Geography of log models:theory and applications

Cone theorem and Mori hyperbolicity

We discuss the cone theorem for quasi-log schemes and the Mori hyperbolicity. In particular, we establish that the log canonical divisor of a Mori hyperbolic projective normal pair is nef if it is nef when restricted to the non-lc locus. This answers Svaldi's question completely. We also treat the uniruledness of the degenerate locus of an extremal contraction morphism for quasi-log schemes. Furthermore, we prove that every fiber of a relative quasi-log Fano scheme is rationally chain connected modulo the non-qlc locus.

关于有限域代数闭包上对数典范三维簇的基点自由定理与Mori dream空间

On base point free theorem and Mori dream spaces for log canonical threefolds over the algebraic closure of a finite field

本文在特征大于5时去除了大性条件，将有限域代数闭包上三维对数典范射影对的基点自由定理推广至一般线丛，并讨论了此类域上的Mori dream空间。

A characterization of projective spaces from the Mori theoretic viewpoint

We give a characterization of projective spaces for quasi-log canonical pairs from the Mori theoretic viewpoint.

Mori dream spaces of Calabi-Yau type and the log canonicity of the Cox rings

We prove that a Mori dream space over a field of characteristic zero is of Calabi-Yau type if and only if its Cox ring has at worst log canonical singularities. By slightly modifying the arguments we also reprove the characterization of the varieties of Fano type by the log terminality of the Cox rings.

根据Shokurov记录极小模型

Log minimal models according to Shokurov

根据Shokurov的想法，我们给出了一个简短的证明以下klt版本的他的结果：终止终端日志翻转在维度d意味着任何klt对维度d有一个日志极小模型或森纤维空间。因此，特别地，维度为4的任何klt对具有对数极小模型或Mori纤维空间。

关于对数典范体积的无限聚点

On infinite accumulation points of log canonical volumes

本文证明了对任意整数d≥2，d维射影对数典范簇的体积集存在无限聚点，揭示了体积地理学的新现象。

The dual complex of log Calabi-Yau pairs on Mori fibre spaces

In this paper we show that the dual complex of a dlt log Calabi-Yau pair $(Y, Δ)$ on a Mori fibre space $π: Y \to Z$ is a finite quotient of a sphere, provided that either the Picard number of $Y$ or the dimension of $Z$ is $\leq 2$. This is a partial answer to Question 4 in "The dual complex of Calabi-Yau pairs" by Kollár and Xu.

Log canonical pairs with boundaries containing ample divisors

Let $(X,Δ)$ be a projective log canonical pair such that $Δ\geq A$ where $A \geq 0$ is an ample $\mathbb{R}$-divisor. We prove that either $(X,Δ)$ has a good minimal model or a Mori fibre space. Moreover, if $X$ is $\mathbb{Q}$-factorial, then any Log Minimal Model Program on $K_X+Δ$ with scaling terminates. As an application we prove that a log Fano type variety $X$ with $\mathbb{Q}$-factorial log canonical singularities is a Mori dream space.

森梦空间与GIT

Mori Dream Spaces and GIT

本文的主要目标是研究具有最好的Mori理论性质（通过有效因子锥的某种分解的存在性来衡量）的品种。我们称之为森梦空间。有很多例子，包括准光滑投射环面（或更一般地说，球面）品种，许多GIT的平行，和log Fano 3-folds。将Cox的环面簇构造推广为仿射空间的GIT簇，并将Mori梦空间刻画为仿射簇的GIT簇.通过商描述，将Mori理论中因子锥的chamber分解与几何不变理论中G-样本锥的分解自然地统一起来。特别地，Mori梦空间的每一个理性收缩都来自GIT，并且理性收缩的所有可能的因式分解都可以从chamber分解中读出。

Toric Mori理论简介

Introduction to the toric Mori theory

本文的主要目的是给出复曲面Mori理论的一个简单的非组合证明。在这里，复曲面Mori理论是指复曲面品种的（log）极小模型程序（简称MMP）。我们尽量减少球迷和他们的分解的参数。我们向以下人员推荐本文：（A）那些对操作扇形及其分解感到不舒服的人，（B）那些熟悉复曲面几何但不熟悉MMP的人。类别（A）中的人将从几个问题的繁琐的组合论证中解脱出来。那些属于（B）类的人将发现复曲面森理论的潜力。作为应用，我们讨论了复曲面簇上的Zapriki分解和非退化超曲面奇点的部分分解。通过这些应用，读者将学会使用Toric Mori理论。

Mori's program for the moduli space of pointed stable rational curves

We prove that, assuming the F-conjecture, the log canonical model of the pair $(\bar{M}_{0,n}, \sum a_i ψ_i)$ is the Hassett's moduli space of weighted pointed stable rational curves without any modification of weight coefficients. For the boundary weight cases, we prove that the birational model is the GIT quotient of the product of the projective lines. This is a generalization of Simpson's theorem for symmetric weight cases.

