Xiaojun Wu
The theory of the Oukounkov body is a useful tool for studying the asymptotic behaviour of the canonical ring of a line bundle over a projective manifold. In this note, combined with the algebraic reduction, we study the asymptotic behaviour of the canonical ring of a line bundle over an arbitrary compact normal irreducible complex space.
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Additional Explanatory Notes on the Analytic Proof of the Finite Generation of the Canonical Ring
这一组注释提供了一些额外的解释材料的解析证明有限生成的典型环的一般类型的紧凑复代数流形。
A note on log canonical thresholds
我们证明了所有三维对数正则阈值$c(X,F)$的集合$\mathcal{T}_3$的最大聚点是5/6。
A Note on Projectivized Tangent Cone Quadrics of a Prym Canonical Curve
在本文中，我们对秩等于或小于4的射影切锥二次曲面，对Prymians，给出了定理1.1的部分类似.
A note on the ampleness of numerically positive log canonical and anti-log canonical divisors
在这篇短文中，我们考虑了一个猜想，即对数典范除子（分别是反对数典范除子）$K_X + Δ$（分别由复射影流形$X$和$X$上的约简法向交叉因子$Δ$组成的对$(X, Δ)$上的一个约简法向交叉因子（$-(K_X + Δ)$）是充分的，如果它是数值正的。更准确地说，我们证明了猜想$K_X + Δ$与$Δ\neq 0$在4维和$-(K_X + Δ)$与$Δ\neq 0$在3或4维。
A note on the canonical divisor of quasi-homogeneous affine algebraic varieties
给出了拟齐次仿射代数簇上典范除子为零的一个充要条件。
Log canonical inversion of adjunction
本文是一篇简短注记，研究伴随的对数典范反演问题。
IMPANGA lecture notes on log canonical thresholds
这些是IMPANGA 2010暑期学校的讲义。他们介绍了对数典型阈值，包括一些基本性质，例子，和一些最近的结果和开放的问题。
Note on quasi-numerically positive log canonical divisors
本文提出了一个子猜想，它包含了拟数值正对数典范除子的半满猜想，并在一些初等情形下证明了该猜想的半满性。
A note on the canonical divisor of the generalised affine Stiefel algebraic varieties
本文研究一类齐性空间，我们称之为广义仿射Stiefel代数簇.主要目的是用群表示的方法研究广义仿射Stiefel代数簇的典范除子.仿射Stiefel代数簇和特别是$S^{n}$是广义仿射Stiefel代数簇的两个特例。
We study projective manifolds with nonamenable and non-residually finite fundamental groups. We generalize the uniformization theorem of our earlier note. We generalize a classical theorem of Maltsev about finitely generated subgroups of GL(m).
A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring
2005年8月5日，在西雅图的美国数学学会代数几何暑期研究所以及后来的几次会议上，我就一般类型情况下典范环的有限生成的解析证明进行了演讲。在我的讲座结束后，许多人向我要一份我在讲座中使用的幻灯片。由于演讲时间的限制，我的幻灯片非常粗略，我答应稍后在预印服务器上发布我的幻灯片摘录的详细笔记。这里是我的详细笔记，给出了技术和证明。
Let $(X,L)$ be a quasi-polarized canonical Calabi-Yau threefold. In this note, we show that $\vert mL\vert$ is basepoint free for $m\geq 4$. Moreover, if the morphism $Φ_{\vert 4L\vert}$ is not birational onto its image and $h^0(X,L)\geq 2$, then $L^3=1$. As an application, if $Y$ is a $n$-dimensional Fano manifold such that $-K_Y=(n-3)H$ for some ample divisor $H$, then $\vert mH\vert$ is basepoint free for $m\geq 4$ and if the morphism $Φ_{\vert 4H\vert}$ is not birational onto its image, then $Y$ is either a weighted hypersurface of degree $10$ in the weighted projective space $\mathbb{P}(1…
In this note, we describe the structure of regular foliations with semi-positive anti-canonical bundle on smooth projective varieties.
This article focuses on the study of toric algebraic statistical models which correspond to toric Del Pezzo surfaces with Du Val singularities. A closed-form for the Maximum Likelihood Estimate of algebraic statistical models which correspond to cubic and quartic toric Del Pezzo surfaces with Du Val singular points is given. Also, we calculate the ML degrees of some toric Del Pezzo surfaces of degree less than or equal to six, which equals the degree of the surface in all the case but one, namely the quintic with two points of type $\mathbb{A}_1$.
Note on the 3-dimensional log canonical abundance in characteristic $>3$
本文在特征 $>3$ 的代数闭域上证明了射影对数典范三维偶对的非消没性及丰度的若干特殊情形，并应用于对数典范环的有限生成及丰度成立的充分条件。
In this note, we obtain various formulas for the curvature of the $L^2$ metric on the direct image of the relative canonical bundle twisted by a holomorphic line bundle endowed with a positively curved metric with analytic singularities, generalizing some of Berndtsson's seminal results in the smooth case. When the twist is assumed to be relatively big, we further provide a very explicit lower bound for the curvature of the $L^2$ metric.
We show that the $\ell$-adic Tate conjecture for divisors on smooth proper varieties over finitely generated fields of positive characteristic follows from the $\ell$-adic Tate conjecture for divisors on smooth projective surfaces over finite fields. Similar results for cycles of higher co-dimension are given.
Canonical abelianization of finite group actions
在这些注记中，我们提出了一个修正的算法的阿贝尔化的稳定器的有限群作用在光滑流形上使用等变爆破。
In this note, we construct three new infinite families of surfaces of general type with canonical map of degree 2 onto a surface of general type. For one of these families the canonical system has base points.
The purpose of this note is to show that in characteristic two there exists a smooth Fano variety for which the square of the anti-canonical bundle violates Kodaira vanishing. This is used in turn to construct non-Cohen-Macaulay canonical singularities in characteristic two.
In this note, we construct nine families of projective complex minimal surfaces of general type having the canonical map of degree 8 and irregularity 0 or 1. For six of these families the canonical system has a non trivial fixed part.
In this note we construct an unlimited family of irregular algebraic surfaces of general type with canonical map of degree $ 8 $, irregularity $ 1 $ and arbitrarily large geometric genus such that the image of the canonical map is not a surface of minimal degree.
Finite generation of a canonical ring
本文的目的是回顾Birkar-Cascini-Hacon-McKernan有限生成定理的代数证明，他的方法是基于极小模型程序。2007年《数学发展现状》（Current Development in Mathematics）
In this note we present examples of complex algebraic surfaces of general type with canonical maps of degree $10$, $11$ and $14$. They are constructed as quotients of a product of two Fermat septics using certain free actions of the group $\mathbb Z_7^2$.
In this note we provide a two-dimensional family of smooth minimal threefolds of general type with canonical map of degree 96, improving the previous known bound of 72.
In this note, we construct a minimal surface of general type with geometric genus p g = 4, self-intersection of the canonical divisor K^2 = 32 and irregularity q = 1 such that its canonical map is an abelian cover of degree 16 of P^1 x P^1.
Negative Holomorphic curvature and positive canonical bundle
本文证明了如果射影流形有一个具有负全纯截面曲率的Kähler度量，则该流形的典范丛是充分的.这证实了第二作者的一个猜想。
A note on the projective varieties of almost general type
一个射影簇$M$上的$\mathbf{Q}$因子$D$是{ \it几乎nup}，如果对$M$上的每一条非常一般的曲线$C$有$(D , C) > 0$.一个代数簇$X$是{ \it几乎一般型}的，如果存在一个只具有终端奇点的射影簇$M$，使得标准因子$K_M$几乎为nup，并且使得$M$与$X$是双有理等价的.在极小模型猜想下，证明了复代数簇是几乎一般型的当且仅当它既不是uniruled的，也不是被双有理等价于具有数值平凡标准因子的极小簇的簇族所覆盖.在极小模型猜想下，证明了对于一个仅有终端奇点的射影簇$X$，$X$是几乎一般型的当且仅当其标准因子$K_X$几乎为nup.
In this note, we construct some minimal smooth surfaces of general type with canonical map of degree $ 13, 15, 17, 18, 21, 22 $. These surfaces are constructed as $ \mathbb{Z}_{3}^2$-covers of a blow-up of $ \mathbb{P}^1 \times \mathbb{P}^1 $.
Canonical curves and varieties of sums of powers of cubic polynomials
本文证明了亏格至少为11的标准曲线的余维两个线性截面的非极性三次型是特殊的，它们表示为立方和。
On quasi-Albanese morphisms for log canonical Calabi-Yau pairs
本文研究了log canonical Calabi-Yau对的准阿贝尔 morphisms，提出了toric pairs的特征化结果。
In this note, we construct two minimal surfaces of general type with geometric genus p_g= 3, irregularity q = 0, self-intersection of the canonical divisor K^22 =20,24 such that their canonical map is of degree 20. In one of these surfaces, the canonical linear system has a non-trivial fixed part. These surfaces, to our knowledge, are the first examples of minimal surfaces of general type with canonical map of degree 20.
The Bogomolov-Gieseker-Koseki inequality on surfaces with canonical singularities in arbitrary characteristic
本文将Bogomolov-Gieseker不等式从特征零光滑射影曲面推广到任意特征、具典范奇点的射影曲面，并推广其若干经典应用。
On canonical threefolds near the Noether line
本文是关于典范体积接近下界4/3p_g - 10/3的典范三维簇的研讨会扩展摘要，为与S. Coughlan等学者合作的研究项目。
Some surfaces with canonical map of degree 4
本文构造了具有4次典范映射的一般型极小曲面的无界族，并证明其斜率极限在6+2/3和8之间取可数多个不同值。
The purpose of this short note is to give a remark on the decomposition theorem for direct images of canonical sheaves tensorized with Nakano semipositive vector bundles. Although our result is a direct consequence of Takegoshi's work, it was not stated explicitly in his article. Here we give the precise statement and the proof.
