On the boundedness of $n$-folds with $κ(X)=n-1$
Stefano Filipazzi
本文研究极小 klt 对在 $κ(X,Δ)=\dim(X)-1$ 条件下的有界性，给出了若干充分条件。
本文研究一组极小 klt 对 $(X,Δ)$ 在满足 $κ(X,Δ)=\dim(X)-1$ 时有界的若干充分条件。
In this note we study certain sufficient conditions for a set of minimal klt pairs $(X,Δ)$ with $κ(X,Δ)=\dim(X)-1$ to be bounded.
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Boundedness of stable minimal models with klt singularities
本文研究了由 Birkar 提出的稳定极小模型的奇点与有界性，证明了在固定维数、Iitaka 体积和 DCC 集等条件下，klt 稳定极小模型的奇点可被一致控制，且在特定有界数据下构成有界族。
Boundedness of klt good minimal models
本文证明在固定适当数值不变量后，由相对 nef 且大的 Weil 除子极化的 klt 良好极小模型构成有界族，并由此构造其分离粗模空间。
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。
We prove a boundedness result for klt pairs $(X,B)$ such that $K_X+B\equiv 0$ and $B$ is big. As a consequence we obtain a positive answer to the Effective Iitaka Fibration Conjecture for klt pairs with big boundary.
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。
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 $d$.
We prove that rationally connected Calabi--Yau 3-folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected $3$-folds of $ε$-CY type form a birationally bounded family for $ε>0$. Moreover, we show that the set of $ε$-lc log Calabi--Yau pairs $(X, B)$ with coefficients of $B$ bounded away from zero is log bounded modulo flops. As a consequence, we deduce that rationally connected klt Calabi--Yau $3$-folds with mld bounded away from $1$ are bounded modulo flops.
A boundedness conjecture for minimal log discrepancies on a fixed germ
本文提出klt芽上极小对数差异的有界性猜想，证明其蕴含Shokurov的ACC猜想，并给出部分结果。
We will prove the following results for $3$-fold pairs $(X,B)$ over an algebraically closed field $k$ of characteristic $p>5$: log flips exist for $\Q$-factorial dlt pairs $(X,B)$; log minimal models exist for projective klt pairs $(X,B)$ with pseudo-effective $K_X+B$; the log canonical ring $R(K_X+B)$ is finitely generated for projective klt pairs $(X,B)$ when $K_X+B$ is a big $\Q$-divisor; semi-ampleness holds for a nef and big $\Q$-divisor $D$ if $D-(K_X+B)$ is nef and big and $(X,B)$ is projective klt; $\Q$-factorial dlt models exist for lc pairs $(X,B)$; terminal models exist for klt pair…
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。
We show the boundedness of B-pluricanonical representations of lc log Calabi-Yau pairs in dimension $2$. As applications, we prove the boundedness of indices of slc log Calabi-Yau pairs up to dimension $3$ and that of non-klt lc log Calabi-Yau pairs in dimension $4$.
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…
In this article we prove two cases of the abundance conjecture for $3$-folds in characteristic $p>5$: $(i)$ $(X, Δ)$ is KLT and $κ(X, K_X+Δ)=1$, and $(ii)$ $(X, 0)$ is KLT, $K_X\equiv 0$ and $X$ is not uniruled.
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 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。
Birational boundedness of rationally connected log Calabi-Yau pairs with fixed index
本文研究固定维数的有理连通 klt 对数 Calabi–Yau 对，证明在反对数典范类的固定 Cartier 指数与 nef 条件下，相应射影簇的集合模 flops 有界。
We prove that the log canonical ring of a klt pair of dimension $3$ with $\mathbb{Q}$-boundary over an algebraically closed field of characteristic $p>5$ is finitely generated. In the process we prove log abundance for such pairs in the case $κ=2$.
We prove the following results for projective klt pairs of dimension $3$ over an algebraically closed field of char $p>5$: the cone theorem, the base point free theorem, the contraction theorem, finiteness of minimal models, termination with scaling, existence of Mori fibre spaces, etc.
The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $x\in (X,Δ)$ satisfies the ACC if the coefficients of $Δ$ belong to a DCC set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of $δ$-plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hol…
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.
