Osamu Fujino
The definition of the weighted version of difficulty in "Termination of 4-fold canonical flips" contains some errors. In this paper, we describe these errors and how to correct them. Anyway, any sequence of 4-fold canonical flips terminates.
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We prove the termination of 4-fold log flips for klt pairs of Kodaira dimension $κ\ge 2$.
We prove the termination of 4-fold canonical flips.
We prove that termination of lower dimensional flips for generalized klt pairs implies termination of flips for log canonical generalized pairs with a weak Zariski decomposition. Moreover, we prove that the existence of weak Zariski decompositions for pseudo-effective generalized klt pairs implies the existence of minimal models for such pairs.
In this paper, we prove the termination of 4-fold semi-stable log flips under the assumption that there always exist 4-fold (semi-stable) log flips.
On the termination of flips for log canonical generalized pairs
该研究在代数几何领域证明了4维伪有效NQC对数典范广义配对的翻转终止性，通过验证3维对应情形的结论，并建立低维翻转终止性到高维容许弱扎里斯基分解情形的推导关系。
This paper has been withdrawn by the author due to a crucial error in Lemma 3.5.
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.
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 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。
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。
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.
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。
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…
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。
We prove the existence of flips in dimension n, contingent on the termination of real flips in dimension n-1.
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…
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.
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.
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.
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…
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