Strominger--Yau--Zaslow geometry, Affine Spheres and Painlevé III

Monge-Ampère equations on compact Hessian manifolds

We consider degenerate Monge-Ampère equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Ampère operators. We then use the Perron method to solve Monge-Ampère equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delanoë, Caffarelli-Viaclovsky and Hultgren-Önnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow…

通过Morse同伦的环面Fano曲面的同调镜像对称

Homological mirror symmetry of toric Fano surfaces via Morse homotopy

应用SYZ构造于环面Fano曲面，通过考虑矩多面体的Morse同伦而非Fukaya范畴来讨论同调镜像对称。

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 a smooth manifold with a natural $L^2$ metric. It is shown that the metric is induced from a local Lagrangian immersion into the product of cohomology groups $H^1(L)\times H^{n-1}(L)$. Using this approach, an interpretation of the mirror symmetry discussed by Strominger, Yau and Zaslow is given in terms of the classical Legendre transform.

Homological mirror symmetry for local Calabi-Yau manifolds via SYZ

This is a write-up of the author's talk in the conference "Algebraic Geometry in East Asia 2016" held at the University of Tokyo in January 2016. We give a survey on a series of papers of the author and his collaborators Daniel Pomerleano and Kazushi Ueda where we show how Strominger-Yau-Zaslow (SYZ) transforms can be applied to understand the geometry of Kontsevich's homological mirror symmetry (HMS) conjecture for certain local Calabi-Yau manifolds.

A Nakai-Moishezon type criterion for supercritical deformed Hermitian-Yang-Mills equation

The deformed Hermitian-Yang-Mills equation is a complex Hessian equation on compact Kähler manifolds that corresponds to the special Lagrangian equation in the context of the Strominger-Yau-Zaslow mirror symmetry. Recently, Chen proved that the existence of the solution is equivalent to a uniform stability condition in terms of holomorphic intersection numbers along test families. In this paper, we establish an analogous stability result not involving a uniform constant in accordance with a recent work on the $J$-equation by Song, which makes further progress toward Collins-Jacob-Yau's origina…

Collapsing of abelian fibred Calabi-Yau manifolds

We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of pola…

Mirror Symmetry via Logarithmic Degeneration Data I

This paper is the first arising from our project announced in math.AG/0211094, "Affine manifolds, log structures, and mirror symmetry." We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certain sorts of degenerations of Calabi-Yau manifolds a log Calabi-Yau space, which is a log structure on the degenerate fibre. Then many statements about mirror symmetry which one hopes to be true for the general fibre should first be proved for this log CY space. In this paper we be…

del Pezzo 曲面与有理椭圆曲面的 SYZ 镜像对称猜想

The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces

本文证明了 del Pezzo 曲面与有理椭圆曲面在非紧 Calabi-Yau 情形下的 SYZ 镜像对称猜想，通过建立度量存在性、唯一性及特殊拉格朗日纤维化定理，构造了镜像映射并验证了 $T$-对偶性。