The minimal log discrepancy

Mather discrepancy and the arc spaces

The goal of this paper is a classification theorem of the singularities according to a new invariant, Mather discrepancy. On the other hand, we show some evidences convincing us that Mather discrepancy is a considerable invariant: By introducing new log-canonical threshold and minimal log-discrepancy by means of Mather discrepancy instead of usual discrepancy of canonical divisors, we obtain the formulas of the new log-canonical threshold in terms of arc spaces, inversion of adjunction for wider class of singularities than the known one, lower seimicontinuity of the new minimal log-discrepancy…

光滑三维簇上计算最小对数判别式的加权爆破次数的一个界

A bound of the number of weighted blow-ups to compute the minimal log discrepancy for smooth 3-folds

本文证明光滑三维簇与一般实理想配对的最小对数判别式可由至多两次加权爆破计算，并推广了 Mustata-Nakamura 猜想，同时得出 ACC 成立。

光滑三维簇上的极小对数差异

Minimal log discrepancies on smooth threefolds

本文证明了光滑三维簇上极小对数差异的ACC，并推导出a-lc阈值ACC、一致m-adic半连续性及计算除子有界性等推论。

Reeb orbits and the minimal discrepancy of an isolated singularity

Let A be an affine variety inside a complex N dimensional vector space which has an isolated singularity at the origin. The intersection of A with a very small sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is torsion then we can assign an invariant of our singularity called the minimal discrepancy, which is an important invariant in birational geometry. We define an invariant of the link up to contactomorphism using Conley-Zehnder indices of Reeb orbi…

On equivalent conjectures for minimal log discrepancies on smooth threefolds

On smooth threefolds, the ACC for minimal log discrepancies is equivalent to the boundedness of the log discrepancy of some divisor which computes the minimal log discrepancy. We reduce it to the case when the boundary is the product of a canonical part and the maximal ideal to some power. We prove the reduced assertion when the log canonical threshold of the maximal ideal is either at most one-half or at least one.

一般排列簇的最小对数歧变的上界

Bounding minimal log discrepancies of general arrangement varieties

本文证明了在一般排列簇上，最小对数歧变可由簇的维数上界限制，扩展了关于该猜想的先前结果。

Divisors computing minimal log discrepancies on lc surfaces

Let $(X\ni x,B)$ be an lc surface germ. If $X\ni x$ is klt, we show that there exists a divisor computing the minimal log discrepancy of $(X\ni x,B)$ that is a Kollár component of $X\ni x$. If $B\not=0$ or $X\ni x$ is not Du Val, we show that any divisor computing the minimal log discrepancy of $(X\ni x,B)$ is a potential lc place of $X\ni x$.

Computations of Mather Minimal Log Discrepancies

We compute the Mather minimal log discrepancy via jet schemes and arc spaces for toric varieties and very general hypersurfaces.

Towards boundedness of minimal log discrepancies by Riemann--Roch theorem

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 …

超曲面镜像的最小对数差异

Minimal log discrepancies of hypersurface mirrors

本文研究加权射影空间中拟光滑 Calabi-Yau 超曲面商的最小对数差异，证明其与 BHK 镜像的权和次数相关，并构造出已知最小对数差异的 klt Calabi-Yau 簇。

A gap theorem for minimal log discrepancies of non-canonical singularities in dimension three

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.

固定芽上极小对数差异的有界性猜想

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。

Finite determination conjecture for Mather-Jacobian minimal log discrepancies and its applications

In this paper we study singularities in arbitrary characteristic. We propose Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies in terms of jet schemes of a singularity. The conjecture is equivalent to the boundedness of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy. We also show that this conjecture yields some basic properties of singularities; e.g. openness of Mather-Jacobian (log) canonical singularities, stability of these singularities under small deformations and lower semi-continuity of Mather-…

An optimal gap of minimal log discrepancies of threefold non-canonical singularities

We show that the minimal log discrepancy of any $\mathbb Q$-Gorenstein non-canonical threefold is $\leq\frac{12}{13}$, which is an optimal bound.

Singularities with respect to Mather-Jacobian discrepancies

As is well known, the "usual discrepancy" is defined for a normal Q-Gorenstein variety. By using this discrepancy we can define a canonical singularity and a log canonical singularity. In the same way, by using a new notion, Mather-Jacobian discrepancy introduced in recent papers we can define a "canonical singularity" and a "log canonical singularity" for not necessarily normal or Q-Gorenstein varieties. In this paper, we show basic properties of these singularities, behavior of these singularities under deformations and determine all these singularities of dimension up to 2.

Minimal log discrepancies of regularity one

In this article, we use the cone of nef curves to study minimal log discrepancies. The first result is an improvement of the nef cone theorem in the case of log Calabi-Yau dlt pairs. Then, we prove that the ascending chain condition for $n$-dimensional minimal log discrepancies of regularity one holds around zero. Furthermore, we show that there exists an upper bound for the minimal log discrepancy of any $n$-dimensional klt singularity of regularity one.

Blow-ups of three-dimensional terminal singularities: cA case

Divisors with minimal discrepancy over cA points are classified

Relations between two log minimal models of log canonical pairs

We study relations between two log minimal models of a fixed lc pair. For any two log minimal models of an lc pair constructed with log MMP, we prove that there are small birational models of the log minimal models which can be connected by a sequence of flops, and the two log minimal models share some properties. We also give examples of two log minimal models of an lc pair which have different properties.

