Termination of 4-fold canonical flips

Termination of (many) 4-dimensional log flips

We prove that any sequence of 4-dimensional log flips that begins with a klt pair (X,D) such that -(K+D) is numerically equivalent to an effective divisor, terminates. This implies termination of flips that begin with a log Fano pair and termination of flips in a relative birational setting. We also prove termination of directed flips with big K+D. As a consequence, we prove existence of minimal models of 4-dimensional dlt pairs of general type, existence of 5-dimensional log flips, and rationality of Kodaira energy in dimension 4.

ACC for log canonical thresholds and termination of log flips

We prove that the ascending chain condition (ACC) for log canonical (lc) thresholds in dimension $d$ and Special Termination in dimension $d$ imply the termination of any sequence of log flips starting with a $d$-dimensional lc pair of nonnegative Kodaira dimension. In particular, in characteristic zero, the latter termination in dimension 4 follows from Alexeev-Borisov's conjecture in dimension 3.

Nakayama-Zariski 分解与翻转的终止性

Nakayama-Zariski decomposition and the termination of flips

本文证明在伪有效射影对中，假设关于Nakayama-Zariski分解在极小模型程序操作下行为的自然猜想，一次翻转序列的终止性蕴含所有翻转的终止性。

伪有效四维翻转移除的终止性

Termination of pseudo-effective 4-fold flips

本文证明特征零代数闭域上对数典范四维簇的任意伪有效翻转移除序列必然终止。

On termination of flips and fundamental groups

In this article, we propose a boundedness conjecture for the regional fundamental group of klt singularities. We prove that this boundedness conjecture, the Zariski closedness of the diminished base locus of $K_X$, and an upper bound for minimal log discrepancies, imply the termination of flips with scaling. We prove the boundedness conjecture in the case of toric singularities, quotient singularites, isolated $3$-fold singularities, and exceptional singularities.

关于减弱乘子理想与翻转的终止

On diminished multiplier ideal and the termination of flips

本文发展了奇异簇上的减弱乘子理想理论，并证明了在Cartier指数有界等条件下带缩放的特定翻转的终止。

Letters of a Bi-Rationalist: VII. Ordered termination

To construct a resulting model in LMMP is sufficient to prove existence of log flips and their termination for certain sequences. We prove that LMMP in dimension $d-1$ and termination of terminal log flips in dimension $d$ imply, for any log pair of dimension $d$, the existence of a {\em resulting} log model: a strictly log minimal model or a strictly log terminal Mori log fibration, and imply existence of log flips in dimension $d+1$. As consequence, we prove existence of a resulting model of 4-fold log pairs, existence of log flips in dimension 5, and some fragments of Geography of log model…

关于flip的终止与例外非典范奇点

On termination of flips and exceptionally non-canonical singularities

本文引入并研究了例外非典范（enc）奇点，将flip的终止问题归结为终端flip的终止及enc对mlds的ACC猜想，并在维度3和4中证明了相关结论。

On the existence of flips

We prove the existence of flips in dimension n, contingent on the termination of real flips in dimension n-1.

Mld's vs thresholds and flips

Minimal log discrepancies (mld's) are related not only to termination of log flips, and thus to the existence of log flips but also to the ascending chain condition (acc) of some global invariants and invariants of singularities in the Log Minimal Model Program (LMMP). In this paper, we draw clear links between several central conjectures in the LMMP. More precisely, our main result states that the LMMP, the acc conjecture for mld's and the boundedness of canonical Mori-Fano varieties in dimension $\le d$ imply the following: the acc conjecture for $a$-lc thresholds, in particular, for canonic…

Factoring 3-fold flips and divisorial contractions to curves

We show that 3-fold terminal flips and divisorial contractions to a curve may be factored by a sequence of weighted blow-ups, flops, blow-downs to a locally complete intersection curve in a smooth 3-fold or divisorial contractions to a point.

Normality of general elephants on 3-fold terminal flips

We prove that the general elephant $E_{X^+}\in |-K_{X^+}|$ is normal where $X--> X^+$ is a 3-fold terminal flip. Hence $E_{X^+}$ has at worst Du Val singularities. As a corollary, there exists no non-Gorenstein singularity of type $cE/2$, $cD/3$, nor $cAx/4$ on the flipped curve $C^+$.

3-fold log flips according to V.V.Shokurov

In this paper, I review the section 8 of V.V.Shokurov's paper '3-fold log flips'.

Fano 4-folds, flips, and blow-ups of points

In this paper we study smooth, complex Fano 4-folds X with large Picard number rho(X), with techniques from birational geometry. Our main result is that if X is isomorphic in codimension one to the blow-up of a smooth projective 4-fold Y at a point, then rho(X) is at most 12. We give examples of such X with Picard number up to 9. The main theme (and tool) is the study of fixed prime divisors in Fano 4-folds, especially in the case rho(X)>6, in which we give some general results of independent interest.

