Examples of non-flat bundles of rank one

Chern invariants of some flat bundles in the arithmetic Deligne cohomology

In this note, we investigate the Chern classes of flat bundles in the arithmetic Deligne Cohomology, introduced by Green-Griffiths, Asakura-Saito. We show nontriviality of the Chern classes in some cases and the proof also indicates that generic flat bundles (in its moduli space) can be expected to have nontrivial classes, which is in contrast with some Gauss-Manin systems.

Unipotent flat bundles and Higgs bundles over compact Kaehler manifolds

We characterize those unipotent representations of the fundamental group $π_1(X,x)$ of a compact Kaehler manifold $X$, which correspond to a Higgs bundle whose underlying Higgs field is equal to zero. The characterization is parallel to the one that R. Hain gave of those unipotent representations of $π_1(X,x)$ that can be realized as the monodromy of a flat connection on the holomorphically trivial vector bundle. We see that Hain's result and ours follow from a careful study of Simpson's correspondence between Higgs bundles and local systems.

Ricci-flat Deformations of Holomorphic Vector Bundles

In this paper we give a criterion for a deformation of a hermitian vector bundle to be Ricci-flat. As an application we show that on a Kähler manifold, every deformation of a vector bundle can be made Ricci-flat whereas on some Hopf manifolds, the non-existence of a Ricci-flat deformation is related to non-trivial vector bundles on the universal cover ${\mathbb C}^n\setminus\{0\}$. On a surface with $b_1(X)\not=0$ filtrable Ricci-rigid vector bundles prove to be very special. We apply this to Inoue and Hopf surfaces.

p-Buchsbaum rank 2 bundles on the projective space

It has been proved by various authors that a normalized, 1-Buchsbaum rank 2 vector bundle on P3 is a nullcorrelation bundle, while a normalized, 2-Buchsbaum rank 2 vector bundle on P3 is an instanton bundle of charge 2. We find that the same is not true for 3-Buchsbaum rank 2 vector bundles on P3, and propose a conjecture regarding the classification of such objects.

Extensions of line bundles and Brill--Noether loci of rank-two vector bundles on a general curve

In this paper we study Brill-Noether loci for rank-two vector bundles and describe the general member of some components as suitable extensions of line bundles.

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.

Local structure of Brill-Noether strata in the moduli space of flat stable bundles

We study the Brill-Noether stratification of the coarse moduli space of locally free stable and flat sheaves of a compact Kahler manifold, proving that these strata have quadratic algebraic singularities.

Rank two Fano bundles on G(1,4)

We classify rank two Fano bundles over the Grassmannian of lines $\G(1,4)$. In particular we show that the only non-split rank two Fano bundle over $\G(1,4)$ is, up to a twist, the universal quotient bundle $\cQ$. This completes the classification of rank two Fano bundles over Grassmannians of lines.

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.

曲面平坦丛的上同调有界性及其应用

Cohomological boundedness for flat bundles on surfaces and applications

本文证明曲面及任意维数下平坦丛的上同调维数、Lefschetz识别及转向轨迹度数的普遍有界性，并引入部分差异运算以给出特征循环的闭式公式。

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.

On the classification of rank 2 almost Fano bundles on projective space

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.

Monads and Rank Three Vector Bundles on Quadrics

In this paper we give the classification of rank 3 vector bundles without "inner" cohomology on a quadric hypersurface \Q_n (n>3) by studying the associated monads.

ACM bundles of rank 2 on quartic hypersurfaces in $\mathbb{P}^3$ and Lazarsfeld-Mukai bundles

Let $X$ be a smooth quartic hypersurface in $\mathbb{P}^3$. By the Brill-Noether theory of curves on K3 surfaces, if a rank 2 aCM bundle on $X$ is globally generated, then it is the Lazarsfeld-Mukai bundle $E_{C,Z}$ associated with a smooth curve $C$ on $X$ and a base point free pencil $Z$ on $C$. In this paper, we will focus on the classification of such bundles on $X$ to investigate aCM bundles of rank 2 on $X$. Concretely, we will give a necessary condition for a rank 2 vector bundle of type $E_{C,Z}$ to be indecomposable initialized and aCM, in the case where the class of $C$ in Pic($X$) i…

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 $X \geq 9$.

