Frederic Campana, Thomas Peternell
We investigate the universal cover of projective threefolds whose tangent bundle is a direct sum of subbundles in case the Kodaira dimension is not 1 and 2. We also prove results on Fano manifolds with splitting tangent bundles in any dimension.
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In this short article, we determine the bigness of the tangent bundle $T_X$ of the projective bundle $X=\mathbb{P}_C(E)$ associated to a vector bundle $E$ on a smooth projective curve $C$.
The variety of minimal rational tangents associated to Hecke curves was used by J.-M.Hwang [8] to prove the simplicity of the tangent bundle on the moduli of vector bundles over a curve. In this paper, we use the tangent maps of the symplectic and orthogonal Hecke curves to prove an analogous result for symplectic and orthogonal bundles. In particular, we show the nondegeneracy of the associated variety of minimal rational tangents, which implies the simplicity of the tangent bundle on the moduli spaces of symplectic and orthogonal bundles over a curve. We also show that for large enough genus…
We show that smooth projective horospherical varieties with nef tangent bundles are rational homogeneous spaces.
We prove that Fano 5-folds with nef tangent bundles are rational homogeneous manifolds.
We determine the stability/instability of the tangent bundles of the Fano varieties in a certain class of two orbit varieties, which are classified by Pasquier in 2009. As a consequence, we show that some of these varieties admit unstable tangent bundles, which disproves a conjecture on stability of tangent bundles of Fano manifolds.
In this note, we give a simple necessary condition for the Zariski relative tangent space and the Grothendieck relative tangent space to be isomorphic.
We prove that the tangent bundle of a manifold of K$3^{[2]}$-type is rigid.
We prove that Fano n-folds with nef tangent bundle and Picard number greater than n-5 are rational homogeneous manifolds.
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.
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 give a simple combinatorial proof of the toric version of Mori's theorem that the only $n$-dimensional smooth projective varieties with ample tangent bundle are the projective spaces $\mathbb{P}^n$.
We prove that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one are rational homogeneous manifolds.
In this paper, we classify Fano manifolds with elementary contractions of birational type such that the second or third exterior power of tangent bundles are numerically effective.
In this paper, we classify Fano fourfolds whose the second exterior power of tangent bundles are numerically effective with Picard number greater than one.
Let X be a Fano manifold with Picard number one such that the tangent bundle T_X is big. If X admits a rational curve with trivial normal bundle, we show that X is isomorphic to the del Pezzo threefold of degree five.
The tangent bundle of a generalized Kummer fourfold is obstructed
针对广义库默尔型超凯勒四元簇，构造了切丛的无穷小形变，得到首个具有阻碍形变的稳定超全纯丛例子
We give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope. Furthermore, we show that for a smooth toric surface and a smooth toric variety of Picard rank 2, there exists an ample line bundle with respect to which the tangent bundle is stable if and only if it is an iterated blow-up of projective space.
Positivity of exterior products of tangent bundles and their subsheaves
该论文研究切丛外幂及其子层的正性，证明了相关射影流形的结构定理、丰富子层的刚性定理，并探讨了外幂严格nef的正则叶状结构。
Recently, Kanemitsu has discovered a counterexample to the long-standing conjecture that the tangent bundle of a Fano manifold of Picard number one is (semi)stable. His counterexample is a smooth horospherical variety. There is a weaker conjecture that the tangent bundle of a Fano manifold of Picard number one is simple. We prove that this weaker conjecture is valid for smooth horospherical varieties of Picard number one. Our proof follows from the existence of an irreducible family of unbendable rational curves whose tangent vectors span the tangent spaces of the horospherical variety at gene…
Positivity of tangent bundle of weak Del Pezzo surfaces of degree $4$
本文证明了度数至少为4的弱Del Pezzo曲面的切包几乎是 nef 的，通过利用由二次面束结构诱导的总双 VMRTs 进行证明。
In this note we give a complete description of all the hyperplane section of the projective bundle associated to the tangent bundle of $\mathbb{P}^2$ under its natural embedding in $\mathbb{P}^7.$ As an application one obtains a description of all possible deformations, in $\text{SL_3(\mathbb{C}}/B,$ of the co-dimension one sub-scheme which is the union of two fundamental Schubert divisors.
