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).
Similar content being viewed by others
References
Chan Y.-M.: Desingularizations of Calabi–Yau 3-folds with a conical singularity. Quart. J. Math. 57, 151–181 (2006)
Chan, Y.-M.: Desingularizations of Calabi–Yau 3-folds with conical singularities. II. The obstructed case. Quart. J. Math. (to appear)
Chan, Y.-M.: Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities. II. Examples. Ann. Global Anal. Geom. (to appear)
Friedman R.: Simultaneous resolution of threefold double points. Math. Ann. 274, 671–689 (1986)
Harvey R., Lawson H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)
Joyce D.D.: Compact Manifolds with Special Holonomy. OUP, Oxford (2000)
Joyce D.D.: Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications. J. Differential Geom. 63, 299–347 (2003)
Joyce D.D.: Special Lagrangian submanifolds with isolated conical singularities. I. Regularity. Ann. Global Anal. Geom. 25, 201–251 (2004)
Joyce D.D.: Special Lagrangian submanifolds with isolated conical singularities. II. Moduli Spaces. Ann. Global Anal. Geom. 25, 301–352 (2004)
Joyce D.D.: Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case. Ann. Global Anal. Geom. 26, 1–58 (2004)
Joyce D.D.: Special Lagrangian submanifolds with isolated conical singularities. IV. Desingularization, obstructions and families. Ann. Global Anal. Geom. 26, 117–174 (2004)
Lockhart R.B., McOwen R.C.: Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Super. Pisa, Cl. Sci. 12, 409–447 (1985)
McDuff D., Salamon D.: Introduction to Symplectic Topology, 2nd edn. OUP, Oxford (1998)
Strominger A., Yau S.-T., Zaslow E.: Mirror symmetry is T-duality. Nucl. Phys. B 479, 243–259 (1996)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chan, YM. Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities: I. The gluing construction. Ann Glob Anal Geom 35, 91–114 (2009). https://doi.org/10.1007/s10455-008-9124-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-008-9124-x