On weighted Bochner-Martinelli residue currents (Q2879611)

Let \(f=(f_1, \dots, f_m)\) be a sequence of germs of holomorphic functions at \(0\in\mathbb{C}^n\) whose common zero set equals the origin. In [Publ. Mat., Barc. 44, No. 1, 85--117 (2000; Zbl 0964.32003)] \textit{M. Passare}, \textit{A.Tsikh} and \textit{A. Yger} constructed a residue current associated to \(f\) based on the Bochner-Martinelli kernel which gives a natural generalization of the Coleff-Herrera product. This idea is further developed in [\textit{C. A. Berenstein} and \textit{A. Yger}, J. Reine Angew. Math. 527, 203--235 (2000; Zbl 0960.32004)], where the authors introduced weighted Bochner-Martinelli residue currents \(R^p(f)\) associated to \(f\) and a weight \(p=(p_1, \dots, p_m)\in\mathbb{N}^m\). NEWLINENEWLINENEWLINE The main results of the paper under review are a description of \(R^p(f)\) in terms of the Rees valuations of the ideal generated by \((f_1^{p_1}, \dots , f_m^{p_m})\) and an explicit description of \(R^p(f)\) when \(f\) is monomial. For a monomial sequence \(f\) it is shown that \(R^p(f)\) is independent of \(p\) if and only if \(f\) is a regular sequence, that is \(m=n\).