Kähler-Ricci solitons on toric Fano orbifolds
Yalong Shi, Xiaohua Zhu
本文证明了复曲面Fano轨道上Kähler-Ricci孤子的存在性，从而将Wang和Zhu [WZ]的定理推广到复曲面Fano轨道上.
We prove the existence of Kähler-Ricci solitons on toric Fano orbifolds, hence extend the the theorem of Wang and Zhu [WZ] to the orbifold case.
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We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension 2 provided the corresponding lattice is arithmetic and small enough. The method is to show that the Albanese mapping on an {é}tale covering space generates jets on the interior, if the lattice is small enough. We also explore some specific examples of Picard-Eisenstein type.
Positive Self Dual Einstein Orbifolds with One-Dimensional Isometry Group
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We show that the orbifold fundamental group of an effective compact K{ä}hler orbifold with nef anticanonical bundle has polynomial growth, which generalizes M.P ăun's results for manifolds [P ău97, Theorem 1,Theorem 2]
Ricci flow on Orbifold
本文研究了具有有限奇点的紧致轨道上的里奇流的行为。我们证明了Perelman伪局部性定理在Orbifold里奇流上也成立.利用这一性质，在某些自然的技术条件下，我们得到了轨道上里奇流的一个弱紧定理.这推广了流形上的相应定理。作为应用，我们可以利用Kähler里奇流在某些轨道Fano曲面上寻找新的Kähler Einstein度量.例如，如果$Y$是一个只有一个普通的二重点的三次曲面，或者$Y$是一个1次的Orbifold Fano曲面，其上的每个奇点都是$\A_k (1 \leq k \leq 6)$型的有理二重点，那么我们可以通过运行Kähler里奇流来找到$Y$的KE度量。
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本文研究五维紧对数法诺Sasaki流形上的锥形Sasaki-Ricci流，得到曲率一致界，证明流的解收敛到奇异轨形孤立子，并在横向对数K-多稳定条件下得到锥形Sasaki-Einstein度量的存在性。
We study the behavior of the Kähler-Ricci flow on some Fano bundle which is a trivial bundle on one Zariski open set. We show that if the fiber is $\mathbb{P}^{m}$ blown up at one point or some weighted projective space blown up at the orbifold point and the initial metric is in a suitable kähler class, then the fibers collapse in finite time and the metrics converge sub-sequentially in Gromov-Hausdorff sense to a metric on the base.
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本文确认了在紧致横贯Fano准正则萨克森5流形中Sasaki-里奇流的Hamilton-天猜想，并推导了横贯Fano准正则萨克森5流形上Sasaki-里奇溶洞的紧致性定理，进一步通过第二萨克森结构定理确认了紧致横贯Fano萨克森5流形中的Hamilton-天猜想，并展示了梯度Sasaki-里奇溶洞轨道度量在紧致萨克森5流形上为Sasaki-爱因斯坦的条件。
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 $K\times K$.
In this paper, we show that the uniform L^4-bound of the transverse Ricci curvature along the Sasaki-Ricci flow on a compact quasi-regular transverse Fano Sasakian (2n+1)-manifold M. When M is dimension up to seven and the space of leaves of the characteristic foliation is well-formed, we first show that any solution of the Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold Sasaki-Ricci soliton on the limit space which is a S^1-orbibundle over the unique singular Kaehler-Ricci soliton on a normal projective variety with codimension two orbifold singularitie…
Resolutions of non-regular Ricci-flat Kahler cones
我们提出了明确的建设完整的Ricci平坦的Kahler度量是渐近锥上的非正则Sasaki-爱因斯坦流形。度量是从一个完备的正里奇曲率的Kahler-Einstein流形（V，g_V）中构造的，并且允许一个二阶Hamilton二形式.我们得到了V上全纯C^2/Z_p orbifold纤维化，加权射影空间WCP ^[1]上全纯orbifold纤维化（一般纤维是V上的典型复锥）和Fano orbifold族上的典型orbifold线丛的全空间上的Ricci平坦Kahler度量。作为特例，我们还得到了V上秩为2的全纯向量丛和V上的正则线丛的全空间上的光滑完备Ricci平坦Kahler度量.当V= CP ^1时，我们的结果给出了Sasaki-Einstein流形Y^{p，q}上锥的各种环面部分分解上的Ricci平坦Kahler轨道重度量.
Singular Kähler--Ricci shrinkers and polarized Fano fibrations
本文证明任何作为 Kähler-Ricci 流非塌缩极限的奇异 Kähler-Ricci 收缩子具有局部代数复解析簇结构，并推导出单连通、唯一端、唯一切触锥等几何性质。
Let $S$ be a smooth projective variety and $Δ$ a simple normal crossing $\mathbb{Q}$-divisor with coefficients in $(0,1]$. For any ample $\mathbb{Q}$-line bundle $L$ over $S$, we denote by $\mathscr{E}(L)$ the extension sheaf of the orbifold tangent sheaf $T_S(-\log(Δ))$ by the structure sheaf $\mathcal{O}_S$ with the extension class $c_1(L)$. We show the following two results: (i) If $-(K_S+Δ)$ is ample and $(S, Δ)$ is K-semistable, then for any $λ\in \mathbb{Q}_{>0}$, the extension sheaf $\mathscr{E}({λc_1(-(K_S+Δ))})$ is slope semistable with respect to $-(K_S+Δ)$; (ii) If $K_S+Δ\equiv 0$, …
We prove that if a $\mathbb{Q}$-Fano variety $V$ specially degenerates to a Kähler-Einstein $\mathbb{Q}$-Fano variety $V$, then for any ample Cartier divisor $H=-r^{-1} K_V$ with $r\in \mathbb{Q}_{>0}$, the normalized volume $\widehat{\rm vol}(v)=A_{\mathcal{C}}^n(v)\cdot {\rm vol}(v)$ is globally minimized at the canonical valuation ${\rm ord}_V$ among all real valuations which are centered at the vertex of the affine cone $\mathcal{C}:=C(V,H)$. This is also generalized to the logarithmic and the orbifold setting. As a consequence, we complete the confirmation of a conjecture in [arXiv:1511.0…
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。
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