Extensions and rank-2 vector bundles on irreducible nodal curves

We generalize Bertram's work on rank two vector bundles to an irreducible projective nodal curve C. We use extensions of a line bundle L by O_C and the associated `forgetful' map to study a compactification of the moduli space of semi-stable vector bundles of rank 2 and determinant L on C.

Lagrangian subbundles of orthogonal bundles of odd rank over an algebraic curve

An orthogonal bundle over a curve has an isotropic Segre invariant determined by the maximal degree of a Lagrangian subbundle. This invariant, and the induced stratifications on moduli spaces of orthogonal bundles, were studied for bundles of even rank in [4]. In this paper, we obtain analogous results for bundles of odd rank. We obtain a sharp upper bound on the isotropic Segre invariant. We show the irreducibility of the induced strata on the moduli spaces of orthogonal bundles of odd rank, and compute their dimensions. As a key ingredient of the proofs, we study the correspondence between L…

On the Buchsbaum index of rank two vector bundles on P3

We classify rank two vector bundles on P3 with Buchsbaum index equal to three and also give some results on the H1-module of "negative instanton"bundles.

Chow groups of moduli spaces of rank 2 vector bundles on curves with determinant of odd degree

We compute Chow groups of moduli spaces of rank 2 vector bundles on curves with determinant of odd degree in terms of generators and relations.

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.

Slope semistability of rank 2 Lazarsfeld-Mukai bundles on K3 surfaces and ACM line bundles

Previously, many people have studied a stability of vector bundles of given rank and Chern classes on algebraic varieties. Recently, we are interested in the slope stability of the rank 2 Lazarsfeld-Mukai bundle $E_{C,Z}$ on a K3 surface $X$ associated to a very ample smooth curve $C$ on $X$ and a base point free pencil $Z$ on $C$ with respect to $\mathcal{O}_X(C)$. In this paper, we will give a sufficient condition for such a Lazarsfeld-Mukai bundle $E_{C,Z}$ to be $\mathcal{O}_X(C)$-slope semistable by ACM line bundles with respect to $\mathcal{O}_X(C)$.

An ampleness criterion for rank 2 vector bundles on surfaces

We observe that the proof of the Bogomolov stable restriction theorem can be adapted to give an ampleness criterion for globally generated rank 2 vector bundles on certain surfaces. This applies to the Lazarsfeld-Mukai bundles, to congruences of lines in P^3, and possibly to the construction of surfaces with ample cotangent bundle (help welcome!).

On higher rank globally generated vector bundles over a smooth quadric threefold

We give a complete classification of globally generated vector bundles of rank 3 on a smooth quadric threefold with $c_1\leq 2$ and extend the result to arbitrary higher rank case. We also investigate the existence of globally generated indecomposable vector bundles, and give the sufficient and necessary conditions on numeric data of vector bundles for indecomposability.

On the Maximal Rank Conjecture for Line Bundles of Extremal Degree

We propose a new method, using deformation theory, to study the maximal rank conjecture. For line bundles of extremal degree, which can be viewed as the first case to test the conjecture, we prove that maximal rank conjecture holds by our new method.

四维二次超曲面Q^4上的秩二弱法诺丛

Rank-two weak Fano bundles on a four-dimensional quadric hypersurface $Q^4$

本文对四维二次超曲面Q^4上的秩2弱法诺丛分类，经线丛扭转后这类丛分为三类，含文献[APW94]构造的特定稳定丛。

Existence of Rank Two Vector Bundles on Higher Dimensional Toric Varieties

In the mid 70's, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on P^n is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain P^n

Vector bundles of finite rank on complete intersections of finite codimension in ind-Grassmannians

In this article we establish an analogue of the Barth-Van de Ven-Tyurin-Sato theorem. We prove that a finite rank vector bundle on a complete intersection of finite codimension in a linear ind-Grassmannian is isomorphic to a direct sum of line bundles.

Rank two globally generated vector bundles with c_1 \leq 5

We classify rank two globally generated vector bundles on P^n, n > 2, with c_1 \leq 5. The classification is complete but for one case (n = 3, c_1 = 5, c_2 = 12)

Rank-3 stable bundles on rational ruled surfaces

In this paper, we compare the moduli spaces of rank-3 vector bundles stable with respect to different ample divisors over rational ruled surfaces. We also discuss the irreducibility, unirationality, and rationality of these moduli spaces.