In this paper we show that a uniruled manifold with a split tangent bundle admits almost holomorphic fibrations that are related to the splitting. We analyse these fibrations in detail in several special cases, this yields new results about the integrability of the direct factors and the universal covering of the manifold.
We study ample vector bundles on smooth projective stacks. In particular, we prove that the tangent bundle for the weighted projective stack $\mathbb{P}(a_0,...,a_n)$ is ample. A result of Mori shows that the only smooth projective varieties with ample vector bundle are isomorphic to $\mathbb{P}^N$ for some $N$. Extending our geometric spaces from varieties to projective stacks, we are able to provide a new example. This leaves the open question of knowing if the only smooth projective stacks are the weighted projective stacks.
Let $G$ be a complex linear algebraic group which is simple of adjoint type. Let $\overline G$ be the wonderful compactification of $G$. We prove that the tangent bundle of $\overline G$ is stable with respect to every polarization on $\overline G$.
I prove a crystalline characterization of abelian varieties in characteristic $p>0$ amongst the class of varieties with trivial tangent bundle. I show using my characterization that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion free. I also show that a conjecture of Ke-Zheng Li about characteristic $p>0$ varieties with trivial tangent bundles is true for surfaces. I give a new proof of a result of Li and prove a refinement of it. Based on my characterization of abelian varieties I propose mod…
In this article, we give numerical restrictions on the Chern classes of Ulrich bundles on higher-dimensional manifolds, which are inspired by the results of Casnati in the case of surfaces. As a by-product, we prove that the only projective manifolds whose tangent bundle is Ulrich are the twisted cubic and the Veronese surface. Moreover, we prove that the cotangent bundle is never Ulrich.
Smooth projective surfaces with pseudo-effective tangent bundles
该数学代数几何研究刻画了非单直纹光滑射影曲面切丛伪有效的等价条件，并探讨了非有理直纹曲面吹胀的相关性质。
Let M be a complex projective manifold with the property that for any compact Riemann surface C and holomorphic map f: C -> M the pullback of the tangent bundle of M is semistable. We prove that in this case M is a curve or a finite etale quotient of an abelian variety answering a conjecture of I. Biswas.
Let $Y$ be a smooth projective surface defined over an algebraically closed field $k$ with ${\rm Char}\ k\nmid n$, and let $π:X\rightarrow Y$ be a $n$-cyclic covering branched along a smooth divisor $B$. We show that under some conditions $\mathscr{T}_X$ is semi-stable with respect to $π^*\mathcal {H}$ if the tangent bundle $\mathscr{T}_Y$ is semi-stable with respect to some ample line bundle $\mathcal {H}$ on $Y$.
Let $X$ be a Fano manifold. While the properties of the anticanonical divisor $-K_X$ and its multiples have been studied by many authors, the positivity of the tangent bundle $T_X$ is much more elusive. We give a complete characterisation of the pseudoeffectivity of $T_X$ for del Pezzo surfaces, hypersurfaces in the projective space and del Pezzo threefolds.
We give an explicit parametrization of the Hilbert schemes of rational curves C in P^n having a given splitting type of the restricted tangent bundle from P^n to C. The adopted technique uses the description of such curves as projections of a rational normal curve from a suitable linear vertex and a classification of those vertices that correspond to the required splitting type of the restricted tangent bundle. This classification involves the study of a suitable PGL(2) action on the relevant Grassmannian variety.
Isomorphisms and automorphisms of multiprojective bundles and symmetric powers of projective bundles
本文研究射影空间上多重射影丛及射影丛相对对称幂的同构分类问题，并进一步刻画了这些对象的自同构群。
Let X be a compact Kaehler manifold. We expect that any direct sum decomposition of the tangent bundle T(X) comes from a splitting of the universal covering space of X as a product of manifolds, in such a way that the given decomposition of T(X) lifts to the canonical decomposition of the tangent bundle of a product. We prove this assertion when X is a Kaehler-Einstein manifold or a Kaehler surface. Simple examples show that the Kaehler hypothesis is necessary.
