Algebraic Geometry Seminar

Title: Gopakumar–Vafa invariants and a support theorem for stable pairs

Abstract: The Gopakumar–Vafa/Pandharipande–Thomas correspondence relates two approaches to counting curves on Calabi–Yau threefolds, using moduli spaces of one-dimensional sheaves and stable pairs. For smooth curves, its cohomological form is suggested by the classical Macdonald formula relating symmetric powers and Jacobians. I will present the cohomological correspondence for local (\mathbb P^2) and local (\mathbb P^1\times\mathbb P^1), in every effective curve class and Euler characteristic. The main ingredient is a support theorem for the direct images of stable-pair vanishing-cycle sheaves: their strict supports are closures of loci of transverse unions of smooth connected curves. I will explain how wall-crossing and duality reduce the problem to a finite range of Euler characteristics, and how a commutation relation for point modifications excludes additional supports. Applying the family Macdonald formula on the reduced locus then gives the correspondence as an identity of semisimplified perverse direct images on the Chow variety.