Sections of homogeneous bundles

Arithmetically Cohen--Macaulay bundles on homogeneous varieties of Picard rank one

In this paper, we study arithmetically Cohen--Macaulay (ACM) bundles on homogeneous varieties $G/P$. Indeed we characterize the homogeneous ACM bundles on $G/P$ of Picard rank one in terms of highest weights. This is a generalization of the result on $G/P$ of classical types presented by Costa and Miró-Roig for type $A$, and Du, Fang, and Ren for types $B,C$ and $D$. As a consequence we prove that only finitely many irreducible homogeneous ACM bundles, up to twisting line bundles, exist over all such $G/P$. Moreover, we derive the list of the highest weights of the irreducible homogeneous ACM …

某些例外型齐性簇上的齐性Ulrich丛

Homogeneous Ulrich bundles on homogenous varieties of certain exceptional types

本文针对有理齐性簇上的齐性向量丛，提出了适用于任意Picard数的Ulrich性判别准则，推广了Fonarev的Picard数1情形，并证明某些例外型齐性簇在最小丰沛类下不存在齐性Ulrich丛。

Homogeneous principal bundles and stability

Let G/P be a rational homogeneous variety, where P is a parabolic subgroup of a simple and simply connected linear algebraic group G defined over an algebraically closed field of characteristic zero. A homogeneous principal bundle over G/P is semistable (respectively, polystable) if and only if it is equivariantly semistable (respectively, equivariantly polystable). A stable homogeneous principal H-bundle (E_H ,ρ) is equivariantly stable, but the converse is not true in general. If a homogeneous principal H-bundle (E_H ,ρ) is not equivariantly stable but not stable, then E_H admits an action ρ…

The endomorphisms monoid of a homogeneous vector bundle

In this paper we give some properties of the algebraic and geometric structure of the endomorphisms monoid of a homogeneous vector bundle.

例外格拉斯曼流形上的齐次ACM丛

Homogeneous ACM bundles on exceptional Grassmannians

本文通过关联数据刻画例外格拉斯曼流形上的齐次ACM丛，证明其仅有有限多个不可约对象，并由此推出某些例外格拉斯曼流形具有野表示型。

Fano 3-folds from homogeneous vector bundles over Grassmannians

We rework the Mori-Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians.

Homogeneous Ulrich bundles on Flag manifolds

Let $V$ be a $K$-vector space of dimension $n+1$. In this paper, we focus our attention into the existence of irreducible homogeneous Ulrich bundles on flag manifolds $\FF(p, q,n)$ which parameterizes all chains of linear subspaces $L_{p} \subset L_{q} \subset \PP(V)$ of dimension $p< q$, respectively. We determine all irreducible homogeneous Ulrich bundles on $\FF(0,n-1,n)$ and we prove that there are exactly $2^{n-1}$. Similarly, we prove that $\FF(0,n-2,n)$ and $\FF(1,n-1,n)$ are also the support of irreducible homogeneous Ulrich bundles. On the other hand, we prove that $\FF(0,1,n)$ do not…

Second Infintesimal Neighborhoods of Projective Bundle Sections

This short paper shows that a projective bundle over a smooth variety with a section can be uniquely recovered from the embedding of the section in its second infinitesimal neighborhood within the bundle.

Homogeneous projective bundles over abelian varieties

We consider those projective bundles (or Brauer-Severi varieties) over an abelian variety that are homogeneous, i.e., invariant under translation. We describe the structure of these bundles in terms of projective representations of commutative algebraic groups; the irreducible bundles correspond to Heisenberg groups and their standard representations. Our results extend those of Mukai on semi-homogeneous vector bundles, and yield a geometric view of the Brauer group of abelian varieties.

均匀非齐次向量丛：对Drézet和Ballico-Ellia猜想的反例

Uniform non-homogeneous vector bundles: A counterexample to the conjecture of Drézet and Ballico-Ellia

在${\mathbb P}^5$上构造出秩9的均匀非齐次向量丛，成为${\mathbb P}^n$上秩$r<2n$的均匀向量丛必为齐次这一猜想的首个反例。

Equivariant Ulrich bundles on exceptional homogeneous varieties

We prove that the only rational homogeneous varieties with Picard number 1 of the exceptional algebraic groups admitting irreducible equivariant Ulrich vector bundles are the Cayley plane $E_6/P_1$ and the $E_7$-adjoint variety $E_7/P_1$. From this result, we see that a general hyperplane section $F_4/P_4$ of the Cayley plane also has an equivariant but non-irreducible Ulrich bundle.

