Weichung Chen
We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from $0$ and the coefficients of the boundaries belonging to a fixed DCC set $S$ form a bounded family. Moreover, such pairs admit a strong $ε$-lc $\mathbb{R}$-complement for some fixed $ε>0$.
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In this short note, we show that K-semistable Fano manifolds with the smallest alpha invariant are projective spaces. Singular cases are also investigated.
We compute global log canonical thresholds, or equivalently alpha invariants, of birationally rigid orbifold Fano threefolds embedded in weighted projective spaces as codimension two or three. As an important application, we prove that most of them are weakly exceptional, $K$-stable and admit Kähler--Einstien metric.
We compute global log canonical thresholds of certain birationally bi-rigid Fano 3-folds embedded in weighted projective spaces as complete intersections of codimension 2 and prove that they admit an orbifold Kähler-Einstein metric and are K-stable. As an application, we give examples of super-rigid affine Fano 4-folds.
We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from $0$ and the coefficients of the boundaries belong to the set of hyperstandard multiplicities $Φ(\mathscr{R})$ associated to a fixed finite set $\mathscr{R}$ form a bounded family. We also show $α(X,B)^{d-1}\mathrm{vol}(-(K_X+B))$ are bounded from above for all klt weak Fano pairs $(X,B)$ of a fixed dimension $d$.
K-polystable toric Fano varieties with small alpha invariants
研究中展示了n维K-多稳定复曲面有理法诺簇\(X_n\)，由明确格多面体的面扇定义，α不变量恰为\(\frac{2}{2n + 1}\)，回答了相关问题。
We generalize the definition of alpha invariant to arbitrary codimension. We also give a lower bound of these alpha invariants for K-semistable Q-Fano varieties and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. Our results demonstrate the relation between alpha invariants of any codimension and volumes of Fano manifolds in the characterization of projective spaces.
The alpha spectrum of K-polystable toric $\mathbb{Q}$-Fano varieties
该研究明确确定n维K-多稳定环面Q-法诺簇的α不变量数值谱，建立完整环面实现定理，强化Liu等的问题并改进其构造结果。
Birational rigidity and alpha invariants of Fano varieties
本文构造双有理超刚性及带有限群作用的 Fano 簇，证明其 alpha 不变量可任意接近 $1/2$，群版本 alpha 不变量可任意小。
We show that any $n$-dimensional Fano manifold $X$ with $α(X)=n/(n+1)$ and $n\geq 2$ is K-stable, where $α(X)$ is the alpha invariant of $X$ introduced by Tian. In particular, any such $X$ admits Kähler-Einstein metrics and the holomorphic automorphism group of $X$ is finite.
Alpha invariant and K-stability of Q-Fano varieties
本文给出了一个纯代数几何证明：如果Q-Fano簇X的α-不变量大于dim X/（dim X+1），则（X，O（-K_X））是K-稳定的.我们证明的关键是Seshadri常数、α-不变量和K-稳定性之间的关系。并给出了自同构群的一些应用。
In this paper, we generalise the theory of complements to log canonical log fano varieties and prove boundedness of complements for them in dimension less than or equal to 3. We also prove some boundedness results for the canonical index of sdlt log Calabi-Yau varieties in dimension 2.
On existence of bounded relative-global complements for Fano fibrations
本文证明了固定维数且具 $ε$-lc 奇点的 Fano 纤维化存在有界相对全局补，并在基为一维时进一步证明了有界相对全局 klt 补的存在性。
We show that $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.
Boundedness of complements for fibered Fano threefolds in positive characteristic
本文在正特征大特征情形下证明了Fano型纤维化的典范丛公式，并解决了具有纤维化结构的Fano型三维配对的Shokurov补集有界性猜想，特别覆盖了$-(K_X+B)$为nef且非big的情形。
We show the existence of $(ε,n)$-complements for $(ε,\Rr)$-complementary projective generalized pairs of Fano type $(X,B+M)$ when either the coefficients of $B$ and $μ_j$ belong to a finite set or the coefficients of $B$ belong to a DCC set and $M'\equiv 0$, where $M'=\sumμ_jM_j'$ and $M_j'$ are b-Cartier nef divisors.
We provide a sufficient condition for polarisations of Fano varieties to be K-stable in terms of Tian's alpha invariant, which uses the log canonical threshold to measure singularities of divisors in the linear system associated to the polarisation. This generalises a result of Odaka-Sano in the anti-canonically polarised case, which is the algebraic counterpart of Tian's analytic criterion implying the existence of a Kähler-Einstein metric. As an application, we give new K-stable polarisations of a general degree one del Pezzo surface. We also prove a corresponding result for log K-stability.
