Division on a complex space with arbitrary singularities (Q1032751)

The main result is the following division theorem on Stein spaces. Let \(X\) be a pure \(n\)-dimensional Stein space and \(A\subset X\) a lower dimensional complex analytic subset with empty interior such that \(X_{\text{ sing}}\subset A.\) Let \(\Omega \) be an open relatively compact Stein domain in \(X,\) \( \Omega ^{\ast }:=\Omega \setminus A,\) and \(f_{1},\ldots ,f_{m}\) be holomorphic functions in \(\Omega \) such that \[ \sup_{z\in \Omega ^{\prime}}\text{dist}{}^{\tilde N}(z,A)/||f||^{2}<+\infty \text{ \;\;for any }\Omega ^{\prime}\Subset \Omega \text{ and some }\tilde N\in \mathbb{N}. \] Then for every \(F\) holomorphic in \(\Omega \) which ``vanish to sufficiently high order near the set \(A\)'' (i.e., \(\int_{\Omega}\left| F(z)\right| ^{2}\text{dist}{}^{-N}(z,A)dV<+\infty \)), there exist holomorphic functions \(g_{1},\ldots ,g_{m}\) such that \(F=\sum f_{i}g_{i}\) in \(\Omega ^{\ast }\) with some estimations on the seminorm of \(g_{j}.\)