Enrique Arrondo, Francesco Malaspina
We introduce a notion of regularity for coherent sheaves on Grassmannians of lines. We use this notion to prove some extension of Evans-Griffith criterion to characterize direct sums of line bundles. We also give a cohomological characterization of exterior and symmetric powers of the universal bundles of the Grassmannian.
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In this note, we give a cohomological characterization of all rank 2 split vector bundles on Hirzebruch surfaces.
In this paper, we characterize homogeneous arithmetically Cohen-Macaulay (ACM) bundles over isotropic Grassmannians of types $B$, $C$ and $D$ in term of step matrices. We show that there are only finitely many irreducible homogeneous ACM bundles by twisting line bundles over these isotropic Grassmannians. So we classify all homogeneous ACM bundles over isotropic Grassmannians combining the results on usual Grassmannians by Costa and Mir{ó}-Roig. Moreover, if the irreducible initialized homogeneous ACM bundles correspond to some special highest weights, then they can be characterized by succinc…
In this paper, we give a full classification of all homogeneous Ulrich bundles on a Grassmannian $\Gr(k,n)$ of $k$-planes on $\PP^n$.
This paper investigates the cohomological property of vector bundles on biprojective space. We will give a criterion for a vector bundle to be isomorphic to the tensor product of pullbacks of exterior products of differential sheaves.
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.
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。
Normal bundles of rational curves in Grassmannians
本研究针对格拉斯曼流形中有理曲线法丛的平衡性质问题，证明了一般有理曲线的法丛可分解为次数相差至多2的线丛直和，补充了射影空间外该类性质的研究结论。
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.
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.
We show that Horrock's criterion for the splitting of vector bundles on $\PP^n$ can be extended to vector bundles on multiprojective spaces and to smooth projective varieties with the weak CM property (see Definition 3.11). As a main tool we use the theory of $n$-blocks and Beilinson's type spectral sequences. Cohomological characterizations of vector bundles are also showed.
We study fourfolds with trivial canonical bundle which are zero loci of sections of homogeneous, completely reducible bundles over ordinary and classical complex Grassmannians. We prove that the only HyperK{ä}hler fourfolds among them are the example of Beauville and Donagi, and the example of Debarre and Voisin. In doing so, we give a complete classification of those varieties. We include also the analogous classification for surfaces and threefolds.
Cohomological rank functions and syzygies of projective bundles on abelian varieties
本文利用Jiang–Pareschi上同调秩函数及Caucci–Ito技术研究阿贝尔簇上射影丛的合冲，从而推广并细化了Chiantipalli关于性质$N_p$的结果。
Twisted ind-Grassmannians are ind-varieties $\GG$ obtained as direct limits of Grassmannians $G(r_m,V^{r_m})$, for $m\in\ZZ_{>0}$, under embeddings $ϕ_m:G(r_m,V^{r_m})\to G(r_{m+1}, V^{r_{m+1}})$ of degree greater than one. It has been conjectured in \cite{PT} and \cite{DP} that any vector bundle of finite rank on a twisted ind-Grassmannian is trivial. We prove this conjecture under the assumption that the ind-Grassmannian $\GG$ is sufficiently twisted, i.e. that $\lim_{m\to\infty}\frac{r_m}{°ϕ_1...\degϕ_m}=0$.
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…
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.
The simplest example of an ind-Grassmannian is the infinite projective space $\mathbf P^\infty$. The Barth-Van de Ven-Tyurin (BVT) Theorem, proved more than 30 years ago \cite{BV}, \cite{T}, \cite{Sa} (see also a recent proof by A. Coandă and G. Trautmann, \cite{CT}), claims that any vector bundle of finite rank on $\mathbf P^\infty$ is isomorphic to a direct sum of line bundles. In the last decade natural examples of infinite flag varieties (or flag ind-varieties) have arisen as homogeneous spaces of locally linear ind-groups, \cite{DPW}, \cite{DiP}. In the present paper we concentrate our at…
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。
If $¶^\infty$ is the projective ind-space, i.e. $¶^\infty$ is the inductive limit of linear embeddings of complex projective spaces, the Barth-Van de Ven-Tyurin (BVT) Theorem claims that every finite rank vector bundle on $¶^\infty$ is isomorphic to a direct sum of line bundles. We extend this theorem to general sequences of morphisms between projective spaces by proving that, if there are infinitely many morphisms of degree higher than one, every vector bundle of finite rank on the inductive limit is trivial. We then establish a relative version of these results, and apply it to the study of …
Uniform (d+1)-bundle over the Grassmannian G(d,n)
本文在特征0的代数闭域上，证明了格拉斯曼流形G(d,n)上秩为d+1的均匀向量丛均为齐次丛，完成了该类丛的分类。
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.
