The HyperKähler Geometry Associated to Wolf Spaces

Holomorphic Lagrangian fibrations of toric hyperkahler manifolds

For the sake of hyperk{ä}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{ä}hler manifolds are real dimension $4n$ non-compact hyperk{ä}hler manifolds which are quaternion analog of toric varieties. The $n$ dimensional residue circle action on it admitting a hyperk{ä}hler moment map. We use the complex part of this moment map to construct a holomorphic Lagrangian fibration with generic fiber diffeomorphic to $(\mathbb{C}^*)^n$, and study the singular fibers.

Rozansky-Witten invariants of hyperkähler manifolds

We investigate invariants of compact hyperk{ä}hler manifolds introduced by Rozansky and Witten: they associate an invariant to each graph homology class. It is obtained by using the graph to perform contractions on a power of the curvature tensor and then integrating the resulting scalar-valued function over the manifold, arriving at a number. For certain graph homology classes, the invariants we get are Chern numbers, and in fact all characteristic numbers arise in this way. We use relations in graph homology to study and compare these hyperk{ä}hler manifold invariants. For example, we show t…

Cotangent bundles of toric varieties and coverings of toric hyperkähler manifolds

Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable points. Then based on these criteria, we show that the cotangent bundles of compact toric varieties in the core of toric hyperk{ä}hler manifold are sufficient to glue the desired toric hyperk{ä}hler manifold.

$\mathbb{P}^1$ 上 K3 曲族的模空间及由变形 twistor 圈诱导的复 hyperkähler 度量

Moduli of K3 families over $\mathbb{P}^1$ and complex-hyperkähler metrics induced by deformed twistor cycles

本文回答了关于 K3 周期域中 twistor 圈小变形几何意义的问题，通过 Penrose 非线性引力子构造证明其能诱导复 hyperkähler 度量，并构造了带标记 K3 曲面族的 Hausdorff 精细模空间，将结果推广至高维不可约全纯辛流形。