Theorems A and B for Compact Blocks in ℂ^m

Abstract

In this chapter the main results of the theory of coherent analytic sheaves for compact blocks Q in ℂ^m are proved (see Paragraph 3.2). The standard techniques for coherent sheaves and cohomology theory are used, in particular the fact that H ^q(Q, S) = 0 for large q (see Chapter B.2.5 and 3.4). Moreover we will bring into play the fact that H ^q(Q, O) = 0 for q ≥ 1. The basic tool which is derived in this chapter is an attaching lemma for analytic sheaf epimorphisms (Theorem 2.3). The proof of this lemma is based on an attaching lemma of H. Cartan for matrices near the identity (Theorem 1.4) and the Runge approximation theorem (Theorem 2.1).