Coleff-Herrera Currents Revisited

Abstract

In the present paper, we describe the recent approach to residue currents by Andersson, Björk, and Samuelsson (Andersson in Ann. Fac. Sci. ToulouseMath. Sér. 18(4):651-661, 2009

Björk in The Legacy of Niels Henrik Abel, pp. 605-651, Springer, Berlin, 2004

Björk and Samuelsson in J. Reine Angew. Math. 649:33- 54, 2010), focusing primarily on the methods inspired by analytic continuation (that were initiated in a quite primitive form in Berenstein et al. in Residue Currents and Bézout Identities. Progress in Mathematics, vol. 114, Birkhäuser, Basel, 1993). Coleff-Herrera currents (with or without poles) play indeed a crucial role in Lelong- Poincaré-type factorization formulas for integration currents on reduced closed analytic sets. As revealed by local structure theorems (which can also be understood as global when working on a complete algebraic manifold due to the GAGA principle), such objects are of algebraic nature (antiholomorphic coordinates playing basically the role of "inert" constants). Thinking about division or duality problems instead of intersection ones (especially in the "improper" setting, which is certainly the most interesting), it happens then to be necessary to revisit from this point of view the multiplicative inductive procedure initiated by Coleff and Herrera (Lecture Notes in Mathematics, vol. 633, Springer, Berlin, 1978), this being the main objective of this presentation. In homage to the pioneering work of Leon Ehrenpreis, to whom we are both deeply indebted, and as a tribute to him, we also suggest a currential approach to the so-called Noetherian operators that remain the key stone in various formulations of Leon's Fundamental Principle. © Springer-Verlag Italia 2012.