## Stein open subsets with analytic complements in compact complex spaces

### Volume 113 / 2015

Annales Polonici Mathematici 113 (2015), 43-60
MSC: Primary 32E10; Secondary 14E05.
DOI: 10.4064/ap113-1-2

#### Abstract

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 $H^0(X, {\mathcal {O}}_X(nD))$.