Caucher Birkar
We prove a conjecture due to V.V. Shokurov on the boundedness of $ε$-log canonical complements on surfaces. As an application we give a new proof to the boundedness of weak log Fano surfaces.
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Failure of Boundedness for Generalised Log Canonical Surfaces
本文通过构造反例证明了具有固定适当不变量且Kodaira维数任意的曲面广义对数典范模型不满足有界性，从而回答了Birkar与第一作者提出的问题。
In this paper, we prove the boundedness of foliated surfaces $(X,\mathscr{F})$ which are minimal partial du Val resolutions of canonical models $(X_c,\mathscr{F}_c)$ of general type. For applications, we show the boundedness of non-cusp singularities on canonical models of foliated surfaces of general type and the effective generation on the complement of the cusp singularities.
We prove Weak Borisov--Alexeev--Borisov Conjecture in dimension three which states that the anti-canonical volume of an $ε$-klt log Fano pair of dimension three is bounded from above.
Boundedness results for families of non-canonically polarized projective varieties
本文证明在光滑拟射影曲线上，具有固定希尔伯特多项式且半丰富典范丛的非等平凡、光滑射影极化簇族是有界的，将Arakelov、Parshin和Kovács–Lieblich的有界性结果推广到非典范极化情形。
Theory of $n$-complements with applications is presented.
Addendum: On generalized canonical bundle formula and boundedness of complements in complex analytic setting
本文在复解析环境下，无需nef部分假设，建立了广义lc平凡纤维化的广义典范丛公式，并给出了相应的代数版本。
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。
On the boundedness of canonical models
本文证实了在固定 Iitaka 体积下非一般型簇典范模型有界性的猜想，条件是相应 Iitaka 纤维化的一般纤维属于极化 Calabi-Yau 对的固定有界族。
We show that the family of semi log canonical pairs with ample log canonical class and with fixed volume is bounded.
On generalized canonical bundle formula and boundedness of complements in complex analytic setting
本文研究了复分析设定中广义lc-平凡纤维化中的广义canonical包公式，探讨了判别b-除子和模b-除子的兼容性及补集的有界性。
We show the existence of $n$-complements for generalized pairs with additional Diophantine approximation properties when the coefficients of boundaries belong to a DCC set.
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。
We show the existence of $(ε,n)$-complements for $(ε,\mathbb{R})$-complementary surface pairs when the coefficients of boundaries belong to a DCC set.
We show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. As a corollary we show that a direct generalization of the geometric version of Shafarevich's original conjecture holds for infinitesimally rigid families of canonically polarized varieties.
Let $X$ be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by $F$ a smooth model of a generic irreducible component in fibers of the canonical map of $X$ and so $F$ is a smooth curve or a smooth surface. The main result of the paper is that there is a computable constant $K$ (independent of $X$) such that $g(F)\leq 647$ or $p_g(F)\leq 38$ whenever $p_g(X)\geq K$. The method heavily relies on both a Noether type of inequality and a Miyaoka-Yau inequality and that is the reason we only treat a Goren…
We answer an open question concerning the boundedness of canonical fiber spaces in high dimensions and prove the following: for any set of integers $n\geq 3$, $0<d<n$ and $N>0$, there exists a nonsingular projective $n$-fold $X$ of general type so that $X$ is canonically fibred by $d$-dimensional varieties $F$ with $p_g(F)\geq N$. This disproves the desired boundedness parallel to Beauville's boundedness theorem in the surface case.
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.
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.