For a hypersurface in a projective space, we consider the set of pairs of a point and a line in the projective space such that the line intersects the hypersurface at the point with a fixed multiplicity. We prove that this set of pairs forms a smooth variety for a general hypersurface.
We complete the proof of the fact that the moduli space of rank two bundles with trivial determinant embeds into the linear system of divisors on $Pic^{g-1}C$ which are linearly equivalent to $2Θ$. The embedded tangent space at a semi-stable non-stable bundle $ξ\oplusξ^{-1}$, where $ξ$ is a degree zero line bundle, is shown to consist of those divisors in $|2Θ|$ which contain $Sing(Θ_ξ)$ where $Θ_ξ$ is the translate of $Θ$ by $ξ$. We also obtain geometrical results on the structure of this tangent space.
In this note we study the image of $\PP^2$ in $\text{Gr}(2, \CC^{4})$ given by tangent bundle of $\PP^2. $ We show that there is component $\mathcal{H}$ of the Hibert scheme of surfaces in $\text{Gr}(2, \CC^{4})$ with no point of it corresponds to a smooth surface.
Stability of the canonical extension of tangent bundles on Picard-rank-1 Fano varieties
本文研究Picard数为1的Fano簇上切丛典范扩张的斜率稳定性，证明在指数整除维数或维数加一时切丛稳定性蕴含典范扩张的（半）稳定性，并给出其在曲线稳定向量丛模空间上的应用推论。
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.
Tangent bundle of punctual Hilbert scheme and distinguishing products of varieties
研究光滑项目表面点希尔伯特方案的切包不可分解成分，并证明关于光滑项目表面点希尔伯特方案乘积分类的猜想，同时确定两个对称幂乘积的同构条件。
On the classification of polarized surfaces with weakly Ulrich or initialized aCM tangent bundles
本文分类了具有弱Ulrich或初始化aCM切丛的光滑极化曲面，确定其为度至少5的Del-Pezzo曲面，并给出嵌入分类及两个无限极化系列。
We study the positivity of exterior powers of the tangent sheaf on toric varieties in order to generalize results by Campana and Peternell about 3-folds with nef second exterior power of the tangent bundle. Using the theory of equivariant vector bundles and the toric MMP, we establish in the smooth case a criterion for the positivity of $Λ^m\mathcal T_X$ in terms of wall relations. As an application, we classify smooth toric varieties of arbitrary dimension $n\ge3$ with $Λ^2\mathcal T_X$ nef and those of dimension $n\ge 4$ with $Λ^3\mathcal T_X$ ample.
In this paper we present a way of computing the degree of the secant (resp., tangent) variety of a smooth projective surface, under the assumption that the divisor giving the embedding in the projective space is $3$-very ample. This method exploits the link between these varieties and the Hilbert scheme $0$-dimensional subschemes of length $2$ of the surface.
This paper generalises Mori's famous theorem about "Projective manifolds with ample tangent bundles" to normal projective varieties in the following way: A normal projective variety over $\mathbb{C}$ with ample tangent sheaf is isomorphic to the complex projective space.
We study global deformations of certain projective bundles over projective spaces. We show that any projective global deformation of a projective bundle over $\bP^1$ carries the structure of a projective bundle over some projective space. Furthermore, we construct examples in arbitrary dimension $\ge 3$ of almost homogeneous Fano projective bundles over $\bP^2$ which can be globally deformed to non-almost homogeneous manifolds.
We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane sub-bundle H of a projective space bundle of rank r-1 over the projective line depends only on the the r-fold self-intersection of H . In particular it depends neither on the ambient bundle nor on a particular ample hyperplane sub-bundle with given r-fold self-intersection. Our proof exploits the unexpected property that every such complement comes equipped with the structure of a non trivial torsor under a vector bundle on the affine line with a double origin.
