A smooth, complete threefold with no nontrivial nef line bundles

Strictly nef divisors on singular threefolds

Let $X$ be a normal projective threefold with mild singularities, and $L_X$ a strictly nef $\mathbb{Q}$-divisor on $X$. First, we show the ampleness of $K_X+tL_X$ with sufficiently large $t$ if either the Kodaira dimension $κ(X)\neq 0$ or the augmented irregularity $q^{\circ}(X)\neq 0$. Second, we show that, if $(X,Δ)$ is a projective klt threefold pair with the anti-log canonical divisor $-(K_X+Δ)$ being strictly nef, then $X$ is rationally connected.

第一陈类为二的二次三维簇上的 nef 向量丛

Nef vector bundles on a quadric threefold with first Chern class two

本文分类了特征零代数闭域上三维光滑超二次簇上第一陈类为二的 nef 向量丛，并证明它们都是整体生成的。

Rational curves and strictly nef divisors on Calabi--Yau threefolds

We give a criterion for a nef divisor $D$ to be semiample on a Calabi--Yau threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\neq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $ν(D)\neq 1$, then $D$ is ample; we also show that if there exists a nef non-ample divisor $D$ with $D\not\equiv 0$, then $X$ contains a rational curve when its topological Euler characteristic is not $0$.

Smooth complete toric threefolds with no nontrivial nef line bundles

We describe all of the smooth complete toric threefolds of Picard number 5 with no nontrivial nef line bundles, and show that no such examples exist with Picard number less than 5.

On the canonical threefolds with strictly nef anticanonical divisors

Any ample Cartier divisor D on a projective variety X is strictly nef (i.e. D.C>0 for any effective curve C on X). In general, the converse statement does not hold. But this is conjectured to be true for anticanonical divisors. The present paper establishes this fact for normal complex projective threefolds with canonical singularities. This result extends several previously known special cases. The proof rests mainly on sophisticated techniques of three dimensional birational geometry developed in the last two decades.

Threefolds with nef anticanonical bundles

We prove that the Albanese map of a smooth projective threefold, whose anticanonical bundle is nef, is a surjective submersion. We also investigate morphisms of threefolds to curves and surfaces whose relative anticanonical bundle are nef.

Compact Kähler threefolds with non-nef canonical bundle and symplectic geometry

We show that the nefness of the canonical bundle of compact Kähler threefolds is invariant under deformed symplectic diffeomorphisms.

具有nef反典范除子的整体$F$-正则型三维簇

Threefolds of globally $F$-regular type with nef anti-canonical divisor

本研究针对Schwede和Smith的猜想特例，证明具nef反典范除子的光滑复射影三维簇若为整体$F$-正则型则是弱Fano簇，推进了相关代数几何猜想的验证。

Smooth projective toric varieties whose nontrivial nef line bundles are big

For any $n\geq 3$, we explicitly construct smooth projective toric $n$-folds of Picard number $\geq 5$, where any nontrivial nef line bundles are big.

On base point free theorem of threefolds in positive characteristic

Let $(X, Δ)$ be a projective klt three dimensional pair defined over an algebraically closed field characteristic larger than 5. Let $L$ be a nef and big line bundle on $X$ such that $L-K_X-Δ$ is big and nef. We show that $L$ is indeed semi-ample.

三重拟正则簇在正特征下的Mumford消失定理

Mumford vanishing for threefolds in positive characteristic

研究三重拟正则簇在正特征下的Mumford消失定理，证明当K_X不大的情况下L为大除子或-L_K_X为拟正则且L的数值维数为二时，H^1(X,-L)=0。

On the base point free theorem for klt threefolds in large characteristic

In this article we present a refinement of the base point free theorem for threefolds in positive characteristic. If $L$ is a nef Cartier divisor of numerical dimension at least one on a projective Kawamata log terminal threefold $(X,Δ)$ over a perfect field $k$ of characteristic $p \gg 0$ such that $L-(K_X+Δ)$ is big and nef, then we show that the linear system $|mL|$ is base point free for all sufficiently large integer $m>0$.

