Manifolds with nef cotangent bundle

Arithmetically nef line bundles

Let $L$ be a line bundle on a scheme $X$, proper over a field. The property of $L$ being nef can sometimes be "thickened", allowing reductions to positive characteristic. We call such line bundles arithmetically nef. It is known that a line bundle $L$ may be nef, but not arithmetically nef. We show that $L$ is arithmetically nef if and only if its restriction to its stable base locus is arithmetically nef. Consequently, if $L$ is nef and its stable base locus has dimension $1$ or less, then $L$ is arithmetically nef.

Fano n-folds with nef tangent bundle and Picard number greater than n-5

We prove that Fano n-folds with nef tangent bundle and Picard number greater than n-5 are rational homogeneous manifolds.

A smooth, complete threefold with no nontrivial nef line bundles

We present a smooth, complete toric threefold with no nontrivial nef line bundles. This is a counterexample to a recent conjecture of Fujino.

Nef reduction and anticanonical bundles

We investigate the structure of smooth projective 3-folds X with -K_X nef and K_X^3=0.

On projective space bundle with nef normalized tautological divisor

In this paper, we study the structure of projective space bundles whose relative anti-canonical line bundle is nef. As an application, we get a characterization of abelian varieties up to finite etale covering.

Toric Fano manifolds with nef tangent bundles

In this note we prove that any toric Fano manifold with nef tangent bundle is a product of projective spaces. In particular, it implies that Campana-Peternell conjecture hold for toric manifolds.

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.

Stability conditions on threefolds with nef tangent bundles

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.

Strictly nef Bundles

In this short note we will show that every homogeneous strictly nef vector bundle on a complex flag variety is ample. Following this, we consider whether ampleness of a bundle on an abelian variety can be tested on curves.

Fano 5-folds with nef tangent bundles and Picard numbers greater than one

We prove that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one are rational homogeneous manifolds.

Nef cone of flag bundles over a curve

Let $X$ be a smooth projective curve defined over an algebraically closed field $k$, and let $E$ be a vector bundle on $X$. We compute the nef cone of any flag bundle associated to $E$.

On foliations with nef anti-canonical bundle

In this paper we prove that the anti-canonical bundle of a holomorphic foliation $\mathcal{F}$ on a complex projective manifold cannot be nef and big if either $\mathcal{F}$ is regular, or $\mathcal{F}$ has a compact leaf. Then we address codimension one regular foliations whose anti-canonical bundle is nef with maximal Kodaira dimension.

第一陈类为$(2,2)$的二次曲面上的Nef向量丛

Nef vector bundles on a quadric surface with first Chern class $(2,2)$

本研究针对特征零代数闭域上光滑二次曲面，完成了第一陈类为$(2,2)$的nef向量丛的分类工作，填补了相关代数几何研究的内容。

抛物 ample 和抛物 nef 向量丛的逆像与直接像

Pullback and Direct Image of Parabolic Ample and Parabolic Nef Vector Bundles

本文研究了抛物 ample 和抛物 nef 向量丛在有限满射映射下的逆像与直接像性质，证明其保持抛物 ample 和抛物 nef 的特性。

第一陈类为2的超二次曲面上的Nef向量丛

Nef vector bundles on a hyperquadric with first Chern class two

本研究在特征零的代数闭域下，完成了维数不小于4的光滑超二次曲面上第一陈类为2的nef向量丛的分类工作。

Generically nef vector bundles on ruled surfaces

The present paper concerns the invariants of generically nef vector bundles on ruled surfaces. By Mehta - Ramanathan Restriction Theorem and by Miyaoka characterization of semistable vector bundles on a curve, the generic nefness can be considered as a weak form of semistability. We establish a Bogomolov type inequality for generically nef vector bundles with nef general fiber restriction on ruled surfaces with no negative section. This gives an affermative answer in this case to a problem posed by Th. Peternell. Concerning ruled surfaces with a negative section, we prove a a similar result fo…

Nef vector bundles on a projective space with first Chern class three

We classify nef vector bundles on a projective space with first Chern class three over an algebraically closed field of characteristic zero; we see, in particular, that these nef vector bundles are globally generated if the second Chern class is less than eight, and that there exist nef but non-globally generated vector bundles with second Chern class eight and nine on a projective plane.

Nef line bundles on Calabi-Yau threefolds, I

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.

Nef and abundant divisors, semiampleness and canonical bundle formula

In this paper, we use canonical bundle formulas to prove some generalizations of an old theorem of Kawamata on the semiampleness of nef and abundant log canonical divisors. In particular, we show that for klt pairs $(X,B)$ with $K_X+B$ effective, $L \in Pic (X)$ nef, nefness and abundance of $K_X+B+L$ implies semiampleness. This essentially generalizes Kawamata's theorem to the setting of generalized abundance.

Base divisors of big and nef line bundles on irreducible symplectic varieties

Under some conditions on the deformation type, which we expect to be satisfied for arbitrary irreducible symplectic varieties, we describe which big and nef line bundles on irreducible symplectic varieties have base divisors. In particular, we show that such base divisors are always irreducible and reduced. This is applied to understand the behaviour of divisorial base components of big and nef line bundles under deformations and for K3$^{[n]}$-type and Kum$^n$-type.

