Vector bundles on G(1,4) without intermediate cohomology
Enrique Arrondo, Beatriz Grana
刻画了G（1，4）上无中间上同调的向量丛.我们从万有丛的扩张和与之相关的其它扩张中得到它们。特别地，我们从泛向量丛的上同调得到了它们的一个刻画。
We characterize the vector bundles on G(1,4) that have no intermediate cohomology. We obtain them from extensions of the universal bundles and others related with them. In particular, we get a characterization of the universal vector bundles from their cohomology.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
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Vanishing theorem for the cohomology of line bundles on Bott-Samelson varieties
利用Bott-Samelson簇的Toric退化和Toric簇上线丛的上同调的描述，导出了Bott-Samelson簇上线丛的上同调的消失结果.
Cohomology of line bundles on a toric variety and constructible sheaves on its polytope
本文利用复曲面簇上某些可构造层的上同调，给出了计算复曲面簇上线丛的上同调的一种方法。我们通过一些例子展示了它的有效使用。
Vector bundles on Hirzebruch surfaces whose twists by a non-ample line bundle have natural cohomology
本文研究Hirzebruch曲面$F_e$上的向量丛$E$，使得它们被一个张成的但不充分的线丛$M = \mathcal {O}_{F_e}(h+ef)$扭曲具有自然上同调，即$h^0(F_e,E(tM)) >0$蕴涵$h^1(F_e,E(tM)) = 0$ .
A horospherical variety is a normal algebraic variety where a connected reductive algebraic group acts with an open orbit isomorphic to a torus bundle over a flag variety. In this article we study the cohomology of line bundles on complete horospherical varieties.
Higher cohomology of the pluricanonical bundle is not deformation invariant
本文计算了Hirzebruch曲面所有复正则丛的第一和第二上同调群的维数，以及射影平面沿有限多个不同点爆破的第二复正则丛的第一和第二上同调群的维数.这两篇文章都给出了多正则丛的高阶上同调不是形变不变的例子。
Vector bundles on Fano 3-folds without intermediate cohomology
研究了指数为2，度d = 3，4，5，贝蒂数为1的Fano三重空间上无中间上同调的向量丛.在秩为2的向量丛的情况下，我们得到了一个完整的刻画。在一般条件下，对任意秩，给出了这类向量丛的所有可能的Chern类.
Positivity in the cohomology of flag bundles (after Graham)
我们基于横截性论点，给出了格雷厄姆关于旗簇等变上同调正性定理的简短几何证明。
Mirror symmetry and quantum cohomology of projective bundles
在以前的文章中，我们证明了光滑簇$X$的量子$\mathcal D$ -模与其上的线丛直和的投影之间的关系，本文证明了当$X$是环面簇中的完全交时的猜想。我们还利用这个猜想证明了由微分算子生成的小量子上同调环$X$的关系提升到投射丛。射影丛的基本上同调关系变形为小量子上同调关系。
Cohomology of Moduli Stacks of Principal $\Bbb C^*$-Bundles Over Nodal Algebraic Curves
研究结点曲线上主$\Bbb C^*$-丛模栈，用陈类确定其有理上同调代数。
Hodge cohomology of étale Nori finite vector bundles
Étale Nori有限向量丛是由有限étale群方案的表示以通常方式定义的丛。本文证明了在许多情况下，这样的向量丛的Hodge上同调群的维数与它被基域的自同构扭曲的维数相同。这将Pink-Roessler的一个结果推广到高阶情形。
We find an algorithm to compute the cohomology groups of spherical vector bundles on complex projective K3 surfaces, in terms of their Mukai vectors. In many good cases, we give significant simplifications of the algorithm. As an application, when the Picard rank is one, we show a numerical condition that is equivalent to weak Brill-Noether for a spherical vector bundle.
