Gopakumar-Vafa invariants and Macdonald formula II

RESUMO

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.