对数极小模型的nef维数

On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles

We apply Ueda theory to a study of singular Hermitian metrics of a (strictly) nef line bundle $L$. Especially we study minimal singular metrics of $L$, metrics of $L$ with the mildest singularities among singular Hermitian metrics of $L$ whose local weights are plurisubharmonic. In some situations, we determine a minimal singular metric of $L$. As an application, we give new examples of (strictly) nef line bundles which admit no smooth Hermitian metric with semi-positive curvature.

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.

Reid型无基点定理II

A Base Point Free Theorem of Reid Type, II

设$X$是{ \bf C}上的完备代数簇。考虑弱Kawamata对数终端的对数簇$(X,Δ)$.我们假设$K_X+Δ$是{ \bf Q}-Cartier { \bf Q}-约数，并且$\lfloor Δ\rfloor$的每个不可约分量都是{ \bf Q}-Cartier.如果对于$(X,Δ)$的每个非“Kawamata log terminal”奇点的中心$B$，$H |_B$是nef和big，则$X$上的nef和big Cartier除数$H$称为$(X,Δ)$上的{ \it nef和log big}。证明了：若$L$是一个nef Cartier因子，使得$aL-(K_X+Δ)$对$a \in$ { \bf N}在$(X,Δ)$上是nef和log big，则完备线性系统$| mL |$对$m \gg 0$是无基点的.

Albanese morphism of log smooth klt compact Kähler manifold with nef log anticanonical divisor

Let $(X, ω)$ be an n-dimensional compact Kähler manifold. Let $D=\sum (1-β_j) Y_j=\sum (1-β_j) [s_j=0]$ a divisor with simple normal crossings with $β_j \in ]0,1[$ such that $-(K_X+D)$ is nef. We show that its Albanese map is submersion outside an analytic set of codimension larger than two with connected fibres.

On log minimal models and Zariski decompositions II

We continue our study of the relation between log minimal models and various types of Zariski decompositions. Let $(X,B)$ be a projective log canonical pair. We will show that $(X,B)$ has a log minimal model if either $K_X+B$ birationally has a Nakayama-Zariski decomposition with nef positive part, or that $K_X+B$ is big and birationally it has a Fujita or CKM Zariski decomposition. Along the way we introduce polarized pairs $(X,B+P)$ where $(X,B)$ is a usual projective pair and $P$ is nef, and study the birational geometry of such pairs.

$P$-平凡MMP、Zariski分解与广义配对的极小模型

$P$-trivial MMP, Zariski decompositions and minimal models for generalised pairs

本研究建立$P$-平凡MMP理论，证明满足特定Nakayama-Zariski分解及数值有效条件的广义klt配对存在极小模型，并推广至三维广义lc配对，拓展了相关领域的经典结论。

Equivariant vector bundles on complete symmetric varieties of minimal rank

Let $X$ be the wonderful compactification of a complex symmetric space $G/H$ of minimal rank. For a point $x\,\in\, G$, denote by $Z$ be the closure of $BxH/H$ in $X$, where $B$ is a Borel subgroup of $G$. The universal cover of $G$ is denoted by $\widetilde{G}$. Given a $\widetilde{G}$ equivariant vector bundle $E$ on $X,$ we prove that $E$ is nef (respectively, ample) if and only if its restriction to $Z$ is nef (respectively, ample). Similarly, $E$ is trivial if and only if its restriction to $Z$ is so.

Note on the semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant

We give another alternative proof to the Kawamata semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant. After the first version of this article was posted to the e-print Arxiv, Prof. Fujino notified the author that the quick and essential proof ([Fujino. On Kawamata's theorem.(EMS 2011), Rem 2.7]) is already known. The author would like to thank him. More precisely, Prof. Fujino already gave the quick and essential proof ([Fujino. On Kawamata's thm.(EMS 2011), Rem 2.7], [Fujino. Finite generation of the lc ring in dim 4. (Kyoto J. Math. 50 (2010)), R…

The cone of pseudo-effective divisors of log varieties after Batyrev

In these notes we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective $\mathbb Q$-factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev's structure theorem for the cone of nef curves of projective terminal threefolds.