Ulrich bundles of arbitrary rank on Segre-Veronese varieties

We generalize the results by Eisenbud and Schreyer about Ulrich bundles over Veronese varieties to Segre-Veronese varieties. We discuss the range where we have natural cohomology and we construct multigraded resolutions and monads for Ulrich bundles of any rank. Moreover we give cohomological characterizations for significant families of bundles.

正特征下平向量丛的完全旗

Complete flags on flat vector bundles in positive characteristic

本文研究正特征下平向量丛的相关问题，证明亏格g≤1等价于X上任意平向量丛存在完全旗，并探索该结果的更广泛推广。

Rank 3 arithmetically Cohen-Macaulay bundles on hypersurfaces

Let $X$ be a smooth projective hypersurface of dimension $\geq 5$ and let $E$ be an arithmetically Cohen-Macaulay bundle on $X$ of any rank. We prove that $E$ splits as a direct sum of line bundles if and only if $H^i_*(X, \wedge^2 E) = 0$ for $i = 1,2,3,4$. As a corollary this result proves a conjecture of Buchweitz, Greuel and Schreyer for the case of rank 3 arithmetically Cohen-Macaulay bundles.

皮卡秩1的阿贝尔簇上经线过滤的秩2 ACM丛

Line-filtered rank-two ACM bundles on abelian varieties of Picard rank one

针对皮卡秩1的维数≥2的复阿贝尔簇，分类其上空亏格的秩2算术Cohen-Macaulay丛，明确其结构形式并证明这些丛均非Ulrich丛。

Rank 2 ACM bundles on complete intersection Calabi-Yau threefolds

The aim of this paper is to classify indecomposable rank 2 arithmetically Cohen-Macaulay (ACM) bundles on compete intersection Calabi-Yau (CICY) threefolds and prove the existence of some of them. New geometric properties of the curves corresponding to rank 2 ACM bundles (by Serre correspondence) are obtained. These follow from minimal free resolutions of curves in suitably chosen fourfolds (containing Calabi-Yau threefolds as hypersurfaces). Also the existence of an indecomposable vector bundle of higher rank on a CICY threefold of type (2,4) is proved.

Globally generated vector bundles of rank 2 on a smooth quadric threefold

We investigate the existence of globally generated vector bundles of rank 2 with $c_1\leq 3$ on a smooth quadric threefold and determine their Chern classes. As an automatic consequence, every rank 2 globally generated vector bundle on $Q$ with $c_1=3$ is an odd instanton up to twist.

On the cohomology of rank two vector bundles on P2 and a theorem of Chiantini and Valabrega

We show that a normalized rank two vector bundle, E, on P2 splits if and only if h1(E(-1)) = 0. Using this fact we give another proof of a theorem of Chiantini and Valabrega. Finally we describe the normalized bundles with h1(E(-1)) <= 4.

Rank four vector bundles without theta divisor over a curve of genus two

We show that the locus of stable rank four vector bundles without theta divisor over a smooth projective curve of genus two is in canonical bijection with the set of theta-characteristics. We give several descriptions of these bundles and compute the degree of the rational theta map.

Rank 2 Ulrich bundles on general double plane covers

We prove that a double cover of $\mathbb{P}^2$ ramified along a general smooth curve B of degree $2s$, for $s \geq 3$, supports a rank 2 special Ulrich bundle.

Picard 秩为一的 Fano 三维流形上的秩二弱 Fano 丛

Rank two Weak Fano bundles on Fano threefolds of Picard rank one

本文对 Picard 秩为一的 Fano 三维流形上的秩二向量丛进行分类，确定其射影化为弱 Fano 的情形，并证明各类例子的存在性，同时给出在二次三维流形上的详细消解。

Double covers and vector bundles of rank two

In 2017, Catanese--Perroni gave a natural correspondence between the Picard group of a double cover and a set of pairs of a vector bundle of rank two and a certain morphism of vector bundles on the base space. In this paper, we describe the group structure of the latter set induced from the Picard group in terms of transition functions of vector bundles of rank two. This study is derived from the study of the embedded topology of plane curves. It also proposes approaches to the study of Picard groups of double covers, and to the construction of vector bundles of rank two.

Rank two vector bundles with canonical determinant

Denote by B^k_{2,K} the locus of vector bundles of rank two and canonical determinant. We show that for a generic curve of genus g, B^k_{2,K} is non-empty if g is sufficiently large.