Effective non-vanishing conjectures for projective threefolds

Let X be a smooth projective threefold, and let A be an ample line bundle such that $K_X+A$ is nef. We show that if $K_X$ or $-K_X$ is pseudoeffective, the adjoint bundle $K_X+A$ has global sections. We also give a very short proof of the Beltrametti-Sommese conjecture in dimension three, recently proven by Fukuma: if A is an ample line bundle such that $K_X+2A$ is nef, the adjoint bundle $K_X+2A$ has global sections.

具有nef反典范除子的终端三维簇的Miyaoka型不等式

Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors

本文研究带nef反典范除子的终端射影3-簇的Miyaoka型不等式，给出陈类乘积的下界及紧性结果，通过部分分类与例子完成证明，还拓展了终端弱Fano簇的相关非零性与Miyaoka–Kawamata型不等式研究。

Calabi-Yau threefolds of quotient type

First, we classify Calabi-Yau threefolds with infinite fundamental group by means of their minimal splitting coverings introduced by Beauville, and deduce that the nef cone is a rational simplicial cone and any rational nef divisor is semi-ample if the second Chern class is identically zero. We also derive a sufficient condition for the fundamental group to be finite in terms of the Picard number in an optimal form. Next, we give a concrete structure Theorem concerning $c_{2}$-contractions of Calabi-Yau threefolds as a generalisation and also a correction of our earlier works for simply connec…

奇异三重簇上的严格 nef 除子

Strictly nef divisors on singular threefolds

本文研究奇异三重簇上 Serrano 猜想，证明在 $\mathbb{Q}$-factorial Gorenstein terminal 条件下，除弱 Calabi–Yau 且第二陈类相交为零的情形外，$K_X+tL_X$ 对充分大的 $t$ 丰饶。

Big and nef classes, Futaki Invariant and resolutions of cubic threefolds

In this note we revisit and extend few classical and recent results on the definition and use of the Futaki invariant in connection with the existence problem for Kaehler constant scalar curvature metrics on polarized algebraic manifolds, especially in the case of resolution of singularities. The general inspiration behind this work is no doubt the beautiful 1992 paper by Ding and Tian which contains the germs of a huge amount of the successive developments in this fundamental problem, and it is a great pleasure to dedicate this to Professor G. Tian on the occasion of his birthday!

On Pseudo-Effectivity of the Second Chern Classes for Smooth Threefolds

We prove that for smooth projective threefolds whose anticanonical divisors are nef, the second Chern classes are pseudo-effective under a weak assumption. As an application, the pseudo-effectivity of the second Chern classes implies that Kawamata's Effective Non-vanishing Conjecture holds for such threefolds.

Curves disjoint from a nef divisor

On a projective surface it is well-known that the set of curves orthogonal to a nef line bundle is either finite or uncountable. We show that this dichotomy fails in higher dimension by constructing a nef line bundle on a threefold which is trivial on countably infinitely many curves. This answers a question of Totaro. As a pleasant corollary, we exhibit a quasi-projective variety with only a countably infinite set of complete, positive-dimensional subvarieties.

具有nef反典范丛的代数簇的非零问题

The Nonvanishing problem for varieties with nef anticanonical bundle

该研究针对具有nef反典范丛的代数簇非零问题，证明了带温和奇点的三维配对中，反典范除子的nef数值类必为有效类，推进了代数几何相关理论。

New constructions of nef classes on self-products of curves

We study the nef cone of self-products of a curve. When the curve is very general of genus $g>2$, we construct a nontrivial class of self-intersection 0 on the boundary of the nef cone. Up to symmetry, this is the only known nontrivial boundary example that exists for all $g > 2$. When the curve is general, we identify nef classes that improve on known examples for arbitrary curves. We also consider self-products of more than two copies of the curve.

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 $n \gg 0$.

有正伴随典范线丛的形变

Deformation of nef adjoint canonical line bundles

本文研究射影与Kähler族中典范线丛的nef性、半充足性形变性质，证明伴随丛nef性与半充足性的形变不变性，推广广义多重亏格不变性，并部分证实三维Kähler形变下典范线丛半充足性的稳定性猜想。

Abundance for Kaehler threefolds

Let X be a compact Kaehler threefold with terminal singularities such that K\_X is nef. We prove that K\_X is semiample.

