## Geometry & Topology

### The $3$–fold vertex via stable pairs

#### 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
Accepted: 25 February 2009
First available in Project Euclid: 20 December 2017

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

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

#### References

• K Behrend, B Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997) 45–88
• J Bryan, R Pandharipande, The local Gromov–Witten theory of curves, J. Amer. Math. Soc. 21 (2008) 101–136 With an appendix by Bryan, C Faber, A Okounkov and Pandharipande
• S K Donaldson, R P Thomas, Gauge theory in higher dimensions, from: “The geometric universe (Oxford, 1996)”, (S A Huggett, L J Mason, K P Tod, S T Tsou, N M J Woodhouse, editors), Oxford Univ. Press (1998) 31–47
• T Graber, R Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999) 487–518
• D Huybrechts, M Lehn, The geometry of moduli spaces of sheaves, Aspects of Math. E31, Friedr. Vieweg & Sohn, Braunschweig (1997)
• J Le Potier, Systèmes cohérents et structures de niveau, Astérisque (1993) 143
• J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998) 119–174
• D Maulik, Gromov–Witten theory of $\mathcal{A}_n$–resolutions, Geom. Topol. 13 (2009) 1729–1773
• D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory. I, Compos. Math. 142 (2006) 1263–1285
• D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. II, Compos. Math. 142 (2006) 1286–1304
• D Maulik, A Oblomkov, Donaldson–Thomas theory of $A_n\times \PP^1$
• D Maulik, A Oblomkov, A Okounkov, R Pandharipande, The Gromov–Witten/Donaldson–Thomas correspondence for toric threefolds, in preparation
• A Okounkov, R Pandharipande, The local Donaldson–Thomas theory for curves
• R Pandharipande, R P Thomas, Curve counting via stable pairs in the derived category
• R P Thomas, A holomorphic Casson invariant for Calabi–Yau $3$–folds, and bundles on $K3$ fibrations, J. Differential Geom. 54 (2000) 367–438