Residues in the not necessarily complete intersection case (Q2745751)

Let \(Z\) be an irreducible analytic subvariety of codimension \(q\) in an open subset \(U\) of \(\mathbb C^n\), defined by equations \(f_1=\dots =f_N = 0\) with \(f_j\in\mathcal O(U)\). Let \(m\) be the multiplicity of the ideal of \(\mathcal O_{U,x}\) generated by the germs of \(f_1,\dots,f_N\) at a generic point \(x\in Z\). NEWLINENEWLINENEWLINEThe author obtains explicitly a decomposition of Serre-Grothendieck residue type for the integration current associated to \(Z\) in terms of \(f=(f_1,\dots,f_N)\): the differential form \(T= d^c(\log|f|(dd^c\log|f|)^{q-1})\) has \(L_{\text{loc}}^1\) coefficients on \(U\) and \(C^\infty\) ones in \(U\setminus Z\), and its residue current is \(m[Z]=dd^c\left(\log|f|(dd^c\log|f|)^{q-1}\right) -(dd^c\log|f|)^q_{|U\setminus Z}.\) To this end, the author proves a generalized King formula from which the Poincaré-Lelong formula follows by blowing up. The obtained decomposition allows to extend to the general case the expressions for the current of integration already known in the case of a complete intersection, such as the Berenstein-Yger formula using the Gelfand-Atiyah-Kashiwara meromorphic continuation and the Coleff-Herrera-Passare formula in terms of residual integrals.