An elementary proof that vector bundles on $\mathbb P^1$ split

A cohomological splitting criterion for rank 2 vector bundles on Hirzebruch surfaces

In this note, we give a cohomological characterization of all rank 2 split vector bundles on Hirzebruch surfaces.

A Few Splitting Criteria for Vector Bundles

We prove a few splitting criteria for vector bundles on a quadric hypersurface and Grassmannians. We give also some cohomological splitting conditions for rank 2 bundles on multiprojective spaces. The tools are monads and a Beilinson's type spectral sequence generalized by Costa and Miró-Roig.

Splitting criteria for vector bundles on higher dimensional varieties

We generalize Horrocks' criterion for the splitting of vector bundles on projective space. We establish an analogous splitting criterion for vector bundles on a class of smooth complex projective varieties of dimension at least four, over which every extension of line bundles splits.

Splitting of low rank ACM bundles on hypersurfaces of high dimension

Let $X$ be a smooth projective hypersurface. In this note we show that any rank 3 arithmetically Cohen-Macaulay vector bundle over $X$ splits when dim $X \geq 7$. We also find a splitting criterion for rank 4 arithmetically Cohen-Macaulay vector bundles on $X$ when dim $X \geq 9$.

On the loci of morphisms from $\mathbb{P}^1$ to $G(r,n)$ with fixed splitting type of the restricted universal sub-bundle or quotient bundle

Let $n\geq 4$, $2 \leq r \leq n-2$ and $e \geq 1$. We show that the intersection of the locus of degree $e$ morphisms from $\mathbb{P}^1$ to $G(r,n)$ with the restricted universal sub-bundles having a given splitting type and the locus of degree $e$ morphisms with the restricted universal quotient-bundle having a given splitting type is non-empty and generically transverse. As a consequence, we get that the locus of degree $e$ morphisms from $\mathbb{P}^1$ to $G(r,n)$ with the restricted tangent bundle having a given splitting type need not always be irreducible.

Splitting criteria for vector bundles induced by restrictions to divisors

In this article we deduce criteria for the splitting and the triviality of vector bundles, by restricting them to partially ample divisors. This allows to study the problem of splitting on the total space of fibre bundles. The statements are illustrated with a number of examples. For products of minuscule homogeneous varieties, our results allow to test the splitting of vector bundles by restricting them to products of Schubert 2-planes. The triviality criteria obtained inhere are particularly suited to Frobenius split varieties, whose splitting is defined by a section in the anti-canonical li…

A noncombinatorial proof that toric rank 2 bundles on projective space split

Hartshorne's conjecture about vector bundles on projective space states that any rank 2 vector bundle on n-dimensional projective space splits as soon as n is at least 7. Klyachko has shown that Hartshorne's conjecture is true when the vector bundles are torus equivariant. Moreover, recent work of Ilten and Süss generalizes Klyachko's work to the case of a smaller rank torus action on projective space. In this note we give a new, direct proof that torus rank 2 bundles split that avoids a description of the category of torus equivariant vector bundles.

Triviality and Split of Vector Bundles on Rationally Connected Varieties

In this paper, we give a simple proof of a triviality criterion due to I.Biswas and J.Pedro and P.Dos Santos. We also prove a vector bundle on a homogenous space is trivial if and only if the restrictions of the vector bundle to Schubert lines are trivial. Using this result and Chern classes of vector bundles, we give a general criterion of a uniform vector bundle on a homogenous space to be splitting. As an application, we prove a uniform vector bundle on classical Grassmannians and quadrics of low rank is splitting.

The structure of uniruled manifolds with split tangent bundle

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.

