Annals of Mathematics· 2026Q1
The Gopakumar--Vafa finiteness conjecture
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- Q1SCImago
- 2026year
Short summary
The finiteness part of the Gopakumar--Vafa conjecture, predicting that BPS invariants of a symplectic 6-manifold are integers and all but finitely many vanish in every homology class, has been proven.
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Key points
- The finiteness of BPS invariants in the Gopakumar--Vafa conjecture is now proven.
- This result confirms that for a symplectic 6-manifold, only a finite number of BPS invariants are non-zero in each homology class.
- The proof builds upon prior work on integrality and employs a modified cluster formalism from geometric measure theory.
AI-generated from the title and abstract; the full text is not read.
Abstract
The Gopakumar--Vafa conjecture predicts that the BPS invariants of a symplectic 6-manifold, defined in terms of the Gromov--Witten invariants, are integers and all but finitely many vanish in every homology class. The integrality part of this conjecture was proved earlier by Ionel and Parker. This article proves the finiteness part. The proof relies on a modification of Ionel and Parker's cluster formalism using results from geometric measure theory.
The authors' abstract, as published at the source. Annals of Mathematics, 2026 · DOI ↗
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Field: Geometry and Topology
Geometry and TopologyMathematics