A remark on compact Kähler manifolds with nef anticanonical bundles and its applications

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.

Nef vector bundles on a projective space with first Chern class 3 and second Chern class 8

We describe nef vector bundles on a projective space with first Chern class three and second Chern class eight over an algebraically closed field of characteristic zero by giving them a minimal resolution in terms of a full strong exceptional collection of line bundles.

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。

A reduction map for nef line bundles

In a recent preprint, H. Tsuji gave a number of interesting assertions on the structure of pseudo-effective line bundles on projective manifolds. In particular, he postulated the existence of an almost-holomorphic "reduction map", whose fibers are maximal subvarieties on which the line bundle is numerically trivial. The purpose of this note is to give a simple, algebraic proof of the existence of a reduction map in the case where the line bundle is nef.

Threefolds with big and nef anticanonical bundles II

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.

Threefolds with big and nef anticanonical bundles I

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.

Semistability vs. nefness for (Higgs) vector bundles

According to Miyaoka, a vector bundle E on a smooth projective curve is semistable if and only if a certain numerical class in the projectivized bundle PE is nef. We establish a similar criterion for the semistability of Higgs bundles: namely, such a bundle is semistable if and only if for every integer s between 0 and the rank of E, a suitable numerical class in the scheme parametrizing the rank s locally-free Higgs quotients of E is nef. We also extend this result to higher-dimensional complex projective varieties by showing that the nefness of the above mentioned classes is equivalent to th…

On The Projective Normality And Normal Presentation On Higher Dimensional Varieties With Nef Canonical Bundle

In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results.

Projective varieties with nef tangent bundle in positive characteristic

Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties with nef tangent bundle and $T_M$ is numerically flat. We also prove that extremal contractions exist as smooth morphisms. As an application, we prove that, if the Frobenius morphism can be lifted modulo $p^2$, then $X$ admits, up to a finite étale Galois cover, a smooth morphism onto an ordinary abelian variety whose fibers are products of projective spaces.

A Nonvanishing Conjecture for Cotangent Bundles

In this paper we study the positivity of the cotangent bundle of projective manifolds. We conjecture that the cotangent bundle is pseudoeffective if and only the manifold has non-zero symmetric differentials. We confirm this conjecture for most projective surfaces that are not of general type.

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.

Nef cones of projective bundles over surfaces and Seshadri constants

In this article, we give a description of the closed cone of curves of the projective bundle $\mathbb{P}(E)$ over a smooth projective variety $X$. Using duality, we then calculate the nef cone of divisors in $\mathbb{P}(E)$ over some special surfaces $X$ and for some special bundles on $X$. As an application, we also calculate the Seshadri constants of semistable ample vector bundles with vanishing discriminant on some special ruled surfaces at special points.

Complete intersection varieties with ample cotangent bundles

Any smooth projective variety contains many complete intersection subvarieties with ample cotangent bundles, of each dimension up to half its own dimension.

On the non-divisorial base locus of big and nef line bundles of K3^{[2]}-type varieties

We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized …

Projective manifolds containing a large linear subspace with nef normal bundle

We classify smooth complex projective varieties $X \subset \proj^N$ of dimension $2s+1$ containing a linear subspace $Λ$ of dimension $s$ whose normal bundle $N_{Λ/X}$ is numerically effective.

The Cotangent Bundle of a Cominuscule Grassmannian

A theorem of the first author states that the cotangent bundle of the type $A$ Grassmannian variety can be embedded as an open subset of a smooth Schubert variety in a two-step affine partial flag variety. We extend this result to cotangent bundles of cominuscule generalized Grassmannians of arbitrary Lie type.

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。

Nef and big divisors on toric weak Fano 3-folds

We show that a nef and big line bundle whose adjoint bundle has non-zero global sections on a nonsingular toric weak Fano 3-fold is normally generated. As a consequence, we see that all ample line bundles on a nonsingular toric weak Fano 3-fold are normally generated. We also show that an ample line bundle whose adjoint bundle has non-zero global sections on a Gorensein toric Fano 3-fold is normally generated. In our proof we do not use full classifications of Fano polytopes.

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。

Twisted cotangent sheaves and a Kobayashi-Ochiai theorem for foliations

Let X be a normal projective variety, and let A be an ample Cartier divisor on X. We prove that the twisted cotangent sheaf $Ω_X \otimes A$ is generically nef with respect to the polarisation A unless X is a projective space. As an application we prove a Kobayashi-Ochiai theorem for foliations: if $F \subsetneq T_X$ is a foliation of rank r such that $det F \equiv i_F A$, then we have $i_F \leq r$.

Nef cone and Seshadri constants on products of projective bundles over curves

Let $X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$ where $C$ is a smooth curve and let $E_1$, $E_2$ be vector bundles over $C$. In this paper, we extend the results in \cite{K-M-R} by computing the nef cone of $X$ without restriction on the rank or semistability of $E_1$ and $E_2$. We also study the Seshadri constants of ample line bundles on $X$. We calculate the Seshadri constants in some cases and give bounds in some of the remaining cases.

