Analytic sheaves of local cohomology

In this paper we are interested in the following problem: Suppose Fis an analytic subvariety of a (not necessarily reduced) complex analytic space X, IF is a coherent analytic sheaf on X-V, and 6: X-V-> X is the inclusion map.When is 6JÍJF) coherent (where 6q(F) is the qth direct image of 3F under 0)?The case ¿7 = 0 is very closely related to the problem of extending rF to a coherent analytic sheaf on X.This problem of extension has already been dealt with in Frisch-Guenot [1], Serre [9], Siu [11]-[14], Thimm [17], and Trautmann [18]-[20].So, in our investigation we assume that !F admits a coherent analytic extension on all of X.In réponse to a question of Serre {9, p. 366], Tratumann has obtained the following in [21]: Theorem T. Suppose V is an analytic subvariety of a complex analytic space X, q is a nonnegative integer, andrF is a coherent analytic sheaf on X.Let 6: X-V-*-X be the inclusion map.Then a sufficient condition for 80(^-V),..., dq(^-V) to be coherent is: for xeV there exists an open neighborhood V of x in X such that codh(3F-V)^dimx V+q+2.Trautmann's sufficiency condition is in general not necessary, as is shown by the following example.Let X=C3, F={z1=z2=0}, ^=0, and J^the analytic idealsheaf of {z2=z3=0}.d^LF-V)=3F is coherent on X.However, codh &=2 at (a, 0, 0) for all a. Hence there exists no neighborhood If of (0, 0, 0) such that codh(^-V)^3.In this paper we derive a necessary and sufficient condition for the coherence of 60(^-V),..., d<l(&'-V).This necessary and sufficient condition is then applied to give an affirmative answer to a question raised by Serre [9,.To state the main results, we introduce the following notations: If ¿F is a coherent analytic sheaf on a complex analytic space X, then SkLF) denotes the analytic subvariety {x e X \ codh F"x^k}, [6, Satz 5, p. 81].If /) is an open subset of X, then SkLF) denotes the topological closure of Sk(&) in X.If V is an analytic subvariety of X, then ¿Cyi^) denotes the sheaf defined by the presheaf U m-Hk(U, SP), where, Hk(U,.F) is the ^-dimensional cohomology group of U with coefficients in & and supports in V.If X, IF, and V are complex algebraic instead of analytic, we use the same notations and !Fh denotes the coherent analytic sheaf canonically associated with !F.