In this paper, we study the behavior of the sets of volumes of the form $\mathrm{vol}(X,K_X+B+M)$, where $(X,B)$ is a log canonical pair, and $M$ is a nef $\mathbb{R}$-divisor. After a first analysis of some general properties, we focus on the case when $M$ is $\mathbb{Q}$-Cartier with given Cartier index, and $B$ has coefficients in a given DCC set. First, we show that such sets of volumes satisfy the DCC property in the case of surfaces. Once this is established, we show that surface pairs with given volume and for which $K_X+B+M$ is ample form a log bounded family. These generalize results …
We study log canonical thresholds (also called global log canonical threshold or $α$-invariant) of $\mathbb{R}$-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds, that is, given a natural number $d$ and a positive real number $ε$, the set of Fano varieties of dimension $d$ with $ε$-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which in particular answers a…
Boundedness of complements for log Calabi-Yau threefolds
本研究针对系数集为[0,1]的对数Calabi-Yau型簇，证明三维簇对若存在局部klt的ℝ-补集，则对应有限正整数集中的某个n有n-补集，同时证得ℝ-互补曲面对的补集有界性。
In this article we prove the following version of the Weak-BAB conjecture for $3$-folds in char $p>5$: Fix a DCC set $I\subset [0, 1)$ and an algebraically closed field $k$ of characteristic $p>5$. Let $\mathfrak{D}$ be a collection of klt pairs $(X, Δ)$ satisfying the following properties: (1) $X$ is a projective $3$-fold, (2) $Δ$ is an $\mathbb{R}$-divisor with coefficients in $I$, (3) $K_X+Δ\equiv 0$, and (4) $-K_X$ is ample. Then the set $\{\mbox{vol}_X(-K_X) \ | \ (X, Δ)\in\mathfrak{D}\mbox{ for some }Δ\}$ is bounded from above.
We show the birational boundedness of anti-canonical irreducible hypersurfaces which form 3-fold plt pairs. We also treat a collection of Du Val K3 surfaces which is birationally bounded but unbounded.
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。
Numerical conditions for the boundedness of foliated surfaces
本文证明固定三个数值不变量即可限定二维叶状典范模型的 Hilbert 函数，并推出其有效双有理性与有界性，还通过例子分析条件的可减弱性。
We show some inductive statements for the index conjecture for log canonical Calabi-Yau pairs. Using it, we show that boundedness of log canonical index for log canonical Calabi Yau pairs with rational DCC coefficients in dimension 3. We also show boundedness of log canonical index for klt log Calabi-Yau pairs in dimension 4 with $B\neq 0$.
Boundedness of polarized foliated surfaces
本文建立klt伴随叶状曲面的极小模型程序，证明满足条件的ε-lc伴随叶状曲面构成有界族，应用于一般类型伴随叶状曲面，还得到秩1叶状结构典范除子体积的离散性结果。
Let $X$ be a terminal weak $\bQ$-Fano threefold. We prove that $P_{-6}(X)>0$ and $P_{-8}(X)>1$. We also prove that the anti-canonical volume has a universal lower bound $-K_X^3 \geq 1/330$. This lower bound is optimal.
The main result of this note is an effective uniform bound for the number of deformation types of certain nonisotrivial families of canonically polarized manifolds. It extends the author's earlier such bound for the classical Shafarevich Conjecture over function fields to the case of higher dimensional fibers, but without the disadvantageous iterated use of Chow or Hilbert varieties that was the core of the proofs in the earlier approaches.
On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$
本文研究了正特征下 Fano 型簇的双有理几何与有界性问题，证明了有界性在正规化下不变，给出了更易用的典范丛公式，并证实了具有良好奇点的弱 Fano 簇的有效双有理性。
Boundedness of polarized log Calabi-Yau fibrations
该文研究带对数Calabi-Yau纤维化结构的对数对的有界性，证明在对数典范除子Iitaka体积有界、一般纤维属有界极化对数Calabi-Yau对族的条件下，纤维化全空间在crepant双有理等价下有界。
On birational boundedness of foliated surfaces
本文研究一般型叶状曲面的双有理有界性，在固定 Hilbert 多项式下证明多重典范线性系自某个有效界起给出双有理映射，并建立典范奇点叶状曲面的 Grauert–Riemannschneider 型消灭定理。
A cone singularity is a normal affine variety $X$ with an effective one-dimensional torus action with a unique fixed point $x\in X$ which lies in the closure of any orbit of the $k^*$-action. In this article, we prove a boundedness theorem for cone singularities in terms of their dimension, singularities, and isotropies. Given $d$ and $N$ two positive integers and $ε$ a positive real number, we prove that the class of $d$-dimensional $ε$-log canonical cone singularities with isotropies bounded by $N$ forms a bounded family.
