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Averaging Residue Currents and the Stückrad–Vogel Algorithm

  • Alain Yger
Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 16)

Abstract

Trace formulas (Lagrange, Jacobi–Kronecker, Bergman–Weil) play a key role in division problems in analytic or algebraic geometry (including arithmetic aspects, see, for example, Berenstein and Yger in Am. J. Math. 121(4):723–796, 1999). Unfortunately, they usually hold within the restricted frame of complete intersections. Besides the fact that it allows one to carry local or semi-global analytic problems to a global geometric setting (think about Crofton’s formula), averaging the Cauchy kernel (from ℂ n ∖{z 1z n =0}⊂ℙ n (ℂ)), in order to get the Bochner–Martinelli kernel (in ℂ n+1∖{0}⊂ℙ n+1(ℂ)=ℂ n+1∪ℙ n (ℂ)), leads to the construction of explicit candidates for the realization of Grothendieck’s duality, namely BM residue currents (Passare et al. in Publ. Mat. 44:85–117, 2000; Andersson in Bull. Sci. Math. 128(6):481–512, 2004; Andersson and Wulcan in Ann. Sci. École Norm. Super. 40:985–1007, 2007), extending thus the cohomological incarnation of duality which appears in the complete intersection or Cohen–Macaulay cases. We will recall here such constructions and, in parallel, suggest how far one could take advantage of the multiplicative inductive construction introduced by Coleff and Herrera (Lecture Notes in Mathematics, vol. 633, Springer, Berlin, 1978), by relating it to the Stückrad–Vogel algorithm developed in Stückrad and Vogel (Queen Pap. Pure Appl. Math. 61:1–32, 1982), Tworzewski (Ann. Pol. Math. 62:177–191, 1995), Andersson et al. (arXiv:1009.2458, 2010) toward improper intersection theory. Results presented here were initiated all along my long-term collaboration with Carlos Berenstein. To both of us, the mathematical work of Leon Ehrenpreis certainly remained a constant and very stimulating source of inspiration. This presentation relies also deeply on my collaboration over the past years with M. Andersson, H. Samuelsson, and E. Wulcan in Göteborg.

Keywords

Complete Intersection Global Section Holomorphic Section Cartier Divisor Integration Current 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Italia 2012

Authors and Affiliations

  1. 1.Institut de MathématiquesUniversité de BordeauxTalenceFrance

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