Cox环的nef反典范有理曲面

Cox rings of nef anticanonical rational surfaces

本文研究具有nef反典范类的光滑射影有理曲面的Cox环生成集，刻画生成元次数并应用于计算Halphen指数大于1的有理椭圆曲面的极小生成集。

Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points

We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each K-trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the K3 case, we extend recent constructions and results of Bini, Boissière and Flamini from the Hilbert scheme of 2 and 3 points to an arbitrary number of points. Among the K-trivial surfaces, the case of Enriques surfaces is the most involved. Our techniques apply to other smooth projective surfaces, including blowups of K3s and minimal surfaces of general type,…

Lobachevsky空间中具有对数终端奇点和nef反正则类的代数曲面和反射群。I（图解法基础）

Algebraic Surfaces with Log-Terminal Singularities and nef Anticanonical Class and Reflection Groups in Lobachevsky Spaces. I (Basics of the Diagram Method)

在我们的预印本中：“Algebras surfaces with log-terminal singularities and nef anticanonical class and reflection groups in Lobachevsky spaces”，Preprint Max-Planck-Institut für Mathematik，Bonn，（1989）MPI/89-28（Russian），我们已经将图解方法推广到具有nef anticanonical class的代数曲面（之前，它被应用于Del Pezzo曲面）。这里的主要问题是，森多面体可能不是有限多面体在这种情况下。在本预印本的§4中，我们已经证明了这种方法对具有nef反正则类和对数终端奇点的代数曲面有效。最近，Valery Alexeev（Boundedness and K^2 for log surfaces，Preprint（1994），alg-geom/942007）特别利用我们的这些结果，对具有半对数正则奇点的一般类型曲面获得了强有力的重要结果。原因是德尔佩佐曲面（图方法是特别有效的）和一般类型的表面“连接”通过表面的数字零规范类。这是我们决定将上面预印本的第一部分（§1-§3）翻译成英语的主要原因，其中有一些扩展（我们省略了证明）和小的更正。这一部分包含了罗巴切夫斯基空间中的广义反射群的图解方…

On base point freeness in positive characteristic

We prove that if $(X,A+B)$ is a pair defined over an algebraically closed field of positive characteristic such that $(X,B)$ is strongly $F$-regular, $A$ is ample and $K_X+A+B$ is strictly nef, then $K_X+A+B$ is ample. Similarly, we prove that for a log pair $(X,A+B)$ with $A$ being ample and $B$ effective, $K_X+A+B$ is big if it is nef and of maximal nef dimension. As an application, we establish a rationality theorem for the nef threshold and various results towards the minimal model program in dimension three in positive characteristic.

基本群与余正则性：具有平凡且数值有效反典范类的klt对

Fundamental Groups and Coregularity: Klt Pairs with Trivial and Nef Log-Anticanonical Class

该研究探讨射影klt对数对的绝对余正则性与基本群的关系，证明其基本群存在有限指数正规阿贝尔子群，结合相关定理得到任意维数下的余正则性界，还给出反典范类数值有效时orbifold基本群的性质。

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…

On Hodge-Riemann Cohomology Classes

We prove that Schur classes of nef vector bundles are limits of classes that have a property analogous to the Hodge-Riemann bilinear relations. We give a number of applications, including (1) new log-concavity statements about characteristic classes of nef vector bundles (2) log-concavity statements about Schur and related polynomials (3) another proof that normalized Schur polynomials are Lorentzian.

Cone theorem and Mori hyperbolicity

We discuss the cone theorem for quasi-log schemes and the Mori hyperbolicity. In particular, we establish that the log canonical divisor of a Mori hyperbolic projective normal pair is nef if it is nef when restricted to the non-lc locus. This answers Svaldi's question completely. We also treat the uniruledness of the degenerate locus of an extremal contraction morphism for quasi-log schemes. Furthermore, we prove that every fiber of a relative quasi-log Fano scheme is rationally chain connected modulo the non-qlc locus.

