Related papers: SYZ mirror symmetry for hypertoric varieties
In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic…
We construct pseudotoric structures (\`{a} la Tyurin) on the two-step flag variety $F\ell_{1, n-1; n}$, and explain a general relation between pseudotoric structures and special Lagrangian torus fibrations, the latter of which are important…
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 $H$ in a toric variety $V$ we construct…
In this article we construct Lagrangian torus fibrations for general quintic \cy hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for…
By the SYZ construction, a mirror pair $(X,\check{X})$ of a complex torus $X$ and a mirror partner $\check{X}$ of the complex torus $X$ is described as the special Lagrangian torus fibrations $X \rightarrow B$ and $\check{X} \rightarrow B$…
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…
For a toric Calabi-Yau (CY) orbifold $\mathcal{X}$ whose underlying toric variety is semi-projective, we construct and study a non-toric Lagrangian torus fibration on $\mathcal{X}$, which we call the Gross fibration. We apply the…
We give a mathematically precise statement of the SYZ conjecture between mirror space pairs and prove it for any toric Calabi-Yau manifold with the Gross Lagrangian fibration. To date, it is the first time we realize the SYZ proposal with…
The SYZ Conjecture explains Mirror Symmetry between mirror Calabi-Yau 3-folds M,M' in terms of special Lagrangian fibrations f : M --> B and f' : M' --> B over the same base B, whose fibres are dual 3-tori, except for singular fibres. One…
We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex manifold…
For a local Calabi-Yau manifold which is a smoothing of toric Gorenstein singularity, this paper computes the open Gromov-Witten invariants of a generic fiber of the special Lagrangian fibration constructed by Gross and thereby constructs…
We give a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. The main result of this article is the proof that the topological…
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 the existence of dual special Lagrangian…
We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the quintic threefold. We interpret our results as…
In this paper, we study the geometry of the SYZ transform on a semi-flat Lagrangian torus fibration. Our starting point is an investigation on the relation between Lagrangian surgery of a pair of straight lines in a symplectic 2-torus and…
The purpose of this paper is to exhibit a natural construction between complex geometry and symplectic geometry following the idea of mirror symmetry. Suppose we are given a family of pairs of 2-dimensional K\"ahler tori and stable…
Given a lattice polytope $Q\subset \mathbb{R}^n$, we can consider the cone $\sigma=C(Q)=\{\lambda(q,1)\in \mathbb{R}^{n+1}|\lambda \in \mathbb{R}_{\geq0}, q\in Q\} \subset \mathbb{R}^{n+1}$, and the affine toric variety $Y_{\sigma}$…
The Strominger-Yau-Zaslow (SYZ) approach to mirror symmetry constructs a mirror space and a superpotential from the data of a Lagrangian torus fibration on a K\"ahler manifold with effective first Chern class. For K\"ahler manifolds whose…
Let $(X,\check{X})$ be a mirror pair of a complex torus $X$ and its mirror partner $\check{X}$. This mirror pair is described as the trivial special Lagrangian torus fibrations $X\rightarrow B$ and $\check{X}\rightarrow B$ on the same base…
Using a construction by Yamamoto of tropical contractions, we construct a non-archimedean SYZ fibration on the Berkovich analytification of a class of maximally degenerate hypersurfaces in projective space. We furthermore prove that under a…