Vol. 65, No. 2, 1976

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ISSN: 0030-8730
On extending regular holomorphic maps from Stein manifolds

Chester Cornelius Seabury

Vol. 65 (1976), No. 2, 499–515

In this paper the following results are proved. Theorem. Let S,M be complex manifolds, S a Stein manifold, and f : S M a holomorphic embedding. Let K S be compact, and let Nf be the normal bundle of f. We identify S with the zero section of Nf. Then in Nf, there is a neighborhood U of K, and a holomorphic embedding F : U M such that F|U S = f. If f above is an immersion, then there is an immersion F as above. There is also an analogous result for holomorphic maps f which are regular at some point p in S.

The idea of the proof is to construct a function ϕ on a neighborhood of f(K) M such that ϕ is strictly plurisubharmonic and ϕ1((−∞,c]) is compact for all c in R. Then a result of Forster and Ramspott is applied to get the final results. To construct ϕ, special coordinates are obtained near f(S) in M.

Mathematical Subject Classification 2000
Primary: 32C25
Secondary: 32D20
Received: 12 April 1976
Published: 1 August 1976
Chester Cornelius Seabury