Stability of the conical Kähler-Ricci flows on Fano manifolds

关于锥形卡拉比-丘势的正则性

On the regularity of conical Calabi-Yau potentials

本文利用退化Sasakian流形上的多重位势理论，证明了Fano锥上局部有界的锥形Calabi-Yau势在正则轨迹上光滑，且证明不依赖额外对称性，推广了Berman的环面情形结果。

The Continuity Method to Deform Cone Angle

The continuity method is used to deform the cone angle of a weak conical Kähler-Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldson's Openness Theorem on deforming cone angle \cite{Don} by combining it with the regularity result of Guenancia-P$\breve{\text{a}}$un \cite{GP} and Chen-Wang \cite{CW}. This continuity method uses relatively less regularity of the metric (only weak conical Kähler-Einstein) and bypasses the difficult Banach space set up; it is also generalized to deform the cone angle…

Kähler-Einstein metrics on group compactifications

We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a $G\times G$-equivariant Fano compactification of a complex connected reductive group $G$ in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the continuity method and its translation into a real Monge-Amp{è}re equation, using the invariance under the action of a maximal compact subgroup $K\times K$.

渐近锥形Kähler-里奇收缩者的K-极稳定性

K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers

本文证明了渐近锥形Kähler-里奇收缩者度量的存在性与K-极稳定性之间的等价关系。

秩一horosymmetric空间上锥Kähler-Einstein度量的极限

Limits of conical Kähler-Einstein metrics on rank one horosymmetric spaces

本文研究秩一horosymmetric Fano流形上锥Kähler-Einstein度量在锥角递减时的极限行为，证明其在补集上收敛到底空间的Kähler-Einstein度量，在对称空间纤维上重标定后收敛到Stenzel的Ricci平坦度量。

渐近锥形收缩梯度Kähler-Ricci孤立子的唯一性

Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons

本文证明了非紧复流形上在无穷远处Ricci曲率趋于零的收缩梯度Kähler-Ricci孤立子在双全纯同构下的唯一性，且该方法可广泛应用于锥形Calabi-Yau度量唯一性及极化Fano纤维化同构的证明。

Conical Kahler-Einstein metric revisited

In this paper we introduce the "interpolation-degneration" strategy to study Kahler-Einstein metrics on a smooth Fano manifold with cone singularities along a smooth divisor that is proportional to the anti-canonical divisor. By "interpolation" we show the angles in $(0, 2π]$ that admit a conical Kahler-Einstein metric form an interval; and by "degeneration" we figure out the boundary of the interval. As a first application, we show that there exists a Kahler-Einstein metric on $P^2$ with cone singularity along a smooth conic (degree 2) curve if and only if the angle is in $(π/2, 2π]$. When th…

Asymptotically conical Calabi-Yau metrics on quasi-projective varieties

Let X be a compact Kahler orbifold without \C-codimension-1 singularities. Let D be a suborbifold divisor in X such that D \supset Sing(X) and -pK_X = q[D] for some p, q \in \N with q > p. Assume that D is Fano. We prove the following two main results. (1) If D is Kahler-Einstein, then, applying results from our previous paper, we show that each Kahler class on X\D contains a unique asymptotically conical Ricci-flat Kahler metric, converging to its tangent cone at infinity at a rate of O(r^{-1-ε}) if X is smooth. This provides a definitive version of a theorem of Tian and Yau. (2) We introduce…

Calabi-Yau manifolds with isolated conical singularities

Let $X$ be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let $L$ be an ample line bundle on $X$. Assume that the pair $(X,L)$ is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point $x \in X$ there exist a Kahler-Einstein Fano manifold $Z$ and a positive integer $q$ dividing $K_Z$ such that $-\frac{1}{q}K_Z$ is very ample and such that the germ $(X,x)$ is locally analytically isomorphic to a neighborhood of the vertex of the blow-down of the zero section of $\frac{1}{q}K_{Z}$. We prove that …

Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor

We study the scalar curvature of Kähler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of Kähler--Einstein Fano manifolds with the second Betti number 1, momentum-constructed constant scalar curvature Kähler metrics with cone singularities along the $\infty$-section exist if and only if the log Futaki invariant vanishes on the fibrewise $\mathbb{C}^*$-action, giving a supporting evidence t…

$\u2102^n$的Kähler紧化与Reeb动力学

Kähler compactification of $\mathbb{C}^n$ and Reeb dynamics

该研究证明了以Kähler子流形为余集的$\mathbb{C}^n$光滑紧化必为标准射影空间对，还利用$S^1$-等变正辛同调刻画奇点偏差，证明$\mathbb{C}^3$上满足无穷远锥链光滑的渐近锥形Ricci平坦Kähler度量仅有平坦度量。

非塌缩凯勒 - 里奇流的格罗莫夫 - 豪斯多夫极限与里奇收缩子的几何

Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers

研究非塌缩凯勒 - 里奇流有限时间奇点及里奇收缩子无穷远处几何，证明凯勒 - 里奇流时间切片收敛等结论，还将其与法诺纤维化联系，应用于证明特定收缩子无穷远处切空间唯一性。

直和丛上的显式完备Ricci平坦度量与Kähler-Ricci孤立子

Explicit complete Ricci-flat metrics and Kähler-Ricci solitons on direct sum bundles

该研究在Kähler-Einstein Fano流形上由典范根丛幂次直和构成的向量丛全空间中，利用哈密顿2-形式理论构造出多种完备Kähler-Ricci孤立子与Calabi-Yau度量，得到了具有不同体积增长特性的新几何例子。