Large Complex Structure Limits of K3 Surfaces

@article{Gross2000LargeCS,
  title={Large Complex Structure Limits of K3 Surfaces},
  author={Mark Gross and Pelham M. H. Wilson},
  journal={Journal of Differential Geometry},
  year={2000},
  volume={55},
  pages={475-546}
}
Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat K\"ahler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsevich, Soibelman and Todorov. Roughly stated, the conjecture says that, if the metrics are normalized to have constant diameter, then this limit is the base of the conjectural… Expand
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References

SHOWING 1-10 OF 44 REFERENCES
Homological mirror symmetry and torus fibrations
In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the FukayaExpand
Special Lagrangian Fibrations II: Geometry
We continue the study of the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, this states that if two Calabi-Yau manifolds X and Y are mirror partners, then X and Y have specialExpand
Special Lagrangian Fibrations I: Topology
In 1996, Strominger, Yau and Zaslow made a conjecture about the geometric relationship between two mirror Calabi-Yau manifolds. Roughly put, if X and Y are a mirror pair of such manifolds, then XExpand
Topological mirror symmetry
This paper focuses on a topological version on the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that mirror pairs of Calabi-Yau manifolds are related byExpand
Kähler–Einstein metrics on Kummer threefold and special Lagrangian tori
The concept of special Lagrangian submanifolds is introduced by Harvey and Lawson in the seminal paper [HL]. In [SYZ], Strominger, Yau and Zaslow propose a construction of the mirror threefold Z of aExpand
Compactifications of moduli spaces inspired by mirror symmetry
We study moduli spaces of nonlinear sigma-models on Calabi-Yau manifolds, using the one-loop semiclassical approximation. The data being parameterized includes a choice of complex structure on theExpand
TheL2 structure of moduli spaces of Einstein metrics on 4-manifolds
In this note, we announce some results showing unexpected similarities between the moduli spaces of constant curvature metrics on 2-manifolds (the Riemann moduli space) and moduli spaces of EinsteinExpand
Examples of Special Lagrangian Fibrations
We explore a number of examples of special Lagrangian fibrations on non-compact Calabi-Yau manifolds invariant under torus actions. These include fibrations on crepant resolutions of canonical toricExpand
The moduli space of special Lagrangian submanifolds
This paper considers the natural geometric structure on the moduli space of deformations of a compact special Lagrangian submanifold $L^n$ of a Calabi-Yau manifold. From the work of McLean this is aExpand
Mirror symmetry via 3-tori for a class of Calabi-Yau threefolds
We give an example of the recent proposed mirror construction of Strominger, Yau and Zaslow in ``Mirror Symmetry is T-duality,'' hep-th/9606040. The paper first considers mirror symmetry for K3Expand
...
1
2
3
4
5
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