Gopakumar-Vafa invariants associated to $cA_n$ singularities

This paper describes Gopakumar-Vafa (GV) invariants associated to $cA_n$ singularities. We (1) generalize GV invariants to crepant partial resolutions of $cA_n$ singularities, (2) show that generalized GV invariants also satisfy Toda’s formula and are determined by their associated contraction algebra, (3) give filtration structures on the parameter space of contraction algebras associated to $cA_n$ crepant resolutions with respect to generalized GV invariants, and (4) numerically constrain the possible tuples of GV invariants that can arise. We further give all the tuples that arise from GV invariants of $cA_2$ crepant resolutions.

💡 Research Summary

The paper develops a systematic framework for Gopakumar‑Vafa (GV) invariants in the setting of compound Aₙ (cAₙ) singularities, extending the classical theory which is traditionally defined only for smooth Calabi‑Yau threefolds. The author first introduces a generalized invariant N₍β₎(π) for any crepant partial resolution π : X → Spec R of a (possibly non‑isolated) cAₙ singularity. Here β denotes a chain of exceptional curves Cₛ + … + Cₜ, and N₍β₎(π) is defined as the complex dimension of the quotient ℂ