Elementary Transformations of Pfaffian Representations of Plane Curves

Let $C$ be a smooth curve in $\PP^2$ given by an equation F=0 of degree $d$. In this paper we consider elementary transformations of linear pfaffian representations of $C$. Elementary transformations can be interpreted as actions on a rank 2 vector bundle on $C$ with canonical determinant and no sections, which corresponds to the cokernel of a pfaffian representation. Every two pfaffian representations of $C$ can be bridged by a finite sequence of elementary transformations. Pfaffian representations and elementary transformations are constructed explicitly. For a smooth quartic, applications t…

On the splitting of Lazarsfeld-Mukai bundles on K3 surfaces II

In this paper, we say that a rank 2 bundle splits if it is given by an extension of two line bundles. In the previous works, we gave a necessary condition for Lazarsfeld-Mukai bundles of rank 2 to split, under a numerical condition ([W2], Theorem 3.1). We gave the splitting types of them on a smooth quartic hypersurface in P3 ([W2], Proposition 3.1) as a corollary of it. However, the assertion of it contains a few mistakes. In this paper, we correct them, and give an application of the results in [W2].

A splitting criterion for rank 2 vector bundles on hypersurfaces in P^4

We show that Horrocks' criterion for the splitting of rank two vector bundles in P^3 can be extended, with some assumptions on the Chern classes, on non singular hypersurfaces in P^4. Extension of other splitting criterion are studied.

A splitting criterion for rank 2 vector bundles on a general sextic threefold

In this paper we show that on a general sextic hypersurface $X\subset \bf P^4$, a rank 2 vector bundle $E$ splits if and only if $h^1(E(n))=0$ for any $n \in \bf Z$. We get thus a characterization of complete intersection curves in $X$

A splitting criterion for vector bundles on blowing ups of the plane

Let $f_s: X_s \to {\bf {P}}^2$ be the blowing-up of $s$ distinct points and $E$ a vector bundle on $X_s$. Here we give a cohomological criterio which is equivalent to $E \cong f_s^\ast (A)$ with $A$ a direct sum of line bundles. We also some cohomological characterizations of very particular rank 2 vector bundles on ${\bf {P}}^2$.

射影超空间上超向量丛的分裂

Splitting of supervector bundles on projective superspaces

该研究给出射影超空间$\boldsymbol{\rm{P}}^{n|m}$上超向量丛的分裂判据，证明n≥2时满足中间上同调消失的秩p|q超向量丛可分解为奇偶线丛直和，并给出n=1时的反例。

Vector bundles on projective varieties which split along q-ample subvarieties

Let Y be a subvariety of a smooth projective variety X, and V a vector bundle on X. Given that the restriction of V to Y splits into a direct sum of line bundles, we ask whether V splits on X. I answer this question in affirmative if holds: Y is a q-ample subvariety of X (for appropriate q), it admits sufficiently many embedded deformations, and is very general within its own deformation space. The result goes beyond the previously known splitting criteria for vector bundles corresponding to restrictions. It allows to treat in a unified way examples arising in totally different situations. I d…

Weakly uniform rank two vector bundles on multiprojective spaces

Here we classify the weakly uniform rank two vector bundles on multiprojective spaces. Moreover we show that every rank $r>2$ weakly uniform vector bundle with splitting type $a_{1,1}=...=a_{r,s}=0$ is trivial and every rank $r>2$ uniform vector bundle with splitting type $a_1>...>a_r$, splits.

Echelon modification of vector bundles

We study a filtered generalization of the operation of elementary modification of vector bundles. The generalization is motivated by applications to the degeneration theory of linear systems.

Complex manifolds with split tangent bundle

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.

Maps between moduli spaces of vector bundles and the base locus of the theta divisor

Given a vector bundle $E$ of rank $r$ and degree $d$ on a curve $C$ of genus $g$, one can associate to $E$ in a natural way several other vector bundles. For example, one can take wedge powers of $E$. If $E$ is generated by global sections, the kernel of the evaluation map of sections is again a vector bundle. Also, new vector bundles can be produced by taking elementary transformations centered at a fixed point. Under suitable conditions on degree and rank, these constructions can be carried out globally. While all this processes seem quite elementary, very little is known about the resulting…

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.

Another proof of Grothendieck's theorem on the splitting of vector bundles on the projective line

This note contains another proof of Grothendieck`s theorem on the splitting of vector bundles on the projective line over a field $k$. Actually the proof is formulated entirely in the classical terms of a lattice $Λ\cong k[T]^d$, discretely embedded into the vector space $V \cong K_\infty^d$, where $K_\infty \cong k((1/T))$ is the completion of the field of rational functions $k(T)$ at the place $\infty$ with the usual valuation.