On Mori cone of Bott towers

A Bott tower of height $r$ is a sequence of projective bundles $$X_r \overset{π_r}\longrightarrow X_{r-1} \overset{π_{r-1}}\longrightarrow \cdots \overset{π_2}\longrightarrow X_1=\mathbb P^1 \overset{π_1} \longrightarrow X_0=\{pt\}, $$ where $X_i=\mathbb P (\mathcal O_{X_{i-1}}\oplus \mathcal L_{i-1})$ for a line bundle $\mathcal L_{i-1}$ over $X_{i-1}$ for all $1\leq i\leq r$ and $\mathbb P(-)$ denotes the projectivization. These are smooth projective toric varieties and we refer to the top object $X_{r}$ also as a Bott tower. In this article, we study the Mori cone and numerically effective …

Birational geometry in the study of dynamics of automorphisms and Brody/Mori/Lang hyperbolicity

We survey our recent papers (some being joint ones) about the relation between the geometry of a compact Kähler manifold and the existence of automorphisms of positive entropy on it. We also use the language of log minimal model program (LMMP) in biraitonal geometry, but not its more sophisticated technical part. We give applications of LMMP to positivity of log canonical divisor of a Mori / Brody / Lang hyperbolic (quasi-) projective (singular) variety.

Adjoint rings are finitely generated

This paper proves finite generation of the log canonical ring without Mori theory.

On weak Fano varieties with log canonical singularities

We prove that the anti-canonical divisors of weak Fano 3-folds with log canonical singularities are semiample. Moreover, we consider semiampleness of the anti-log canonical divisor of any weak log Fano pair with log canonical singularities. We show semiampleness dose not hold in general by constructing several examples. Based on those examples, we propose sufficient conditions which seem to be the best possible and we prove semiampleness under such conditions. In particular we derive semiampleness of the anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers are at most 1-…

Special MMP for log canonical generalised pairs

We show that minimal models of $\mathbb{Q}$-factorial NQC log canonical generalised pairs exist, assuming the existence of minimal models of smooth varieties. More generally, we prove that on a $\mathbb{Q}$-factorial NQC log canonical generalised pair $ (X,B+M) $ we can run an MMP with scaling of an ample divisor which terminates, assuming that it admits an NQC weak Zariski decomposition or that $K_X+B+M$ is not pseudoeffective. As a consequence, we establish several existence results for minimal models and Mori fibre spaces.

稳定映射Kontsevich空间的对数极小模型程序$\bar{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$

Log Minimal Model Program for the Kontsevich Space of Stable Maps $\bar{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$

这部作品的灵感来自于与Izzet Coskun和Joe Harris的对话。我们对稳定映射的Kontsevich空间$\bar{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$运行了log极小模型程序，并对在此过程中出现的所有中间空间给出了模解释.特别是，我们表明，希尔伯特计划$\mathcal H_{3,0,3}$的一个组成部分是翻转的$\bar{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$在周品种。最后作为一个简单的推论，我们得到$\bar{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$是Mori梦空间.

Remarks on log Calabi--Yau structure of varieties admitting polarized endomorphism

We discuss the Calabi--Yau type structure of normal projective surfaces and Mori dream spaces admitting a non-trivial polarized endomorphism.

Minimal models, flips and finite generation : a tribute to V.V. SHOKUROV and Y.-T. SIU

In this paper, we discuss a proof of existence of log minimal models or Mori fibre spaces for klt pairs $(X/Z,B)$ with $B$ big$/Z$. This then implies existence of klt log flips, finite generation of klt log canonical rings, and most of the other results of Birkar-Cascini-Hacon-McKernan paper.

