Boundedness of $(ε, n)$-Complements for Surfaces

Boundedness of ($ε, n$)-Complements for projective generalized pairs of Fano type

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.

关于广义canonical包公式和复分析设定中补集的有界性

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。

附录：关于广义典范丛公式和复解析环境下补集有界性

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。

对数 Calabi-Yau 三维簇的补集有界性

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。

曲面上 Fano 纤维化的 klt 补的有界性

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 on log canonical Fano varieties

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 the boundedness of SLC surfaces of general type

The purpose of this note is to give a new proof of Alexeev's boundedness result for stable surfaces which is independent of the base field and to highlight some important consequences of this result.

广义对数典范曲面的有界性失效

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。

Boundedness of minimal partial du Val resolutions of canonical surface foliations

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.

K3曲面的布劳尔群的p-准素挠子的有界性

Boundedness of the p-primary torsion of the Brauer groups of K3 surfaces

本研究针对有限生成域上K3曲面的超越布劳尔群展开，证明其通常有限，仅在域为正特征p且曲面超奇异时被p零化，完善了相关代数几何理论。

极化叶状曲面的有界性

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。

几何整的 $\varepsilon$-klt del Pezzo 曲面的有界性

Boundedness of geometrically integral $ε$-klt del Pezzo surfaces

本文证明了任意域上几何整的 $\varepsilon$-klt del Pezzo 曲面在正特征下的 Borisov-Alexeev-Borisov 猜想，通过证明不规则性的一致界，补全了所有剩余特征情形。

Boundedness for surfaces in weighted P^4

Ellingsrud and Peskine (1989) proved that there exists a bound on the degree of smooth non general type surfaces in P^4. The latest proven bound is 52 by Decker and Schreyer in 2000. In this paper we consider bounds on the degree of a quasismooth non-general type surface in weighted projective 4-space. We show that such a bound in terms of the weights exists, and compute an explicit bound in simple cases.

Strong $(δ,n)$-complements for semi-stable morphisms

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…

非完备域上正则 del Pezzo 曲面的有界性

Boundedness of regular del Pezzo surfaces over imperfect fields

本文证明了非完备域上正则 del Pezzo 曲面的 $|-12K_X|$ 非常丰富性，给出了体积 $K_X^2$ 仅依赖于基域的显式上界，从而得出几何整正则 del Pezzo 曲面的有界性。

Boundedness of log-pluricanonical maps for surfaces of log-general types in positive characteristic

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 the boundedness of the denominators in the Zariski decomposition on surfaces

Zariski decompositions play an important role in the theory of algebraic surfaces. For making geometric use of the decomposition of a given divisor, one needs to pass to a multiple of the divisor in order to clear denominators. It is therefore an intriguing question whether the surface has a 'universal denominator' that can be used to simultaneously clear denominators in all Zariski decompositions on the surface. We prove in this paper that, somewhat surprisingly, this condition of bounded Zariski denominators is equivalent to the bounded negativity of curves that is addressed in the Bounded N…

Boundedness of Log Canonical Surface Generalized Polarized Pairs

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 …

On boundedness of divisors computing minimal log discrepancies for surfaces

Let $Γ$ be a finite set, and $X\ni x$ a fixed klt germ. For any lc germ $(X\ni x,B:=\sum_{i} b_iB_i)$ such that $b_i\in Γ$, Nakamura's conjecture, which is equivalent to the ACC conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor $E$ over $X\ni x$, such that $a(E,X,B)={\rm{mld}}(X\ni x,B)$, and $a(E,X,0)$ is bounded from above. We extend Nakamura's conjecture to the setting that $X\ni x$ is not necessarily fixed and $Γ$ satisfies the DCC, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of $a(E,X…

最小对指标的界——曲面

On boundedness of indices of minimal pairs -- surfaces

本文研究射影 klt 对的指标有界性，证明在给定维数和 Cartier 指标时，无法仅由这些参数确定一个整数使得线性等价成立。

叶状曲面有界性的数值条件

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。

A boundedness result for toric log Del Pezzo surfaces

In this paper we give an upper bound for the Picard number of the rational surfaces which resolve minimally the singularities of toric log Del Pezzo surfaces of given index $\ell$. This upper bound turns out to be a quadratic polynomial in the variable $\ell$.

Toward the second main theorem on complements: from local to global

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.

曲面平坦丛的上同调有界性及其应用

Cohomological boundedness for flat bundles on surfaces and applications

本文证明曲面及任意维数下平坦丛的上同调维数、Lefschetz识别及转向轨迹度数的普遍有界性，并引入部分差异运算以给出特征循环的闭式公式。

叶状曲面的双有理有界性

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。

Birational boundedness for holomorphic symplectic varieties, Zarhin's trick for $K3$ surfaces, and the Tate conjecture

We investigate boundedness results for families of holomorphic symplectic varieties up to birational equivalence. We prove the analogue of Zarhin's trick by for $K3$ surfaces by constructing big line bundles of low degree on certain moduli spaces of stable sheaves, and proving birational versions of Matsusaka's big theorem for holomorphic symplectic varieties. As a consequence of these results, we give a new geometric proof of the Tate conjecture for $K3$ surfaces over finite fields of characteristic at least $5$, and a simple proof of the Tate conjecture for $K3$ surfaces with Picard number a…

Calabi–Yau 三维簇的有界性问题

Boundedness questions for Calabi-Yau threefolds

本文研究光滑 Calabi–Yau 三维簇由三次上积形式和第二陈类线性形式决定后的模空间有界性，借助刚性有理曲面分析，并在 Picard 数 2 时证明一般有界性。

Complements on surfaces

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.

