} for xeX.\f0: {X, 3f) -* {X', Jf') is a holomorphic map (i.e. a morphism of ringed spaces) from {X, ¿f) to another complex space (A", 30"), then R°0(y) denotes the zeroth direct image of 3~ under 0. life T(X, ¿f) and xel.we say that/vanished at x iffx is not a unit in ¿fx.