Analytic sheaf cohomology groups of dimension $n$ of $n-dimensional$ $complex$ $spaces.$
Yum-Tong Siu · Transactions of the American Mathematical Society · 1969
In this paper we prove the following: Main Theorem.Suppose & is a coherent analytic sheaf on a a-compact complex space X (not necessarily reduced).Z/dim X = n and X has no compact n-dimensional branch, ( )n then Hn(X, &) = 0.Z/dim X = n and X has only a finite number of compact n-dimensional branches, then dim Hn(X, ¿F) < oo.(A)n with the additional assumption that A' is a manifold and !F is locally free was proved by Malgrange [12, p. 236, Problème 1].In [9] Komatsu proved the following related result: if Sf is a coherent analytic sheaf on an n-dimensional complex manifold U such that Hn(U, Sf) = Q, then H\V, y)=0 for any open subset F of U (p. 83, Theorem 7).The author in [16] proved (A)n with the additional assumption that X is a manifold.The paper is divided into five sections.In §1 some Lemmas about Fréchet spaces and LF-spaces are proved.In §11 a duality concerning distributions with restricted supports is established.In §111 by partial normalizations and results of [13] the proof of the Main Theorem is reduced to the proof of (A)n with the additional assumption that X is reduced and normal and F is torsion-free.In §IV we prove by local resolutions of singularities that Hn(G, J^) = 0 for Gc ç x and also obtain a result on the approximation of (n-l)-cocycles with coefficients in &. §V sews up the proof of the Main Theorem.All complex spaces here are a-compact and are in the sense of Grauert [3, p. 9, Definition 2].The structure sheaf of a complex space X is denoted by x<3 unless specified otherwise.The set of all singular points of X is denoted by o(X).The inverse image [4, p. 410, Definition 8] and the qth direct image [4, p. 413, Definition 9] of an analytic sheaf J5" under a holomorphic map / of complex spaces are denoted respectively byf'XF) and R"f(3P).If S? is a subsheaf of F, then R°f(&) is regarded as a subsheaf of R°f(F).A covering 11 of a complex space X is called a Stein covering if It is countable and every member of U is a Stein open subset.If Fis an open subset of X, then 111 Y={U e 111 £/<= Y} is called the restriction of 11