Advertisement

Inventiones mathematicae

, Volume 178, Issue 2, pp 407–447 | Cite as

Curve counting via stable pairs in the derived category

  • R. Pandharipande
  • R. P. Thomas
Article

Abstract

For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where CX is an embedded curve and DC is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of X. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category.

Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described.

The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.

Keywords

Modulus Space Hilbert Scheme Stable Pair Topological Vertex Obstruction Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alexeev, V., Knutson, A.: Complete moduli spaces of branchvarieties. math.AG/0602626
  2. 2.
    Aganagic, M., Klemm, A., Mariño, M., Vafa, C.: The topological vertex. Commun. Math. Phys. 254, 425–478 (2005). hep-th/0305132 zbMATHCrossRefGoogle Scholar
  3. 3.
    Aspinwall, P.S., Morrison, D.R.: Topological field theory and rational curves. Commun. Math. Phys. 151, 245–262 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bayer, A.: Polynomial Bridgeland stability conditions and the large volume limit. arXiv:0712.1083
  5. 5.
    Behrend, K.: Donaldson-Thomas invariants via microlocal geometry. Ann. Math. (2009, to appear). math.AG/0507523
  6. 6.
    Behrend, K., Bryan, J.: Super-rigid Donaldson-Thomas invariants. Math. Res. Lett. 14, 559–571 (2007). math.AG/0601203 zbMATHMathSciNetGoogle Scholar
  7. 7.
    Behrend, K., Fantechi, B.: The intrinsic normal cone. Invent. Math. 128, 45–88 (1997). math.AG/9601010 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Behrend, K., Fantechi, B.: Symmetric obstruction theories and Hilbert schemes of points on threefolds. Algebra Number Theory 2, 313–345 (2008). math.AG/0512556 zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bridgeland, T.: Flops and derived categories. Invent. Math. 147, 613–632 (2002). math.AG/0009053 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bridgeland, T.: Stability conditions on triangulated categories. Ann. Math. 166, 317–345 (2007). math.AG/0212237 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bryan, J., Pandharipande, R.: The local Gromov-Witten theory of curves. J. Am. Math. Soc. 21, 101–136 (2008). math.AG/0411037 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Candelas, P., de la Ossa, X.C., Green, P.S., Parkes, L.: A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys. B 359, 21–74 (1991) zbMATHCrossRefGoogle Scholar
  13. 13.
    Denef, F., Moore, G.W.: Split states, entropy enigmas, holes and halos. hep-th/0702146
  14. 14.
    Diaconescu, D.E.: Moduli of ADHM sheaves and local Donaldson-Thomas theory. arXiv:0801.0820
  15. 15.
    Diaconescu, D.E., Moore, G.W.: Crossing the wall: Branes vs. bundles. arXiv:0706.3193 [hep-th]
  16. 16.
    Donaldson, S.K., Thomas, R.P.: Gauge theory in higher dimensions. In: The Geometric Universe, Oxford, 1996, pp. 31–47. Oxford Univ. Press, Oxford (1998) Google Scholar
  17. 17.
    Faber, C., Pandharipande, R.: Hodge integrals and Gromov-Witten theory. Invent. Math. 139, 173–199 (2000). math.AG/9810173 zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Graber, T., Pandharipande, R.: Localization of virtual classes. Invent. Math. 135, 487–518 (1999). alg-geom/9708001 zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Gopakumar, R., Vafa, C.: M-theory and topological strings—I, hep-th/9809187
  20. 20.
    Gopakumar, R., Vafa, C.: M-theory and topological strings—II. hep-th/9812127
  21. 21.
    Honsen, M.: A compact moduli space parameterizing Cohen-Macaulay curves in projective space. PhD thesis, MIT (2004) Google Scholar
  22. 22.
    Hosono, S., Saito, M.-H., Takahashi, A.: Relative Lefschetz actions and BPS state counting. IMRN 15, 783–816 (2001) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Shaves. Aspects of Mathematics, vol. E31. Vieweg, Braunschweig (1997) Google Scholar
  24. 24.
    Huybrechts, D., Thomas, R.P.: Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes. arXiv:0805.3527
  25. 25.
    Inaba, M.: Moduli of stable objects in a triangulated category. math.AG/0612078
  26. 26.
