Analytic Residue Theory in the Non-complete Intersection Case

In previous work of the authors and their collaborators (see, e.g., Progress in Math. 114, Birkhäuser (1993)) it was shown how the equivalence of several constructions of residue currents associated to complete intersection families of (germs of) holomorphic functions in C could be profitably used to solve algebraic problems like effective versions of the Nullstellensatz. In this work, the authors explain how such ideas can be transposed to the non-complete intersection situation, leading to an explicit way to construct a Green current attached to a purely dimensional cycle in P. This construction extends a previous result of the authors done in the complete intersection case. When the cycle is defined over Q, they give a closed expression for the analytic contribution in the definition of its logarithmic height (as the residue at λ = 0 of a ζ-function attached to a system of generators of the ideal which defines the cycle). They also introduce an extension of the Cauchy-Weil division process and apply it in order to make explicit the membership of the Jacobian determinant of n elements fj ∈ On, j = 1, ..., n, (which fail to define a regular sequence) in the ideal (f1, ..., fn).