通过 nef 复杂性特征化射影空间的积

Characterization of products of projective spaces via nef complexity

本文通过定义 nef 复杂性，特征化射影空间的积，并对光滑 Fano 三次簇进行分类，同时证明了 Mukai 猜想并研究了射影空间积的像。

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.

Calabi--Yau三重折叠中的曲线计数通用方法

Universally counting curves in Calabi--Yau threefolds

本文证明了具有nef反典范丛的复三重折叠的曲线计数不变量由局部曲线的值决定，从而在初级插入下为所有此类三重折叠（含Calabi--Yau三重折叠）确立了MNOP猜想，关键技术是复流形中全纯曲线的一般横截性结果。

关于具有最小非平凡单值群的三维流形

On threefolds with the smallest nontrivial monodromy group

本文利用伴随理论与SGA7命题，完整分类了光滑超平面截面上H^2单值群为Z/2Z的光滑三维流形，证实了Zak于1991年提出的分类可能性。

On irregular threefolds and fourfolds with numerically trivial canonical bundle

We prove that for a smooth projective irregular $3$-fold $X$ with $K_X\equiv 0$ and a nef and big divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\geq 3$ and all $P\in \text{Pic}^0(X)$. We also use the same method to deal with $4$-folds, and prove that for a smooth projective irregular $4$-fold $X$ with $K_X\equiv 0$ and an ample divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\geq 5$ and all $P\in \text{Pic}^0(X)$. These results are also optimal.

Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links

We characterise smooth curves in P^3 whose blow-up produces a threefold with anticanonical divisor big and nef. These are curves C of degree d and genus g lying on a smooth quartic, such that (i) $4d-30 \le g\le 14$ or $(g,d) = (19,12)$, (ii) there is no 5-secant line, 9-secant conic, nor 13-secant twisted cubic to C. This generalises the classical similar situation for the blow-up of points in P^2. We describe then Sarkisov links constructed from these blow-ups, and are able to prove the existence of Sarkisov links which were previously only known as numerical possibilities.

Nef line bundles over finite fields

We use Totaro's examples of non-semiample nef line bundles on smooth projective surfaces over finite fields to construct nef line bundles for which the first cohomology group cannot be killed by any generically finite covers. This is used to show a similar example of a nef and big line bundle on a smooth projective threefold over a finite field. This improves some examples of Bhatt and answers some of his questions. Finally, we prove a new vanishing theorem for the first cohomology group of strictly nef line bundles on projective varieties defined over finite fields.

A Kawamata-Viehweg Vanishing Theorem on compact Kahler manifolds

We prove a Kawamata-Viehweg vanishing theorem on a normal compact Kahler space X: if L is a nef line bundle with numerical dimension at least equal to 2, then the q-th cohomology group of K_X+L vanishes for q at least equal to the dimension of X minus 1. As an application, a special case of the abundance conjecture for minimal Kahler threefolds is proven: if X is a minimal Kahler threefold (in the usual sense that K_X is nef), then the Kodaira dimension kappa(X) is nonnegative.

法诺三维簇的导出范畴与退化

Derived categories of Fano threefolds and degenerations

本文利用奇点的范畴吸收技术，证明了特定度数德尔佩佐三维簇导出范畴非平凡分支及相关分解可光滑形变至对应亏格素法诺三维簇导出范畴非平凡分支，修正并证明了相关猜想，还描述了模空间紧化及边界分支。

On a conjecture of Cox and Katz

This note shows that a certain toric quotient of the quintic Calabi-Yau threefold in projective four-space provides a counterexample to a recent conjecture of Cox and Katz concerning nef cones of toric hypersurfaces.

Conic bundles with nontrivial unramified Brauer group over threefolds

We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over P^3 where this formula applies and which have nontrivial unramified Brauer group. The construction uses the theory of contact surfaces and, at least implicitly, matrix factorizations and symmetric arithmetic Cohen--Macaulay sheaves, as well as the geometry of special arrangements of rational curves in P^2. We also prove the existence of universally CH_0-trivial resolutions for the general class…

Remarks on Ein-Lazarsfeld criterion of spannedness of adjoint bundles of polarized threefolds

Let B be a nef and big line bundle on a smooth complex threefold X with canonical bundle K. Let x be a point on X and suppose that BC\ge3 for any curve C passing x, B^2S\ge7 for any surface S containing x, and B^3\ge51. Then K+B is spanned at x. (Ein-Lazarsfeld proved the assertion assuming B^3\ge92.) Corollary: K+3L is spanned if L is an ample line bundle with L^3>1.