On Log Canonical MMP for Kähler 3-Fold
该数学研究证明对数极小模型程序适用于仅含klt奇点的紧Kähler三维簇对应的$\u211a$-阶乘lc对，推进了复几何领域相关理论发展。
We show that there exists a positive real number $δ>0$ such that for any normal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, that is, the minimal log discrepancy of $X$ is less than $1$, then the minimal log discrepancy of $X$ is not greater than $1-δ$. As applications, we show that the set of all non-canonical klt Calabi-Yau $3$-folds are bounded modulo flops, and the global indices of all klt Calabi-Yau $3$-folds are bounded from above.
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。
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 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 $B\neq 0$.
On the boundedness of elliptic Calabi-Yau 4-folds
本研究解决了一大类椭圆Calabi-Yau 4-fold的有界性问题，证明非等平凡纤维化的此类4-fold属于有限代数族，还为Calabi-Yau 3-fold中间Betti数有界性提供部分证据并研究相关4-fold的指标。
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。
We prove the termination of 4-fold log flips for klt pairs of Kodaira dimension $κ\ge 2$.
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…
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。
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。
On boundedness of singularities and minimal log discrepancies of Kollár components, II
本文证明K-半稳定log Fano锥奇点的有界性由局部体积下界和Kollár分量极小对数差异上界决定，并据此确认了三维情形的有界性猜想及klt奇点局部体积的聚集性质。
Boundedness of log Fano cone singularities and discreteness of local volumes
本文证明固定维度下体积有下界的K-半稳定log Fano锥奇点有界，进而得出klt奇点局部体积集合仅以零为聚点。
We introduce an approach of Riemann--Roch theorem to the boundedness problem of minimal log discrepancies in fixed dimension. After reducing it to the case of a Gorenstein terminal singularity, firstly we prove that its minimal log discrepancy is bounded if either multiplicity or embedding dimension is bounded. Secondly we recover the characterisation of a Gorenstein terminal three-fold singularity by Reid, and the precise boundary of its minimal log discrepancy by Markushevich, without explicit classification. Finally we provide the precise boundary for a special four-fold singularity, whose …
Kawamata boundedness for Fano threefolds and the Graded Ring Database
本研究实现了法诺三维簇的川又有界性，给出39550个候选希尔伯特级数列表，为法诺三维簇分类提供初步近似，明确了相关数据的含义及与分类计划的关联。
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。
We show boundedness of $3$-folds of $ε$-Fano type with Mori fibration structures. The proof is based on the birational boundedness result in our previous work arXiv:1509.08722 combining with arguments in Kawamata \cite{K} and Kollár--Miyaoka--Mori--Takagi \cite{KMMT}.
We consider a class of stable smoothable n-dimensional varieties, the analogs of stable curves. Assuming the minimal model program in dimension n+1, we prove that this class is bounded. From Kollar's method of constructing projective moduli spaces we get as a corollary that minimal model program in dimension n+1 implies the existence of a projective coarse moduli space for stable smoothable n-folds.
Boundedness and volume of generalised pairs
本文研究广义配对的有界性与体积，证明 nef 除子下降结果，应用得广义 lc 配对体积满足 DCC、klt 情形配对有界族等结论，定义强稳定极小模型并证明其有界，为后续模空间构造提供关键基础。
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 prove the general diagram method theorem valid for the quite large class of 3-folds with Q-factorial singularities (see Basic Theorem 1.3.2 and also Theorem 2.2.6). This gives the generalization of our results about Fano 3-folds with Q-factorial terminal singularities (Preprint alg-geom/9311007). As an application, we get the following result about 3-dimensional Calabi-Yau manifolds X: Assume that the Picard number ρ(X) > 93. Then one of two cases (i) or (ii) holds: (i) there exists a small extremal ray on X; (ii) There exists a nef element h such that h^3=0 (thus, the nef cone NEF(X) and t…
An effective upper bound for anti-canonical volumes of singular Fano threefolds
本文研究奇异 Fano 三体簇的反典范体积上界，证明对 $0<ε<1/3$ 的 $ε$-klt Fano $3$-维簇，其反典范体积至多为 $3200/ε^4$，并证明该阶 $O(1/ε^4)$ 不可改进。
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…
On the $4$-dimensional minimal model program for Kähler varieties
本文证明 $\mathbb Q$-factorial Kähler 4-fold 上 dlt 对的对数极小模型存在性及半稳定 klt 情形的 MMP 终止结果，并建立任意维数解析簇 flip 与相对 MMP。
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