Yukinobu Toda (MIT, IPMU), Gopakumar-Vafa invariants via perverse sheaves

The Gopakumar-Vafa invariants are integer valued invariants on Calabi-Yau 3-folds which are expected to be related to other curve counting invariants on them such as Gromov-Witten invariants, Pandharipande-Thomas invariants. However a mathematically rigorous definition of GV invariants is not obvious. In 2012, Kiem-Li used perverse sheaves of vanishing cycles to define GV invariants, which modify the previous definition by Hosono-Saito-Takahashi. In this talk, I will explain that Kiem-Li's definition is not deformation invariant, so not a correct one. I will modify the definition of Kiem-Li, by introducing the notion of Calabi-Yau d-critical structures. The main result is that our definition of GV invariants with irreducible curve classes match with PT invariants for local surfaces. This is a joint work with Davesh Maulik.