In this note, we will show that delta invariant of a log Fano pair can be approximated by lc places of complements of plt type if it is no greater than one. Give a log Fano pair with delta invariant no greater than one, under the assumption that delta invariant of the log Fano pair can be approximated by lc places of bounded complements of plt type, we show the existence of the divisorial valuation computing delta invariant of this log Fano pair.
We prove that every birationally superrigid Fano variety whose alpha invariant is greater than (resp. no smaller than) $\frac{1}{2}$ is K-stable (resp. K-semistable). We also prove that the alpha invariant of a birationally superrigid Fano variety of dimension $n$ is at least $\frac{1}{n+1}$ (under mild assumptions) and that the moduli space (if exists) of birationally superrigid Fano varieties is separated.
Polarized endomorphisms of Fano varieties with complements
本文研究带补的Fano簇的极化自同态问题，证明了存在极化自同态的对数Calabi-Yau对是环面对的有限商，且带标准系数的klt Calabi-Yau对若有极化自同态则是阿贝尔簇的商。
Boundedness of klt complements on Fano fibrations over surfaces
本文证明 Birkar–Shokurov 关于 ε-lc 对局部有效 Cartier 除子补的猜想等价于 Shokurov 的 Fano 纤维化 klt 补有界性猜想，并在二维情形加以证实，从而推出曲面上 Fano 纤维化的 klt 补有界性。
Complements and coregularity of Fano varieties
本研究探讨法诺簇的上正则性、对数Calabi-Yau对指标与补的关系，扩展了相关结果，证明不同绝对上正则性的法诺簇和klt奇点的补的存在性，并得到新典范丛公式与假设下的类似结论。
Weakly special test configurations of log canonical Fano varieties
该研究针对严格对数典范Fano簇，证明补的所有lc位置均为dreamy型，并建立了(X,-K_X)的弱特殊测试构型与补的lc位置的对应关系，丰富了对数典范Fano簇的相关理论。
Tian's criterion for K-stability states that a Fano variety of dimension $n$ whose alpha invariant is greater than $\frac{n}{n+1}$ is K-stable. We show that this criterion is sharp by constructing singular Fano varieties with alpha invariants $\frac{n}{n+1}$ that are not K-polystable for sufficiently large $n$. We also construct K-unstable Fano varieties with alpha invariants $\frac{n-1}{n}$.
Existence of complements for foliations
本文证明了klt环境簇上代数可积log-Fano叶状结构存在$\mathbb{Q}$-补，并研究了其伴随反转与连通性原理。
The first main theorem on complements: from global to local
本文的目的是澄清和推广工作alg-geom/9711024（也见数学AG/9810097和数学AG/9901004）的技术。粗略地说，我们证明了局部Fano压缩的补的存在性可以减少到低维射影Fano簇的补的存在性。
We study three-dimensional Fano varieties with $\mathbb{C}^*$-action. Complementing recent results [13], we give classification results in the canonical case, where the maximal orbit quotient is $\mathbb{P}_2$ having a line arrangement of five lines in special position as its critical values.
On explicit bounds of Fano threefolds
本文研究Fano三维流形的显式几何，找到可计算的正整数N使得除有界族外所有Fano三维流形有N-补，并给出多个代数不变量的首批显式下界及反典范体积的显式上界。
An explicit bound for complements on toric varieties
该研究针对环面法诺簇上的klt补问题，推导得出其指标的有效上界，为代数几何中环面簇相关研究提供了显式的量化结果。
Boundedness of n-complements on surfaces
本文在无重数限制且不假设Fano型的广义情形下，证明了曲面对的n-补有界性。
Dynamic alpha-invariants of del Pezzo surfaces with boundary
全局对数典范阈值（或田氏α-不变量）在Fano簇的几何中起着重要的作用。Tian证明了具有大α-不变量的Fano流形可以配备Kahler-Einstein度量。近年来唐纳森起草了一个计划，以确定当一个顺利法诺品种X承认卡勒-爱因斯坦度量。证明了这样一个度量的存在等价于X是K-稳定的，这是一个代数几何性质。唐纳森计划中的一个关键步骤是找到一个卡勒-爱因斯坦度规，它具有沿光滑反正则边界的小角度边缘奇点。Jeffres，Mazzeo和Rubinstein表明，α不变的动态版本可以用来找到这样的度量。全局对数规范阈值测量反规范对如何不能成为对数规范。在这篇论文中，我们计算了各种设置下的del Pezzo曲面的全局对数正则阈值。本文将Cheltsov关于复del Pezzo曲面的全局对数正则阈值的计算推广到代数闭且具有任意特征的基域上的非奇异del Pezzo曲面。然后，我们研究了反正则对未能日志正则，给出了一个分类的非常奇异的反正则对的del Pezzo表面的小程度。我们猜想在何种情况下，这样的分类是合理的任意法诺品种和衍生的后果。作为应用，我们计算了以任意光滑椭圆曲线为边界的小次光滑del Pezzo曲面的动态α-不变量。本文的主要结果是计算了以任意光滑椭圆曲线C为边界的所有光滑del Pezzo曲面的动力学α-不变量。α不变式的值取决于C的选择。我们应用我们的计算找到Kahler-Ei…
Boundedness of weak Fano threefolds with fixed Gorenstein index in positive characteristic
该研究在特征p>5的不可数代数闭域上，通过引入Gorenstein指数、补与Kodaira型零化条件，部分证明了三维簇的BAB猜想，同时给出了射影三维簇上nef大可逆层Seshadri常数的一致下界。
