Gopakumar-Vafa不变量与Macdonald公式 II
Gopakumar-Vafa invariants and Macdonald formula II
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- Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo（东京大学宇宙物理数学研究所）
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中文总结 AI 辅助
本文证明了局部平面和二次曲面上所有有效曲线类及欧拉特征数的上同调GV/PT对应，通过点修正交换关系排除其他支撑，并应用族Macdonald公式在Chow簇上建立恒等式。
中文摘要 AI 辅助
我们证明了在局部平面和二次曲面上，对每个有效曲线类和欧拉特征数，上同调Gopakumar-Vafa/Pandharipande-Thomas对应成立。稳定对消失循环直像的简单反常层的严格支撑集是光滑连通曲线横截并的轨迹闭包。墙交叉和对偶性将需考虑的欧拉特征数限制在有限范围内。在此范围内，有界曲面偶堆上点修正的交换关系排除了所有其他支撑集。在约化轨迹上应用族Macdonald公式，给出了Chow簇上半简化反常直像的恒等式形式的对应关系。
英文摘要
We prove the cohomological Gopakumar-Vafa/Pandharipande-Thomas correspondence for the local plane and quadric in every effective curve class and Euler characteristic. The strict supports of the simple perverse constituents of the stable pair vanishing cycle direct images are closures of loci of transverse unions of smooth connected curves. Wall-crossing and duality leave a finite range of Euler characteristics to consider. In this range, a commutation relation for point modifications on bounded stacks of surface pairs excludes all other supports. Applying the family Macdonald formula on the reduced locus gives the correspondence as an identity of semisimplified perverse direct images on the Chow variety.
