The Oka-Grauert principle without induction over the base dimension.

*(English)*Zbl 0955.32019From the introduction: We give a new proof of Grauert’s theorem on Oka’s principle in the case of (smooth) Stein manifolds, which does not use induction over the base dimension. Instead we use induction over the levels of a strictly plurisubharmonic exhausting function (Grauert’s bump method).

The present paper is an edited version of our preprint [‘Proof of Grauter’s Oka-principle without induction over the basis dimension’, preprint Karl-Weierstrass Inst. Math. Berlin P-Math-05186 (1986)] from 1986 (which is difficult to find and of bad printing quality). We did not publish the paper in a journal at that time, because we planned to write a book containing it. But the book has not been written until now. On the other hand, in the meantime, some interest to this proof appeared. Therefore we think a publication of our proof could be useful even with a delay of 10 years.

The proof gives also some Oka principle for arbitrary pseudoconvex manifolds (see Theorem 1.3 below).

The present paper is an edited version of our preprint [‘Proof of Grauter’s Oka-principle without induction over the basis dimension’, preprint Karl-Weierstrass Inst. Math. Berlin P-Math-05186 (1986)] from 1986 (which is difficult to find and of bad printing quality). We did not publish the paper in a journal at that time, because we planned to write a book containing it. But the book has not been written until now. On the other hand, in the meantime, some interest to this proof appeared. Therefore we think a publication of our proof could be useful even with a delay of 10 years.

The proof gives also some Oka principle for arbitrary pseudoconvex manifolds (see Theorem 1.3 below).