Stability conditions and curve counting invariants on Calabi–Yau 3-folds
Yukinobu Toda
Source: Kyoto J. Math. Volume 52, Number 1
(2012), 1-50.
Abstract
The purpose of this paper is twofold. First we give a survey on the recent developments of curve counting invariants on Calabi–Yau 3-folds, for example, Gromov–Witten theory, Donaldson–Thomas theory, and Pandharipande–Thomas theory. Next we focus on the proof of the rationality conjecture of the generating series of PT invariants and discuss its conjectural Gopakumar–Vafa form.
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Permanent link to this document: http://projecteuclid.org/euclid.kjm/1329684741
Digital Object Identifier: doi:10.1215/21562261-1503745
Zentralblatt MATH identifier: 06026361
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References
[1] A. Bayer, Polynomial Bridgeland stability conditions and the large volume limit, Geom. Topol. 13 (2009), 2389–2425.
Mathematical Reviews (MathSciNet): MR2515708
[2] K. Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170 (2009), 1307–1338.
Mathematical Reviews (MathSciNet): MR2600874
Zentralblatt MATH: 1191.14050
Digital Object Identifier: doi:10.4007/annals.2009.170.1307
[3] K. Behrend and J. Bryan, Super-rigid Donaldson-Thomas invariants, Math. Res. Lett. 14 (2007), 559–571.
Mathematical Reviews (MathSciNet): MR2335983
Zentralblatt MATH: 1137.14041
[4] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), 45–88.
Mathematical Reviews (MathSciNet): MR1437495
Zentralblatt MATH: 0909.14006
Digital Object Identifier: doi:10.1007/s002220050136
[5] K. Behrend and B. Fantechi, Symmetric obstruction theories and Hilbert schemes of points on threefolds, Algebra Number Theory 2 (2008), 313–345.
Mathematical Reviews (MathSciNet): MR2407118
[6] K. Behrend and E. Getzler, Chern-Simons functional, in preparation.
[7] F. Bittner, The universal Euler characteristic for varieties of characteristic zero, Compos. Math. 140 (2004), 1011–1032.
Mathematical Reviews (MathSciNet): MR2059227
Zentralblatt MATH: 1086.14016
Digital Object Identifier: doi:10.1112/S0010437X03000617
[8] L. Bodnarchuk, I. Burban, Y. Drozd, and G. Greuel, “Vector bundles and torsion free sheaves on degenerations of elliptic curves” in Global Aspects of Complex Geometry, Springer, Berlin, 2006, 83–128.
Mathematical Reviews (MathSciNet): MR2264108
Zentralblatt MATH: 1111.14027
Digital Object Identifier: doi:10.1007/3-540-35480-8_3
[9] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), 317–345.
Mathematical Reviews (MathSciNet): MR2373143
Zentralblatt MATH: 1137.18008
Digital Object Identifier: doi:10.4007/annals.2007.166.317
[10] T. Bridgeland, Hall algebras and curve-counting invariants, J. Amer. Math. Soc. 24 (2011), 969–998.
Mathematical Reviews (MathSciNet): MR2813335
Zentralblatt MATH: 05950698
Digital Object Identifier: doi:10.1090/S0894-0347-2011-00701-7
[11] T. Bridgeland and A. Maciocia, Fourier-Mukai transforms for K3 and elliptic fibrations, J. Algebraic Geom. 11 (2002), 629–657.
Mathematical Reviews (MathSciNet): MR1910263
Zentralblatt MATH: 1066.14047
Digital Object Identifier: doi:10.1090/S1056-3911-02-00317-X
[12] C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math 139 (2000), 173–199.
Mathematical Reviews (MathSciNet): MR1728879
Zentralblatt MATH: 0960.14031
Digital Object Identifier: doi:10.1007/s002229900028
[13] R. Gopakumar and C. Vafa, M-theory and topological strings, II, preprint, arXiv:hep-th/9812127v1
[14] L. Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990), 193–207.
Mathematical Reviews (MathSciNet): MR1032930
Zentralblatt MATH: 0679.14007
Digital Object Identifier: doi:10.1007/BF01453572
[15] D. Happel, I. Reiten, and S. O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575.
Mathematical Reviews (MathSciNet): MR1327209
Zentralblatt MATH: 0849.16011
[16] D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, 2nd ed., Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 2010.
Mathematical Reviews (MathSciNet): MR2665168
[17] D. Joyce, Configurations in abelian categories, II: Ringel-Hall algebras, Adv. Math. 210 (2007), 635–706.
Mathematical Reviews (MathSciNet): MR2303235
Zentralblatt MATH: 1119.14005
Digital Object Identifier: doi:10.1016/j.aim.2006.07.006
[18] D. Joyce, Motivic invariants of Artin stacks and “stack functions”, Q. J. Math. 58 (2007), 345–392.