On the second and third exterior power of tangent bundles of Fano manifolds with birational contractions

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.

向量丛在$\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$上的分裂的上同调准则

A Cohomological criterion for the splitting of vector bundles on $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$

本文提出了一种判断向量丛在乘积射影空间上是否分裂的上同调准则，并推广了在对称乘积射影空间上的消失上同调准则。

关于具有平凡典范丛的拟\(F -\)分裂簇的整体和局部几何

On the global and local geometry of quasi-$F$-split varieties with trivial canonical bundle

研究具有平凡典范丛的拟\(F -\)分裂簇，通过多种方法证明正则拟\(F^{\infty}-\)分裂簇性质、解答相关曲面问题、证明正规簇几何正规性，以及在奇点解消假设下证明特定拟\(F^e -\)纯正规簇是对数典范的。

Splitting criteria for vector bundles on the symplectic isotropic Grassmannian

We extend a theorem of Ottaviani on cohomological splitting criterion for vector bundles over the Grassmannian to the case of the symplectic isotropic Grassmanian. We find necessary and sufficient conditions for the case of the Grassmanian of symplectic isotropic lines. For the general case the generalization of Ottaviani's conditions are sufficient for vector bundles over the symplectic isotropic Grassmannian. By a calculation in the program LiE, we find that Ottaviani's conditions are necessary for Lagrangian Grassmannian of isotropic $k$-planes for $k\leq 6$, but they fail to be necessary f…

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.

$\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$ 上向量丛的上同调分裂条件

Cohomological splitting conditions of vector bundles on $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$

该研究将 Ballico 和 Malaspina 关于多射影空间上向量丛正则性与分裂条件的结论推广到更一般情形。

非分裂环面丛的镜像定理

A mirror theorem for non-split toric bundles

该研究构造了非分裂向量丛经纤维式GIT商得到的环面丛的I-函数，推广了相关已有结果，并通过建立环面丛Givental拉格朗日锥点的刻画等工作证明了非分裂环面丛的镜像定理。

Infinitesimally Equivariant Bundles on Complex Manifolds

We show that any continuous $\mathbf{C}$-linear Lie algebra splitting of the symbol map from the Atiyah algebra of a vector bundle on a complex manifold is given by a differential operator of order at most the rank of the bundle plus one. Bundles equipped with such a splitting can be thought of as \emph{infinitesimally equivariant} bundles, and our theorem implies these are, in a certain sense, in a categorical formal neighbourhood of vector bundles with a flat connection.

Regularity and Cohomological Splitting Conditions for Vector Bundles on Multiprojective Spaces

Here we give a definition of regularity on multiprojective spaces which is different from the definitions of Hoffmann-Wang and Costa-Miró Roig. By using this notion we prove some splitting criteria for vector bundles.

On plane rational curves and the splitting of the tangent bundle

Given an immersion $ϕ: P^1 \to ¶^2$, we give new approaches to determining the splitting of the pullback of the cotangent bundle. We also give new bounds on the splitting type for immersions which factor as $ϕ: P^1 \cong D \subset X \to P^2$, where $X \to P^2$ is obtained by blowing up $r$ distinct points $p_i \in P^2$. As applications in the case that the points $p_i$ are generic, we give a complete determination of the splitting types for such immersions when $r \leq 7$. The case that $D^2=-1$ is of particular interest. For $r \leq8$ generic points, it is known that there are only finitely m…

Vector Bundles on Products of Varieties with $n$-blocks Collections

Here we consider the product of varieties with $n$-blocks collections . We give some cohomological splitting conditions for rank 2 bundles. A cohomological characterization for vector bundles is also provided. The tools are Beilinson's type spectral sequences generalized by Costa and Miró-Roig. Moreover we introduce a notion of Castelnuovo-Mumford regularity on a product of finitely many projective spaces and smooth quadric hypersurfaces in order to prove two splitting criteria for vector bundle with arbitrary rank.