In this paper we collect some results on the obstruction spaces for rational surface singularities and minimally elliptic surface singularities. Based on the known results we calculate higher obstruction spaces for such surface singularities. The results imply that in general the higher obstruction spaces of deforming semi-log-canonical surfaces do not vanish. We apply the calculation result to show that there is no Li-Tian and Behrend-Fantechi style virtual fundamental class on such moduli space of semi-log-canonical surfaces.
Few explicit families of 3-folds are known for which the computation of the canonical ring is accessible and the birational geometry non-trivial. In this note we investigate a family of determinantal 3-folds in $\mathbb P^2 \times \mathbb P^3$ where this is the case.
A Bound for the Castelnuovo-Mumford Regularity of Log Canonical Varieties
本文给出了齐次理想$I$的Castelnuovo-Mumford正则性的一个关于生成元次数的界.我们假设$I$定义了一个与对数正则奇点的局部完全交集。
In this note, we extend the theories of the canonical bundle formula and adjunction to the case of generalized pairs. As an application, we study a particular case of a conjecture by Prokhorov and Shokurov.
A note on Mather-Jacobian multiplier ideals
利用Mather-Jacobian乘子理想，证明了特征为零的代数闭域上的Grauert-Riemenschneider标准层与簇的标准层的比较公式.然后我们研究了代数曲线上的Mather-Jacobian乘子理想，在这种情况下，Mather-Jacobian乘子理想的定义可以推广到任意特征的基域.我们证明了曲线上的Mather-Jacobian乘子理想与整闭理想本质上是相同的。最后通过比较导体理想和Mather-Jacobian乘子理想，给出了代数曲线是局部完全交的一个判别准则。
In this note we present examples of complex algebraic surfaces with canonical maps of degree $12$, $13$, $15$, $16$ and $18$. They are constructed as quotients of a product of two curves of genus $10$ and $19$ using certain non-free actions of the group $S_3\times \mathbb Z_3^2$. To our knowledge there are no other examples in literature of surfaces with canonical map of degree $13$, $15$ and $18$.
Let $k$ be an $F$-finite field containing an infinite perfect field of positive characteristic. Let $(X, Δ)$ be a projective log canonical pair over $k$. In this note we show that, for a semi-ample divisor $D$ on $X$, there exists an effective $\mathbb{Q}$-divisor $Δ' \sim_{\mathbb Q} Δ+D$ such that $(X, Δ')$ is log canonical if there exists a log resolution of $(X, Δ)$.
Let $N$ be a lattice of rank $n$ and let $M = N^{\vee}$ be its dual lattice. In this note we show that given two compact, bounded, full-dimensional convex sets $K_1 \subseteq K_2 \subseteq M_{\R} \coloneqq M \otimes_{\Z} \R$, there is a canonical convex decomposition of the difference $K_2 \setminus K_1$ and we interpret the volume of the pieces geometrically in terms of intersection numbers of toric $b$-divisors.
This note is the sequel to [A note on secondary K-theory. Algebra and Number Theory 10 (2016), no. 4, 887-906]. Making use of the recent theory of noncommutative motives, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injectivity properties: in the case of a regular integral quasi-compact quasi-separated scheme, it is injective; in the case of an integral normal Noetherian scheme with a single isolated singularity, it distinguishes any two derived Brauer classes whose difference is of infinite order. As an application, we show…
Let $f:X\to Y$ be a fibration from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field $k$ of characteristic $p >5$. We prove that if the generic fiber $X_η$ has big canonical divisor $K_{X_η}$, then $$κ(X)\geκ(Y) + κ(X_η).$$
A celebrated conjecture of Kobayashi and Lang says that the canonical line bundle $K_X$ of a Kobayashi hyperbolic compact complex manifold $X$ is ample. In this note we prove that $K_X$ is ample if $X$ is projective and satisfies a stronger condition of nondegenerate negative total $k$-jet curvature. We use positivity of direct image sheaves and decomposition of jets in order to produce pluridifferentials on $X$.
This note gives a simple formula for the unique asymptotically conical Calabi-Yau metrics on the canonical bundle of a flag variety known to exist by the work of R. Goto and others. This is done by generalizing the well known Calabi Ansatz to general Kähler classes. We give some examples of explicit families, in particular, a formula for the two dimensional family of asymptotically conical metrics on the canonical bundle of $F_{1,2}$.
Canonical p-dimensions of algebraic groups and degrees of basic polynomial invariants
本文利用V. Kac所描述的基本多项式不变量的次数，给出了一种新的计算分裂代数群G的规范p-维数的统一方法。作为应用，我们计算了所有分裂例外代数群的规范p-维数.
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