On Shokurov's Log Flips: The 3-dimensional Case

非klt情形下Shokurov指标猜想的一个反例

A non-klt counterexample to Shokurov's index conjecture

该研究针对非klt情形的Shokurov指标猜想，构造了反例，为相关代数几何问题提供了否定性依据。

On a conjecture of Shokurov: Characterization of toric varieties

We verify a special case of V. V. Shokurov's conjecture about characterization of toric varieties. More precisely, let $(X,D=\sum d_iD_i)$ be a three-dimensional log variety such that $K_X+D$ is numerically trivial and $(X,D)$ has only purely log terminal singularities. In this situation we prove the inequality \{center} $\sum d_i\le \rk\Weil(X)/(\operatorname{algebraic equivalence}) +\dim(X)$. \{center} We describe such pairs for which the equality holds and show that all of them are toric.

Boundedness of $ε$-log Canonical Complements on Surfaces

We prove a conjecture due to V.V. Shokurov on the boundedness of $ε$-log canonical complements on surfaces. As an application we give a new proof to the boundedness of weak log Fano surfaces.

Log canonical thresholds on smooth varieties: the Ascending Chain Condition

Building on results of Kollár, we prove Shokurov's ACC Conjecture for log canonical thresholds on smooth varieties, and more generally, on varieties with quotient singularities.

A geometric characterisation of toric varieties

We prove a conjecture of Shokurov which characterises toric varieties using log pairs.

Non-vanishing theorem for log canonical pairs

We obtain a correct generalization of Shokurov's non-vanishing theorem for log canonical pairs. It implies the base point free theorem for log canonical pairs. We also prove the rationality theorem for log canonical pairs. As a corollary, we obtain the cone theorem for log canonical pairs. We do not need Ambro's theory of quasi-log varieties.

The Ascending Chain Condition for log canonical thresholds on l.c.i. varieties

Shokurov's ACC Conjecture says that the set of all log canonical thresholds on varieties of bounded dimension satisfies the Ascending Chain Condition. This conjecture was proved for log canonical thresholds on smooth varieties in [EM1]. Here we use this result and inversion of adjunction to establish the conjecture for locally complete intersection varieties.

On the Log Discrepancies in Toric Mori Contractions

It was conjectured by McKernan and Shokurov that for all Mori contractions from X to Y of given dimensions, for any positive epsilon there is a positive delta, such that if X is epsilon-log terminal, then Y is delta-log terminal. We prove this conjecture in the toric case and discuss the dependence of delta on epsilon, which seems mysterious.

Toward an inductive description of singularities of pairs

Motivated by Shokurov's ACC Conjecture for log canonical thresholds, we propose an inductive point of view on singularities of pairs, in the case when the ambient variety is smooth. Our main result characterizes the log canonicity of a pair in dimension N+1 by the log canonicity of another pair (not effective, in general) in dimension N.

Toward the equivalence of the ACC for $a$-log canonical thresholds and the ACC for minimal log discrepancies

In this paper, we show that Shokurov's conjectures on the ACC for $a$-lc thresholds and the ACC for minimal log discrepancies are equivalent in the interval $[0,1)$. That is, the conjecture on ACC for $a$-lc thresholds holds for every $0\leq a<1$ if and only if the set of minimal log discrepancies for pairs with DCC coefficients do not have an accumulation point from below which belongs to $[0,1)$.

On existence of log minimal models

In this paper, we prove that the log minimal model program in dimension $d-1$ implies the existence of log minimal models for effective lc pairs (eg of nonnegative Kodaira dimension) in dimension $d$. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension $d-1$. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log minimal models for effective lc pairs in dimension four and the Shokurov reduction theorem. Other applications …

正特征中 Stein 次数的有界性

Boundedness of Stein degrees in positive characteristics

本文在足够大的特征下证明了相对维数1的Shokurov猜想，并由此得出对数Calabi-Yau纤维化中水平边界Stein次数的有界性。

On termination of log flips in dimension four

We prove the termination of 4-fold log flips for klt pairs of Kodaira dimension $κ\ge 2$.

On termination of 4-fold semi-stable log flips

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.

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.

Termination of log flips in dimension 4

This paper has been withdrawn by the author due to a crucial error in Lemma 3.5.

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.

关于对数典范广义配对的翻转终止性

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。

Letters of a Bi-Rationalist IV: Geometry of log flips

For a birational log Fano contraction, it is conjectured an inequality between the dimension of its exceptional locus and the minimal log discrepancy over the locus. The conjecture follows from the existence of the flip for the contraction and another conjecture about the maximum of minimal log discripancy in given dimension; in particular, it holds in small dimensions (up to 4), and some special cases. It is illustrated by some examples.

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…

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

We prove an existence theorem for good moduli spaces, and use it to construct the second flip in the log minimal model program for the moduli space of stable curves. In fact, our methods give a uniform, self-contained construction of the first three steps of the log minimal model program for the moduli spaces of stable pointed curves.

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…

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。

Existence of log canonical flips and a special LMMP

Let $(X/Z,B+A)$ be a $\Q$-factorial dlt pair where $B,A\ge 0$ are $\Q$-divisors and $K_X+B+A\sim_\Q 0/Z$. We prove that any LMMP$/Z$ on $K_X+B$ with scaling of an ample$/Z$ divisor terminates with a good log minimal model or a Mori fibre space. We show that a more general statement follows from the ACC for lc thresholds. An immediate corollary of these results is that log flips exist for log canonical pairs.

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。

VGIT presentation of the second flip of $\overline{M}_{2,1}$

We perform a variation of geometric invariant theory stability analysis for 2nd Hilbert points of bi-log-canonically embedded pointed curves of genus two. As a result, we give a GIT construction of the last three non-trivial log canonical models of the moduli space of pointed genus two curves, and obtain a VGIT presentation of the second flip in its Hassett-Keel program.

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

We give a geometric invariant theory (GIT) construction of the log canonical model $\bar M_g(α)$ of the pairs $(\bar M_g, αδ)$ for $α\in (7/10 - ε, 7/10]$ for small $ε\in \mathbb Q_+$. We show that $\bar M_g(7/10)$ is isomorphic to the GIT quotient of the Chow variety bicanonical curves; $\bar M_g(7/10-ε)$ is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic t…

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…

Stable pairs and log flips

This paper has two parts. In the first part, we review stable pairs and triples on curves, leading up to Thaddeus' diagram of flips and contractions starting from the blow-up of projective space along a curve embedded by a complete linear series of the form K + ample. In the second part, we identify log canonical divisors which exhibit Thaddeus' flips and contractions as "log" flips and contractions in the sense of the log-minimal-model program.

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.

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…

Existence of minimal models for varieties of log general type II

We prove the existence of pl-flips.

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.