On some generalizations of Jacobi's residue formula (Q2725498)

Let \(X\) be a compact complex analytic variety of dimension \(n,\) and let \(D_1, \ldots, D_n\) be effective divisors which intersect properly. Let \(\omega\) be a meromorphic form whose poles are contained in the intersection \(Z\) of the supports \(|D_i|\), \(i = 1, \ldots, n.\) NEWLINENEWLINENEWLINEThe well-known theorem of Griffiths implies that the total sum of the Grothendieck residues of \(\omega\) over all the points of \(Z\) is equal to zero [\textit{Ph. Griffiths} and \textit{J. Harris,} `Principles of algebraic geometry'. New York, Wiley (1978; Zbl 0408.14001)]. NEWLINENEWLINENEWLINEIn the paper under review the analogous result is proved in the case when \(X\) is a projective space or toric variety under the assumption that the divisors intersect properly in the affine part of \(X.\) The proof is based on the multidimensional residue theory with the use of residual currents of Bochner-Martinelli type. NEWLINENEWLINENEWLINEAs a consequence some applications concerning effectivity questions related to the algebraic Nullstellensatz in the projective case or the sparse Nullstellensatz in the toric case are considered. These results improves previous ones from \textit{C. A. Berenstein} and \textit{A. Yger} [Am. J. Math. 121, No. 4, 723-796 (1999; Zbl 0944.14002)] and \textit{A. Fabiano, G. Pucci} and \textit{A. Yger} [Acta Arith. 78, No. 2, 165-187 (1996; Zbl 0892.14002)]. Among other things the authors also discuss applications to some results of Cayley-Bacharach type [\textit{D. Eisenbud, M. Green} and \textit{J. Harris}, Bull. Am. Math. Soc., New Ser. 33, No. 3, 295-324 (1996; Zbl 0871.14024)] in the context of improper intersections on \({\mathbb P}^n\) or on a smooth toric variety, and some others.