Gopakumar-Vafa Invariants for Local Calabi-Yau Orbifolds of $A_N$-type

Abstract

Let $\mathcal{X}$ be a local orbifold Calabi-Yau threefold whose coarse space $X$ has transverse $A_N$ singularities along a smooth non-compact curve. We define orbifold Gopakumar-Vafa invariants, and we prove they are integers and satisfy a finiteness property. We compute our invariants for local orbifold $K3$ surfaces, where we prove an orbifold version of the classical Yau-Zaslow formula: we show that for an orbifold $K3$ surface $\mathcal{S}$, the genus zero Gopakumar-Vafa invariant of $\mathcal{S}$ in a class of square $2n$ is the Euler characteristic of the Hilbert scheme of $n+1$ points on the singular surface $S$.

41 pages