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Homogeneous Contact Manifolds and Resolutions of Calabi–Yau Cones

  • Eder M. CorreaEmail author
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Abstract

In the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds. The main feature of our approach is to describe the Cartan–Ehresmann connection (gauge field) for principal U(1)-bundles over complex flag manifolds by using elements of representation theory of simple Lie algebras. This description allows us to compute explicitly the expression of the contact form for any Boothby–Wang fibration over complex flag manifolds (Boothby and Wang in Ann Math 68:721–734, 1958) as well as their underlying Sasaki structures. By following Conlon and Hein (Duke Math J 162:2855–2902, 2013), Van Coevering (Math Ann, 2009.  https://doi.org/10.1007/s00208-009-0446-1) and Goto (J Math Soc Jpn 64:1005–1052, 2012), as an application of our results we use the Cartan–Remmert reduction (Grauert in Math Ann 146:331–368, 1962) and the Calabi Ansatz technique (Calabi in Ann Sci École Norm Sup (4) 12:269–294, 1979) to provide many explicit examples of crepant resolutions of Calabi–Yau cones with certain homogeneous Sasaki–Einstein manifolds realized as links of isolated singularities. These concrete examples illustrate the existence part of the conjecture introduced in Martelli and Sparks (Phys Rev D 79(6):065009, 2009).

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Acknowledgments

The authorwould like to thank the anonymous reviewers for their helpful and constructive comments that greatly contributed to improving the final version of the paper.