相对Nakayama-Zariski分解与广义对的最小模型

Relative Nakayama-Zariski decomposition and minimal models of generalized pairs

本文证明了相对Nakayama-Zariski分解的基本性质，并将其应用于研究lc广义对，证明了由丰富除子极化的lc广义对的log极小模型或Mori纤维空间的存在性。

关于三重形上秩为一的叶形的Log Canonical极小模型程序

Log Canonical Minimal Model Program for corank one foliations on Threefolds

本文研究了三重形上秩为一的叶形的Log Canonical极小模型程序，证明了通过运行$(K_\mcF+\Delta)$-MMP可以得到极小模型或Mori纤维空间，并探讨了极小模型之间的连接及有限数量性质。

Kähler 3维流形的对数极小模型纲领

The log minimal model program for Kähler $3$-folds

本文针对紧 Kähler 3维流形上的 Q-factorial dlt 配对，证明经有限次除子收缩和翻转可得到对数极小模型或 Mori 纤维空间，并建立无基点定理。

稳定曲线模空间的对数极小模型程序：第一次翻转

Log minimal model program for the moduli space of stable curves: The first flip

我们给出了一个几何不变理论（GIT）的对$(\bar M_g, αδ)$为$α\in (7/10 - ε, 7/10]$为小$ε\in \mathbb Q_+$的对数正则模型$\bar M_g(α)$的建设。证明了$\bar M_g(7/10)$同构于Chow型双正则曲线的GIT商，$\bar M_g(7/10-ε)$同构于双正则曲线的渐近线性化Hilbert格式的GIT商.在每种情况下，我们完全分类的（半）稳定曲线和他们的轨道封闭。Chow半稳定曲线有普通的尖点和三角点作为奇点，但不允许椭圆尾。希尔伯特半稳定曲线满足进一步的条件，例如，它们不包含椭圆桥。我们发现，有一个小的收缩$Ψ: \bar M_g(7/10+ε) \to \bar M_g(7/10)$，收缩椭圆桥的轨迹。此外，通过使用对数正则模型的GIT解释，我们构造了一个小的收缩$Ψ^+ : \bar M_g(7/10-ε) \to \bar M_g(7/10)$，这是$Ψ$的Mori翻转。

关于 log Calabi-Yau 曲面的锥猜想

A cone conjecture for log Calabi-Yau surfaces

本文研究了带奇异边界的 log Calabi-Yau 曲面，证明了其自同构群和单值群在 nef 有效锥上的作用容许有理多面体基本域，并给出了 Mori Dream Spaces 的新示例。

Deformations of canonical pairs and Fano varieties

This paper is devoted to the study of various aspects of deformations of log pairs, especially in connection to questions related to the invariance of singularities and log plurigenera. In particular, using recent results from the minimal model program, we obtain an extension theorem for adjoint divisors in the spirit of Siu and Kawamata and more recent works of Hacon and McKernan. Our main motivation however comes from the study of deformations of Fano varieties. Our first application regards the behavior of Mori chamber decompositions in families of Fano varieties: we prove that, in the case…

Birational geometry of varieties, fibred into complete intersections of codimension two

In this paper we prove the birational superrigidity of Fano-Mori fibre spaces $π\colon V\to S$, every fibre of which is a complete intersection of type $d_1\cdot d_2$ in the projective space ${\mathbb P}^{d_1+d_2}$, satisfying certain conditions of general position, under the assumption that the fibration $V/S$ is sufficiently twisted over the base (in particular, under the assumption that the $K$-condition holds). The condition of general position for every fibre guarantees that the global log canonical threshold is equal to one. This condition bounds the dimension of the base $S$ by a consta…

Note on the semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant

We give another alternative proof to the Kawamata semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant. After the first version of this article was posted to the e-print Arxiv, Prof. Fujino notified the author that the quick and essential proof ([Fujino. On Kawamata's theorem.(EMS 2011), Rem 2.7]) is already known. The author would like to thank him. More precisely, Prof. Fujino already gave the quick and essential proof ([Fujino. On Kawamata's thm.(EMS 2011), Rem 2.7], [Fujino. Finite generation of the lc ring in dim 4. (Kyoto J. Math. 50 (2010)), R…

Positivity criteria for log canonical divisors and hyperbolicity

Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. By adjunction and running the log minimal model program, natural to our setting, we obtain a geometric criterion for K_X+D to be numerically effective as well as a geometric version of the cone theorem, generalizing to the context of log pairs these results of Mori. A criterion f…