A Gromov-Witten approach to $G$-equivariant birational invariants

TL;DR

This paper develops a Gromov-Witten theoretical framework to study G-equivariant birational invariants, connecting Chen-Ruan cohomology with classical and new invariants, and proposing a theory of equivariant atoms.

Contribution

It introduces a novel approach linking Gromov-Witten theory with G-birational invariants and extends the theory of atoms to the equivariant setting.

Findings

Established a connection between Chen-Ruan cohomology and G-birational invariants.

Proposed a new theory of equivariant atoms.

Provided applications illustrating the theory's utility.

Abstract

In arXiv:2404.19088, we initiated a program linking birational invariants with smooth ones and offering new interpretations of classical invariants, such as the Kervaire-Milnor invariants. Here, we rely on the profound geometric reasoning provided by Lupercio and Uribe in the early 00s to establish a connection between Chen-Ruan cohomology and several $G$-birational invariants introduced in the pioneering works Kontsevich, Kresch, Pestun, Tschinkel, along with presenting applications. Combined with the theory of atoms by Katzarkov, Kontsevich, Pantev, and Yu, the proposal in this paper program will lead to a theory of equivariant atoms.