Gopakumar-Vafa invariants via vanishing cycles

Abstract

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.

63 pages, many improvements of the exposition following referee comments, final version to appear in Inventiones

References in corpus (5)

• Donaldson-Thomas invariants via microlocal geometry
• Relative Calabi-Yau structures
• An overview of Morihiko Saito's theory of mixed Hodge modules
• A young person's guide to mixed Hodge modules
• Generalized Donaldson-Thomas theory over fields K $\ne$ C

Cited by in corpus (6)

• Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds
• Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds
• Semiorthogonal decompositions of stable pair moduli spaces via d-critical flips
• Gopakumar-Vafa invariants and wall-crossing
• Topological strings, quiver varieties and Rogers-Ramanujan identities
• The moduli space of stable coherent sheaves via non-archimedean geometry