Boundedness and $K^2$ for log surfaces

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…

界定几何整的 del Pezzo 曲面

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。

叶状曲面奇点的补集、指标定理与极小对数差异

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。

关于对数曲面的有限性

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。

Complements on log surfaces

More strong version of the main inductive theorem about the complements on surfaces is proved and the models of exceptional log del Pezzo surfaces with $δ=0$ are constructed

Q-complements on log surfaces

In this paper the log surfaces without $\QQ$-complement are classified. In particular, they are non-rational always. This result takes off the restriction in the theory of complements and allows one to apply it in the most wide class of log surfaces.

Lectures on complements on log surfaces

The aim of these notes is to explain main ideas of the theory of complements. Basically we will follow Shokurov's work alg-geom/9711024.

Effective log Iitaka fibrations for surfaces and threefolds

We prove an analogue of Fujino and Mori's ``bounding the denominators'' in the log canonical bundle formula (see also Prokhorov and Shokurov) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a klt pair $(X,Δ)$ of Kodaira codimension one and dimension at most three such that the coefficients of $Δ$ are in a DCC set $\mathcal{A}$, there is a natural number $N$ that depends only on $\mathcal{A}$ for which the round down of $\N(K_X+Δ)$ induces the Iitaka fibration. We also prove a birational boundedness result for klt surfaces of general type.

带有受限分量类的光滑曲面上可约曲线补集的基本群

Fundamental groups of the complements to reducible curves on smooth surfaces with restricted classes of components

本文分类光滑单连通射影曲面上满足对偶簇次数≤6等奇异曲线束族，进而描述不可约分量为超平面截影类且基本群可满射到秩>6自由群的曲面上曲线补集的基本群。

法诺簇的补与上正则性

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。

关于曲面上补集的注记

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。

格极化 K3 曲面的模空间与 Brauer 群的有界性

Moduli of lattice-polarized K3 surfaces and boundedness of Brauer groups

本文定义了任意基上格极化 K3 曲面的模栈并引入极小丰富锥，通过添加水平结构参数化 Brauer 类，证明了秩为 19 时模空间分支的亏格和度数随阶数增长，从而给出了 K3 曲面 Brauer 群挠部分一致有界的新证明。

On homological mirror symmetry for the complement of a smooth ample divisor in a K3 surface

We introduce a conjecture on homological mirror symmetry relating the symplectic topology of the complement of a smooth ample divisor in a K3 surface to algebraic geometry of type III degenerations, and prove it when the degree of the divisor is either 2 or 4.

Weak analytic hyperbolicity of complements of generic surfaces of high degree in projective 3-space

In this article we prove that every entire curve in the complement of a generic hypersurface of degree $d\geq 586$ in $\mathbb{P}_{\mathbb{C}}^{3}$ is algebraically degenerate i.e there exists a proper subvariety which contains the entire curve.

Braid Group, Algebraic Surfaces and Fundamental Groups of complements of Branch Curves

This paper will appear in the Santa Cruz proceedings. An overview of the braid group techniques in the theory of algebraic surfaces, from Zariski to the latest results, is presented. An outline of the Van Kampen algorithm for computing fundamental groups of complements of curves and the modification of Moishezon-Teicher regarding branch curves of generic projections are given. The paper also contains a description of a quotient of the braid group, namely $\tilde B_n$ which plays an important role in the description of fundamental groups of complements of branch curves.It turns out that all suc…

紧曲面上半正反正则除子补集的$\overline{\partial}$上同调

$\overline{\partial}$ cohomology of the complement of a semi-positive anticanonical divisor of a compact surface

本文研究具有半正曲率反正则线丛的紧复曲面上，其反正则除子补集的$\overline{\partial}$上同调群$H^1(M,\mathcal{O}_M)$的结构与性质。

On the Hyperbolicity of the Complements of Curves in Algebraic Surfaces: The Three Component Case

The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a line and three quadrics. The main results are Let C be the union of three curves in P_2 whose degrees are at least two, one of which is at least three. Then for generic such configurations the complement of C is hyperbolic and hyperbolically embedded. The same statement holds for complements of curves in generic hypersurfaces X of degree at least five and curv…

二维有限生成代数的互反补

The reciprocal complement of a surface

研究二维有限生成K代数D的互反补R(D)及其性质，给出R(D)维数为2的充分条件，并通过实例验证，特别确定当D为K[X,Y,Z]除以二次不可约多项式时R(D)的维数。