Geometry & Topology

The $3$–fold vertex via stable pairs

Rahul Pandharipande and Richard P Thomas

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800281

Digital Object Identifier
doi:10.2140/gt.2009.13.1835

Mathematical Reviews number (MathSciNet)
MR2497313

Zentralblatt MATH identifier
1195.14073

Subjects
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]

Keywords
curve threefold Gromov–Witten toric

Citation

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. https://projecteuclid.org/euclid.gt/1513800281


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