Der Kodairasche Verschwindungssatz auf streng pseudokonvexen Räumen. II
This is the second of two papers [for part I see the review above Zbl 0538.32022] in which the Kodaira vanishing theorem on an irreducible reduced strongly pseudoconvex complex space X is proved: \(H^ q(X,{\mathcal E}\otimes {\mathcal K}_ X)=0\) for \(q>0\) and any semi-positive locally free sheaf \({\mathcal E}\) on X with a suitable canonical sheaf \({\mathcal K}_ X\). The first part dealt with a special - namely the algebraic - case. Now in this second part the theorem in the general non-algebraic case is proved using essentially the solution of the special case. The main tool is the method of algebraic approximation. If X is a normal strongly pseudoconvex space with exceptional set A and ideal sheaf \({\mathcal J}\) defining A in X, then an algebraic strongly pseudoconvex X' (i.e. X' has a Moishezon compactification) with exceptional set A' and ideal sheaf \({\mathcal J}'\) defining A' is called an algebraic approximation of order \(\mu\in {\mathbb{N}}\) if \((A,0_ X/{\mathcal J}^{\mu +1})\simeq(A',\quad 0_{X'}/{\mathcal J}^{\prime\mu +1})\). Then the proof of the existence of algebraic approximations of arbitrary high order and with sufficiently nice properties constitutes the technical heart of the paper. - At the end of the paper some generalizations for coherent sheaves are proved.
irreducible reduced strongly pseudoconvex complex space
scientific article; zbMATH DE number 3855496
scientific article