Flips from semi-stable 4-folds whose degenerate fibers are unions of Cartier divisors which are terminal factorial 3-folds

We shall investigate flipping contractions from a semi-stable 4-fold $X$ whose degenerate fiber is a union of Cartier divisors which are terminal factorial 3-folds. Especially we shall prove that $X$ is smooth along the flipping locus, and that the flip exists for such contractions.

Semistable 3-fold flips

We piece together ingredients, which are well known and documented in the literature, into a new proof of the existence of semistable 3-fold flips

Existence of flips and minimal models for 3-folds in char p

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…

New outlook on the Minimal Model Program, II

We prove that the finite generation of adjoint rings proved in [Cascini and Lazić] implies all the foundational results of the Minimal Model Program: the Rationality, Cone and Contraction theorems, the existence of flips, and termination of flips with scaling in the presence of a big boundary.

Flips from 4-folds with isolated complete intersection singularities whose downstairs have rational bi-elephants

We shall investigate a flipping contraction g : X -> Y from a 4-fold X with at most isolated complete intersection singularities. If Y has an anti-bi-canonical divisor (=bi-elephant) with only rational singularities, then g carries an inductive structure chained up by blow-ups (La Torre Pendente), and in particular the flip exists. This naturally contains Miles Reid's `Pagoda' as an anti-canonical divisor (=elephant) and its proper transforms.

Existence of Mori fibre spaces for 3-folds in char $p$

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.

Birational maps of 3-folds

We show that 3-fold terminal flips and divisorial contractions may be factored into a sequence of flops, blow-downs to a smooth curve in a smooth 3-fold or divisorial contractions to points with minimal discrepancies.

MMP for algebraically integrable foliations

We show that termination of flips for $\mathbb Q$-factorial klt pairs in dimension $r$ implies existence of minimal models for algebraically integrable foliations of rank $r$ with log canonical singularities over a $\mathbb Q$-factorial klt projective variety.

The Existence of Gorenstein terminal fourfold flips

This is a revised version of my paper which appeared in J. Math. Sci. Univ. Tokyo. We prove that for a flipping contraction from a Gorenstein terminal 4-fold, the pull back of a general hyperplane section of the down-stair has only canonical singularities. Based on this fact and using Siu-Kawamata-Nakayama's extension theorem [Si], [Kaw4], [Kaw5] and [Nak2], we prove the existence of the flip of such a flipping contraction. Furthermore we classify such flipping contractions and the flips under some additional assumptions.

Semi-stable minimal model program for varieties with trivial canonical divisor

We give a sufficient condition for the termination of flips. Then we discuss a semi-stable minimal model program for varieties with (numerically) trivial canonical divisor as an application. We also treat a slight refinement of dlt blow-ups.

The LMMP for log canonical 3-folds in char $p$

We prove that one can run the log minimal model program for log canonical $3$-fold pairs in characteristic $p>5$. In particular we prove the Cone Theorem, Contraction Theorem, the existence of flips and the existence of log minimal models for pairs with log divisor numerically equivalent to an effective divisor. These follow from our main results, which are that certain log minimal models are good.

关于 Kähler 3-fold 的极小模型纲领

On the Minimal Model Program for Kähler 3-folds

本文证明了紧 Kähler 簇在维度 n 下的 pl-flipping、除子收缩及 pl flip 的存在性，并给出了维度 3 下锥定理、flip 及极小模型存在的自包含证明。

Positivity of the Moduli Part

We prove the Cone Theorem for algebraically integrable foliations. As a consequence, we show that termination of flips implies the b-nefness of the moduli part of a log canonical pair with respect to a contraction, generalising the case of lc trivial fibrations.

Two Two-dimensional Terminations

Varieties with log terminal and log canonical singularities are considered in the Minimal Model Program, see \cite{...} for introduction. In \cite{shokurov:hyp} it was conjectured that many of the interesting sets, associated with these varieties have something in common: they satisfy the ascending chain condition, which means that every increasing chain of elements terminates. Philosophically, this is the reason why two main hypotheses in the Minimal Model Program: existence and termination of flips should be true and are possible to prove. In this paper we prove that the following two sets s…

On the log minimal model program for $3$-folds over imperfect fields of characteristic $p>5$

We prove that many of the results of the LMMP hold for $3$-folds over fields of characteristic $p>5$ which are not necessarily perfect. In particular, the existence of flips, the cone theorem, the contraction theorem for birational extremal rays, and the existence of log minimal models. As well as pertaining to the geometry of fibrations of relative dimension $3$ over algebraically closed fields, they have applications to tight closure in dimension $4$.

Kähler 3折上广义对的MMP

MMP for Generalized Pairs on Kähler 3-folds

本文研究Kähler 3折的广义对MMP，证明了极值定理、翻转存在性、log终端模型等核心结果。

三维簇上余秩一叶状结构的极小模型计划（MMP）

MMP for co-rank one foliations on threefolds

该研究针对Q-阶乘射影三维簇上余秩一的F-dlt叶状配对，证明了翻转存在、特殊终止、基点自由定理及对数一般型下极小模型存在，推导了相关修正与奇点性质，验证了数值平凡情形的丰饶性，推进了叶状结构极小模型计划研究。