Existence of cscK metrics on smooth minimal models

Given a compact Kähler manifold $X$ it is interesting to ask whether it admits a constant scalar curvature Kähler (cscK) metric. In this short note we show that there always exist cscK metrics on compact Kähler manifolds with nef canonical bundle, thus on all smooth minimal models, and also on the blowup of any such manifold. This confirms an expectation of Jian-Shi-Song \cite{JianShiSong} and extends their main result from $K_X$ semi-ample to $K_X$ nef, with a direct proof that does not appeal to the Abundance conjecture. As a byproduct we obtain that the connected component $\mathrm{Aut}_0(X…

Hopf流形上的非半正 nef 线丛

Nef but Non-Semi-positive Line Bundles on Hopf Manifolds

该研究刻画了Hopf流形的相关锥，证明其不变椭圆曲线为nef非半正，并借助Bott–Chern上同调在任意维Hopf流形上构造了这类线丛的新例子。

Reid-Fukuda型无基点定理

Base point free theorem of Reid-Fukuda type

设$(X,Δ)$是一个真dlt对，$L$是一个nef Cartier因子，使得对于某些$a\in {\mathbb Z}_{>0}$，$aL-(K_X+Δ)$在$(X,Δ)$上是nef和log大的.则$|mL|$对于每个$m\gg 0$都是基点自由的。

具有严格 nef 相对反 log 典范除子的代数纤维空间

Algebraic fibre spaces with strictly nef relative anti-log canonical divisor

本文证明具有严格 nef 相对反 log 典范除子的纤维化必为纤维有理连通的局部常值纤维化，底空间为典范极化双曲流形，并在三维情形下证实了 Campana-Peternell 猜想的奇异版本。

关于 log Calabi-Yau 曲面的锥猜想

A cone conjecture for log Calabi-Yau surfaces

本文研究了带奇异边界的 log Calabi-Yau 曲面，证明了其自同构群和单值群在 nef 有效锥上的作用容许有理多面体基本域，并给出了 Mori Dream Spaces 的新示例。

On the Kodaira vanishing theorem for log del Pezzo surfaces in positive characteristic

We investigate the vanishing of $H^1(X,\mathcal{O}_X(-D))$ for a big and nef $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor $D$ on a log del Pezzo surface $(X,Δ)$ over an algebraically closed field of positive characteristic $p$.

紧致Kähler品种的fundamental groups与 nef反canonical bundle

Fundamental groups of compact Kahler varieties with nef anti canonical bundle

本文研究了紧致Kähler品种的fundamental groups，证明在特定条件下，其fundamental group几乎为Abelian，并扩展了相关定理至log canonical对和klt对。

Structure of projective varieties with nef anticanonical divisor: the case of log terminal singularities

In this article we study the structure of klt projective varieties with nef anticanonical divisor (and more generally, varieties of semi-Fano type), especially the canonical fibrations associated to them. We show that: 1. the Albanese map for such variety is a locally constant fibration (that is, an analytic fibre bundle with connected fibres that $X$ is equal to the product of the universal cover of the Albanese torus by the fibre of the Albanese map quotient by a diagonal action of the fundamenatl group of the Albanese torus); 2. if the smooth locus is simply connected, the MRC fibration of …

On the Seshadri constants of equivariant bundles over Bott-Samelson varieties and wonderful compactifications

We study torus-equivariant vector bundles $E$ on a complex projective variety $X$ which is either a Bott-Samelson-Demazure-Hansen variety or a wonderful compactification of a complex symmetric variety of minimal rank. We show that $E$ is nef (respectively, ample) if and only if its restriction to every torus--invariant curve in $X$ is nef (respectively, ample). We also compute the Seshadri constants $\varepsilon(E,x)$, where $x\, \in\, X$ is any point fixed by the action of a maximal torus.

