Residue currents associated with weakly holomorphic functions

We construct Coleff-Herrera products and Bochner-Martinelli type residue currents associated with a tuple $f$ of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case, as the transformation law, the Poincar\'e-Lelong formula and the equivalence of the Coleff-Herrera product and the Bochner-Martinelli type residue current associated with $f$ when $f$ defines a complete intersection.Comment: 28 pages. Updated with some corrections from the revision process. In particular, corrected and clarified some things in Section 5 and 6 regarding products of weakly holomorphic functions and currents, and the definition of the Bochner-Martinelli type current

Averaging Residue Currents and the Stückrad–Vogel Algorithm

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 [10]). Unfortunately, they usually hold within the restricted frame of complete intersections. Besides the fact that it allows 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 \{z1... zn = 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 ([27, 3, 6]), 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 in [13] by N. Coleff and M. Herrera, by relating it to the Stückrad-Vogel algorithm developed in ([30],[31],[8]) towards 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 how much stimulating source of inspiration. This presentation relies also deeply on my collaboration with M. Andersson, H. Samuelsson and E. Wulcan in Göteborg, through the past years. 1 Coleff-Herrera residue currents for complete intersections and the Transformation Law Let X be a n-dimensional (ambient) complex manifold and V ⊂ X be