Complete intersection Calabi--Yau manifolds with respect to homogeneous vector bundles on Grassmannians

Based on the method by [Küc95], we give a procedure to list up all complete intersection Calabi--Yau manifolds with respect to direct sums of irreducible homogeneous vector bundles on Grassmannians for each dimension. In particular, we give a classification of such Calabi--Yau 3-folds and determine their topological invariants. We also give alternative descriptions for some of them.

同向旗簇上的齐次 Ulrich 丛

Homogeneous Ulrich bundles on isotropic flag varieties

本文研究了类型为 $B$、$C$ 或 $D$ 的有理齐次空间 $G/P$ 上 Ulrich 丛的存在性问题，证明了当 $G/P$ 的 Picard 数大于等于 2 时，不存在关于极小丰富类的不可约齐次 Ulrich 丛。

Balanced line bundles and equivariant compactifications of homogeneous spaces

Manin's conjecture predicts an asymptotic formula for the number of rational points of bounded height on a smooth projective variety in terms of its global geometric invariants. The strongest form of the conjecture implies certain inequalities among geometric invariants of the variety and of its subvarieties. We provide a general geometric framework explaining these phenomena, via the notion of balanced line bundles, and prove the required inequalities for a large class of equivariant compactifications of homogeneous spaces.

Hyperplane sections of projective bundle associated to the tangent bundle of $\mathbb{P}^2.$

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.

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.

具有 1 - 齐次射影化的曲面上的斜率不稳定丛

A slope-unstable bundle on a surface with 1-homogeneous projectivization

回答了Fulger和Langer关于判别式为零的射影丛斜率半稳定性与正性相互作用的问题，核心方法和主要结果由Chatgpt 5.5 pro及基于Rethlas系统的Danus系统得出。

Existence of vector bundles of rank two with fixed determinant and sections

Consider the scheme B_{2,L}^k of stable vector bundles of rank two and fixed determinant L which have at least k sections. Under suitable numerical conditions and for generic L, we show the existence of a component of the expected dimension of B_{2,L}^k.

二次曲面上的一致非齐次层

Uniform non-homogeneous bundles on quadrics

该研究针对广义格拉斯曼流形与光滑二次曲面，界定了一致层为齐次层的最大秩k(X)，分类了特定二次曲面上的一致层，并以一致层刻画射影空间。

On degenerate sections of vector bundles

We consider the locus of sections of a vector bundle on a projective scheme that vanish in higher dimension than expected. We show that after applying a high enough twist, any maximal component of this locus consists entirely of sections vanishing along a subscheme of minimal degree. In fact, we will give a more refined description of this locus, which will allow us to deduce its limit in the Grothendieck ring of varieties.

On finite generation of the section ring of the determinant of cohomology line bundle

For $C$ a stable curve of arithmetic genus $g\ge 2$, and $\mathcal{D}$ the determinant of cohomology line bundle on $\operatorname{Bun}_{\operatorname{SL}(r)}(C)$, we show the section ring for the pair $(\operatorname{Bun}_{\operatorname{SL}(r)}(C), \mathcal{D})$ is finitely generated. Three applications are given.

Resolutions of homogeneous bundles on P^2

In this paper we characterize minimal free resolutions of homogeneous bundles on P^2. Besides we study stability and simplicity of homogeneous bundles on P^2 by means of their minimal free resolutions; in particular we give a criterion to see when a homogeneous bundle is simple by means of its minimal free resolution in the case the first bundle of the resolution is irreducible.

On the intersection theory of Quot schemes and moduli of bundles with sections

We consider a class of tautological top intersection products on the moduli space of stable pairs consisting of semistable vector bundles together with N sections on a smooth complex projective curve C. We show that when N is large, these intersection numbers can equally be computed on the Grothendieck Quot scheme of coherent sheaf quotients of the rank N trivial sheaf on C. The result has applications to the calculation of the intersection theory of the moduli space of semistable bundles on C.