In this {paper}, we consider holomorphic embeddings of $Gr(2,m)$ into $Gr(2,n)$. We study such embeddings by finding all possible total Chern classes of the pull-back of the universal bundles under these embeddings. Using the relations between Chern classes of the universal bundles and Schubert cycles, together with properties of holomorphic vector bundles of rank $2$ on complex Grassmannians, we find conditions on $m$ and $n$ for which any holomorphic embedding of $Gr(2,m)$ into $Gr(2,n)$ is linear.
Uniform bundles on generalised Grassmannians
本文研究广义Grassmannian上的均匀向量丛，证明当秩不超过VMRT的e.d.值时丛必分裂，并给出最优上界及最小秩非分裂丛的分类，同时部分解决Kumar猜想。
Let G be a simple simply-connected group scheme over a regular local scheme U. Let E be a principal G-bundle over A^1_U trivial away from a subscheme finite over U. We show that E is not necessarily trivial and give some criteria of triviality. To this end we define affine Grassmannians for group schemes and study their Bruhat decompositions for semi-simple group schemes. We also give examples of principal G-bundles over A^1_U with split G such that the bundles are not isomorphic to pull-backs from U.
Cohomological characterization of Universal Bundles of G(1,n)
本文通过一种适用于任意簇且优于 Beilinson 谱序列标准技术的方法，刻画了线格拉斯曼流形上通用商丛对称幂的扭曲直和。
We show that intersection numbers on the moduli space of stable bundles of coprime rank and degree over a smooth complex curve can be recovered as highest-degree asymptotics in formulas of Vafa-Intriligator type. In particular, we explicitly evaluate all intersection numbers appearing in the Verlinde formula. Our results are in agreement with previous computations of Witten, Jeffrey-Kirwan and Liu. Moreover, we prove the vanishing of certain intersections on a suitable Quot scheme which can be interpreted as giving equations between counts of maps to the Grassmannian.
Here we define the concept of $L$-regularity for coherent sheaves on the Grassmannian G(1,4) as a generalization of Castelnuovo-Mumford regularity on ${\bf{P}^n}$. In this setting we prove analogs of some classical properties. We use our notion of $L$-regularity in order to prove a splitting criterion for rank 2 vector bundles with only a finite number of vanishing conditions. In the second part we give the classification of rank 2 and rank 3 vector bundles without "inner" cohomology (i.e. $H^i_*(E)=H^i(E\otimes\Q)=0$ for any $i=2,3,4$) on G(1,4) by studying the associated monads.
We construct a full exceptional collection consisting of vector bundles in the derived category of coherent sheaves on the so-called Cayley Grassmannian, the subvariety of the Grassmannian $\mathrm{Gr}(3, 7)$ parameterizing 3-subspaces that are annihilated by a general 4-form. The main step in the proof of fullness is a construction of two self-dual vector bundles which is obtained from two operations with quadric bundles that might be interesting in themselves.