Chern invariants of some flat bundles in the arithmetic Deligne cohomology
本文研究了Green-Griffiths，Asakura-Saito提出的算术Deligne上同调中平坦丛的Chern类。我们证明了在某些情况下陈类的非平凡性，证明也表明一般平坦丛（在其模空间）可以预期有非平凡类，这是在某些Gauss-Manin系统。
Quantum cohomology of projective bundles over P^n
本文试图确定射影空间P^n上射影丛的量子上同调。与前面的例子相反，在我们的例子中，相关的模空间通常没有期望的维数。这使得计算更加困难。我们通过使用过度相交理论克服了这个困难。
The quantum cohomology of moduli space of $\PGL_2$-bundles on curves
本文针对亏格不小于 2 的光滑曲线，计算其上稳定 $\PGL_2$-丛模空间的量子上同调。
Cohomology of line bundles on compactified Jacobians
设C是具有曲面奇点的整射影曲线。证明了C的紧化Jacobian上的拓扑平凡线丛与C上的线丛一一对应（自对偶猜想），并计算了线丛的上同调.我们还表明，自然之间的导出范畴的准相干层的雅可比和紧雅可比函子是完全忠实的。
We study Givental's Lagrangian cone for the quantum orbifold cohomology of toric stack bundles and prove that the I-function gives points in the Lagrangian cone, namely we construct an explicit slice of the Lagrangian cone defined by the genus $0$ Gromov-Witten theory of a toric stack bundle.
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.
Generators for the cohomology of the moduli space of irregular parabolic Higgs bundles
该研究证明非正则抛物Higgs丛模空间上同调环纯部分的生成元构成，并将类似结果推广到正则完全旗情形下的抛物与强抛物Higgs丛模空间。
A Reconstruction Theorem for Quantum Cohomology of Fano Bundles on Projective Space
给出了射影空间上Fano向量丛的一个重构定理，该定理从少量的低次Gromov-Witten不变量中恢复了丛的射影化的小量子上同调.我们提供了一个扩展的例子，在其中我们计算的量子上同调的某些法诺9倍，并从中推导出，使用量子莱夫谢茨定理，量子周期序列的法诺3倍的皮卡德秩2和度24。这个例子是新的，对Fanosearch程序很重要。
Simple Cohomology bundles on multiprojective spaces
本文研究了多重射影空间上由单调关联的向量束所形成的上同调束的稳定性与简单性。
Complete sets of relations in the cohomology rings of moduli spaces of holomorphic bundles and parabolic bundles over a Riemann surface
亏格g>1的Riemann曲面上秩为n，度为d的稳定全纯向量丛的模空间的上同调环，当n和d互素时，有一个标准生成元集.当n=2这些发电机之间的关系是很好地理解，特别是一个猜想芒福德，即一定的一套关系是一个完整的一套，是众所周知的是真实的。本文将芒福德关系推广到n>2和抛物丛的情形，并证明了这些关系是完备的。
Cohomology jump loci in the moduli spaces of vector bundles
二十年前，作为他们的工作的一般消灭定理的一部分，绿色拉扎斯菲尔德表明，在一个紧凑的卡勒流形$X$，上同调跳转轨迹在$Pic^τ(X)$都是subtori的翻译。本文将这一结果推广到高维向量丛。Nadel证明了Chern类为零的向量丛的模空间局部正则同构于一点的Zapriki切空间中的二次锥.我们证明了在同构下，上同调跳轨迹是由线性方程定义的。
This note corrects a certain construction in (the old version of) the paper [GL]. The construction in question starts from a Brylinski-Deligne datum, which is an extension of a group G by K_2, and produces a factorization line bundle on the affine Grassmannian of the group G.
Line bundles, connections, Deligne-Beilinson and absolute Hodge cohomology
已知复流形的Picard群可以表示为Deligne上同调群。人们可能想知道，如果同样的情况下，皮卡德群的光滑代数簇和Deligne-Beilinson上同调，但这是不正确的，因为已经指出M。齐藤我们解释了如何修改后者，证明了Picard群也可以用绝对Hodge上同调表示，并在Picard群和通常的Deligne-Beilinson上同调群之间引入了一个中间对象.与Deligne-Beilinson上同调的情况类似，人们可以将具有正则联络的线丛与（修正的）Deligne-Beilinson上同调联系起来。为了考虑不规则联络，我们必须进一步改变Deligne-Beilinson上同调的定义。
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.
In this paper, we show that the motive of the quaternionic Grassmannian $HP^n$ (as defined by I. Panin and C. Walter) splits in the category of effective MW-motives (as defined by B. Calmès, F. Déglise and J. Fasel). Moreover, we extend this result to an arbitrary symplectic bundle, obtaining the so-called quaternionic projective bundle theorem. Finally, we give the Gysin triangle in MW-motivic cohomology.