Mathematical Reviews (MathSciNet): MR2354923
Zentralblatt MATH: 1131.14005
Digital Object Identifier: doi:10.1093/qmath/ham019
arXiv: 0810.5645v6
[20] S. Katz, “Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds” in Snowbird Lectures on String Geometry, Contemp. Math. 401, Amer. Math. Soc., Providence, 2006, 43–52.
Mathematical Reviews (MathSciNet): MR2222528
Zentralblatt MATH: 1171.14305
Digital Object Identifier: doi:10.1090/conm/401/07552
[21] S. Katz, Genus zero Gopakumar-Vafa invariants of contractible curves, J. Differential Geom. 79 (2008), 185–195.
Mathematical Reviews (MathSciNet): MR2420017
Zentralblatt MATH: 1142.32011
Project Euclid: euclid.jdg/1211512639
[22] M. Kontsevich, “Enumeration of rational curves via torus actions” in The Moduli Space of Curves (Texel Island, Netherlands, 1994), Progr. Math. 129, Birkhäuser, Boston, 1995, 335–368.
Mathematical Reviews (MathSciNet): MR1363062
Zentralblatt MATH: 0885.14028
Digital Object Identifier: doi:10.1007/978-1-4612-4264-2_12
arXiv: 0811.2435v1
[24] A. Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), 495–536.
Mathematical Reviews (MathSciNet): MR1719823
Zentralblatt MATH: 0938.14003
Digital Object Identifier: doi:10.1007/s002220050351
[25] G. Laumon and L. Moret-Bailly, Champs algébriques, Ergeb. Math. Grenzeb. (3) 39, Springer, Berlin, 2000.
Mathematical Reviews (MathSciNet): MR1771927
[26] M. Levine and R. Pandharipande, Algebraic cobordism revisited, Invent. Math. 176 (2009), 63–130.
Mathematical Reviews (MathSciNet): MR2485880
Zentralblatt MATH: 1210.14025
Digital Object Identifier: doi:10.1007/s00222-008-0160-8
[27] J. Li, Zero dimensional Donaldson-Thomas invariants of threefolds, Geom. Topol. 10 (2006), 2117–2171.
Mathematical Reviews (MathSciNet): MR2284053
Digital Object Identifier: doi:10.2140/gt.2006.10.2117
[28] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), 119–174.
Mathematical Reviews (MathSciNet): MR1467172
Zentralblatt MATH: 0912.14004
Digital Object Identifier: doi:10.1090/S0894-0347-98-00250-1
[29] M. Lieblich, Moduli of complexes on a proper morphism, J. Algebraic Geom. 15 (2006), 175–206.
Mathematical Reviews (MathSciNet): MR2177199
Zentralblatt MATH: 1085.14015
Digital Object Identifier: doi:10.1090/S1056-3911-05-00418-2
[30] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory, I; Compos. Math. 142 (2006), 1263–1285.
Mathematical Reviews (MathSciNet): MR2264664
Zentralblatt MATH: 1108.14046
Digital Object Identifier: doi:10.1112/S0010437X06002302
[31] A. Okounkov and R. Pandharipande, The local Donaldson-Thomas theory of curves, Geom. Topol. 14 (2010), 1503–1567.
Mathematical Reviews (MathSciNet): MR2679579
[32] R. Pandharipande and R. P. Thomas, Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009), 407–447.
Mathematical Reviews (MathSciNet): MR2545686
Zentralblatt MATH: 1204.14026
Digital Object Identifier: doi:10.1007/s00222-009-0203-9
arXiv: 0903.1444v3
[34] R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds and bundles on K3-fibrations, J. Differential Geom. 54 (2000), 367–438.
Mathematical Reviews (MathSciNet): MR1818182
Zentralblatt MATH: 1034.14015
Project Euclid: euclid.jdg/1214341649
[35] Y. Toda, Limit stable objects on Calabi-Yau 3-folds, Duke Math. J. 149 (2009), 157–208.
Mathematical Reviews (MathSciNet): MR2541209
Zentralblatt MATH: 1172.14007
Digital Object Identifier: doi:10.1215/00127094-2009-038
Project Euclid: euclid.dmj/1246453791
[36] Y. Toda, Curve counting theories via stable objects, I: DT/PT correspondence, J. Amer. Math. Soc. 23 (2010), 1119–1157.
Mathematical Reviews (MathSciNet): MR2669709
Zentralblatt MATH: 1207.14020
Digital Object Identifier: doi:10.1090/S0894-0347-10-00670-3
[37] Y. Toda, “Generating functions of stable pair invariants via wall-crossings in derived categories” in New Developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry (Kyoto, 2008), Adv. Stud. Pure Math. 59, Math. Soc. Japan, Tokyo, 2010, 389–434.
Mathematical Reviews (MathSciNet): MR2683216
Zentralblatt MATH: 1216.14009
[38] Y. Toda, On a computation of rank two Donaldson-Thomas invariants, Commun. Number Theory Phys. 4 (2010), 49–102.
Mathematical Reviews (MathSciNet): MR2679377
Zentralblatt MATH: 1225.14031
arXiv: 0909.5129v3
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