Extremal metrics on toric manifolds and homogeneous toric bundles

In this paper, we prove the Yau-Tian-Donaldson conjecture of the filtration version for toric manifolds and homogeneous toric bundles.

Effective non-vanishing of global sections of multiple adjoint bundles for polarized 4-folds

Let X be a smooth complex projective variety of dimension 4 and let L be an ample line bundle on X. In this paper, we study a natural number m such that h^{0}(m(K_{X}+L))>0 for any polarized 4-folds (X,L) with κ(K_{X}+L)\geq 0.

直纹曲面上的高秩优先丛及其整体截面

Higher rank prioritary bundles on ruled surfaces and their global sections

本文研究亏格非负曲线的直纹曲面，构造任意秩的简单优先向量丛，并给出其整体截面模空间维数的有效界。

Absolute Chow-Kuenneth decomposition for rational homogeneous bundles and for log homogeneous varieties

In this paper, we investigate Murre's conjecture on the existence of a Chow--Kuenneth decomposition for a rational homogeneous bundle $Z\to S$ over a smooth variety, defined over complex numbers. Chow-Künneth decomposition is exhibited for $Z$ whenever $S$ has a Chow--Kuenneth decomposition. The same conclusion holds for a class of log homogeneous varieties, studied by M. Brion.

Global sections of twisted normal bundles of K3 surfaces and their hyperplane sections

Let $S \subset \mathbb{P}^g$ be a smooth $K3$ surface of degree $2g-2$, $g \geq 3$. We classify all the cases for which $h^0(\mathcal{N}_{S/\mathbb{P}^g}(-2)) \neq 0$ and the cases for which $h^0(\mathcal{N}_{S/\mathbb{P}^g}(-2)) < h^0(\mathcal{N}_{C/\mathbb{P}^{g-1}}(-2))$ for $C \subset \mathbb{P}^{g-1}$ a general canonical curve section of $S$.

Homogeneous Vector Bundles on Symplectic Grassmannians

In this paper the K-Theory and the category of homogeneous vector bundles on the symplectic Grassmannian SpGr(2,N) of isotropic 2-planes are discussed.

Ample vector bundles with sections vanishing on special varieties

Let E be an ample vector bundle of rank r on a complex projective manifold X such that there exists a section $s \in Γ(\cal E)$ whose zero locus Z = (s = 0) is a smooth submanifold of the expected dimension dim X - r: = n -r. Assume that Z is not minimal; we investigate the hypothesis under which the extremal contractions of Z can be lifted to X. Finally we study in detail the cases in which Z is a scroll, a quadric bundle or a del Pezzo fibration.

Generalised Kostka-Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles

Let $G$ be a semisimple algebraic group and $B$ a Borel subgroup. We consider generalisations of Lusztig's q-analogues of weight multiplicity, where the set of positive roots is replaced with the multiset of weights of a $B$-submodule of an arbitrary finite-dimensional $G$-module.

On the slopes of the lattice of sections of Hermitian line bundles

In this paper we study the distribution of successive minima of global sections of powers of a metrized ample line bundle on a variety over a number field. We develop criteria for there to exist a measure on the real line describing the limiting behavior of this distribution as one considers increasing powers of the bundle. When this measure exists, we develop methods for determining it explicitly. We present applications to the distribution of Petersson norms of cusp forms of increasing weight for SL_2(Z) and to the minimal sup norm of algebraic functions on adelic subsets of curves arising i…

立方超曲面的奇点与射影化切丛的超平面截面

Singularity of cubic hypersurfaces and hyperplane sections of projectivized tangent bundle of projective space

研究立方超曲面的奇点性质，并通过线性代数方法描述射影空间射影化切丛的超平面截面，扩展了Mazouni-Nagaraj的高维结果。

Stability of homogeneous bundles on P^3

We study the stability of some homogeneous bundles on P^3 by using their representations of the quiver associated to the homgeneous bundles on P^3. In particular we show that homogeneous bundles on P^3 whose support of the quiver representation is a parallelepiped are stable, for instance the bundles E whose minimal free resolution is of the kind 0 --> S^{l_1, l_2, l_3} V (t) --> S^{l_1 +s, l_2, l_3} V (t+s) --> E --> 0 are stable.