Extensions of vector bundles and rationality of certain moduli spaces of stable bundles

In this paper, it is proved that certain stable rank-3 vector bundles can be written as extensions of line bundles and stable rank-2 bundles. As an application, we show the rationality of certain moduli spaces of stable rank-3 bundles over the projective plane P^2.

非零度情形下Higgs丛模空间的半平坦度量

Semi-flat metrics of the moduli spaces of Higgs bundles in the non-zero degree case

本文研究非零度情形下谱曲线光滑的Higgs丛，定义Hitchin纤维化的可积联络，引入半平坦度量并比较其与Hitchin度量的渐近行为。

Chern classes of rank two globally generated vector bundles on P^2

We determine the Chern classes of globally generated rank two vector bundles on P^2.

来自射影空间上向量丛的常秩线性形式矩阵

Matrices of linear forms of constant rank from vector bundles on projective spaces

本文利用射影空间上小第一陈类的全局生成向量丛分类，完全刻画了秩不超过六的不可分解常秩线性形式矩阵，并揭示了其与Steiner丛、Drézet丛、刚性齐次向量丛及instanton丛的联系。

On rank 2 vector bundles on Fano manifolds

In this work we deal with vector bundles of rank two on a Fano manifold $X$ with $b_2=b_4=1$. We study the nef and pseudoeffective cones of the corresponding projectivizations and how these cones are related to the decomposability of the vector bundle. As consequences, we obtain the complete list of $\mathbb{P}^1$-bundles over $X$ that have a second $\mathbb{P}^1$-bundle structure, classify all the uniform rank two vector bundles on this class of Fano manifolds and show the stability of indecomposable Fano bundles (with one exception on $\mathbb{P}^2$).

卡拉比-丘流形上数值平坦希格斯丛与曲线半稳定希格斯丛的滤过

Filtrations of numerically flat Higgs bundles and curve semistable Higgs bundles on Calabi-Yau manifolds

研究卡拉比-丘流形上的希格斯丛，证明数值平坦希格斯丛存在商为稳定局部自由H-nflat的Jordan-Hölder滤过，进而推出单连通卡拉比-丘流形上曲线半稳定希格斯丛判别式为零。

A restriction theorem for stable rank two vector bundles on P3

Let E be a stable rank two normalized vector bundle on P3. If H is a general plane we show that h^0(E_H(1)) \leq 2+c_1. It follows that h^0(E(1)) \leq 2+c_1.

Further examples of stable bundles of rank 2 with 4 sections

In this paper we construct new examples of stable bundles of rank 2 of small degree with 4 sections on a smooth irreducible curve of maximal Clifford index. The corresponding Brill-Noether loci have negative expected dimension of arbitrarily large absolute value.

Geometry and Moduli Space of Certain Rank-4 Vector Bundles on ${\mathbb P}^4$

Mathematical instanton bundles of rank 4 and $c_2=2$ on ${\mathbb P}^4$ have a smoothquasiprojective moduli space, which is shown via a direct GIT construction. A complete classification of jumping lines of these vector bundles is obtained. The duals of these bundles are described. Dimension computations and irreducibility proofs for some "interesting" subsets of the moduli space are presented. Restrictions of vector bundles from ${\mathbb P}^4$ to ${\mathbb P}^3$ are shown to produce mathematical instanton bundles on ${\mathbb P}^3$.

Semistable Higgs bundles of small ranks are strongly Higgs semistable

We give a proof of the conjecture that a semistable Higgs bundle is strongly Higgs semistable in the case of small ranks, based upon the fact that there exists a gr-semistable Griffiths-transverse filtration on a $\nabla$-invariant semistable vector bundle. The latter is a generalization of a result of Carlos Simpson.

Stability of Rank 2 Ulrich Bundles on Projective K3 Surfaces

Let $F\subseteq\mathbb P ^{a+1}$ be a non-degenerate $K3$ surface of degree $2a$, where $a\ge2$. In this paper we deal with Ulrich bundles on $F$ of rank $2$. We deal with their stability and we construct $K3$ surfaces endowed with families of non-special Ulrich bundles of rank $2$ for each $a\ge2$.

关于秩为2的稳定丛的广义Verschiebung的generic degree的一个上界

An upper bound on the generic degree of the generalized Verschiebung for rank two stable bundles

本文针对具有平凡行列式的秩为2稳定丛模空间，给出了广义Verschiebung的generic degree的一个上界。

Criteria for very ampleness of rank two vector bundles over ruled surfaces

Very ampleness criteria for rank 2 vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.