Dual complexes of qdlt Fano type models and strong complete regularity
本文提出两种新的数值不变量，用于研究Fano类型对的性质，并揭示其与K-稳定性中qdlt Fano类型模型的关系。
Boundedness of complements for generalized pairs
本文研究广义配对补集的有界性问题，继Shokurov之后证明了边界系数集为[0,1]的法诺型广义配对的补集具有有界性，推进了代数几何领域相关理论研究。
A. Borisov classified into finitely many series the set of isomorphism classes of germs of toric $\Q$-factorial singularities, of fixed dimension and with minimal log discrepancy over the special point bounded from below by a fixed real number. We extend this classification to germs of toric Fano fibrations, possibly not $\Q$-factorial. As an application, we verify in the toric setting a conjecture proposed by V. V. Shokurov on the existence of bounded complements.
K-stability of adjoint foliated structures
本文引入了adjoint foliated structures的K稳定性概念，通过测试配置和叶状Donaldson-Futaki不变量进行定义，并证明了adjoint Fano foliated structures在特殊测试配置下的约简性。同时，作者引入了Ding稳定性概念，并证明其与K稳定性等价。此外，文章还引入了混合alpha、beta和delta不变量，并利用约简定理建立了adjoint Fano foliated structures K稳定性的估值标准。最后，作者证明了具有有界体积的K半稳定adjoint Fano foliated structures构成一个有界族。
We study Kawamata log terminal singularities of full rank, i.e., $n$-dimensional klt singularities containing a large finite abelian group of rank $n$ in its regional fundamental group. The main result of this article is that klt singularities of full rank degenerate to cones over log crepant equivalent toric quotient varieties. To establish the main theorem, we reduce the proof to the study of Fano type varieties with large finite automorphisms of full rank. We prove that such Fano type varieties are log crepant equivalent toric. Furthermore, any such Fano variety of dimension $n$ contains an…
We construct klt projective varieties with ample canonical class and the smallest known volume. We also find exceptional klt Fano varieties with the smallest known anticanonical volume. We conjecture that our examples have the smallest volume in every dimension, and we give low-dimensional evidence for that. In order to improve on earlier examples, we are forced to consider weighted hypersurfaces that are not quasi-smooth. We show that our Fano varieties are exceptional by computing their global log canonical threshold (or $α$-invariant) exactly; it is extremely large, roughly $2^{2^n}$ in dim…
We prove boundedness of global strong $(δ,n)$-complements for generalized $ε$-log canonical pairs of Fano type. We also prove some partial results towards boundedness of local strong $(δ,n)$-complements for semi-stable morphisms. As applications, we prove an effective generalized canonical bundle formula for generalized klt pairs and an effective generalized adjunction formula for exceptional generalized log canonical centers. Moreover, we prove that the existence of strong $(δ,n)$-complements implies a conjecture due to M$^{\rm c}$Kernan concerning the singularities of the base of a Mori fibe…
An equivariant fixed-level Demailly identity for Fano manifolds
本文完全解决了 Jin 和 Rubinstein 提出的关于等变 Tian alpha 不变量与等变全局对数规范阈值在固定水平上相等的问题。
A Frobenius Version of Tians Alpha-Invariant
本文在正特征射影簇上引入Tian α不变量的类比α_F，利用全局F-正则性刻画其性质，证明其正性、Fano情形上界1/2，并与复情形一致。
Remark on complements on surfaces
本文给出任意维数例外对奇点的显式刻画，证明例外 Fano 曲面是 1/42-lc，并由此首次给出曲面的 n-补集和 Tian alpha 不变量的显式上界。
Birational boundedness of stable families
本文证明了固定维数和有界余体积的稳定族的理查德森有界性，通过研究代数可积的叶结构，建立了叶状结构的理查德森几何框架，解决经典纤维化有界性问题。
On the uniform positivity of $F$-signature under reduction modulo $p$
研究复klt型奇点模\(p\)约化的\(F\)-特征标是否一致正，对正则局部环纯子环情形给出肯定回答，证明可归结为Gorenstein情形，并讨论与\(F\)-α不变量联系。
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