# Homological Mirror Symmetry for local SYZ singularities

@inproceedings{Abouzaid2021HomologicalMS, title={Homological Mirror Symmetry for local SYZ singularities}, author={Mohammed Abouzaid and Zachary Sylvan}, year={2021} }

Gross and Siebert identified a class of singular Lagrangian torus fibrations which arise when smoothing toroidal degenerations, and which come in pairs that are related by mirror symmetry. We identify an immersed Lagrangian in each of these local models which supports a moduli space of objects that is isomorphic to the mirror space, and prove a homological mirror statement along the way.

## 2 Citations

### Local mirror symmetry via SYZ

- Physics
- 2021

In this note, we explain how mirror symmetry for basic local models in the Gross-Siebert program can be understood through the non-toric blowup construction described by Gross-Hacking-Keel. This is…

### Family Floer mirror space for local SYZ singularities

- Mathematics
- 2022

A BSTRACT : We state a non-archimedean and Floer-theoretic version of the SYZ conjecture and verify it for Gross’s special Lagrangian ﬁbration in any toric Calabi-Yau manifold. By combining the ideas…

## References

SHOWING 1-10 OF 32 REFERENCES

### Homological mirror symmetry for punctured spheres

- Mathematics, Physics
- 2011

We prove that the wrapped Fukaya category of a punctured sphere (S with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg…

### Homological mirror symmetry for the genus two curve

- Mathematics
- 2008

Katzarkov has proposed a generalization of Kontsevich's mirror symmetry conjecture, covering some varieties of general type. We prove a version of this conjecture in the simplest example, relating…

### Homogeneous coordinate rings and mirror symmetry for toric varieties

- Mathematics
- 2006

Given a smooth toric variety X and an ample line bundle O(1), we construct a sequence of Lagrangian submanifolds of (C^*)^n with boundary on a level set of the Landau-Ginzburg mirror of X. The…

### Functoriality and homological mirror symmetry for toric varieties

- Mathematics
- 2020

We study homological mirror symmetry for toric varieties, exploring the relationship between various Fukaya-Seidel categories which have been employed for constructing the mirror to a toric variety,…

### Mirror symmetry for very affine hypersurfaces

- MathematicsActa Mathematica
- 2022

We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasiprojective DM toric stack is equivalent to the wrapped Fukaya category of a hypersurface in a complex…

### Lagrangian Torus Fibrations and Homological Mirror Symmetry for the Conifold

- Mathematics
- 2016

We discuss homological mirror symmetry for the conifold from the point of view of the Strominger–Yau–Zaslow conjecture.

### Mirror Symmetry for Truncated Cluster Varieties

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2022

In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on work of Shende-Treumann-Williams-Zaslow in…

### An open string analogue of Viterbo functoriality

- Mathematics
- 2007

In this preprint, we look at exact Lagrangian submanifolds with Legendrian boundary inside a Liouville domain. The analogue of symplectic cohomology for such submanifolds is called “wrapped Floer…

### Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces

- Mathematics
- 2012

We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface…