The set of toric minimal log discrepancies

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。

Classification of log smooth toric del Pezzo pairs

We give a description of all log-Fano pairs (X, D) where X is a smooth toric surface and D a reduced simple normal crossing divisor such that the pair (X, D) is equivariant.

Discreteness of log discrepancies over log canonical triples on a fixed pair

For a fixed pair and fixed exponents, we prove the discreteness of log discrepancies over all log canonical triples formed by attaching a product of ideals with given exponents.

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 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.

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。

Cascades of toric log del Pezzo surfaces of Picard number one

We classify toric log del Pezzo surfaces of Picard number one by introducing the notion, cascades. As an application, we show that if such a surface is Kähler-Einstein, then it should admit a special cascade, and it satisfies the equality of the orbifold Bogomolov-Miyaoka-Yau inequality, i.e., $K^2 = 3e_{orb}.$

Chow Quotients of Toric Varieties as Moduli of Stable Log Maps

Let $X$ be a projective normal toric variety and $T_0$ a rank one subtorus of the defining torus of $X$. We show that the normalization of the Chow quotient $X//T_0$, in the sense of Kapranov-Sturmfels-Zelevinsky, coarsely represents the moduli space of stable log maps to $X$ with discrete data given by $T_0\subset X$.

Real loci in (log-) Calabi-Yau manifolds via Kato-Nakayama spaces of toric degenerations

We study the real loci of toric degenerations of complex varieties with reducible central fibre. We show that the topology of such degenerations can be explicitly described via the Kato-Nakayama space of the central fibre as a log space. We furthermore provide generalities of real structures in log geometry and their lift to Kato-Nakayama spaces. A key point of this paper is a description of the Kato-Nakayama space of a toric degeneration and its real locus, both as bundles determined by combinatorial data. We provide several examples including real toric degenerations of K3-surfaces and a tor…

Nash problem for a toric pair and the minimal log-discrepancy

This paper formulates the Nash problem for a pair consisting of a toric variety and an invariant ideal and gives an affirmative answer to the problem. We also prove that the minimal log-discrepacy is computed by a divisor corresponding to a Nash component, if the minimal log-discrepancy is finite. On the other hand there exists a Nash component such that the corresponding divisor has negative log-discrepancy, if the minimal log-discrepancy is $-/infty$.

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。

Descendant log Gromov-Witten invariants for toric varieties and tropical curves

Using degeneration techniques, we prove the correspondence of tropical curve counts and log Gromov-Witten invariants with general incidence and psi-class conditions in toric varieties for genus zero curves and all non-superabundant higher-genus situations. We also relate the log invariants to the ordinary ones, in particular explaining the appearance of negative multiplicities in the descendant correspondence result of Mark Gross.

Logarithmic geometry, minimal free resolutions and toric algebraic stacks

In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie, and Kato) and investigate their moduli. Then by applying this we define a notion of toric algebraic stacks over arbitrary schemes, which may be regarded as torus embeddings within the framework of algebraic stacks, and study some basic properties. Furthermore, we study the stack-theoretic analogue of toroidal embeddings.

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。

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.

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。

关于有界广义子对的最小对数差异的ACC猜想

On the ACC for Minimal Log Discrepancies for Bounded Generalized Sub-Pairs

本文证明了最小对数差异的ACC猜想在任意维有界广义子对及有界三维簇上成立。

关于固定 Gorenstein 指数簇上的极小对数差异

On minimal log discrepancies on varieties with fixed Gorenstein index

本文推广了对数典范阈值聚点的有理性定理，并将其应用于极小对数差异的 ACC 问题，证明了固定 Gorenstein 指数簇上具有固定系数的三维典范对的极小对数差异满足 ACC。

On a comparison of minimal log discrepancies in terms of motivic integration

We formulate a comparison of minimal log discrepancies of a variety and its ambient space with appropriate boundaries in terms of motivic integration. It was obtained also by Ein and Mustaţǎ independently.

The minimal log discrepancies on a smooth surface in positive characteristic

This paper shows that Mustata-Nakamura's conjecture holds for pairs consisting of a smooth surface and a multiideal with a real exponent over the base field of positive characteristic. As corollaries, we obtain the ascending chain condition of the minimal log discrepancies and of the log canonical thresholds for those pairs. We also obtain finiteness of the set of the minimal log discrepancies of those pairs for a fixed real exponent.

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 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.

Picard秩二的环面簇上等变对数切层的稳定性

Stability of Equivariant Logarithmic Tangent Sheaves on Toric Varieties of Picard Rank Two

本文研究了正规环面簇上等变对数切层的斜率稳定性，并完整刻画了Picard秩为一或二时除子与极化使得该层（半）稳定的条件。

On the classification of toric singularities

For a toric log variety with standard coefficients, we show that the minimal log discrepancy at a closed invariant point bounds the Cartier index of a neighbourhood.

从典范流形的对数Landau-Ginzburg镜中构造原始形式的微扰方法

A perturbative construction of primitive forms from log Landau-Ginzburg mirrors of toric manifolds

本文通过研究典范流形的对数Landau-Ginzburg镜的变形理论，提出了构造原始形式的微扰方法，并揭示了其与bulk变形Lagrangian Floer超势能的关系。

枚举某些环面簇上的对数有理曲线

Enumerating log rational curves on some toric varieties

本文通过模空间上的相交理论计算，完全确定了射影丛X=P_{P^r}(O^s+O(-a))上的亏格0、固定域对数Gromov-Witten不变量，证实了Cela-Iribar López的一个猜想，并否证了关于P^r在r个点处爆破的另一个猜想。

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…

ACC for minimal log discrepancies of exceptional singularities

We prove the existence of $n$-complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities. In order to prove these results, we develop the theory of complements for real coefficients. We introduce $(n,Γ_0)$-decomposable $\mathbb{R}$-complements, and show its existence for pairs with DCC coefficients.

关于最小 log 抛物线差异的 PIA 猜想的反例

A counterexample to the PIA conjecture for minimal log discrepancies

本文通过构造反例，推翻了关于最小 log 抛物线差异的 PIA 猜想，并对家庭中的 LSC 猜想也提供了反例。

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.

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 …

ACC for minimal log discrepancies of terminal threefolds

We prove that the ACC conjecture for minimal log discrepancies holds for threefolds in $[1-δ,+\infty)$, where $δ>0$ only depends on the coefficient set. We also study Reid's general elephant for pairs, and show Shokurov's conjecture on the existence of $(ε,n)$-complements for threefolds for any $ε\geq 1$. As a key important step, we prove the uniform boundedness of divisors computing minimal log discrepancies for terminal threefolds. We show the ACC for threefold canonical thresholds, and that the set of accumulation points of threefold canonical thresholds is equal to $\{0\}\cup\{\frac{1}{n}\…

Numerical characterization of some toric fiber bundles

Given a complex projective manifold $X$ and a divisor $D$ with normal crossings, we say that the logarithmic tangent bundle $T_X(-\log D)$ is R-flat if its pull-back to the normalization of any rational curve contained in $X$ is the trivial vector bundle. If moreover $-(K_X+D)$ is nef, then the log canonical divisor $K_X+D$ is torsion and the maximally rationally chain connected fibration turns out to be a smooth locally trivial fibration with typical fiber $F$ being a toric variety with boundary divisor $D_{|F}$.

A geometric characterisation of toric varieties

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

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-…

Classification of toric log del Pezzo surfaces with few singular points

We give a classification of toric log del Pezzo surfaces with two or three singular points.

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 $X\ni x$.

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.

Jet schemes, log discrepancies and Inversion of Adjunction

We use the theory of motivic integration for singular spaces to give a characterization of minimal log discrepencies in terms of the codimension of certain subsets of spaces of arcs. This is done for arbitrary pairs $(X,Y)$, with $X$ normal and Q-Gorenstein. As a first application, we prove a precise version of Inversion of Adjunction, in the case when the ambient variety is smooth. Another application concerns the semicontinuity of minimal log discrepancies on smooth varieties.

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。

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。

Affine manifolds, log structures, and mirror symmetry

This is an outline of work in progress concerning an algebro-geometric form of the Strominger-Yau-Zaslow conjecture. We introduce a limited type of degeneration of Calabi-Yau manifolds, which we call toric degenerations. For these, the degenerate fibre is a union of toric varieties meeting along toric strata. In addition, the total space must look locally toric outside of some singular set. This is a generalisation of both Mumford-type degenerations of abelian varieties and degenerations of Calabi-Yau hypersurfacesin toric varieties. From such a degeneration, we construct a dual intersection g…

Some examples of Calabi-Yau pairs with maximal intersection and no toric model

It is known that a maximal intersection log canonical Calabi-Yau surface pair is crepant birational to a toric pair. This does not hold in higher dimension: this paper presents some examples of maximal intersection Calabi-Yau pairs that admit no toric model.

Toric log del Pezzo surfaces with one singularity

This paper focuses on the classification of all toric log Del Pezzo surfaces with exactly one singularity up to isomorphism, and on the description of how they are embedded as intersections of finitely many quadrics into suitable projective spaces.

Classification of Toric log Del Pezzo Surfaces having Picard Number 1 and Index $\le 3$

Toric log Del Pezzo surfaces with Picard number 1 have been completely classified whenever their index is $\le 2$: In this paper we extend the classification for those having index 3: We prove that, up to isomorphism, there are exactly 18 surfaces of this kind.

On toric face rings II

We introduce the class of weakly log canonical singularities, a natural generalization of semi-log canonical singularities. Toric varieties (associated to toric face rings, possibly non-normal or reducible) which have weakly (semi-) log canonical singularities are classified. In the toric case, we discuss residues to lc centers of codimension one or higher.

Singularities on toric fibrations

In this paper we investigate singularities on toric fibrations. In this context we study a conjecture of Shokurov (a special case of which is due to M$^\rm{c}$Kernan) which roughly says that if $(X,B)\to Z$ is an $ε$-lc Fano type log Calabi-Yau fibration, then the singularities of the log base $(Z,B_Z+M_Z)$ are bounded in terms of $ε$ and $\dim X$ where $B_Z,M_Z$ are the discriminant and moduli divisors of the canonical bundle formula. A corollary of our main result says that if $X\to Z$ is a toric Fano fibration with $X$ being $ε$-lc, then the multiplicities of the fibres over codimension one…

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.

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.

A boundedness result for toric log Del Pezzo surfaces

In this paper we give an upper bound for the Picard number of the rational surfaces which resolve minimally the singularities of toric log Del Pezzo surfaces of given index $\ell$. This upper bound turns out to be a quadratic polynomial in the variable $\ell$.

Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants

We give a tropical description of the counting of real log curves in toric degenerations of toric varieties. We treat the case of genus zero curves and all non-superabundant higher-genus situations. The proof relies on log deformation theory and is a real version of the Nishinou-Siebert approach to the tropical correspondence theorem for complex curves. In dimension two, we use similar techniques to study the counting of real log curves with Welschinger signs and we obtain a new proof of Mikhalkin's tropical correspondence theorem for Welschinger invariants.

环面、环面型和环几何中实定向吹胀的介绍

An introduction to real oriented blowups in toric, toroidal and logarithmic geometries

本文介绍了复对数空间取整（Kato-Nakayama实现）在奇点理论中的应用，通过极坐标转换引入仿射环面簇、实定向吹胀、对数结构等概念，并涵盖Nakayama-Ogus关于特殊对数态射取整局部平凡性的定理。

局部体积与极小对数偏差之间的严格不等式

A Sharp inequality between local volumes and minimal log discrepancies

该论文解答了Li–Liu–Xu的问题，证明了n维klt芽满足局部体积与极小对数偏差的严格不等式，还证明了相关比值的离散性，并将其应用于得到对数Fano对极小对数偏差的下界。