二次曲面上的均匀向量丛

Uniform bundles on quadrics

本文研究二次曲面上的均匀向量丛，证明了特定条件下态射为常值，并改进了均匀向量丛分裂的秩上界，分类了最小秩不可分裂均匀向量丛，部分回答了Ellia猜想。

Absolutely split locally free sheaves on proper $k$-schemes and Brauer--Severi varieties

We classify absolutely split vector bundles on proper $k$-schemes. More precise, we prove that the closed points of the Picard scheme are in one-to-one correspondence with indecomposable absolutely split vector bundles. Furthermore, we apply the obtained results to study the geometry of (generalized) Brauer--Severi varieties.

环面簇上向量丛的分裂

Splitting of Vector Bundles on Toric Varieties

本文在类似Eisenbud-Erman-Schreyer射影空间乘积情形的附加假设下，证明了任意光滑射影环面簇的Horrocks型分裂准则。

Invariance of quantum rings under ordinary flops: III

The paper is Part III of our ongoing project to study a case of Crepant Transformation Conjecture: K-equivalence Conjecture for ordinary flops. In this paper we prove the invariance of quantum rings for general ordinary flops, whose local models are certain non-split toric bundles over arbitrary smooth base. An essential ingredient in the proof is a quantum splitting principle, which reduces a statement in Gromov--Witten theory on non-split bundles to the case of split bundles.

Uniruled varieties with split tangent bundle

Beauville asked if a compact Kähler manifold with split tangent bundle has a universal covering that is a product of manifolds. We use Mori theory and elementary results about holomorphic foliations to study this problem for projective uniruled varieties. In particular we obtain an affirmative answer for rationally connected varieties in any dimension and uniruled varieties in dimension 4.

A home-made Hartshorne-Serre correspondence

We provide an elementary proof of the Hartshorne-Serre correspondence for constructing vector bundles from local complete intersection subschemes of codimension two. This will be done, as in the correspondence of hypersurfaces and line bundles, by patching together local determinantal equations in order to produce sections of a vector bundle.

Simultaneous diagonalization of vector bundles

Grothendieck-Birkhoff Theorem states that every finite dimensional vector bundle over the projective line splits as the sum of one dimensional vector bundles. In this work we study simultaneous splittings of two dimensional vector bundles over a finite field using the theory of Eichler orders. In the sheaf-theoretical context, an Eichler order in a matrix algebra is the intersection of the sheaves of endomorphisms of two vector bundles, so characterizing split Eichler orders solves the problem of simultaneous splitting. We caracterize both the genera of Eichler orders containing only split Eic…

Non-vanishing theorems for rank 2 bundles on P^3: a simple approach without the speciality lemma

The paper investigates vanishing conditions on the first cohomology module of a normalized rank 2 vector bundle E on P^3 which force E to split, and finds therefore strategic levels of non-vanishing for a non-split bundle. The present conditions improve other conditions known in the literature and are obtained with simple computations on the Euler characteristic function, avoiding the speciality lemma, Barth's restriction theorem, the discriminat property, and other heavy tools.

Simplicity and exceptionality of syzygy bundles over P^n

In this work we will prove results that ensure the simplicity and the exceptionality of vector bundles which are defined by the splitting of pure resolutions. We will call such objects syzygy bundles.

On the splitting types of bundles of logarithmic vector fields along plane curves

We give a formula relating the total Tjurina number and the generic splitting type of the bundle of logarithmic vector fields associated to a reduced plane curve. By using it, we give a characterization of nearly free curves in terms of splitting types. Several applications to free and nearly free arrangements of lines are also given, in particular a proof of a form of Terao's Conjecture for arrangements having a line with at most 4 intersection points.

On the Splitting Type of an Equivariant Vector Bundle over a Toric Manifold

From the work of Lian, Liu, and Yau on "Mirror Principle", in the explicit computation of the Euler data $Q=\{Q_0, Q_1, ... \}$ for an equivariant concavex bundle ${\cal E}$ over a toric manifold, there are two places the structure of the bundle comes into play: (1) the multiplicative characteric class $Q_0$ of $V$ one starts with, and (2) the splitting type of ${\cal E}$. Equivariant bundles over a toric manifold has been classified by Kaneyama, using data related to the linearization of the toric action on the base toric manifold. In this article, we relate the splitting type of ${\cal E}$ t…

Arithmetically Cohen-Macaulay Bundles on Hypersurfaces

We prove that any rank two arithmetically Cohen-Macaulay vector bundle on a general hypersurface of degree at least three in $\mathbb{P}^5$ must be split.

