2025/09/19 by Cavenaghi, Leonardo F., Katzarkov, Ludmil, Kontsevich, Maxim
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2509.15831
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 ℤ/2- and 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.