One base point free theorem for weak log Fano threefolds

Let $(X,D)$ be log canonical pair such $\dim X = 3$ and the divisor $-(K_X + D)$ is nef and big. For a special class of such $(X,D)$'s we prove that the linear system $|-n(K_{X}+D)|$ is free for $n \gg 0$.

On birational geometry of minimal threefolds with numerically trivial canonical divisors

For a minimal $3$-fold $X$ with $K_X\equiv 0$ and a nef and big Weil divisor $L$ on $X$, we investigate the birational geometry inspired by $L$. We prove that $|mL|$ and $|K_X+mL|$ give birational maps for all $m\geq 17$. The result remains true under weaker assumption that $L$ is big and has no stable base components.

射影klt对具有nef反典范除子的结构定理

Structure theorem for projective klt pairs with nef anti-canonical divisor

本文证明射影klt对在有限拟平展覆盖下具有局部平凡有理连通纤维化，基为数值平凡典范除子的klt簇，推广了光滑情形并应用于分解klt Calabi-Yau型簇。

On existence of log minimal models and weak Zariski decompositions

We first introduce a weak type of Zariski decomposition in higher dimensions: an $\R$-Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can be written as the sum of a nef and an effective $\R$-Cartier divisor. We then prove that there is a very basic relation between Zariski decompositions and log minimal models which has long been expected: we prove that assuming the log minimal model program in dimension $d-1$, a lc pair $(X/Z,B)$ of dimension $d$ has a log minimal model if and only if $K_X+B$ has a weak Zariski decomposition$/Z$.

Projective klt pairs with nef anti-canonical divisor

In this paper, we study a projective klt pair $(X, Δ)$ with the nef anti-log canonical divisor $-(K_X+Δ)$ and its maximally rationally connected fibration $ψ: X \dashrightarrow Y$. We prove that the numerical dimension of the anti-log canonical divisor $-(K_X+Δ)$ on $X$ coincides with that of the anti-log canonical divisor $-(K_{X_y}+Δ_{X_y})$ on a general fiber $X_y$ of $ψ: X \dashrightarrow Y$, which is an analogue of Ejiri-Gongyo's result formulated for the Kodaira dimension. As a corollary, we reveal a relation between positivity of the anti-canonical divisor and the rational connectedness…

紧致Kahler流形上的Kawamata-Viehweg消灭定理

A Kawamata-Viehweg Vanishing Theorem on compact Kahler manifolds

本文证明了正规紧Kahler空间X上的Kawamata-Viehweg消灭定理：若L是一个数值维数至少等于2的Nef线丛，则当q至少等于X的维数减1时，K_X+L的第q个上同调群为零.作为应用，证明了关于极小Kahler三重数的丰度猜想的一个特例：如果X是极小Kahler三重数（通常意义下KX是nef），则Kodaira维数kappa（X）是非负的.

On ruled surfaces with big anti-canonical divisor and numerically trivial divisors on weak log Fano surfaces

We investigate the structure of geometrically ruled surfaces whose anti-canonical class is big. As an application we show that the Picard group of a normal projective surface whose anti-canonical class is nef and big is a free abelian group of finite rank.

关于射影平面上点的Hilbert格式的几何

On the geometry of the Hilbert schemes of points in the projective plane

我们在代数曲面上的点的希尔伯特格式中对最小度曲线进行了分类。当代数曲面为射影平面时，确定了这些Hilbert格式的Nef锥和翻转结构。

Semi-positivity in positive characteristics

Let $f : (X, Δ) \to Y$ be a flat, projective family of sharply $F$-pure, log-canonically polarized pairs over an algebraically closed field of characteristic $p >0$ such that $p \nmid \ind(K_{X/Y} + Δ)$. We show that $K_{X/Y} + Δ$ is nef and that $f_* (\sO_X(m (K_{X/Y} + Δ)))$ is a nef vector bundle for $m \gg 0$ and divisible enough. Some of the results also extend to non log-canonically polarized pairs. The main motivation of the above results is projectivity of proper subspaces of the moduli space of stable pairs in positive characteristics. Other applications are Kodaira vanishing free, al…

