Kähler-Ricci 流在 $\mathbf G$-球面 Fano 流形上的收敛性

Horosymmetric limits of Kähler-Ricci flow on Fano $G$-manifolds

In this paper, we prove that on a Fano $\mathbf G$-manifold $(M,J)$, the Gromov-Hausdorff limit of Kähler-Ricci flow with initial metric in $2πc_1(M)$ must be a $\mathbb Q$-Fano horosymmetric variety $M_\infty$, which admits a singular Kähler-Ricci soliton. Moreover, $M_\infty$ is a limit of $\mathbb C^*$-degeneration of $M$ induced by an element in the Lie algebra of Cartan torus of $\mathbf G$. A similar result can be also proved for Kähler-Ricci flows on any Fano horosymmetric manifolds. As an application, we generalize our previous result about the type II singularity of Kähler-Ricci flows…

log Fano纤维芽的稳定退化

Stable Degenerations of log Fano Fibration Germs

本文证明了Sun-Zhang提出的log Fano纤维芽稳定退化猜想，引入了H-不变量，证明存在唯一准单项估值使H-不变量最小，并证明其关联环是有限生成的，诱导出特殊退化到K-semistable极化log Fano纤维芽。

Mabuchi solitons和Fano可容 manifold上的Mabuchi常数

Mabuchi solitons and Mabuchi constants on Fano admissible manifolds

本文研究了Fano可容manifold上的Mabuchi soliton存在性，证明其存在当且仅当Mabuchi常数小于1，并给出显式公式以确定复射影空间P^n上该soliton的存在与非存在性。

Skew-symmetric complex matrices, pure spinors, the twistor space of the conformal $2n$-sphere, and the Fano variety of linear $n$-folds of a non-singular complex quadric hypersurface in $\mathbb{P}^{2n+1}$

For $n \geq 1$, the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ of the conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the complex Grassmannian manifold $\mathbf{G}(n+1, 2n+2)$, of the set of graphs of skew-symmetric linear endomorphism of $\mathbb{C}^{n+1}$. We use this fact to describe a natural stratification of the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ with $n \geq 3$, in terms of what we have called {\it generalised complex orthogonal Stiefel manifolds} of $\mathbb{C}^{n+1}$. In particular, the twistor space $\mathfrak{Z}(\mathbb{S}^{6})$ is biholomorphic…

On sharp lower bounds for Calabi type functionals and destabilizing properties of gradient flows

Let $X$ be a compact Kähler manifold with a given ample line bundle $L$. In \cite{Don05}, Donaldson proved that the Calabi energy of a Kähler metric in $c_1(L)$ is bounded from below by the supremum of a normalized version of the minus Donaldson--Futaki invariants of test configurations of $(X,L)$. He also conjectured that the bound is sharp. In this paper, we prove a metric analogue of Donaldson's conjecture, we show that if we enlarge the space of test configurations to the space of geodesic rays in $\mathcal{E}^2$ and replace the Donaldson--Futaki invariant by the radial Mabuchi K-energy $\…