Varieties With Ample Cotangent Bundle

We study smooth projective complex varieties with ample cotangent bundle. Our main result is that in an abelian variety of dimension n, a complete intersection of at least n/2 general hypersurfaces of sufficiently high degrees has ample cotangent bundle. We discuss the conjecture that the analogous statement should hold in the projective space. Finally, we present a construction due to Bogomolov of varieties with ample cotangent bundle as linear sections of a product of varieties with big cotangent bundle.

Resolutions of surfaces with big cotangent bundle and $A_2$ singularities

We give a new criterion for when a resolution of a surface of general type with canonical singularities has big cotangent bundle and a new lower bound for the values of $d$ for which there is a surface with big cotangent bundle that is deformation equivalent to a smooth hypersurface in $\mathbb{P}^3$ of degree $d$.

Cotangent bundle to the Grassmann variety

We show that there is an affine Schubert variety in the infinite dimensional partial Flag variety (associated to the two- step parabolic subgroup of the Kac-Moody group {\hat SL(n)}, corresponding to omitting α_0,α_d) which is a natural compactification of the cotangent bundle to the Grassmann variety.

A Remark on Projective Embeddings of Varieties with Non-Negative Cotangent Bundles

We prove an elementary but somewhat unexpected result about projective embeddings of smooth varieties X whose cotangent bundles are numerically effective. Specifically, we show that the degree of X in any projective embedding must grow (almost) exponentially in the dimension of X.

Extremal rays and nefness of tangent bundles

In view of Mori theory, rational homogenous manifolds satisfy a recursive condition: every elementary contraction is a rational homogeneous fibration and the image of any elementary contraction also satisfies the same property. In this paper, we show that a smooth Fano $n$-fold with the same condition and Picard number greater than $n-6$ is either a rational homogeneous manifold or the product of $n-7$ copies of $\mathbb{P}^1$ and a Fano $7$-fold $X_0$ constructed by G. Ottaviani. We also clarify that $X_0$ has non-nef tangent bundle and in particular is not rational homogeneous.

CM线丛及沿子概形的big和nef线丛的斜率$\K$-半稳定性

CM-line bundles and slope $\K$-semistability for big and nef line bundles along subschemes

本文利用Chow-Mumford线丛理论，给出了沿子概形及big和nef线丛的Donaldson-Futaki $\K$-稳定性不变量的显式相交理论描述，推广了Ross-Thomas斜率稳定性理论，并利用了线丛关于丰富锥的连续性。

椭圆曲面上余切丛的非消失猜想

A non-vanishing conjecture for cotangent bundles on elliptic surfaces

本文证明了等平凡椭圆曲面上余切丛的非消失猜想，并结合已有结果完全解决了Kodaira维数至多1的曲面上的该问题。

Basepoint freeness for nef and big line bundles in positive characteristic

A necessary and sufficient condition is given for semi-ampleness of a numerically effective (nef) and big line bundle in positive characteristic. One application is to the geometry of the universal stable curve over M_g, specifically, the semi-ampleness of the relative dualizing sheaf, in positive characteristic. An example is given which shows this and the semi-ampleness criterion fail in characteristic zero. A second application is to Mori's program for minimal models of 3-folds in positive characteristic, namely, to the existence of birational extremal contractions.

On Chern ratios for surfaces with ample cotangent bundle

In this paper we study the problem of density in (1,3] for the Chern ratio of surfaces with ample cotangent bundle. In particular we prove density in (1,2) by constructing a family of complete intersection surfaces in a product of varieties with big cotangent bundle. We also analyse the case of complete intersections in a product of curves of genus at least 2.

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,…

The role of the cotangent bundle in resolving ideals of fat points in the plane

We study the connection between the generation of a fat point scheme supported at general points in the plane and the behaviour of the cotangent bundle with respect to some rational curves particularly relevant for the scheme. We put forward two conjectures, giving examples and partial results in support of them.

Varieties in positive characteristic with numerically flat log cotangent bundle

In this paper, we prove that a smooth projective globally $F$-split variety with numerically flat tangent bundle is an étale quotient of an ordinary abelian variety. We also show its logarithmic analog, which contains a characterization of toric varieties. We further prove that, without assumption of global $F$-splitting, a smooth projective separably rationally connected variety of arbitrary characteristic with numerically flat log cotangent bundle is a toric variety.

Geometric stability of the cotangent bundle and the universal cover of a projective manifold

Consider a projective manifold X and suppose that some wedge power of the cotangent bundle contains a subsheaf whose determinant bundle has maximal Kodaira dimension. Then we prove that X is of general type. More generally we compute the Kodaira dimension if the determinant bundle has sufficiently large Kodaira dimension. This is based on the study of the determinant bundle of a quotient of the cotangent bundle of a non-uniruled manifold: this bundle is always pseudo-effective. We apply this to study the universal cover of a projective manifold. Finally we prove the following: if the canonical…