Thomas Peternell, Fernando Serrano
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.
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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.
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.
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 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]
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.
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.
We investigate the structure of smooth projective 3-folds X with -K_X nef and K_X^3=0.
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.
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。
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 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…
We classify all Gorenstein Fano threefolds with at worst canonical singularities for which the anticanonical system has a nonempty base locus.
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。
We treat threefolds with divisorial contractions whose exceptional divisors contract to compound Du Val points. We prove that general elements in their anticanonical systems around the exceptional divisors have at worst Du Val singularities. As applications to classification, we describe divisorial contractions to compound A_n points, and moreover we conclude that discrepancies of divisorial contractions to compound D_n or E_n points are at most four.
Let X be the blow-up of the three dimensional complex projective space along r general points of a smooth elliptic quartic curve B of P^3 and let L be any line bundle of X. The aim of this paper is to provide an explicit algorithm for determining the dimension of H^0(X,L).
In this paper, we prove the Bogomolov-Gieseker type inequality conjecture for threefolds with nef tangent bundles. As a corollary, there exist Bridgeland stability conditions on these threefolds.
We present a smooth, complete toric threefold with no nontrivial nef line bundles. This is a counterexample to a recent conjecture of Fujino.
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。
We prove that a nef line bundle $\mathcal L$ with $c_1(\mathcal L)^2 \ne 0$ on a Calabi-Yau threefold $X$ with Picard number $2$ and with $c_3(X) \ne 0$ is semiample, that is, some multiple of $\mathcal L$ is generated by global sections.
On anticanonical volumes of weak $\mathbb{Q}$-Fano terminal threefolds of Picard rank two
本文证明Picard秩为2的弱$\mathbb{Q}$-Fano terminal三维簇的反典范体积上界为72，并确定等号成立的唯一情形，为一般canonical Fano三维簇的最优上界问题提供了策略。
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.
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。
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 $0$.
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.
We show that the nefness of the canonical bundle of compact Kähler threefolds is invariant under deformed symplectic diffeomorphisms.
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.
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。
Given $ε>0$, we show that over an algebraically closed field of characteristic $p>5$, the anticanonical volume of a Fano threefold $X$ (with arbitrary singularities) whose anticanonical divisor has Seshadri constant $ε(-K_X,x)>2+ε$ at some smooth point $x\in X$ is bounded from above.
A Supplement to the anticanonical Volumes of weak $\mathbb{Q}$-Fano threefolds of Picard rank two
本文补充证明了Picard秩不超过2的弱Q-Fano三维簇的反典范体积要么不超过64，要么等于72且为特定射影丛，完善了先前结果。
We discuss in this note which K3 surfaces appear as anticanonical divisors in a Fano threefold. We prove in particular that a general K3 surface with given Picard lattice P and polarization class h in P is an anticanonical divisor in a Fano threefold if and only if (P,h) is isomorphic to (Pic(V), c_1(V)) for some Fano threefold V, where Pic(V) is equipped with the intersection product (L,M) --> (L.M.c_1(V)).
The map given by the anticanonical bundle of a Fano manifold is investigated with respect to a number of natural notions of higher order embeddings of projective manifolds. This is of importance in the understanding of higher order embeddings of the special varieties of adjunction theory, which are usually fibered by special Fano manifolds. An analysis is carried out of the higher order embeddings of the special varieties of adjunction theory that arise in the study of the first and second reductions. Special attention is given to determining the order of the anticanonical embeddings of the th…
We prove the existence of a Mori contraction on a compact Kaehler threefold whose canonical bundle is (analytically) not nef if the threefold can be approximated by projective threefolds or if the algebraic dimension is 2.
Let X be a compact Kaehler threefold with terminal singularities such that K\_X is nef. We prove that K\_X is semiample.
We classify Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3-folds with canonical Gorenstein singularities which are not intersections of quadrics. We also study the rationality questions for most of these varieties.
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…
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.
Degrees of non-Gorenstein canonical Fano threefolds with Picard number one
研究了Picard数为一的非Gorenstein Q-极小canonical Fano三维簇的反canonical次数上界，得出最佳上界为200/3。
We study the deformation theory of a Q-Fano 3-fold with only terminal singularities. First, we show that the Kuranishi space of a Q-Fano 3-fold is smooth. Second, we show that every Q-Fano 3-fold with only "ordinary" terminal singularities is Q-smoothable, that is, it can be deformed to a Q-Fano 3-fold with only quotient singularities. Finally, we prove Q-smoothability of a Q-Fano 3-fold assuming the existence of a Du Val anticanonical element. As an application, we get the genus bound for primary Q-Fano 3-folds with Du Val anticanonical elements.
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!
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.
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。
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 $m>0$.
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.
Non-smoothable Calabi-Yau threefolds from reflexive polytopes
本文构造了具有孤立Gorenstein典范奇点且不可光滑化的Calabi-Yau三维流形，利用Altmann形变理论与Corti-Filip-Petracci准则，并在Kreuzer-Skarke分类中识别出约8.27%的多胞形存在此阻碍。
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.
We prove that the anti-pluricanonical map $Φ_{|-mK_{X}|}$ is birational when $m\geq 16$ for 5-fold $X$ whose anticanonical divisor is nef and big.
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。
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