Zhining Liu
We show that the orbifold fundamental group of an effective compact K{ä}hler orbifold with nef anticanonical bundle has polynomial growth, which generalizes M.P ăun's results for manifolds [P ău97, Theorem 1,Theorem 2]
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Let X be a simply connected projective manifold with nef anticanonical bundle. We prove that X is a product of a rationally connected manifold and a manifold with trivial canonical bundle. As an application we describe the MRC fibration of any projective manifold with nef anticanonical bundle.
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.
Let $X$ be a projective manifold such that the anticanonical bundle $-K_X$ is nef. We prove that the Albanese map $p: X \rightarrow Y$ is locally isotrivial. In particular, $p$ is a submersion.
Let X be a compact Kähler manifold such that the anticanonical bundle $-K_X$ is nef. A classical conjecture claims that the Albanese map is submersive. We prove this conjecture if the general fibre is a weak Fano manifold. If X is projective we prove the conjecture also for fibres of dimension at most two.
We prove a structure theorem for projective varieties with nef anticanonical divisors.
In this paper we study normal surfaces whose anticanonical divisors are strictly nef, i.e. (-K)C>0 for every curve C.
Let $X$ be a smooth projective rationally connected threefold with nef anticanonical divisor. We give a classification for the case when $-K_X$ is not semi-ample.
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.
We investigate the structure of smooth projective 3-folds X with -K_X nef and K_X^3=0.
Albanese map for Kähler manifolds with nef anticanonical bundle
本文研究具有 nef 反正则丛的 Kähler 流形的 Albanese 映射结构，证明一般纤维为 Calabi-Yau 时流形本身为 Calabi-Yau，并对 Cao–Höring 关于射影空间纤维情形的结果给出更拓扑的证明。
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 …
In continuation of our paper in Math. Ann. 333 we classify smooth complex projective threefolds X with -K_X big and nef but not ample and Picard number 2, whose anticanonical map is small. We assume also that the Mori contraction of X and of its flop are not both birational.
We study the geometry of projective manifolds whose tangent bundles are nef on sufficiently general curves (i.e. the tangent bundle is generically nef) and show that manifolds whose anticanonical bundles are semi-ample have this property. Furthermore we introduce a notion of sufficient nefness and investigate the relation with manifolds whose anticanonical bundles are nef.
Let $(X, ω_X)$ be a compact Kähler manifold such that the anticanonical bundle $-K_X$ is nef. We prove that the slopes of the Harder-Narasimhan filtration of the tangent bundle with respect to a polarization of the form $ω_X^{n-1}$ are semi-positive. As an application, we give a characterization of rationally connected compact Kähler manifolds with nef anticanonical bundles. As another application, we give a simple proof of the surjectivity of the Albanese map.
Let $X$ be a smooth complex projective rationally connected threefold with $-K_X$ nef and not semi-ample. In our previous work, we classified all such threefolds when $|{-}K_X|$ has no fixed divisor. In this paper, we continue our classification when $|{-}K_X|$ has a non-zero fixed divisor.
We start the classification of smooth projective threefolds X whose anticanonical bundles -K_X are big and nef but not ample. In this paper we treat the case b_2(X) = 2 and the morphism associated with the base point free linear system |-mK_X|, m>>0, is divisorial.
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.
Varieties with nef anticanonical divisors and Albanese morphisms of relative dimension one in positive characteristic
本文研究正特征代数闭域上具有 nef 反典范除数的光滑射影簇，并在其到 Albanese 簇的态射相对维数为一时给出精确结构。
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。
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。
In our preprint: "Algebraic 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), we had extended the diagram method to algebraic surfaces with nef anticanonical class (before, it was applied to Del Pezzo surfaces). The main problem here is that Mori polyhedron may not be finite polyhedral in this case. In the §4 of this preprint we had shown that this method does work for algebraic surfaces with nef anticanonical class and log terminal singularities. Rece…
Automorphism groups and Homogeneous fibrations on projective varieties with nef anticanonical divisors
本文研究具有 nef 反典范除子的代数簇的 Albanese 态射与极大有理链连通纤维化的齐次结构，构造自然群同态诱导 Aut^0 的 Chevalley 分解（相差等源），并类比 Nishi-Matsumura 定理，用现代语言重证之。
Kawamata--Miyaoka type inequality for varieties with nef anticanonical divisor
针对n≥2的Q-因子ε-lc弱Fano簇，证明了满足特定系数约束的川又-宫冈型不等式，丰富了代数几何中Fano簇的相关理论。
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。
The paper consists of three parts. In the first of them different kinds stability are discussed. In particular, the stability concept with respect to nef divisor is introduced. A structure of rigid and superrigid vector bundles on smooth projective surfaces with nef anticanonical class is studied in the second part. We prove that any superrigid bundle has a unique exceptional filtration. In the last part we give constructive description of exceptional bundles on these surfaces.
Positivity of the tangent bundle of rational surfaces with nef anticanonical divisor
本文研究了具有nef反典范除子的光滑射影有理曲面切丛的bigness性质，证明了有理椭圆曲面的切丛不是大的，并利用$(-2)$-曲线配置完全或部分确定了弱del Pezzo曲面切丛的bigness。
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。
We show that the tangent bundle of a projective manifold with nef anticanonical class is generically nef. That is, its restriction to a curve cut out by general sufficiently ample divisors is a nef vector bundle. This confirms a conjecture of Peternell. As a consequence, the second Chern class of such a manifold has non-negative intersections with ample divisors. We also investigate under which conditions these positivities are strict, and answer a question of Yau.
We prove the general diagram method theorem valid for the quite large class of 3-folds with Q-factorial singularities (see Basic Theorem 1.3.2 and also Theorem 2.2.6). This gives the generalization of our results about Fano 3-folds with Q-factorial terminal singularities (Preprint alg-geom/9311007). As an application, we get the following result about 3-dimensional Calabi-Yau manifolds X: Assume that the Picard number ρ(X) > 93. Then one of two cases (i) or (ii) holds: (i) there exists a small extremal ray on X; (ii) There exists a nef element h such that h^3=0 (thus, the nef cone NEF(X) and t…
A criterion for associated bundles and the weak Beauville--Bogomolov decomposition
本文给出纤维丛成为伴随纤维丛的判别准则，证明满足特定条件的满射纤维丛均为伴随纤维丛，进而推出丰反典范除子代数簇的弱博韦尔-博戈莫洛夫分解。
Admissible HYM metrics on klt KE varieties and the MY equality for big anticanonical K-stable varieties
本文研究了Kähler-Einstein变种上反射sheaf的稳定性、K稳定变种的Miyaoka-Yau不等式等式以及holomorphic向量包的semistability性质。
Constructing Rational Curves via Jets on Projective Varieties with Non-Nef Canonical Bundle
本文通过射流技术给出Miyaoka--Mori判据的代数证明，构造通过负典范曲线上任意点的有理曲线，并结合BDPP锥对偶性恢复射影无理性判据。
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。
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