Vol. 65, No. 2, 1976

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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