We propose a conjectural list of Fano manifolds of Picard number $1$ with pseudoeffective normalized tangent bundles, which we prove in various situations by relating it to the complete divisibility conjecture of Russo and Zak on varieties with small codegree. Furthermore, the pseudoeffective thresholds and hence the pseudoeffective cones of the projectivized tangent bundles of rational homogeneous spaces of Picard number $1$ are explicitly determined by studying the total dual VMRT and the geometry of stratified Mukai flops. As a by-product, we obtain sharp vanishing theorems on the global tw…
The aim of this note is to investigate the relation between two types of non-singular projective varieties of Picard rank 2, namely the Projective bundles over Projective spaces and certain Blow-up of Projective spaces.
Fano manifolds of Picard number one whose co-tangent bundle is algebraically completely integrable system and its endomorphisms
本文针对皮卡数为一的射影Fano流形，在其余切丛为代数完全可积系统且切丛非nef的条件下证明了非常数自同态必为同构的猜想，并构造了满足余切丛可积条件的射影Fano流形例子，从而为三个已有主要结果提供替代证明。
Let X be a smooth projective surface over C and let L be a line bundle on X generated by its global sections. Let f:X-->P^r be the morphism associated to L and let T be the tangent bundle of P^r; we investigate the μ-stability of f*T with respect to L when X is either a regular surface with p_g=0, a K3 surface or an abelian surface. In particular, we show that it is μ-stable when X is K3 and L is ample and when X is abelian and L^2>13.
Positivity of higher exterior powers of the tangent bundle
该数学代数几何领域研究证明，满足切丛特定高阶外幂严格nef条件的光滑射影簇为Fano簇，并在前两种情形下于ρ(X)≠1假设下完成分类。
Bigness of tangent bundles of blow-ups of ruled surfaces
研究直纹曲面\(S = \mathbb{P}_C(\mathcal{E})\)在点爆破下切丛丰沛性的保持情况，通过第一塞格雷不变量\(s_1(\mathcal{E})\)给出爆破点数的精确界。
Tangent bundles of Quot schemes of vector bundles on curves
本文研究光滑射影曲线上向量丛的 Quot 概形切丛，刻画其不可分解性与单性，并完全描述不可分解分量的自同态环，进而确定两个 Quot 概形乘积的同构条件。
The cohomology of the Hilbert schemes of points on smooth projective surfaces can be approached both with vertex algebra tools and equivariant tools. Using the first tool, we study the existence and the structure of universal formulas for the Chern classes of the tangent bundle over the Hilbert scheme of points on a projective surface. The second tool leads then to nice generating formulas in the particular case of the Hilbert scheme of points on the affine plane.
Isomorphism of Multiprojective Bundles and Projective Towers
本文研究不同维数光滑射影簇上射影丛的同构问题，证明不同射影空间上的多射影丛除平凡情形外不同构，并给出高度3射影塔顶层簇同构的充要条件。
Deformations of the tangent bundle of a projective hypersurface
本文研究了射影超曲面切丛的无穷小形变，证明了其形变空间的维数公式及无阻碍性，并通过构造显式双有理模型证明包含切丛的稳定丛模空间不可约分支是有理簇。
Dynamics of Endomorphisms for Projective Bundles on Elliptic Curves
本文通过丛的转移函数研究椭圆曲线上射影丛满射自同态的动力学，证明了其Kawaguchi--Silverman猜想，并将结果推广至具有限生成Mori锥的光滑射影簇上的射影分裂丛。
An almost Fano bundle is a vector bundle on a smooth projective variety that its projectivization is an almost Fano variety. In this paper, we prove that almost Fano bundles exist only on almost Fano manifolds and study rank 2 almost Fano bundles over projective spaces.
Restricted tangent bundle of rational curves on projective hypersurfaces
本文完全分类了一般射影超曲面包含具有平衡受限切丛的有理曲线的所有次数参数，并给出了有理正规曲线情形下的显式构造方法。
We prove that any Fano manifold of coindex three admitting nef tangent bundle is homogeneous.
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.
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