Gopakumar–Vafa invariants via vanishing cycles
Author(s)Maulik, Davesh; Toda, Yukinobu
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In this paper, we propose an ansatz for defining Gopakumar–Vafa invariants of Calabi–Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal is a modification of a recent approach of Kiem–Li, which is itself based on earlier ideas of Hosono–Saito–Takahashi. We conjecture that these invariants are equivalent to other curve-counting theories such as Gromov–Witten theory and Pandharipande–Thomas theory. Our main theorem is that, for local surfaces, our invariants agree with PT invariants for irreducible one-cycles. We also give a counter-example to the Kiem–Li conjectures, where our invariants match the predicted answer. Finally, we give examples where our invariant matches the expected answer in cases where the cycle is non-reduced, non-planar, or non-primitive.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Maulik, Davesh and Yukinobu Toda. “Gopakumar–Vafa Invariants via Vanishing Cycles.” Inventiones Mathematicae (April 2018): 1 - 81 © 2018 Springer-Verlag