Existence and boundedness of $n$-complements

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.

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…

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。

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.

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 Fano 纤维化有界相对全局补的存在性

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。

叶状结构的补存在性

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。