Virtual Residue and an integral formalism

TL;DR

This paper extends Grothendieck's residue theory to virtual cases with higher-dimensional zero loci and introduces an exponential integral formalism analogous to Mathai-Quillen for localized Euler classes.

Contribution

It generalizes the concept of residues to virtual zero loci and develops an exponential integral formalism similar to Mathai-Quillen.

Findings

Generalization of Grothendieck's residues to virtual cases

Development of an exponential integral formalism for virtual residues

Analogy with Mathai-Quillen formalism for localized Euler classes

Abstract

We generalize Grothendieck's residues $Res\frac{\psi}{s}$ to virtual cases, namely cases when the zero loci of the section $s$ has dimension larger than the expected dimension(zero). We also provide an exponential type integral formalism for the virtual residue, which can be viewed as an analogue of the Mathai-Quillen formalism for localized Euler classes.