近Gorenstein有理表面奇点

Nearly Gorenstein rational surface singularities

本文研究了近Gorenstein有理表面奇点的性质，通过分析canonical trace理想和基本循环，给出了其判据并进行了分类。

第二陈类消失的极小紧 Kähler 流形的丰度定理

Abundance theorem for minimal compact Kähler manifolds with vanishing second Chern class

本文证明极小紧 Kähler 流形上第二陈类消失等价于余切丛 nef 且典范丛数值维数为 0 或 1，并由此推出典范丛半丰富，同时揭示 Iitaka 纤维化纤维变化与余切丛半正性的联系。

一般型光滑极小n维簇上多重典范线性系统的无基点性

Freeness of Pluricanonical Linear System on Smooth Minimal n-folds of General Type

本文证明了一般型光滑极小n维簇上$|\frac{n^2+n+2}{2}K_X|$无基点，并给出Fujita无基点猜想在nef且大除数情形下的界限，同时证明了$K_X+mB$的基点集包含于增广基轨迹。

极化叶状曲面的有界性

Boundedness of polarized foliated surfaces

本文建立klt伴随叶状曲面的极小模型程序，证明满足条件的ε-lc伴随叶状曲面构成有界族，应用于一般类型伴随叶状曲面，还得到秩1叶状结构典范除子体积的离散性结果。

Examples of Mori dream surfaces of general type with $p_g=0$

In this paper we study effective, nef and semiample cones of minimal surfaces of general type with $p_g=0.$ We provide examples of minimal surfaces of general type with $p_g=0, 2 \leq K^2 \leq 9$ which are Mori dream spaces. On these examples we also give explicit description of effective cones and all irreducible reduced curves of negative self-intersection. We also present non-minimal surfaces of general type with $p_g=0$ that are not Mori dream surfaces.

On a connectedness principle of Shokurov-Kollár type

Let $(X,Δ)$ be a log pair over $S$, such that $-(K_X+Δ)$ is nef over $S$. It is conjectured that the intersection of the non-klt (non Kawamata log terminal) locus of $(X,Δ)$ with any fiber $X_s$ has at most two connected components. We prove this conjecture in dimension $\leq 4$ and in arbitrary dimension assuming the termination of klt flips.

对数正则曲面的基点自由定理

A base point free theorem for log canonical surfaces

证明了对数标准曲面上nef和对数大因子的基点自由定理。

Hirzebruch曲面的orbifold退化

Orbifold Degenerations of Hirzebruch Surfaces

该研究探讨Hirzebruch曲面的orbifold退化，通过证明其为环面曲面的部分光滑化，结合三维极小模型程序等，分类了不同k值下的中心纤维及对应奇点，补充了反典范除子丰富情形的相关工作。

Some non-vanishing results on log canonical pairs of dimension 4

Let $(X,Δ)$ be a log canonical pair over $\mathbb{C}$ with $X$ a normal projective variety, $Δ$ an effective $\mathbb{Q}$-divisor, and $K_X+Δ$ nef. We give a non-vanishing criterion for $K_X+Δ$ in dimension $n$ with $X$ uniruled, assuming various conjectures of LMMP in dimensions (up to) $n-1$ or $n$, and a semi-ampleness criterion in the irregular case. In particular, we obtain that if $X$ is a uniruled $4$-fold, then $κ(K_X+Δ)\geq 0$ and if $X$ is a $4$-fold with $q(X)>0$, then $K_X+Δ$ is semi-ample.

