A_N型局部卡拉比-丘轨形的Gopakumar-Vafa不变量
Gopakumar-Vafa Invariants for Local Calabi-Yau Orbifolds of $A_{N}$-type
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中文总结 AI 辅助
该研究针对具有$A_N$型横截奇点的局部轨形卡拉比-丘三维簇定义轨形Gopakumar-Vafa不变量并证明其性质，还计算了局部轨形K3曲面的该类不变量并证明了轨形版本的Yau-Zaslow公式。
中文摘要 AI 辅助
设$\boldsymbol{\textit{X}}$为局部轨形卡拉比-丘三维簇，其粗空间$X$沿一条光滑非紧曲线具有横截$A_N$奇点。我们定义了轨形Gopakumar-Vafa不变量，证明其为整数且满足有限性性质。我们计算了局部轨形K3曲面的此类不变量，证明了经典Yau-Zaslow公式的轨形版本：即对于轨形K3曲面$\boldsymbol{\textit{S}}$，其平方为$2n$的类中的零亏格Gopakumar-Vafa不变量，等于奇异曲面$S$上$n+1$个点的希尔伯特概型的欧拉示性数。
英文摘要
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 $S$.
