Extension of local direct product structures of normal complex spaces (Q1196285)

Let \(X\) be a pure-dimensional complex space. Let \(A\subset X\) be an analytic set containing \(\text{Sing} X\) and let \(\text{codim}_ X A\geq 2\). We define a foliation \({\mathcal F}\) by curves on \(X\) as follows: Consider a pair \(({\mathcal F}_ A,A)\), where \({\mathcal F}_ A\) is a holomorphic foliation by curves on \(X\backslash A\). Two pairs \(({\mathcal F}_ A,A)\), \(({\mathcal F}_{A'},A')\) are said to be equivalent if there is an analytic set \(B\subset X\), \(\text{codim}_ XB\geq 2\), \(B\supset A\cap A'\) and \({\mathcal F}_ A\) coincides with \({\mathcal F}_{A'}\) on \(X\backslash B\). Then \({\mathcal F}\) is an equivalence class of this relation. The simplest foliation by curves is given by \(X\backslash A\cong\) curve \(\times\) complex space. The author considers the following problem: Let \(X\) be an arbitrary complex space. Assume that in a neighbourhood \(U\) of \(x\in X\) there is an analytic set \(A\ni x\), \(\text{codim}_ UA\geq 2\) such that \(U\backslash A\cong\) curve \(W\times\) complex space \(V\). Is it possible to choose a smaller neighbourhood \(U_ 0\) of \(x\) such that the above product intersected with \(U_ 0\backslash A\) extends to a product on the whole \(U_ 0\) (i.e. \(U_ 0\cong\) another curve \(W_ 0\times\) another complex space \(V_ 0)\)? The main theorem says that this is the case if one can find \(U\), \(A\), \(V\), \(W\) as above such that \(U\) is normal and the projection \(U\backslash A\to W\) extends to a holomorphic map \(U\to W'\), where \(W'\) is a curve containing \(W\). Now let \(X\) be normal. In a neighbourhood \(U\) of \(x\in X\) consider an analytic set \(A\ni x\). Assume (I) \(A\) is irreducible, \(\text{codim}_ UA\geq 2\) or (II) \(\text{codim}_ UA\geq d+1\), where \(d\) is a positive integer. Then a corollary says that the author's problem can be solved affirmatively for \(W\) standing for a \(d\)-dimensional polydisc (then \(W_ 0\) is a smaller polydisc). Finally consider a general foliation \({\mathcal F}\) by curves on a normal \(X\). Let \(\lambda\) be a holomorphic vector field tangent to the leaves of \({\mathcal F}\) wherever it does not vanish. For an \(x\in X\) let \(\lambda(x)=0\) and \(\text{codim}_ X \text{zero}(\lambda)\geq 2\). Then another corollary says that it is impossible to find a neighbourhood \(U\) of \(x\) and an analytic set \(A\subset U\) with \(\text{codim}_ U A\geq 2\) and \(A\supset\text{zero}(\lambda)\cap U\) such that \(U\backslash A\cong\) complex space \(\times\) open disc in \(\mathbb{C}\) and \({\mathcal F}\) agrees with this product on \(U\).