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Compact singularities of meromorphic mappings between complex 3-dimensional manifolds. (English) Zbl 0984.32008

The main result of this paper is the following extendibility theorem for equidimensional meromorphic mappings: Let \(M\) be a Stein 3-manifold and \(X\) be a compact complex 3-manifold, let \(K\) be a compact set with connected complement in \(M\) and let \(f: M \setminus K \to X\) be a meromorphic mapping, then there exists a finite set \(A=\{ a_1,\dots,a_d \} \subset K\) such that: 1) \(f\) has a meromorphic extension \(\widehat{f}: M \setminus A \to X\); 2) for every coordinate ball \(B(a_j)\) satisfying \(\partial B(a_j) \bigcap A =\emptyset\), \(\widehat{f}(\partial B(a_j))\) is not homologous to zero in \(X.\) In particular, if \(X\) is simply connected, then the mapping \(f\) extends to all of \(M\).
There are also some generalizations made and open questions raised.

MSC:

32H04 Meromorphic mappings in several complex variables
32J17 Compact complex \(3\)-folds
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