The Futaki invariant and the Mabuchi energy of a complete intersection

CM Stability And The Generalized Futaki Invariant II

The Mabuchi K-energy map is exhibited as a singular metric on the refined CM polarization of any equivariant family $\mathbf{X}\overset{p}{\to} S$. Consequently we show that the generalized Futaki invariant is the leading term in the asymptotics of the reduced K-energy of the generic fiber of the map $p$. Properness of the K-energy implies that the generalized Futaki invariant is strictly negative.

Generalized Futaki Invariant of Almost Fano Toric Varieties, Examples

The interpretation, due to T. Mabuchi, of the classical Futaki invariant of Fano toric manifolds is extended to the case of the Generalized Futaki invariant, introduced by W. Ding and G. Tian, of almost Fano toric varieties. As an application it is shown that the real part of the Generalized Futaki invariant is positive for all degenerations of the Fano manifold V_{38}, obtained by intersection of the Veronese embedding of ${\bf P}^3\times{\bf P}^2 \subset {\bf P}^{11}$ with codimension-two hyperplanes.

Higher K-energy functionals and higher Futaki invariants

This note discusses the higher K-energy functionals which were defined by Bando and Mabuchi, and integrate higher Futaki invariants. Two new formulas for the higher K-energy functionals are given, and the second K-energy is shown to be related to Donaldson's Lagrangian applied to metrics on the tangent bundle.

量子化Mabuchi能量与cscK度量的尖锐参数间隙

Sharp parameter gaps for quantized Mabuchi energies and cscK metrics

该研究针对极化流形，建立了量子化Mabuchi能量半稳定性与一致K-稳定性、cscK度量存在性的等价关系，得到了相关斜率的尖锐比较结果，并构造了具有无界参数的cscK极化，为Kähler几何中的相关问题提供了关键理论依据。

Mabuchi Solitons and Relative Ding Stability of Toric Fano Varieties

As a generalization of Kahler-Einstein metrics for Fano manifolds with nonvanishing Futaki invariant, Mabuchi solitons are critical points of a Calabi-type energy functional. We study their existence on toric Fano varieties and the underlying algebraic stability notion: relative Ding stability. As a toy model for a YTD type correspondence, a new feature is the emergence of a non-uniformly stable case. We show a partial coercivity for the modified Ding functionals in this case, and obtain singular Mabuchi solitons via a variational approach. In the unstable case, we determine the maximal destab…

加权K-稳定性与强制性及其在极值Kähler和Sasaki度量中的应用

Weighted K-stability and coercivity with applications to extremal Kahler and Sasaki metrics

本文证明紧加权极值Kähler流形具有强制的加权Mabuchi能量，统一了特殊曲率条件下的Kähler度量结果，并据此刻画了极值Kähler度量和Sasaki-Einstein度量的存在性。