Indices of 1-forms, intersection indices, and Newton polyhedra

A. I. Èsterov

Sbornik. Mathematics, Tome 197 (2006) no. 7, pp. 1085-1108

Résumé

The intersection indices of a certain kind of analytic set (resultant cycles) are expressed in terms of the Newton polyhedra of the corresponding defining systems of functions, provided that the principal parts of the functions are in general position. Among special cases of resultant cycles are complete intersections and the loci of matrix rank drop. Among special cases of the intersection indices of such sets are the index of a singularity of a Poincaré–Hopf vector field and its generalizations to the case of singular varieties, the index of a system of germs of 1-forms at an isolated singularity of a Guseǐn-Zade–Ebeling complete intersection, and the Suwa residue of a system of germs of sections of a vector bundle. One also obtains as a consequence the well-known Kushnirenko–Oka formula for the Milnor number of the germ of a map in terms of the Newton polyhedra of its components. A generalization of the well-known equality of the above-mentioned invariants of singularities to the dimensions of certain local rings is also presented. Bibliography: 17 titles.