The article has two parts. The first part is devoted to proving a singular version of the logarithmic Kodaira-Akizuki-Nakano vanishing theorem of Esnault and Viehweg. This is then used to prove other vanishing theorems. In the second part these vanishing theorems are used to prove an Arakelov-Parshin type boundedness result for families of canonically polarized varieties with rational Gorenstein singularities.
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…
Boundedness of elliptic Calabi-Yau varieties with a rational section
本文证明了在典范类挠指数有界且椭圆纤维化具有有理截面的条件下，固定维数的 klt 椭圆卡拉比-丘簇在余维一内有界，并将三维情形推广到有理连通的 ε-lc 对。
We prove birational boundedness results on complete intersections with trivial canonical class of base point free divisors in (some version of) Fano varieties. Our results imply in particular that Batyrev-Borisov toric construction produces only a bounded set of Hodge numbers in any given dimension, even as the codimension is allowed to grow.
On pluricanonical boundedness of varieties of general type
本文给出 Chen–Jiang 定理的新证明，修正原证明漏洞，并利用该方法改进 Lacini 的有界性定理，证明一般型 n 维簇的 r-典范映射具有双有理有界纤维，从而肯定回答 Chen 和 Jiang 2014 年提出的问题。
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.
In this article we prove the following boundedness result: Fix a DCC set $I\subset [0, 1]$. Let $\mathfrak{D}$ be the set of all log pairs $(X, Δ)$ satisfying the following properties: (i) $X$ is a projective surface defined over an algebraically closed field, (ii) $(X, Δ)$ is log canonical and the coefficients of $Δ$ are in $I$, and (iii) $K_X+Δ$ is big. Then there is a positive integer $N=N(I)$ depending only on the set $I$ such that the linear system $|\lceil m(K_X+Δ)\rceil|$ defines a birational map onto its image for all $m\geq N$ and $(X, Δ)\in\mathfrak{D}$
On finite generation and boundedness of adjoint foliated structures
本文证明了一般型klt代数可积伴随叶状结构存在良好极小模型、特定Fano代数可积伴随叶状结构构成有界族，并给出环境簇为Fano型的应用，分别类比典范环有限生成性与Fano簇有界性结果。
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。
We prove the boundedness of complements modulo two conjectures: Borisov-Alexeev conjecture and effective adjunction for fibre spaces. We discuss the last conjecture and prove it in two particular cases.
Bounding geometrically integral del Pezzo surfaces
本文证明任意域上几何整正规 del Pezzo 曲面的有界性，给出不规则度锐界，并证明野生典范情形仅存在于特征 2，推广了 Tanaka 的结果并验证了 BAB 猜想。
We expand the theory of log canonical $3$-fold complements. We prove that if $X\rightarrow T$ is a $3$-dimensional contraction of log Calabi-Yau type, then we can find $B\geq 0$ on $X$ for which $(X,B)$ is log canonical and $n(K_X+B)\sim_T 0$, where $n$ is an uniform natural number. This means that every $3$-fold of log Calabi-Yau type can be turned into a log Calabi-Yau pair in an effective way.
