Geometry & Topology

The $3$–fold vertex via stable pairs

Rahul Pandharipande and Richard P Thomas

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The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3–folds. We evaluate the equivariant vertex for stable pairs on toric 3–folds in terms of weighted box counting. In the toric Calabi–Yau case, the result simplifies to a new form of pure box counting. The conjectural equivalence with the DT vertex predicts remarkable identities.

The equivariant vertex governs primary insertions in the theory of stable pairs for toric varieties. We consider also the descendent vertex and conjecture the complete rationality of the descendent theory for stable pairs.

Article information

Geom. Topol., Volume 13, Number 4 (2009), 1835-1876.

Received: 3 June 2008
Accepted: 25 February 2009
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14M25: Toric varieties, Newton polyhedra [See also 52B20] 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 14J30: $3$-folds [See also 32Q25]

curve threefold Gromov–Witten toric


Pandharipande, Rahul; Thomas, Richard P. The $3$–fold vertex via stable pairs. Geom. Topol. 13 (2009), no. 4, 1835--1876. doi:10.2140/gt.2009.13.1835.

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