环面簇的像与弱Fano三维流形的放大自同态

Images of toric variety and amplified endomorphism of weak Fano threefolds

证明Picard秩为2的弱Fano三维流形若为射影环面簇的像则是环面的，并证明具有放大自同态的此类流形也是环面的。

关于紧Kähler三维簇的对数丰度 II

On the Log Abundance for Compact {K{ä}hler} threefolds II

该研究证明了数值维数为2时，具nef对数典范除子的紧Kähler三维簇对的半丰沛性，结合前期工作完整解决了这类三维簇的对数丰度问题。

There Exist Nontrivial Threefolds with Vanishing Hodge Cohomology

We analyze the structure of the algebraic manifolds $Y$ of dimension 3 with $H^i(Y, Ω^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$, by showing the deformation invariant of some open surfaces. Secondly, we show when a smooth threefold with nonconstant regular functions satisfies the vanishing Hodge cohomology. As an application, we prove the existence of nonaffine and nonproduct threefolds $Y$ with this property by constructing a family of a certain type of open surfaces parametrized by the affine curve $\C-\{0\}$ such that the corresponding smooth completion $X$ has Kodai…

Notes on very ample vector bundles on 3-folds

Let $\Cal E$ be a very ample vector bundle of rank two on a smooth complex projective threefold $X$. An inequality about the third Segre class of $\Cal E$ is provided when $K_X+\det \Cal E$ is nef but not big, and when a suitable positive multiple of $K_X+\det \Cal E$ defines a morphism $X\to B$ with connected fibers onto a smooth projective curve $B$, where $K_X$ is the canonical bundle of $X$. As an application, the case where the genus of $B$ is positive and $\Cal E$ has a global section whose zero locus is a smooth hyperelliptic curve of genus $\geq 2$ is investigated, and our previous res…

Fibration and classification of smooth projective toric varieties of low Picard number

In this paper we show that a smooth toric variety $X$ of Picard number $r\leq 3$ always admits a nef primitive collection supported on a hyperplane admitting non-trivial intersection with the cone $\Nef(X)$ of numerically effective divisors and cutting a facet of the pseudo-effective cone $\Eff(X)$, that is $\Nef(X)\cap\partial\overline{\Eff}(X)\neq\{0\}$. In particular this means that $X$ admits non-trivial and non-big numerically effective divisors. Geometrically this guarantees the existence of a fiber type contraction morphism over a smooth toric variety of dimension and Picard number lowe…

正特征下素法诺三维簇的向井（Mukai）丛与布里尔-诺特（Brill-Noether）一般性

Mukai bundles and Brill-Noether generality for prime Fano threefolds in positive characteristic

针对正特征下亏格≥8的素法诺三维簇，证明其存在对应亏格非平凡分解的唯一向井丛，核心方法是证明其光滑超平面截面为布里尔-诺特一般，丰富了代数几何中相关簇的结构理论。

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。

${\bf C}_+$-actions on contractible threefolds

Let $X$ be a smooth contractible affine algebraic threefold with a nontrivial algebraic ${\bf C}_+$-action on it. We show that $X$ is rational and the algebraic quotient $X//{\bf C}_+$ is a smooth contractible surface $S$ which is isomorphic to ${\bf C}^2$ in the case when $X$ admits a dominant morphism from a threefold of form $C \times {\bf C}^2$. Furthermore, if the action is free then $X$ is isomorphic to $S \times {\bf C}$ and the action is induced by translation on the second factor. In particular, we have the following criterion: if a smooth contractible affine algebraic threefold $X$ w…

Automorphisms of blowups of threefolds being Fano or having Picard number $1$

Let $X_0$ be a smooth projective threefold which is Fano or which has Picard number $1$. Let $π:X\rightarrow X_0$ be a finite composition of blowups along smooth centers. We show that for "almost all" of such $X$, if $f\in Aut(X)$ then its first and second dynamical degrees are the same. We also construct many examples of finite blowups $X\rightarrow X_0$, on which any automorphism is of zero entropy. The main idea is that because of the log-concavity of dynamical systems and the invariance of Chern classes under holomorphic automorphisms, there are some constraints on the nef cohomology class…

The 14th case VHS via K3 fibrations

We present a study of certain singular one-parameter subfamilies of Calabi-Yau threefolds realized as anticanonical hypersurfaces or complete intersections in toric varieties. Our attention to these families is motivated by the Doran-Morgan classification of variations of Hodge structure which can underlie families of Calabi-Yau threefolds with $h^{2,1} = 1$ over the thrice-punctured sphere. We explore their torically induced fibrations by $M$-polarized K3 surfaces and use these fibrations to construct an explicit geometric transition between an anticanonical hypersurface and a nef complete in…

On the canonical bundle formula and log abundance in positive characteristic

We show that a weak version of the canonical bundle formula holds for fibrations of relative dimension one. We provide various applications thereof, for instance, using the recent result of Xu and Zhang, we prove the log non-vanishing conjecture for three-dimensional klt pairs over any algebraically closed field $k$ of characteristic $p>5$. We also show the log abundance conjecture for threefolds over $k$ when the nef dimension is not maximal, and the base point free theorem for threefolds over the algebraic closure of any finite field of characteristic $p>2$.

半 log canonical 丰度在紧 Kähler 三维流形上的失效

Abundance for semi log canonical compact Kähler threefolds

本文展示紧 Kähler 变量在三维时半 log canonical 丰度失效，构造了反例表明 K_X 非负但 κ 为零，而 K_X 不是半 ample。同时证明若起始为紧 Kähler 半 dlt 对，则丰度成立。

关于反典范丛的两个非消失性结果

Two Non-Vanishing results concerning the Anti-Canonical Bundle

本文研究了klt三维fold对在反log典范丛为nef时的非消失性问题，证明了其Kodaira维数非负，并通过建立任意维度均成立的等变非消失性结果来实现这一目标。

Fano 三流形与亏格 9 的 K3 曲面上的层模空间

Moduli spaces of sheaves on Fano threefolds and K3 surfaces of genus 9

本文利用射影对偶性研究亏格9的Fano三流形上秩2稳定层到反典范K3曲面的限制，构建了有理Lagrange纤维化，并通过Bridgeland稳定性条件展示了其所有K-平凡双有理模型及锥结构。

Artin-Mazur heights and Yobuko heights of proper log smooth schemes of Cartier type, and Hodge-Witt decompositions and Chow groups of quasi-$F$-split threefolds

In this article we prove a fundamental inequality between Artin-Mazur heights and Yobuko heights of certain proper log smooth schemes of Cartier type over a fine log scheme whose underlying scheme is the spectrum of a perfect field $κ$ of characteristic $p>0$. We also prove that the cohomologies of Witt-sheaves of them are finitely generated ${\cal W}(κ)$-modules if the Yobuko heights of them are finite. As an application, we prove that the $p$-primary torsion parts of the Chow groups of codimension $2$ of proper smooth threefolds over $κ$ is of finite cotype if the Yobuko heights of them are …

广义完全交Calabi-Yau三折叠的镜像对称性(II)：$[3,-1]$块、VGIT与环面相位

Mirror symmetry for gCICY threefolds (II): $[3,-1]$ blocks, VGIT, and toric phases

本文研究具有$[3,-1]$块的gCICY三折叠的镜像对称性，通过VGIT将其约化为Batyrev-Borisov对偶，并构造显式镜像及计算周期系统。

On threefolds without nonconstant regular functions

We consider smooth threefolds $Y$ defined over $\Bbb{C}$ with $H^i(Y, Ω^j_Y)=0$ for all $j\geq 0$, $i>0$. Let $X$ be a smooth projective threefold containing $Y$ and $D$ be the boundary divisor with support $X-Y$. We are interested in the following question: What geometry information of $X$ can be obtained from the regular function information on $Y$? Suppose that the boundary $X-Y$ is a smooth projective surface. In this paper, we analyse two different cases, i.e., there are no nonconstant regular functions on $Y$ or there are lots of regular functions on $Y$. More precisely, if $H^0(Y, {\mat…

On the canonical line bundle and negative holomorphic sectional curvature

We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, ampleness is again obtained. The methods used come from both complex differential geometry and complex algebraic geometry.