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Stein open subsets with analytic complements in compact complex spaces. (English) Zbl 1308.32013

Summary: Let \(Y\) be an open subset of a reduced compact complex space \(X\) such that \(X-Y\) is the support of an effective divisor \(D\). If \(X\) is a surface and \(D\) is an effective Weil divisor, we give sufficient conditions so that \(Y\) is Stein. If \(X\) is of pure dimension \(d\geq 1\) and \(X-Y\) is the support of an effective Cartier divisor \(D\), we show that \(Y\) is Stein if \(Y\) contains no compact curves, \(H^i(Y, {\mathcal {O}}_Y)=0\) for all \(i>0\), and for every point \(x_0\in X-Y\) there is an \(n\in \mathbb {N}\) such that \(\varPhi _{|nD|}^{-1}(\varPhi _{|nD|}(x_0))\cap Y\) is empty or has dimension 0, where \(\varPhi _{|nD|} \) is the map from \(X\) to the projective space defined by a basis of \(H^0(X, {\mathcal {O}}_X(nD))\).

MSC:

32C11 Complex supergeometry
32E10 Stein spaces
14E05 Rational and birational maps
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