Positivity of vector bundles on homogeneous varieties

We study the following question: Given a vector bundle on a projective variety $X$ such that the restriction of $E$ to every closed curve $C \,\subset\, X$ is ample, under what conditions $E$ is ample? We first consider the case of an abelian variety $X$. If $E$ is a line bundle on $X$, then we answer the question in the affirmative. When $E$ is of higher rank, we show that the answer is affirmative under some conditions on $E$. We then study the case of $X \,=\, G/P$, where $G$ is a reductive complex affine algebraic group, and $P$ is a parabolic subgroup of $G$. In this case, we show that th…

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.

Homogeneous bundles and the first eigenvalue of symmetric spaces

We prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kaehler metric on a compact Hermitian symmetric spaces of ABCD--type.

Stable bundles of rank 2 with 4 sections

This paper contains results on stable bundles of rank 2 with space of sections of dimension 4 on a smooth irreducible projective algebraic curve $C$. There is a known lower bound on the degree for the existence of such bundles; the main result of the paper is a geometric criterion for this bound to be attained. For a general curve $C$ of genus 10, we show that the bound cannot be attained, but that there exist Petri curves of this genus for which the bound is sharp. We interpret the main results for various curves and in terms of Clifford indices and coherent systems.

Estimates of sections of determinant line bundles on Moduli spaces of pure sheaves on algebraic surfaces

Let $X$ be any smooth simply connected projective surface. We consider some moduli space of pure sheaves of dimension one on $X$, i.e. $\mhu$ with $u=(0,L,χ(u)=0)$ and $L$ an effective line bundle on $X$, together with a series of determinant line bundles associated to $r[\mo_X]-n[\mo_{pt}]$ in Grothendieck group of $X$. Let $g_L$ denote the arithmetic genus of curves in the linear system $\ls$. For $g_L\leq2$, we give a upper bound of the dimensions of sections of these line bundles by restricting them to a generic projective line in $\ls$. Our result gives, together with Göttsche's computati…

Homogeneous vector bundles over abelian varieties via representation theory

Let $A$ be an abelian variety over a field. The homogeneous (or translation-invariant) vector bundles over $A$ form an abelian category ${\rm HVec}_A$; the Fourier-Mukai transform yields an equivalence of ${\rm HVec}_A$ with the category of coherent sheaves with finite support on the dual abelian variety. In this paper, we develop an alternative approach to homogeneous vector bundles, based on the equivalence of ${\rm HVec}_A$ with the category of finite-dimensional representations of a commutative affine group scheme (the "affine fundamental group" of $A$). This displays remarkable analogies …

Homogeneous Higgs and co-Higgs bundles on Hermitian symmetric spaces

We define homogeneous principal Higgs and co-Higgs bundles over irreducible Hermitian symmetric spaces of compact type. We provide a classification for each type of object up to isomorphism, which in each case can be interpreted as defining a moduli space.

Quantum Hyperplane Section Principle for Concavex Decomposable Vector Bundles

The quantum hyperplane section theorem is explained for nonnegative decomposable concavex bundle spaces over generalized flag manifolds.

Principal elliptic bundles and compact homogeneous l.c.K. manifolds

In this paper, we provide a systematic and constructive description of Vaisman structures on certain principal elliptic bundles over complex flag manifolds. From this description we explicitly classify homogeneous l.c.K. structures on compact homogeneous Hermitian manifolds using elements of representation theory of complex simple Lie algebras. Moreover, we also describe using Lie theory all homogeneous solutions of the Hermitian-Einstein-Weyl equation on compact homogeneous Hermitian-Weyl manifolds. As an application, we provide a huge class of explicit (nontrivial) examples of such structure…

Homogeneous principal bundles over manifolds with trivial logarithmic tangent bundle

Winkelmann considered compact complex manifolds $X$ equipped with a reduced effective normal crossing divisor $D\, \subset\, X$ such that the logarithmic tangent bundle $TX(-\log D)$ is holomorphically trivial. He characterized them as pairs $(X,\, D)$ admitting a holomorphic action of a complex Lie group $\mathbb G$ satisfying certain conditions \cite{Wi1}, \cite{Wi2}; this $\mathbb G$ is the connected component, containing the identity element, of the group of holomorphic automorphisms of $X$ that preserve $D$. We characterize the homogeneous holomorphic principal $H$--bundles over $X$, wher…

On simplicity and stability of tangent bundles of rational homogeneous varieties

Given a rational homogeneous variety G/P where G is complex simple and of type ADE, we prove that all tangent bundles T_{G/P} are simple, meaning that their only endomorphisms are scalar multiples of the identity. This result combined with Hitchin-Kobayashi correspondence implies stability of these tangent bundles with respect to the anticanonical polarization. Our main tool is the equivalence of categories between homogeneous vector bundles on G/P and finite dimensional representations of a given quiver with relations.