具有固定指数的有理连通对数 Calabi–Yau 对的双有理有界性

Birational boundedness of rationally connected log Calabi-Yau pairs with fixed index

本文研究固定维数的有理连通 klt 对数 Calabi–Yau 对，证明在反对数典范类的固定 Cartier 指数与 nef 条件下，相应射影簇的集合模 flops 有界。

On the Log Abundance for Compact Kähler $3$-folds

In this article we show that if $(X, Δ)$ is a log canonical compact Kähler $3$-fold such that $K_X+Δ$ is nef and the numerical dimension $ν(K_X+Δ)\neq 2$, then $K_X+Δ$ is semi-ample.

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 $D_{|F}$.

Positivity of divisors on blown-up projective spaces, II

We construct log resolutions of pairs on the blow-up of the projective space in an arbitrary number of general points and we discuss the semi-ampleness of the strict transforms. As an application we prove that the abundance conjecture holds for an infinite family of such pairs. For $n+2$ points, these strict transforms are F-nef divisors on the moduli space $\overline{\mathcal{M}}_{0,n+3}$ in a Kapranov's model: we show that all of them are nef.

Some remarks on the minimal model program for log canonical pairs

We prove that the target space of an extremal Fano contraction from a log canonical pair has only log canonical singularities. We also treat some related topics, for example, the finite generation of canonical rings for compact Kähler manifolds, and so on. The main ingredient of this paper is the nefness of the moduli parts of lc-trivial fibrations. We also give some observations on the semi-ampleness of the moduli parts of lc-trivial fibrations. For the reader's convenience, we discuss some examples of non-Kähler manifolds, flopping contractions, and so on, in order to clarify our results.

On the number and boundedness of log minimal models of general type

We show that the number of marked minimal models of an n-dimensional smooth complex projective variety of general type can be bounded in terms of its volume, and, if n=3, also in terms of its Betti numbers. For an n-dimensional projective klt pair (X,D) with $K_X+D$ big, we show more generally that the number of its weak log canonical models can be bounded in terms of the coefficients of D and the volume of $K_X+D$. We further show that all n-dimensional projective klt pairs (X,D), such that $K_X+D$ is big and nef of fixed volume and such that the coefficients of D are contained in a given DCC…

关于从拟光滑加权超曲面生成的极小簇

On minimal varieties growing from quasismooth weighted hypersurfaces

本文研究小典范体积极小簇的构造，通过建立加权超曲面加权爆破的有效nef准则构造了多族极小3-折叠，并证明了一般型极小n-折叠典范体积的有效下界，证明了其在低维最优、高维近似最优。

Boundedness of Log Canonical Surface Generalized Polarized Pairs

In this paper, we study the behavior of the sets of volumes of the form $\mathrm{vol}(X,K_X+B+M)$, where $(X,B)$ is a log canonical pair, and $M$ is a nef $\mathbb{R}$-divisor. After a first analysis of some general properties, we focus on the case when $M$ is $\mathbb{Q}$-Cartier with given Cartier index, and $B$ has coefficients in a given DCC set. First, we show that such sets of volumes satisfy the DCC property in the case of surfaces. Once this is established, we show that surface pairs with given volume and for which $K_X+B+M$ is ample form a log bounded family. These generalize results …

On numerical nonvanishing for generalized log canonical pairs

The nonvanishing conjecture for projective log canonical pairs plays a key role in the minimal model program of higher dimensional algebraic geometry. The numerical nonvanishing conjecture considered in this paper is a weaker version of the usual nonvanishing conjecture, but valid in the more general setting of generalized log canonical pairs. We confirm it in dimension two. Under some necessary conditions we obtain effective versions of numerical nonvanishing for surfaces. Several applications are also discussed. In higher dimensions, we mainly consider the conjecture for generalized klt pair…

沿对数典范轨迹的正规配对极小模型纲领

Minimal model program for normal pairs along log canonical locus

本文研究正规配对在非lc轨迹外的极小模型纲领，在非nef轨迹与非lc轨迹不相交且限制半丰富的条件下，证明了MMP终止及极小模型的丰富猜想。