Multiple-point residue formulas for holomorphic maps

TL;DR

This paper introduces a novel method for analyzing multipoint loci of holomorphic maps, connecting them to Hilbert schemes and deriving explicit residue formulas, advancing the understanding of complex map singularities.

Contribution

It establishes a new approach linking multipoint loci to Hilbert schemes and derives explicit residue formulas, addressing conjectures by Kazarian and Rimányi.

Findings

Derived a closed iterated multipoint residue formula

Linked multipoint loci to the curvilinear component of Hilbert schemes

Provided new insights into residual polynomials in complex geometry

Abstract

We develop a new approach to the study of the multipoint loci of holomorphic maps between complex manifolds. We relate the $k$-fold locus to the curvilinear component of the Hilbert scheme of $k$ points on the source space of the map, and using equivariant localisation, we derive a closed iterated multipoint residue formula. Our work is motivated by ideas and conjectures of M. Kazarian and R. Rim\'anyi on residual polynomials.