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Annals of Global Analysis and Geometry

, Volume 35, Issue 1, pp 91–114 | Cite as

Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities: I. The gluing construction

  • Yat-Ming ChanEmail author
Original Paper

Abstract

In this paper we extend our previous results on resolving conically singular Calabi–Yau 3-folds (Chan, Quart. J. Math. 57:151–181, 2006; Quart. J. Math., to appear) to include the desingularizations of special Lagrangian (SL) 3-folds with conical singularities that occur at the same points of the ambient Calabi–Yau. The gluing construction of the SL 3-folds is achieved by applying Joyce’s analytic result (Joyce, Ann. Global. Anal. Geom. 26: 1–58, 2004, Thm. 5.3) on deforming Lagrangian submanifolds to nearby special Lagrangian submanifolds. Our result will in principle be able to construct more examples of compact SL submanifolds in compact Calabi–Yau manifolds. Various explicit examples and applications illustrating the result in this paper can be found in the sequel (Chan, Ann. Global. Anal. Geom., to appear).

Keywords

Calabi–Yau manifolds Special Lagrangian submanifolds Conical singularities Asymptotically conical 

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References

  1. 1.
    Chan Y.-M.: Desingularizations of Calabi–Yau 3-folds with a conical singularity. Quart. J. Math. 57, 151–181 (2006)zbMATHCrossRefGoogle Scholar
  2. 2.
    Chan, Y.-M.: Desingularizations of Calabi–Yau 3-folds with conical singularities. II. The obstructed case. Quart. J. Math. (to appear)Google Scholar
  3. 3.
    Chan, Y.-M.: Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities. II. Examples. Ann. Global Anal. Geom. (to appear)Google Scholar
  4. 4.
    Friedman R.: Simultaneous resolution of threefold double points. Math. Ann. 274, 671–689 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Harvey R., Lawson H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Joyce D.D.: Compact Manifolds with Special Holonomy. OUP, Oxford (2000)zbMATHGoogle Scholar
  7. 7.
    Joyce D.D.: Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications. J. Differential Geom. 63, 299–347 (2003)MathSciNetGoogle Scholar
  8. 8.
    Joyce D.D.: Special Lagrangian submanifolds with isolated conical singularities. I. Regularity. Ann. Global Anal. Geom. 25, 201–251 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Joyce D.D.: Special Lagrangian submanifolds with isolated conical singularities. II. Moduli Spaces. Ann. Global Anal. Geom. 25, 301–352 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Joyce D.D.: Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case. Ann. Global Anal. Geom. 26, 1–58 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Joyce D.D.: Special Lagrangian submanifolds with isolated conical singularities. IV. Desingularization, obstructions and families. Ann. Global Anal. Geom. 26, 117–174 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lockhart R.B., McOwen R.C.: Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Super. Pisa, Cl. Sci. 12, 409–447 (1985)zbMATHMathSciNetGoogle Scholar
  13. 13.
    McDuff D., Salamon D.: Introduction to Symplectic Topology, 2nd edn. OUP, Oxford (1998)zbMATHGoogle Scholar
  14. 14.
    Strominger A., Yau S.-T., Zaslow E.: Mirror symmetry is T-duality. Nucl. Phys. B 479, 243–259 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Tian, G.: Smoothing 3-folds with Trivial Canonical Bundle and Ordinary Double Points. Essay on Mirror Manifolds, pp. 458–479. Internat. Press, Hong Kong (1992)Google Scholar
  16. 16.
    Yau S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equations. I. Commun. Pure Appl. Math. 31, 339–411 (1978)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Hong KongHong KongChina

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