The canonical map of a foliated surface of general type
本文研究一般型叶状曲面的典范映射几何性质，证明其有界性结果，将Beauville关于代数曲面的工作推广至叶状曲面，并得出三个Noether型不等式。
Let $ε, C$ be two positive real numbers, and $\mathcal C \subset \mathbb R$ be a DCC (descending chain condition) set. Let $(X, B = \sum b_j B_j)$ denote a projective surface with an $\mathbb R$-divisor. Then (1) The class $\{X\}$ of surfaces for which there exists a divisor $B$ such that $(X,B)$ is $ε$-log terminal and $-(K_X + B)$ is nef (excluding only those for which at the same time $K_X\equiv 0$, $B=0$, and $X$ has at worst Du Val singularities), is bounded. (2) The set $\{(K_X + B)^2\}$ of squares for the semi log canonical pairs $(X, B)$ with ample $K_X + B$ and $b_j \in \mathcal C$, i…
We show that the number of marked minimal models of an n-dimensional smooth complex projective variety of general type can be bounded in terms of its volume, and, if n=3, also in terms of its Betti numbers. For an n-dimensional projective klt pair (X,D) with $K_X+D$ big, we show more generally that the number of its weak log canonical models can be bounded in terms of the coefficients of D and the volume of $K_X+D$. We further show that all n-dimensional projective klt pairs (X,D), such that $K_X+D$ is big and nef of fixed volume and such that the coefficients of D are contained in a given DCC…
On smooth threefolds, the ACC for minimal log discrepancies is equivalent to the boundedness of the log discrepancy of some divisor which computes the minimal log discrepancy. We reduce it to the case when the boundary is the product of a canonical part and the maximal ideal to some power. We prove the reduced assertion when the log canonical threshold of the maximal ideal is either at most one-half or at least one.
The $abcd$ conjecture, uniform boundedness, and dynamical systems
本文综述了Vojta对$abc$猜想和Szpiro猜想的高维推广及其在算术动力学中的应用，特别指出$abcd$猜想蕴含了挠点一致有界性及典范高度下界的动力学类比猜想。
ACC of plc thresholds
研究定义潜在对数典范阈值并证明其满足ACC，同时考察Fano型簇序列族的有界性等基本性质。
The main result of the paper is a boundedness for $n$-complements on algebraic surfaces. In addition, applications of this theorem to a classification of log Del Pezzo surfaces and of birational contractions for 3-folds are formulated.
Canonical Models of Adjoint Foliated Structures on Surfaces
本文研究代数曲面上形如K_F + D的伴随流形结构及其极小和典范模型，分析线性系|m(K_F + D)|的有效行为，并解决Hacon和Langer提出的流形曲面有界性问题。
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。
On boundedness of singularities and minimal log discrepancies of Kollár components
本文研究klt奇点的有界性问题，证明在三维或Kollár分量极小对数差异有上界时，局部体积远离零的klt奇点有界至多相差特殊退化，并回答了关于对数典范阈值与局部体积关系的一个问题。
Complements, index theorem, and minimal log discrepancies of foliated surface singularities
本文将配对与簇的若干结果推广到叶状曲面对，证明局部补集有界性、局部指标定理及mlds一致有界性，否定Cascini-Spice的两个问题，并恢复叶状曲面mlds升链条件猜想的证明。
We introduce orbifold Euler numbers for normal surfaces with Q-divisors. These numbers behave multiplicatively under finite maps and in the log canonical case we prove that they satisfy the Bogomolov-Miyaoka-Yau type inequality. As a corollary we prove effective versions of Bogomolov's result on boundedness of rational curves in some surfaces of general type. Finally, we give some applications to singularities of plane curves.
On the finiteness of log surfaces
该研究将川又-松木猜想推广到klt对数对并在二维证明，通过归结为极化有界性，证明对数曲面在对数同构下具有有限多个带klt奇点的弱对数典范射影模型。
We prove that the ACC conjecture for minimal log discrepancies holds for threefolds in $[1-δ,+\infty)$, where $δ>0$ only depends on the coefficient set. We also study Reid's general elephant for pairs, and show Shokurov's conjecture on the existence of $(ε,n)$-complements for threefolds for any $ε\geq 1$. As a key important step, we prove the uniform boundedness of divisors computing minimal log discrepancies for terminal threefolds. We show the ACC for threefold canonical thresholds, and that the set of accumulation points of threefold canonical thresholds is equal to $\{0\}\cup\{\frac{1}{n}\…
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