$L^2$-extension theorems for jet sections of nef holomorphic vector bundles on compact Kähler manifolds and rational homogeneous manifolds, I

In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tangent bundles are rational homogeneous spaces.

The log-term of the disc bundle over a homogeneous Hodge manifold

We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kaehler--Einstein manifold of classical type.

Quivers and the cohomology of homogeneous vector bundles

We describe the cohomology groups of a homogeneous vector bundle $E$ on any Hermitian symmetric variety $X=G/P$ of ADE type as the cohomology of a complex explicitly described. The main tool is the equivalence between the category of homogeneous bundles and the category of representations of a certain quiver ${\cal Q}_X$ with relations, whose vertices are the dominant weights of the reductive part of $P$. This equivalence was found in some cases by Bondal, Kapranov and Hille and we find the appropriate relations for any Hermitian symmetric variety, computing them explicitly for Grassmannians.

Vector Bundles on Rational Homogeneous Spaces

We consider a uniform $r$-bundle $E$ on a complex rational homogeneous space $X$ %over complex number field $\mathbb{C}$ and show that if $E$ is poly-uniform with respect to all the special families of lines and the rank $r$ is less than or equal to some number that depends only on $X$, then $E$ is either a direct sum of line bundles or $δ_i$-unstable for some $δ_i$. So we partially answer a problem posted by Muñoz-Occhetta-Solá Conde. In particular, if $X$ is a generalized Grassmannian $\mathcal{G}$ and the rank $r$ is less than or equal to some number that depends only on $X$, then $E$ split…

Exceptional collections of line bundles on projective homogeneous varieties

We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G of rank 2 over a field. We exhibit exceptional collections of the expected length for types A_2 and B_2=C_2 and prove that no such collection exists for type G_2. This settles the question of the existence of full exceptional collections of line bundles on projective homogeneous G-varieties for split linear algebraic groups G of rank at most 2.

$L^2$ extension of adjoint line bundle sections

We prove an $L^2$ extension theorem of Ohsawa-Takegoshi type for extending holomorphic sections of line bundles from a subvariety which is given as a maximal log-canonical center of a pair and is of general codimension in a projective variety. Our method of proof indicates that such a setting is the most natural one in a sense, for general $L^2$ extension of line bundle sections.

Exterior algebra resolutions arising from homogeneous bundles

For the exterior algebra E over a vector space, we consider general maps E^a -> E(1)^b and general symmetric and skew-symmetric maps E^a -> E(1)^a and describe the associated exterior algebra resolutions. Using the theory of homogeneous bundles on homogeneous varieties we also describe the exterior algebra resolutions of much wider classes of maps E^a -> E(r)^b.

正交丛的各向同性截面的Hopf指数

A Hopf index for isotropic sections of orthogonal bundles

将Hopf指数推广到带有二次型的丛的各向同性截面，得到八个类似物，应用于余截面局部化虚拟循环和DT^4虚拟循环。

Balanced metrics on homogeneous vector bundles

Let $E\rightarrow M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, ω)$ and let $E=E_1\oplus... \oplus E_m\rightarrow M$ be its decomposition into irreducible factors. Suppose that each $E_j$ admits a $ω$-balanced metric in Donaldson-Wang terminology. In this paper we prove that $E$ admits a unique $ω$-balanced metric if and only if $\frac{r_j}{N_j}=\frac{r_k}{N_k}$ for all $j, k=1, ..., m$, where $r_j$ denotes the rank of $E_j$ and $N_j=\dim H^0(M, E_j)$. We apply our result to the case of homogeneous vector bundles over a rational homogeneous variety $(M, ω)$ and we show…