On the construction of Weakly Ulrich bundles

I provide a construction of intrinsic weakly Ulrich bundles of large rank on any smooth complete surface in ${\bf P}^3$ over fields of characteristic $p>0$ and also for some classes of surfaces of general type in ${\bf P}^n$. I also construct intrinsic weakly Ulrich bundles on any Frobenius split variety of dimension at most three. The bundles constructed here are in fact ACM and weakly Ulrich bundles and so I call them almost Ulrich bundles. Frobenius split varieties in dimension three include as special cases: (1) smooth hypersurfaces in ${\bf P}^4$ of degree at most four, (2) Frobenius spli…

Arithmetically Cohen-Macaulay Bundles on Threefold Hypersurfaces

We prove that any rank two arithmetically Cohen-Macaulay vector bundle on a general hypersurface of degree at least six in projective four space must be split.

射影化环面向量丛的动力学

Dynamics of Projectivized Toric Vector Bundles

该研究聚焦环面簇上射影丛的满自同态动力学，通过结构定理、动力次数控制分析及显式转移函数法，完成分类、验证猜想并证明相关丛无非自同构交换满自同态，推进代数几何动力学研究。

Split Milnor-Witt Motives and its Applications to Fiber Bundles

We study the Milnor-Witt motives which are a finite direct sum of $\mathbb{Z}(q)[p]$ and $\mathbb{Z}/η(q)[p]$. We show that for MW-motives of this type, we could determine an MW-motivic cohomology class in terms of a motivic cohomology class and a Witt cohomology class. We define the motivic Bockstein cohomology and show that it corresponds to subgroup of Witt cohomology, if the MW-motive splits as above. As an application, we give the splitting formula of Milnor-Witt motives of Grassmannian bundles and complete flag bundles. This in particular shows that the integral cohomology of real comple…

Elementary counterexamples to Kodaira vanishing in prime characteristic

We give elementary counterexamples to Kodaira vanishing in prime characteristic constructed as the projectivization of Frobenius pull-backs of natural bundles on the incidence variety of points and planes in projective n-space (n>2).

Elementary modifications and line configurations in P^2

Associated to an arrangement of projective hyperplanes A is the module D(A), which consists of derivations tangent to A. We study D(A) when A is a configuration of lines in the projective plane. In this setting, we relate the deletion/restriction construction used in the study of hyperplane arrangements to elementary modifications of bundles. This allows us to obtain bounds on the Castelnuovo-Mumford regularity of D(A). We also give simple combinatorial conditions for the associated bundle to be stable, and describe its jump lines. These regularity bounds and stability considerations impose co…

具有分裂陈多项式的圆锥曲线-直线排列的增删法

Addition-deletion for conic-line arrangements with split Chern polynomial

本研究提出复射影平面曲线对数向量场丛陈多项式整分解诱导的组合/几何障碍，推广了射影直线排列与自由曲线的相关结果，并给出秩2向量丛基于光滑圆锥曲线限制的分裂准则。

Harder-Narasimhan Filtrations which are not split by the Frobenius maps

Let $X$ be a smooth projective variety over a perfect field $k$ of characteristic $p>0$, and $V$ be a vector bundle over $X$. It is well known that if $X$ is a curve and $V$ is not strongly semistable, then some Frobenius pullback $(F^t)^*V$ is a direct sum of strongly semistable bundles. A natural question to ask is whether this still holds in higher dimension. Indranil Biswas, Yogish I. Holla, A.J. Parameswaran, and S. Subramanian showed that there is always a counterexample to this over any algebraically closed field of positive characteristic which is uncountable. However, we will produce …

An algorithm for the normal bundle of rational monomial curves

We give an algorithm for calculating the splitting type of the normal bundle of any rational monomial curve. The algorithm is obtained by reducing the calculus to a combinatorial problem and then by solving this problem.