    Ionel, E., Parker, T.H.: Relative Gromov-Witten invariants. Ann. Math. (2) 157, 45–96 (2003). math.SG/9907155 zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Joyce, D.: Configurations in Abelian categories, IV. Invariants and changing stability conditions. Adv. Math. 217, 125–204 (2008). math.AG/0410268 zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Katz, S.: Genus zero Gopakumar-Vafa invariants of contractible curves. J. Differential Geom. 79, 185–195 (2008). math.AG/0601193 zbMATHMathSciNetGoogle Scholar
  29. 29.
    Katz, S., Klemm, A., Vafa, C.: M-theory, topological strings and spinning black holes. Adv. Theor. Math. Phys. 3, 1445–1537 (1999). hep-th/9910181 zbMATHMathSciNetGoogle Scholar
  30. 30.
    Klemm, A., Pandharipande, R.: Enumerative geometry of Calabi-Yau 4-folds. Commun. Math. Phys. 281, 621–653 (2008). math.AG/0702189 zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Kollár, J.: Projectivity of complete moduli. J. Differ. Geom. 32, 235–268 (1990) zbMATHGoogle Scholar
  32. 32.
    Le Potier, J.: Systèmes cohérents et structures de niveau. Astérisque 214, 143 (1993) MathSciNetGoogle Scholar
  33. 33.
    Le Potier, J.: Faisceaux semi-stables et systèmes cohérents. In: Vector Bundles in Algebraic Geometry, Durham, 1993. London Math. Soc. Lecture Note Ser., vol. 208, pp. 179–239. Cambridge Univ. Press, Cambridge (1995) Google Scholar
  34. 34.
    Levine, M., Pandharipande, R.: Algebraic cobordism revisited. math.AG/0605196
  35. 35.
    Li, J.: Stable morphisms to singular schemes and relative stable morphisms. J. Differ. Geom. 57, 509–578 (2001). math.AG/0009097 zbMATHGoogle Scholar
  36. 36.
    Li, J.: A degeneration formula of GW-invariants. J. Differ. Geom. 60, 199–293 (2002). math.AG/0110113 zbMATHGoogle Scholar
  37. 37.
    Li, J.: Zero dimensional Donaldson-Thomas invariants of threefolds. Geom. Topol. 10, 2117–2171 (2006). math.AG/0604490 zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Li, A.-M., Ruan, Y.: Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds I. Invent. Math. 145, 151–218 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Li, J., Tian, G.: Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Am. Math. Soc. 11, 119–174 (1998). math.AG/9602007 zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Li, J., Wu, B.: Degeneration of Donaldson-Thomas invariants (in preparation) Google Scholar
  41. 41.
    Lieblich, M.: Moduli of complexes on a proper morphism. J. Algebraic Geom. 15, 175–206 (2006). math.AG/0502198 zbMATHMathSciNetGoogle Scholar
  42. 42.
    Lowen, W.: Obstruction theory for objects in Abelian and derived categories. Commun. Algebra 33, 3195–3223 (2005). math.KT/0407019 zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov-Witten theory and Donaldson-Thomas theory. I. Compos. Math. 142, 1263–1285 (2006). math.AG/0312059 zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov-Witten theory and Donaldson-Thomas theory. II. Compos. Math. 142, 1286–1304 (2006). math.AG/0406092 zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Pandharipande, R.: Hodge integrals and degenerate contributions. Commun. Math. Phys. 208, 489–506 (1999). math.AG/9811140 zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Pandharipande, R.: Three questions in Gromov-Witten theory. In: Proceedings of the International Congress of Mathematicians, Beijing, 2002, vol. II, pp. 503–512. Higher Ed. Press., Beijing (2002). math.AG/0302077 Google Scholar
  47. 47.
    Pandharipande, R., Thomas, R.P.: The 3-fold vertex via stable pairs. Geom. Topol. 13, 1835–1876 (2009). arXiv:0709.3823 zbMATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Pandharipande, R., Zinger, A.: Enumerative geometry of Calabi-Yau 5-folds. arXiv:0802.1640
  49. 49.
    Szendrői, B.: Non-commutative Donaldson-Thomas theory and the conifold. Geom. Topol. 12, 1171–1202 (2008). arXiv:0705.3419 [math.AG] CrossRefMathSciNetGoogle Scholar
  50. 50.
    Thomas, R.P.: A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations. J. Differ. Geom. 54, 367–438 (2000). math.AG/9806111 zbMATHGoogle Scholar
  51. 51.
    Toda, Y.: Birational Calabi-Yau 3-folds and BPS state counting. Commun. Number Theory Phys. 2, 63–112 (2008). arXiv:0707.1643 [math.AG] zbMATHMathSciNetGoogle Scholar
  52. 52.
    Toda, Y.: Limit stable objects on Calabi-Yau 3-folds. arXiv:0803.2356

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsImperial CollegeLondonUK

Personalised recommendations