Balancedness of Normal Bundles of Rational Curves in Grassmannians
本文证明了 Coskun-Larson-Vogt 关于 Grassmannians 中有理曲线法丛平衡性的猜想在 G(2,4)、G(2,5) 和 G(2,6) 上成立，推广了射影空间中的平衡性结果。
Let Gr(2, E) be the Grassmann bundle of two-planes associated to a general bundle E over a curve X. We prove that an embedding of Gr(2, E) by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the Lagrangian Segre invariant for orthogonal bundles over X, analogous to those given for vector bundles and symplectic bundles in [2, 3]. From the non-defectivity we also deduce an interesting feature of a general orthogonal bundle over X, contrasting with the classical and symplectic cases: Any maximal Lagran…
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。
Let $E$ be the tilting bundle on the Grassmannian $\text{Gr}(n,r)$ of $r$-dimensional quotients of $\Bbbk^n$ constructed by Kapranov. Buchweitz, Leuschke and Van den Bergh introduced a quiver $Q$ and a surjective $\Bbbk$-algebra homomorphism $Φ\colon\Bbbk Q\rightarrow A=\text{End}(E)$, together with a recipe on how the kernel may be computed. In this paper, for the case $r=2$ we give a new, direct proof that $Φ$ is surjective and then complete the picture by calculating the ideal of relations explicitly. As an application, we then use this presentation to show that $\text{Gr}(n,2)$ is isomorph…
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.
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。
In this paper we study smooth complex projective varieties $X$ containing a Grassmannian of lines $G(1,r)$ which appears as the zero locus of a section of a rank two nef vector bundle $E$. Among other things we prove that the bundle $E$ cannot be ample.
Secondary staircase complexes on isotropic Grassmannians
本文在 $\mathrm{IGr}(k,2n)$ 上定义由阶梯复形截断得到的等变向量丛，并构造拟同构于辛外幂的次级阶梯复形，用于研究例外集满性及 Lefschetz 例外集。
In this work we introduce the definition of Schwarzenberger bundle on a Grassmannian. Recalling the notion of Steiner bundle, we generalize the concept of jumping pair for a Steiner bundle on a Grassmannian. After studying the jumping locus variety and bounding its dimension, we give a complete classification of Steiner bundle with jumping locus of maximal dimension, which all are Schwarzenberger bundles.
We classify irreducible equivariant Ulrich vector bundles on isotropic Grassmannians.
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.
Linear sections of Grassmannians and resonance of vector bundles
本文研究给定共振簇是否与某个向量丛相关联，通过构造一族自然态射作为肯定回答的障碍，并分析Grassmann簇横截线性截面上万有秩二商丛的共振，特别证明P^5中十四条不相交直线构成的共振簇是Mukai工作中某丛的共振。
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.
We build compact moduli spaces of Grassmannian framed bundles over a Riemann surface, essentially replacing a group by its bi-invariant compactification. We do this both in the algebraic and symplectic settings, and prove a Hitchin-Kobayashi correspondence between the two. The spaces are universal spaces for parabolic bundles, and the reduction to parabolic bundles commutes with the correspondence. An analogous correspondence is outlined for the generalized parabolic bundles of Bhosle.
This lecture note is devoted to prove that the space of vacua is isomorphic with the space of generalized theta functions (joint work with M.S. Narasimhan, and A. Ramanathan)
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 ${\bf {P}}^2$.
We construct full exceptional collections of vector bundles on the Lagrangian Grassmannians LG(4,8) and LG(5,10).
On uniform and nonhomogeneous vector bundles over Grassmannians
本文研究 Grassmann 流形上均匀向量丛与齐次向量丛的分界，构造了指定秩和 $\mathbb{P}$-齐次度的非齐次均匀丛，并给出均匀—齐次阈值上界及一族不可分解实例。
Let $X$ be a Fano manifold which is the zero scheme of a general global section $s$ in an irreducible homogenous vector bundle over a Grassmannian. We prove that the restriction of the Plücker embedding embeds $X$ projectively normal, and that every small deformation of $X$ comes from a deformation of the section $s$. These results are strengthened in the case of Fano 4-folds.
We study the null fiber of a moment map related to dual pairs. We construct an equivariant resolution of singularities of the null fiber, and get conormal bundles of closed $ K_C $-orbits in the Lagrangian Grassmannian as the categorical quotient. The conormal bundles thus obtained turn out to be a resolution of singularities of the closure of nilpotent $ K_C $-orbits, which is a "quotient" of the resolution of the null fiber.
