# Special Lagrangian Fibrations II: Geometry

@article{Gross1998SpecialLF, title={Special Lagrangian Fibrations II: Geometry}, author={Mark Gross}, journal={arXiv: Algebraic Geometry}, year={1998} }

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 special Lagrangian torus fibrations which are dual to each other. Much work on this conjecture is necessarily of a speculative nature, as in dimension 3 it is still a very difficult problem of how to construct such fibrations. Nevertheless, assuming the existence of such fibrations there are many things one can…

## 60 Citations

### Special Lagrangian Fibrations I: Topology

- Mathematics
- 1997

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 X…

### Topological mirror symmetry

- Mathematics
- 1999

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 by…

### Special Lagrangian fibrations, mirror symmetry and Calabi-Yau double covers

- Mathematics
- 2008

The first part of this paper is a review of the Strominger-Yau-Zaslow conjecture in various settings. In particular, we summarize how, given a pair (X,D) consisting of a Kahler manifold and an…

### Examples of Special Lagrangian Fibrations

- Mathematics
- 2000

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 toric…

### Singularities of special Lagrangian fibrations and the SYZ Conjecture

- Mathematics
- 2000

The SYZ Conjecture explains Mirror Symmetry between Calabi–Yau 3-folds M, ˆ M in terms of special Lagrangian fibrations f : M ! B and ˆ f : ˆ M ! B over the same base B, whose fibres are dual 3-tori,…

### Large Complex Structure Limits of K3 Surfaces

- Mathematics
- 2000

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…

### J an 2 00 1 Examples of Special Lagrangian Fibrations

- Mathematics
- 2000

Of late there has been a great deal of interest in special Lagrangian submanifolds and manifolds fibred in special Lagrangian submanifolds, motivated by the Strominger-Yau-Zaslow conjecture [33]. One…

### Lectures on special Lagrangian geometry

- Mathematics
- 2001

expected to play a role in the eventual explanation of Mirror Symmetry. This article is intended as an introduction to special Lagrangian geometry, and a survey of the author's research on the…

### Lectures on Calabi-Yau and special Lagrangian geometry

- Mathematics
- 2003

This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on…

### N ov 2 00 0 Some families of special Lagrangian tori

- Mathematics
- 2000

We give a simple proof of the local version of Bryant’s result [1], stating that any 3-dimensional Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a…

## 29 References

### Special Lagrangian Fibrations I: Topology

- Mathematics
- 1997

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 X…

### The Geometry Underlying Mirror Symmetry

- Mathematics
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. The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum ﬁeld theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which…

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The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which…

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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 a…

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We consider regular Calabi-Yau hypersurfaces in $N$-dimensional smooth toric varieties. On such a hypersurface in the neighborhood of the large complex structure limit point we construct a fibration…

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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 K3…

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We announce several results of an extensive study [A] of the size of singular sets in oriented m dimensional surfaces which are area minimizing in m + I dimensional Riemannian manifolds. Our…

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Assuming the ambient manifold is Kahler, the theory of complex submanifolds can be placed in the more general context of calibrated submanifolds, see [HL]. It is therefore natural to try to extend…