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Descendents on local curves: rationality

Part of: Families, fibrations Projective and enumerative geometry

Published online by Cambridge University Press:  01 November 2012

R. Pandharipande  and
A. Pixton
R. Pandharipande
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA (email: [email protected], [email protected])
A. Pixton
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA (email: [email protected])
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Abstract

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We study the stable pairs theory of local curves in 3-folds with descendent insertions. The rationality of the partition function of descendent invariants is established for the full local curve geometry (equivariant with respect to the scaling 2-torus), including relative conditions and odd-degree insertions for higher-genus curves. The capped 1-leg descendent vertex (equivariant with respect to the 3-torus) is also proven to be rational. The results are obtained by combining geometric constraints with a detailed analysis of the poles of the descendent vertex.

Keywords

Donaldson–Thomasdescendentmodulisheaf

MSC classification

Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 14D20: Algebraic moduli problems, moduli of vector bundles
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 1 , January 2013 , pp. 81 - 124
Copyright
Copyright © The Author(s) 2012

References

[Beh09]Behrend, K., Donaldson–Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170 (2009), 1307–1338.CrossRefGoogle Scholar
[BF97]Behrend, K. and Fantechi, B., The intrinsic normal cone, Invent. Math. 128 (1997), 45–88.CrossRefGoogle Scholar
[Bri11]Bridgeland, T., Hall algebras and curve-counting invariants, J. Amer. Math. Soc. 24 (2011), 969–998.CrossRefGoogle Scholar
[BP08]Bryan, J. and Pandharipande, R., The local Gromov–Witten theory of curves, J. Amer. Math. Soc. 21 (2008), 101–136.CrossRefGoogle Scholar
[DT98]Donaldson, S. K. and Thomas, R. P., Gauge theory in higher dimensions, in The geometric universe (Oxford, 1996) (Oxford University Press, Oxford, 1998), 31–47.CrossRefGoogle Scholar
[FP00]Faber, C. and Pandharipande, R., Hodge integrals and Gromov–Witten theory, Invent. Math. 139 (2000), 173–199.CrossRefGoogle Scholar
[GP99]Graber, T. and Pandharipande, R., Localization of virtual classes, Invent. Math. 135 (1999), 487–518.CrossRefGoogle Scholar
[HL97]Huybrechts, D. and Lehn, M., The geometry of moduli spaces of sheaves, Aspects of Mathematics, vol. E31 (Vieweg, Braunschweig, 1997).CrossRefGoogle Scholar
[JS12]Joyce, D. and Song, Y., A theory of generalized Donaldson–Thomas invariants, Mem. Amer. Math. Soc. 217 (2012), doi:10.1090/S0065-9266-2011-00630-1.Google Scholar
[LeP93]Le Potier, J., Systèmes cohérents et structures de niveau, Astérisque 214 (1993).Google Scholar
[LT98]Li, J. and Tian, G., Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), 119–174.CrossRefGoogle Scholar
[LW09]Li, J. and Wu, B., Degeneration of Donaldson–Thomas invariants, Preprint (2009).Google Scholar
[MNOP06a]Maulik, D., Nekrasov, N., Okounkov, A. and Pandharipande, R., Gromov–Witten theory and Donaldson–Thomas theory. I, Compositio Math. 142 (2006), 1263–1285.CrossRefGoogle Scholar
[MNOP06b]Maulik, D., Nekrasov, N., Okounkov, A. and Pandharipande, R., Gromov–Witten theory and Donaldson–Thomas theory. II, Compositio Math. 142 (2006), 1286–1304.CrossRefGoogle Scholar
[MOOP11]Maulik, D., Oblomkov, A., Okounkov, A. and Pandharipande, R., The Gromov–Witten/Donaldson–Thomas correspondence for toric 3-folds, Invent. Math. 186 (2011), 435–479.CrossRefGoogle Scholar
[MP06]Maulik, D. and Pandharipande, R., A topological view of Gromov–Witten theory, Topology 45 (2006), 887–918.CrossRefGoogle Scholar
[MPT10]Maulik, D., Pandharipande, R. and Thomas, R., Curves on K3 surfaces and modular forms, J. Topology 3 (2010), 937–996.CrossRefGoogle Scholar
[OP06]Okounkov, A. and Pandharipande, R., Virsoro constraints for target curves, Invent. Math. 163 (2006), 47–108.CrossRefGoogle Scholar
[OP10a]Okounkov, A. and Pandharipande, R., The quantum cohomology of the Hilbert scheme of points of the plane, Invent. Math. 179 (2010), 523–557.CrossRefGoogle Scholar
[OP10b]Okounkov, A. and Pandharipande, R., The local Donaldson–Thomas theory of curves, Geom. Topol. 14 (2010), 1503–1567.CrossRefGoogle Scholar
[OP10c]Okounkov, A. and Pandharipande, R., The quantum differential equation of the Hilbert scheme of points of the plane, Invent. Math. 178 (2010), 523–557.CrossRefGoogle Scholar
[PP10a]Pandharipande, R. and Pixton, A., Descendents on local curves: stationary theory, in Geometry and Arithmetic, EMS Series of Congress Reports, to appear, Preprint (2010).Google Scholar
[PP10b]Pandharipande, R. and Pixton, A., Descendent theory for stable pairs on toric 3-folds, J. Math. Soc. Japan, to appear, Preprint (2010).Google Scholar
[PT09a]Pandharipande, R. and Thomas, R. P., Curve counting via stable pairs in the derived category, Invent Math. 178 (2009), 407–447.CrossRefGoogle Scholar
[PT09b]Pandharipande, R. and Thomas, R. P., The 3-fold vertex via stable pairs, Geom. Topol. 13 (2009), 1835–1876.CrossRefGoogle Scholar
[PT10]Pandharipande, R. and Thomas, R. P., Stable pairs and BPS invariants, J. Amer. Math. Soc. 23 (2010), 267–297.CrossRefGoogle Scholar
[Tho00]Thomas, R. P., A holomorphic Casson invariant for Calabi–Yau 3-folds, and bundles on K3 fibrations, J. Differential Geom. 54 (2000), 367–438.CrossRefGoogle Scholar
[Tod10]Toda, Y., Generating functions of stable pairs invariants via wall-crossings in derived categories, in New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008), Adv. Stud. Pure Math., vol. 59 (Mathematical Society of Japan, Tokyo, 2010), 389–434.CrossRefGoogle Scholar