References

  1. 1.
    Abreu M.: Toric Kähler metrics: cohomogeneity one examples of constant scalar curvature in action-angle coordinates. J. Geom. Symmetry Phys. 17, 1–33 (2010)ADSMathSciNetzbMATHGoogle Scholar
  2. 2.
    Aloff S., Wallach N.R.: An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures. Bull. Am. Math. Soc. 81, 93 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Azad H., Biswas I.: Quasi-potentials and Kähler–Einstein metrics on flag manifolds. II. J. Algebra 269(2), 480–491 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baum H., Friedrich T., Grunewald R., Kath I.: Twistors and Killing Spinors on Riemannian Manifolds, Volume 124 of Teubner-Texte zur Mathematik. B. G. Teubner, Leipzig (1991)zbMATHGoogle Scholar
  5. 5.
    Besse A.L.: Einstein Manifolds, 1987th edn. Springer, Berlin (2007)Google Scholar
  6. 6.
    Billey S., Lakshmibai V.: Singular Loci of Schubert Varieties, Progress in Mathematics, vol. 182. Birkhäuser, Boston (2000)CrossRefzbMATHGoogle Scholar
  7. 7.
    Blair D.E.: Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics, vol. 203. Birkhäuser, Basel (2010)CrossRefGoogle Scholar
  8. 8.
    Boothby W.M., Wang H.C.: On contact manifolds. Ann. Math. 68, 721–734 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bott R., Tu L.W.: Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Book 82. Springer, Berlin (1995)Google Scholar
  10. 10.
    Boyer C.P., Galicki K.: Sasakian–Einstein geometry. Int. J. Math. 11, 873–909 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Boyer C., Galicki K.: Sasakian Geometry, Oxford Mathematical Monographs, 1st edn. Oxford University Press, Oxford (2008)Google Scholar
  12. 12.
    Boyer C.P., Galicki K.: New Einstein metrics in dimension five. J. Differ. Geom. 57(3), 443–463 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Boyer C.P., Galicki K., Nakamaye M.: Sasakian–Einstein structures on \({9\#(S^{2}\times S^{3} )}\). Trans. Am. Math. Soc. 354(8), 2983–2996 (2002)CrossRefzbMATHGoogle Scholar
  14. 14.
    Boyer C.P., Galicki K.: New Einstein metrics on \({8\#(S^{2}\times S^{3} )}\). Differ. Geom. Appl. 19(2), 245–251 (2003)CrossRefzbMATHGoogle Scholar
  15. 15.
    Boyer C.P., Galicki K., Kollár J.: Einstein metrics on spheres. Ann. Math. (2) 162(1), 557–580 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Brylinski J.-L.: Loop Spaces, Characteristic Classes and Geometric Quantization. (Reprint of the 1993 edn.). Birkhäuser, Basel (2007)Google Scholar
  17. 17.
    Calabi E.: Metriques Kählériennes et fibrés holomorphes. Ann. Sci. École Norm. Sup. (4) 12, 269–294 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cap A., Slovák J.: Parabolic Geometries I: Background and General Theory, Mathematical Surveys and Monographs. American Mathematical Society, Providence (2009)CrossRefzbMATHGoogle Scholar
  19. 19.
    Conlon R.J., Hein H.-J.: Asymptotically conical Calabi–Yau manifolds, I. Duke Math. J. 162, 2855–2902 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Correa, E.M., Grama, L.: Calabi–Yau metrics on canonical bundles of complex flag manifolds. arXiv:1709.07956 (2017)
  21. 21.
    Correa, E.M.: Integrable systems in coadjoint orbits and applications. Ph.D. Thesis, Universidade Estadual de Campinas (2017)Google Scholar
  22. 22.
    Eguchi T., Hanson A.J.: Asymptotically flat solutions to Euclidean gravity. Phys. Lett. 74, 249–251 (1978)CrossRefGoogle Scholar
  23. 23.
    Falcitelli M., Pastore A.M., Ianus S.: Riemannian Submersions and Related Topics. World Scientific Pub Co Inc., Hackensack (2004)CrossRefzbMATHGoogle Scholar
  24. 24.
    Fritzsche K., Grauert H.: From Holomorphic Functions to Complex Manifolds, Graduate Texts in Mathematics. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  25. 25.
    Gauntlett J.P., Martelli D., Sparks J., Waldram D.: A new infinite class of Sasaki–Einstein manifolds. Adv. Theor. Math. Phys. 8, 987 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gauntlett J.P., Martelli D., Sparks J., Waldram D.: Sasaki–Einstein metrics on \({S^{2}\times S^{3}}\). Adv. Theor. Math. Phys. 8, 711–734 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gauntlett J.P., Martelli D., Sparks J., Yau S.-T.: Obstructions to the existence of Sasaki–Einstein metrics. Commun. Math. Phys. 273, 803–827 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Geiges H.: A brief history of contact geometry and topology. Expo. Math. 19, 25–53 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Geiges, H.: Contact manifold-definition. In: Bulletin of the Manifold Atlas-Definition (2013)Google Scholar
  30. 30.
    Goto R.: Calabi–Yau structures and Einstein–Sasakian structures on crepant resolutions of isolated singularities. J. Math. Soc. Jpn. 64, 1005–1052 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Grauert H.: Über Modifikationen und exzeptionelle analytische Mengen. Math. Ann. 146, 331–368 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hatakeyama Y.: Some notes on differentiable manifolds with almost contact structures. Osaka Math. J. (2) 15, 176–181 (1963)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Hijazi O., Montiel S., Urbano F.: Spin c geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds. Math. Z. 253(4), 821–853 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Hiller H.: Geometry of Coxeter Groups. Research Notes in Mathematics, vol. 54. Pitman Advanced Publishing Program, Baldwin City (1982)Google Scholar
