Degeneration of Kähler-Einstein Manifolds I: The Normal Crossing Case

解析K-半稳定性与局部壁交叉

Analytic K-semistability and local wall-crossing

本文研究了常标量曲率Kähler流形小极化形变下的K-多稳定性，证明其蕴含cscK度量的存在性，将其归结为Futaki不变量计算，并给出模空间的局部壁交叉现象。

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.

Global smoothings of degenerate K3 surfaces with triple points

Let $X$ be a normal crossing compact complex surface with triple points. We prove that there exists a family of smoothings of $X$ when $X$ satisfies suitable conditions. Since our differential geometric proof also includes the case where $X$ is neither Kählerian nor $H^1(X, \mathcal O_X)=0$, this generalizes Friedman's result on degenerations of $K3$ surfaces in algebraic geometry.