Hyperkahler SYZ conjecture and semipositive line bundles

超双曲SYZ猜想

Hyperkähler SYZ conjecture

本文证明了超双曲SYZ猜想，即非平凡nef迷向线丛的若干次幂的截面可定义拉格朗日纤维化。

超凯勒流形族中的丰度与SYZ猜想

The abundance and SYZ conjectures in families of hyperkahler manifolds

研究超凯勒流形上具有nef且非大第一陈类的全纯线丛的半丰性，通过引入Teichmuller空间并证明全局Torelli定理，建立半丰性的形变不变性。

Kobayashi pseudometric on hyperkahler manifolds

The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture for any hyperkähler manifold that admits a deformation with two Lagrangian fibrations and whose Picard rank is not maximal. The Strominger-Yau-Zaslow (SYZ) conjecture claims that parabolic nef line bundles on hyperkähler manifolds are semi-ample. We prove that the Kobayashi pseudom…

Algebraic non-hyperbolicity of hyperkahler manifolds with Picard rank greater than one

A projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. This is a property which follows from Kobayashi hyperbolicity. We prove that hyperkahler manifolds are non algebraically hyperbolic when the Picard rank is at least 3, or if the Picard rank is 2 and the SYZ conjecture on existence of Lagrangian fibrations is true. We also prove that if the automorphism group of a hyperkahler manifold is infinite then it is algebraically non-hyperbolic.

SYZ Mirror Symmetry of Hitchin's Moduli Spaces Near Singular Fibers I

We study hyperkahler metrics and hyperholomorphic connections of Hitchin's moduli spaces after Gaiotto, Moore and Neitzke. Their construction via the twistor technique produces intricate wall crossing behaviors. For certain four dimensional Hitchin's moduli spaces local models and degeneration to local models near singular fibers of the Hitchin's fibration are understood.

家族 Floer 纲领与非阿基米德 SYZ 镜像构造

Family Floer program and non-archimedean SYZ mirror construction

本文基于家族 Floer 理论，用全纯圆盘计数、非阿基米德分析和 $A_\infty$ 同调代数构造含刚性解析空间与超势的镜像 Landau-Ginzburg 模型，并将假设减弱到半正 Lagrangian 纤维化。