Locus of exceptional points and the multidirectional tangent cone of a closed positive current (Q1972899)

Let \(\Omega\) be an open set in \(\mathbb C^n\), \(n\geqslant 2,\) and let \((X_k)_k\) be a sequence of analytic sets of dimension \(p\leqslant n - 2\) in \(\Omega.\) The author proves that there exists a closed positive \((1,1)\)-current \(T\) on \(\Omega\) such that the tangent cone of \(T\) does not exist at any point of \(\cup_k X_k.\) More precisely, for every open set \(\omega\subset\Omega\) and for every analytic set of dimension \(p\leqslant n - 2\) in \(\Omega,\) there exists a closed positive current \(T\) of bidegree \((1,1)\) on \(\Omega\) such that the tangent cone of \(T\) does not exist on \(X\cap\omega.\) Moreover, for every closed positive current \(\Theta\) of bidegree \((1,1),\) the tangent cone of the current \(T + \Theta\) does not exist at any point of \(X\cap\omega.\) The directional and multidirectional tangent cones associated to a closed positive current are also studied.