The Intrinsic Normal Cone For Artin Stacks

TL;DR

This paper generalizes the concept of the normal cone to higher Artin stacks, providing a unique construction that recovers known cases and enables the definition of a relative virtual fundamental class.

Contribution

It extends the normal cone construction to higher Artin stacks, characterizes it axiomatically, and applies it to define virtual fundamental classes in Chow groups.

Findings

Unified construction of the normal cone for Artin stacks

Recovery of the classical case in Deligne-Mumford stacks

Application to virtual fundamental classes in Chow groups

Abstract

We extend the construction of the normal cone of a closed embedding of schemes to any locally of finite type morphism of higher Artin stacks and show that in the Deligne-Mumford case our construction recovers the relative intrinsic normal cone of Behrend and Fantechi. We characterize our extension as the unique one satisfying a short list of axioms, and use it to construct the deformation to the normal cone. As an application of our methods, we associate to any morphism of Artin stacks equipped with a choice of a global perfect obstruction theory a relative virtual fundamental class in the Chow group of Kresch.