We confirm Beauville's conjecture that claims that if the p-th exterior power of the tangent bundle of a smooth projective variety contains the p-th power of an ample line bundle, then the variety is either the projective space or the p-dimensional quadric hypersurface.
We construct new examples of derived autoequivalences for a family of higher-dimensional Calabi-Yau varieties. Specifically, we take the total spaces of certain natural vector bundles over Grassmannians G(r,d) of r-planes in a d-dimensional vector space, and define endofunctors of the bounded derived categories of coherent sheaves associated to these varieties. In the case r=2 we show that these are autoequivalences using the theory of spherical functors. Our autoequivalences naturally generalize the Seidel-Thomas spherical twist for analogous bundles over projective spaces.
We study parabolic reductions and Newton points of G-bundles on the Fargues-Fontaine curve and the Newton stratification on the $B_{dR}^+$-Grassmannian for any reductive group G. Let $Bun_G$ be the stack of G-bundles on the Fargues-Fontaine curve. Our first main result is to show that under the identification of the points of $Bun_G$ with Kottwitz's set B(G), the closure relations on the topological space $|Bun_G|$ coincide with the opposite of the usual partial order on B(G). Furthermore, we prove that every non-Hodge-Newton decomposable Newton stratum in a minuscule affine Schubert cell in t…
We construct an equivalence of categories from a strong categorical sl(2) action, following the work of Chuang-Rouquier. As an application, we give an explicit, natural equivalence between the derived categories of coherent sheaves on cotangent bundles to complementary Grassmannians.
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.
Cohomological $χ$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles
本文证明了一维半稳定层和半稳定Higgs丛模空间的交上同调独立于欧拉特征，证实了相关猜想，并给出了结合Ngô支撑定理等工具的证明。
Higgs Grassmannians
本文研究 Higgs 格拉斯曼流形，即光滑簇上给定 Higgs 丛的 Higgs 子丛的参数化方案，通过分析 Higgs 场的局部 Jordan 型刻画其结构，并应用于 Simpson 系统的半稳定性证明及秩 1 情形与谱覆盖的关系。
We study the behavior of the Horrocks-Mumford bundle when restricted to a plane P^2 in P^4, looking for all possible minimal free resolutions for the restricted bundle. To each of the 6 resolutions (4 stable and 2 unstable) we find, we then associate a subvariety of the Grassmannian G(2,4) of planes in P^4. We thus obtain a filtration of the Grassmannian, which we describe in the second part of this work.
Weak Brill-Noether bundles on nodal Enriques surfaces
利用节点型Enriques曲面作为Grassmannian中Reye同余的实现，研究向量丛的弱Brill-Noether问题，考虑不可约齐次丛的限制并分析其弱Brill-Noether性质和斜率稳定性。
Ulrich Bundles on certain Grassmann Bundles Over Curves
本文证明曲线上相对Grassmann簇在任意非常丰沛极化下存在Ulrich丛，通过全局化纤维上的齐次Ulrich丛并利用上同调消没与Falting半稳定性准则完成构造。
We describe the monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian in terms of the equivariant K-theory algebra of the cotangent bundle. This description is based on the hypergeometric integral representations for solutions of the equivariant quantum differential equation. We identify the space of solutions with the space of the equivariant K-theory algebra of the cotangent bundle. In particular, we show that for any element of the monodromy group, all entries of its matrix in the standard basis of the equivariant K-theory algebra of the cotang…
A Bott tower is an iterated $\CP ^1$-bundle over a point, where each $\CP ^1$-bundle is the projectivization of a rank $2$ decomposable complex vector bundle. For a Bott tower, the filtered cohomology is naturally defined. We show that isomorphism classes of Bott towers are distinguished by their filtered cohomology rings. We even show that any filtered cohomology ring isomorphism between two Bott towers is induced by an isomorphism of the Bott towers.
On the GIT quotient of Grassmannians by one dimensional torus
本文研究格拉斯曼流形在对应单根的一维环面作用下的GIT商结构，利用外尔群组合刻画半稳定轨迹，分析轨道结构并计算相关几何不变量。
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