  35. 35.
    Huybrechts D.: Complex Geometry: An Introduction. Universitext. Springer, Berlin (2005)zbMATHGoogle Scholar
  36. 36.
    Joyce D.D.: Compact Manifolds with Special Holonomy, Oxford Mathematical Mono Graphs. Oxford University Press, Oxford (2000)Google Scholar
  37. 37.
    Kobayashi S.: On compact Kähler manifolds with positive definite Ricci tensor. Ann. Math. 74, 381–385 (1961)CrossRefzbMATHGoogle Scholar
  38. 38.
    Kobayashi S.: Principal fiber bundles with the 1-dimensional toroidal group. Tôhoku Math. J. (2) 8, 29–45 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Kobayashi S.: Topology of positively pinched Kahler manifolds. Tohoku Math. J. 15, 121–139 (1963)CrossRefzbMATHGoogle Scholar
  40. 40.
    Lakshmibai V., Raghavan K.N.: Standard Monomial Theory, Encyclopaedia of Mathematical Sciences, vol. 137, p. 213. Springer, Berlin (2008)Google Scholar
  41. 41.
    Laufer H.B.: On rational singularities. Am. J. Math. 94, 597–608 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    LeBrun C.: Counterexamples to the generalized positive action conjecture. Commun. Math. Phys. 118, 591–596 (1988)ADSCrossRefzbMATHGoogle Scholar
  43. 43.
    Lie S.: Geometrie der Berhrungstransformationen (dargestellt von S. Lie und G. Scheffers). B. G. Teubner, Leipzig (1896)zbMATHGoogle Scholar
  44. 44.
    Lie, S.: Zur Theorie partieller Differentialgleichungen, p. 480 ff. Göttinger Nachrichten (1872)Google Scholar
  45. 45.
    Lutz, R.: Quelques remarques historiques et prospectives sur la gomtrie de contact. In: Conference on Differential Geometry and Topology (Sardinia, 1988) Rend. Sem. Fac. Sci. Univ. Cagliari, vol. 58(no.suppl.), pp. 361–393 (1988)Google Scholar
  46. 46.
    Maldacena J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Martelli D., Sparks J.: Symmetry-breaking vacua and baryon condensates in AdS/CFT correspondence. Phys. Rev. D 79(6), 065009 (2009)ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    Matsushima Y.: Remarks on Kähler–Einstein manifolds. Nagoya Math. J. 46, 161–173 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    McDuff D., Salamon D.: Introduction to Symplectic Topology, Oxford Graduate Texts in Mathematics, 3rd edn. Oxford University Press, Oxford (2017)CrossRefGoogle Scholar
  50. 50.
    Montgomery D.: Simply-connected homogeneous spaces. Proc. Am. Math. Soc. 1, 467–469 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Morimoto A.: On normal almost contact structures. J. Math. Soc. Jpn. 15, 420–436 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    O’Neill B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459–469 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Ornea L., Verbitsky M.: Embeddings of compact Sasakian manifolds. Math. Res. Lett. 14(4), 703–710 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Palais, R.S.: A global formulation of the Lie theory of transformation groups. Mem. Am. Math. Soc. 22 (1957)Google Scholar
  55. 55.
    Pedersen H., Poon Y.: Hamiltonian Constructions of Kähler–Einstein metrics of constant scalar curvature. Commun. Math. Phys. 136, 309–326 (1991)ADSCrossRefzbMATHGoogle Scholar
  56. 56.
    Reeb, G.: Sur certaines proprits topologiques des trajectoires des systmes dynamiques. Acad. R. Belgique. Cl. Sci. Mm. Coll. in 8 27(9) (1952)Google Scholar
  57. 57.
    Salamon S.: Riemannian Geometry and Holonomy Groups, Pitman Research Notes in Mathematics Series, vol. 201. Longman Scientific & Technical, Harlow (1989)Google Scholar
  58. 58.
    San Martin L.A.B.: Álgebras de Lie, 2a edição. Editora da Unicamp, Campinas (2010)Google Scholar
  59. 59.
    Sasaki S., Hatakeyama Y.: On differentiable manifolds with contact metric structures. J. Math. Soc. Jpn. 14, 249–271 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Sasaki S.: On differentiable manifolds with certain structures which are closely related to almost-contact structure. Tohoku Math. J. 2, 459–476 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Sparks J.: Sasaki–Einstein manifolds. Surv. Differ. Geom. 16, 265 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Takizawa S.: On contact structures of real and complex manifolds. Tôhoku Math. J. (2) 15, 227–252 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Taylor J.L.: Several Complex Variables with Connections to Algebraic Geometry and Lie Groups, Graduate Studies in Mathematics (Book 46). American Mathematical Society, Providence (2002)Google Scholar
  64. 64.
    Van Coevering C.: Examples of asymptotically conical Ricci-flat Kähler manifolds. Math. Z. 267(1-2), 465–496 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Van Coevering, C.: Ricci-flat Kähler metrics on crepant resolutions of Kähler cones. Math. Ann.  https://doi.org/10.1007/s00208-009-0446-1 (2009)
  66. 66.
    Voisin C., Schneps L.: Hodge Theory and Complex Algebraic Geometry I: Volume 1, Cambridge Studies in Advanced Mathematics, 1st edn. Cambridge University Press, Cambridge (2008)Google Scholar
  67. 67.
    Wang H.C.: Closed manifolds with homogeneous complex structure. Am. J. Math. 76, 1–32 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Wang M.Y., Ziller W.: Einstein metrics on principal torus bundles. J. Differ. Geom. 31, 215 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.IMPA - Instituto de Matemática Pura e AplicadaRio de JaneiroBrasil

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