Residue currents of the Bochner-Martinelli type
The main result of the paper is the following: Theorem. Let \(f_1,\dots,f_m\) be holomorphic functions in an open set \(V\) of \(\mathbb{C}^n\). For any ordered subset \(J\subset \{1,\dots,m\}\) the current defined as \[ \langle T_{f,J}, \varphi\rangle =\lim_{\eta \to 0}{c_p \over\eta p}\int_{\{\|f \|^2 =\eta\}} \left(\sum^p_{k=1} (-1)^{k-1}\overline f_{i_k} \wedge_{\ell\neq k}\overline {df}_{i_\ell} \right)\wedge \varphi \tag{*} \] exists and satisfies the following properties: (i) \(\overline hT_{f,J}=0\) for any \(h\in{\mathcal C} (V)\) which vanishes on the common zero set \(\{f_1= \cdots =f_m=0\}\); (ii) \(hT_{f,J}=0\) for any \(h\in{\mathcal C}(V)\) which is locally in the integral closure of the ideal \((f_1,\dots, f_m)^p\); (iii) \(T_{f,J}=0\) if \(p<\text{codim} \{f_1 =\cdots =f_m=0\}\). Moreover, \(T_{f,J}\) depends in an alternating way on the ordering of the elements in \(J\). Here \(p\leq\min \{m,n\}\) and \(\varphi\) in (*) is a test \((n,n-p)\)-form.
scientific article; zbMATH DE number 1505966