相关论文: Toric geometry and integral affine structures in n…

We consider a toric degeneration of Calabi--Yau complete intersections of Batyrev--Borisov in the Gross--Siebert program. The author showed in his previous work that there exists an integral affine contraction map called a tropical…

We construct non-archimedean SYZ fibrations for maximally degenerate Calabi-Yau varieties, and we show that they are affinoid torus fibrations away from a codimension two subset of the base. This confirms a prediction by Kontsevich and…

This is an outline of work in progress concerning an algebro-geometric form of the Strominger-Yau-Zaslow conjecture. We introduce a limited type of degeneration of Calabi-Yau manifolds, which we call toric degenerations. For these, the…

We describe in purely combinatorial terms dual pairs of integral affine structures on spheres which come from the conjectural metric collapse of mirror families of Calabi-Yau toric hypersurfaces. The same structures arise on the base of a…

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…

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…

We show a mathematically precise version of the SYZ conjecture, proposed in the family Floer context, for the conifold with a conjectural mirror relation between smoothing and crepant resolution. The singular T-duality fibers are explicitly…

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…

Let $U$ be an affine log Calabi-Yau variety containing an open algebraic torus. We show that the naive counts of rational curves in $U$ uniquely determine a commutative associative algebra equipped with a compatible multilinear form. This…

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…

This is an extended example of the study of mirror symmetry via log schemes and the discrete Legendre transform on affine manifolds, introduced by myself and Bernd Siebert in "Mirror Symmetry via Logarithmic Degeneration Data I"…

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…

We explicitly construct special Lagrangian fibrations on finite quotients of maximally degenerating abelian varieties, glue with Berkovich retraction in non-Archimedean geometry by using "hybrid" technique. We also study their symmetries…

The main theme of this paper is to use toric degeneration to produce distinct homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms. We focus on the (complex $n$-dimensional) quadric hypersurface and the del Pezzo surfaces,…

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…

Given a polarized tropical affine torus, we show that the fibered Lagrangian cobordism group of the corresponding symplectic manifold admits a natural geometric filtration of finite length. This contrasts with results of Sheridan-Smith in…

We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: a cluster $\mathcal{X}$-variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially…

Let $X$ be a closed symplectic manifold equipped a Lagrangian torus fibration over a base $Q$. A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space $Y$, which can be considered as a…

For K\"ahler K3 surfaces we consider Kulikov models of type III tamed by a symplectic form. Our main result shows that the generic smooth fiber admits an almost toric fibration over the intersection complex, which inherits a natural nodal…

We construct a Lagrangian torus fibration on a smooth hypertoric variety and a corresponding SYZ mirror variety using $T$-duality and generating functions of open Gromov-Witten invariants. The variety is singular in general. We construct a…