Atoms meet symbols

Abstract

This paper introduces a novel framework for constructing invariants in $G$-equivariant birational geometry by unifying two recent approaches: the theory of atoms recently developed by Katzarkov, Kontsevich, Pantev, and Yu, and the theory of modular symbols due to Kontsevich, Tschinkel, and Pestun. We initiate the theory of Chen-Ruan atoms. Assuming the blowup formula for the quantum Chen-Ruan cohomology, we outline how to extend the theory of atoms to global quotient orbifolds and present some striking applications. In addition, we develop a separate class of purely geometric invariants for $\mathbb{Z}/2$- and $\mathbb{Z}/3$-actions on surfaces and threefolds. We provide many examples of non-$G$-linearizable $G$-actions on projective varieties treated with these new techniques.

v4. Minor changes, mostly stylistic and small corrections. References added