The 3–fold vertex via stable pairs

Pandharipande, Rahul¹ ; Thomas, Richard P²

¹ Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
² Department of Mathematics, Imperial College, London SW7 2AZ, UK

Geometry & topology, Tome 13 (2009) no. 4, pp. 1835-1876.

Résumé

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.

Détail

DOI : 10.2140/gt.2009.13.1835

Keywords: curve, threefold, Gromov–Witten, toric

Bibliographie

[1] K Behrend, B Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997) 45

[2] J Bryan, R Pandharipande, The local Gromov–Witten theory of curves, J. Amer. Math. Soc. 21 (2008) 101

[3] S K Donaldson, R P Thomas, Gauge theory in higher dimensions, from: "The geometric universe (Oxford, 1996)" (editors S A Huggett, L J Mason, K P Tod, S T Tsou, N M J Woodhouse), Oxford Univ. Press (1998) 31

[4] T Graber, R Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999) 487

[5] D Huybrechts, M Lehn, The geometry of moduli spaces of sheaves, Aspects of Math. E31, Friedr. Vieweg Sohn (1997)

[6] J Le Potier, Systèmes cohérents et structures de niveau, Astérisque (1993) 143

[7] J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998) 119

[8] D Maulik, Gromov–Witten theory of $\mathcal{A}_n$–resolutions, Geom. Topol. 13 (2009) 1729

[9] D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory. I, Compos. Math. 142 (2006) 1263

[10] D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. II, Compos. Math. 142 (2006) 1286

[11] D Maulik, A Oblomkov, Donaldson–Thomas theory of $A_n{\times}\mathbb{P}^1$

[12] D Maulik, A Oblomkov, A Okounkov, R Pandharipande, The Gromov–Witten/Donaldson–Thomas correspondence for toric threefolds, in preparation

[13] A Okounkov, R Pandharipande, The local Donaldson–Thomas theory for curves

[14] R Pandharipande, R P Thomas, Curve counting via stable pairs in the derived category

[15] R P Thomas, A holomorphic Casson invariant for Calabi–Yau $3$–folds, and bundles on $K3$ fibrations, J. Differential Geom. 54 (2000) 367