On the tangent cone with a closed positive current
Translation of the French review: ``Let \(\text{K}_p\) be the set of \((p,p)\) closed positive and conic currents on \(\mathbb{C}^n\) such that the Lelong number is equal to 1 at the origin \(O\). The convex subset \(\text{K}_p\) is weakly compact metrizable subspace of the space of currents on \(\mathbb{C}^n\). For every current \(\text{T}\) on \(\mathbb{C}^n\), the set of limits of the family \(((h_r)^* \text{T})_{r < 1}\) is connected and weakly compact in \(m.\text{K}_p\) where \(m\) is the Lelong number of \(\text{T}\) at \(\text{O}\) and \(h_r\) is the homothety with the origin as centre and with support \(r\). We prove that if M is any closed connected subset of \(\text{K}_p\) \((n - 1 \geq p \geq 1)\); there exists a closed positive current \(\text{T}\) such that \(\text{M}\) is the limit set of the family \(((h_r)^* \text{T})_{r < 1}\). In the second part we prove that for currents of bidimension \((p,p)\) \((p =1\) or \(n - 1)\) a necessary and sufficient condition such that the tangent cone exists in \(\text{T}\) is that the family \((h_r)^* \text{T} \wedge \beta^n\) admits a limit if \(r\) goes to zero. We prove that this condition is not valid for currents of type \((p,p)\) in the case \(2 \leq p\leq n - 2\)''.
scientific article; zbMATH DE number 811432