Minimal model theory for log surfaces

We discuss the log minimal model theory for log surfaces. We show that the log minimal model program, the finite generation of log canonical rings, and the log abundance theorem for log surfaces hold true under assumptions weaker than the usual framework of the log minimal model theory.

计算曲面上极小对数差异性的除子的差异上界

Upper bound of discrepancies of divisors computing minimal log discrepancies on surfaces

本文研究了曲面上计算极小对数差异性的除子差异上界问题，给出了一个显式上界$\ell(\gamma)\in O(1/\gamma^2)$，并通过实例证明该界是最优的。

Ideal-adic semi-continuity problem for minimal log discrepancies

We discuss the ideal-adic semi-continuity problem for minimal log discrepancies by Mustata. We study the purely log terminal case, and prove the semi-continuity of minimal log discrepancies when a Kawamata log terminal triple deforms in the ideal-adic topology.

Minimal models of semi-log-canonical pairs

We compare the minimal model of a log canonical pair with the minimal model of its reduced boundary. These results are then used to study the existence of the minimal model of a semi-log-canonical pair using its normalization.

环面曲面奇点的分类

Classification of toric surface singularities

本文以最小对数偏差为不变量，完成了二维环面对数芽的分类。

Minimal log discrepancies in positive characteristic

We show the existence of prime divisors computing minimal log discrepancies in positive characteristic except for a special case. Moreover we prove the lower semicontinuity of minimal log discrepancies for smooth varieties in positive characteristic.

Ideal-adic semi-continuity of minimal log discrepancies on surfaces

We prove the ideal-adic semi-continuity of minimal log discrepancies on surfaces.

On isolated log canonical centers

In this paper, we show that the depth of an isolated log canonical center is determined by the cohomology of the -1 discrepancy diviors over it. A similar result also holds for normal isolated Du Bois singularities.

The set of toric minimal log discrepancies

We describe the set of minimal log discrepancies of toric log varities, and study its accumulation points.

固定三维簇上的最小对数差异

Minimal log discrepancies on a fixed threefold

本文证明任意固定三维簇上最小对数差异满足ACC，为三维簇奇点不变量的有界性提供结论。

On minimal log discrepancies and Kollár components

In this article we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$-dimensional $a$-log canonical singularities, with standard coefficients, which admit an $ε$-plt blow-up have minimal log discrepancies belonging to a finite set which only depends on $d,a$ and $ε$. This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities relate…

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 …

On the minimal model theory for dlt pairs of numerical log Kodaira dimension zero

We prove the existence of good log minimal models for dlt pairs of numerical log Kodaira dimension 0.

Remarks on special kinds of the relative log minimal model program

We prove $\mathbb{R}$-boundary divisor versions of results proved by Birkar or Hacon-Xu on special kinds of the relative log minimal model program.

On existence of log minimal models II

We prove that the existence of log minimal models in dimension $d$ essentially implies the LMMP with scaling in dimension $d$. As a consequence we prove that a weak nonvanishing conjecture in dimension $d$ implies the minimal model conjecture in dimension $d$.

On the log minimal model program for irreducible symplectic varieties

We show the termination of any log-minimal model program for a pair $(X,Δ)$ of a symplectic manifold $X$ and an effective $\mathbb R$-divisor $Δ$.

Log minimal models according to Shokurov

Following Shokurov's ideas, we give a short proof of the following klt version of his result: termination of terminal log flips in dimension d implies that any klt pair of dimension d has a log minimal model or a Mori fibre space. Thus, in particular, any klt pair of dimension 4 has a log minimal model or a Mori fibre space.

On minimal log discrepancies

An explanation to the boundness of minimal log discrepancies conjectured by V.V. Shokurov would be that the minimal log discrepancies of a variety in its closed points define a lower semi-continuous function. We check this lower semi-continuity behaviour for varieties of dimension at most 3 and for toric varieties of arbitrary dimension.

关于解析环境下log曲面的最小模型理论的评论

Remarks on the minimal model theory for log surfaces in the analytic setting

本文研究了复曲面在解析环境下log最小模型理论，证明了极小模型程序、充足性定理及log canonical环的有限生成性。

On the existence of minimal models for log canonical pairs

We show that minimal models of log canonical pairs exist, assuming the existence of minimal models of smooth varieties.

Existence of minimal models for varieties of log general type II

We prove the existence of pl-flips.

On the number of minimal models of a log smooth threefold

We give a topological bound on the number of minimal models of a class of three dimensional log smooth pairs of general type.

Minimal model theory for log surfaces in Fujiki's class $\mathcal C$

We establish the minimal model theory for $\mathbb Q$-factorial log surfaces and log canonical surfaces in Fujiki's class $\mathcal C$.

Second largest accumulation point of minimal log discrepancies of threefolds

The second largest accumulation point of the set of minimal log discrepancies of threefolds is $\frac{5}{6}$. In particular, the minimal log discrepancies of $\frac{5}{6}$-lc threefolds satisfy the ACC.

Berkovich log discrepancies in positive characteristic

We introduce and study a log discrepancy function on the space of semivaluations centered on an integral noetherian scheme of positive characteristic. Our definition shares many properties with the analogue in characteristic zero; we prove that if log resolutions exist, then our definition agrees with previous approaches to log discrepancies of semivaluations that these resolutions. We then apply this log discrepancy to a variety of topics in singularity theory over fields of positive characteristic. Strong F-regularity and sharp $F$-purity of Cartier subalgebras are detected using positivity …