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Annals of Global Analysis and Geometry

, Volume 26, Issue 2, pp 117–174 | Cite as

Special Lagrangian Submanifolds with Isolated Conical Singularities. IV. Desingularization, Obstructions and Families

  • Dominic Joyce
Article

Abstract

This is the fourth in a series of five papers studying special Lagrangian submanifolds(SLV m-folds) X in (almost) Calabi–Yau m-foldsM with singularities x1,..., x n locally modelled on special Lagrangian conesC1,..., C n in ℂm with isolated singularities at 0. Readers are advised to begin with Paper V.

Paper III and this one construct desingularizations of X, realizing X as a limitof a family of compact, nonsingular SL m-folds Ct in M for small t > 0. Suppose L1,..., L n are Asymptotically Conical SL m-folds in ℂm, withL i asymptotic to the cone C i at infinity. We shrink L i by a small t > 0, and gluetL i into X at x i for i= 1,..., n to get a 1-parameter family of compact, nonsingularLagrangianm-folds N t for small t> 0.

Then we show using analysis that when t is sufficiently small we can deform N t toa compact, nonsingular special Lagrangianm-fold Ct, via a small Hamiltonian deformation. This Ct depends smoothly on t, and as t→ 0 it converges to the singular SL m-fold X, in the sense of currents.

Paper III studied simpler cases, where by topological conditions on X and L i we avoid obstructions to the existence of Ct. This paper considers more complex cases when theseobstructions are nontrivial, and also desingularization in families of almost Calabi–Yaum-folds M s for sF, rather than in a single almost Calabi–Yau m-fold M.

Calabi–Yau manifold special Lagrangian submanifold singularity 

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References

  1. 1.
    Bredon, G. E.: Topology and Geometry, Graduate Texts in Math. 139, Springer-Verlag, Berlin, 1993.Google Scholar
  2. 2.
    Butscher, A.: Regularizing a singular special Lagrangian variety, math.DG/0110053, 2001.Google Scholar
  3. 3.
    Harvey, R. and Lawson, H. B.: Calibrated geometries, Acta Math. 148 (1982), 4–157.Google Scholar
  4. 4.
    Lee, Y.-I.: Embedded special Lagrangian submanifolds in Calabi-Yau manifolds, Comm. Anal. Geom., to appear.Google Scholar
  5. 5.
    Joyce, D. D.: On counting special Lagrangian homology 3-spheres, in: A. J. Berrick, M. C. Leung and X. W. Xu (eds), Topology and Geometry: Commemorating SISTAG, Contemp. Math. 314, Amer. Math. Soc., Providence, RI, 2002, pp. 12–151.Google Scholar
  6. 6.
    Joyce, D. D.: Lectures on Calabi-Yau and special Lagrangian geometry, math.DG/0108088, 2001. Published, with extra material, as Part I of M. Gross, D. Huybrechts and D. Joyce, Calabi-Yau Manifolds and Related Geometries, Universitext Series, Springer-Verlag, Berlin, 2003.Google Scholar
  7. 7.
    Joyce, D. D.: Special Lagrangian submanifolds with isolated conical singularities. I. Regularity, Ann. Global Anal. Geom. 25, 2004, 20–251.Google Scholar
  8. 8.
    Joyce, D. D.: Special Lagrangian submanifolds with isolated conical singularities. II. Moduli spaces, Ann. Global Anal. Geom. 25, 2004, 30–352.Google Scholar
  9. 9.
    Joyce, D. D.: Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case, Ann. Global Anal. Geom. 26, 2004, –58.Google Scholar
  10. 10.
    Joyce, D. D.: Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications, J. Differential Geom. 63 (2003), 29–347.Google Scholar
  11. 11.
    Marshall, S. P.: Deformations of special Lagrangian submanifolds, Oxford DPhil. Thesis, 2002.Google Scholar
  12. 12.
    McDuff, D. and Salamon, D.: Introduction to Symplectic Topology, 2nd edn, Oxford Univ. Press, Oxford, 1998.Google Scholar
  13. 13.
    McLean, R. C.: Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), 70–747.Google Scholar
  14. 14.
    Salur, S.: A gluing theorem for special Lagrangian submanifolds, math.DG/0108182, 2001.Google Scholar
  15. 15.
    Salur, S.: On constructing special Lagrangian submanifolds by gluing, math.DG/0201217, 2002.Google Scholar
  16. 16.
    Strominger, A., Yau, S.-T. and Zaslow, E.: Mirror symmetry is T-duality, Nuclear Phys. B 479 (1996), 24–259.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Dominic Joyce
    • 1
  1. 1.Lincoln CollegeOxfordU.K.

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