On the quantum cohomology of Fano bundles over projective spaces
在他们的论文“Quantum cohomology of projective bundles over $P^n$“（Trans. Am. Math. Soc.（1998）350：9 3615-3638）Z.Qin和Y.Ruan介绍了计算流形的量子环的有趣技巧，这些流形是射影空间上的射影化丛;特别地，在分裂丛的情况下，他们在一些限制下证明了关于环面流形的量子环的Batyrev公式。在这里，我们证明了Batyrev公式的量子Chern-Leray方程的一大类分裂丛。
The Orbifold Cohomology Ring of Simplicial Toric Stack Bundles
我们引入扩展复曲面Deligne-Mumford堆栈。我们使用一个扩展的环面Deligne-Mumford栈来得到环面栈丛，并计算它的轨道Chow环。最后，我们推广了Borisov，Chen和Smith的一个结果，使得复曲面叠丛的轨道折叠Chow环和它的收缩分解的Chow环是一个平坦族的纤维。
There is a standard method to calculate the cohomology of torus-invariant sheaves $L$ on a toric variety via the simplicial cohomology of associated subsets $V(L)$ of the space $N_{\mathbb R}$ of 1-parameter subgroups of the torus. For a line bundle $L$ represented by a formal difference $Δ^+-Δ^-$ of polyhedra in the character space $M_{\mathbb R}$, [ABKW18] contains a simpler formula for the cohomology of $L$, replacing $V(L)$ by the set-theoretic difference $Δ^- \setminus Δ^+$. Here, we provide a short and direct proof of this formula.
The cohomology of line bundles of splice-quotient singularities
我们提供了几个结果拼接商奇点：所有“自然”线丛的第一上同调维数的组合表达式，线丛相对截面维数的等变Campillo-Delgado-Gusein-Zade型公式（证明了等变的、整除的多变量希尔伯特级数是拓扑的），解析函数-芽的因子的组合描述，并由其分解图给出了奇点重数的表达式。此外，我们建立了一个新的公式的Seiberg-Witten不变量的任何有理同调球奇异链。
The Banks of the Cohomology River
利用因子上同调的消失给出了n维射影空间上向量丛张量积上同调消失的精确界。为了这个目的，我们引入正则性指数推广的Castelnuovo-Mumford正则性。作为应用，我们给出了向量丛具有只依赖于丛的上同调表的无障碍形变理论的一个充分条件。我们用这样的上同调表构造了完全的丛族。
A vector bundle on a projective variety has a natural cohomology if for every twist its cohomology is concentrated in a single degree. Eisenbud and Schreyer conjectured there should be vector bundles on $\mathbb{P}^1 \times \mathbb{P}^1$ with natural cohomology with respect to bundles $\mathcal{O}(1,0),\mathcal{O}(0,1)$ with prescribed Hilbert polynomial. We prove this conjecture.
Hodge structure on the cohomology of the moduli space of Higgs bundles
研究了紧致黎曼曲面上Higgs丛的模空间。Thaddeus和Hausel（秩2）和Markman（秩一般）证明了这个空间的有理上同调环是由泛类生成的。本文利用他们的结果计算了这种模空间上同调的Hodge结构。我们证明了H^k上的自然Hodge结构是纯权k的。
This paper is a companion of the paper "Weil's conjecture for function fields" by J. Lurie and the author. We present a different exposition of essentially the same algebro-geometric proof of the Atiyah-Bott for the cohomology of Bun(G), which subsequently leads to the proof of the Tamagawa number formula for the volume of the automorphic space for function fields.
On the Cohomology of Moduli of Vector Bundles
计算光滑射影曲线上稳定秩$r$度$d$向量丛的模空间的Hodge数和贝蒂数.我们不假定$r$和$d$互质。在这个过程中，我们配备了一个函子混合霍奇结构的任意代数栈的上同调。该霍奇结构是在曲线上的秩$r$、度$d$向量丛的模栈的情况下计算的。我们的方法还产生了一个公式的庞加莱多项式的模量堆栈是有效的任何基域。
On the cohomology of tautological bundles over Quot schemes of curves
研究曲线商上塔利奥格利 bundles 的外幂和对称幂的上同调消失问题，通过构造具有消失上同调的解析，描述其全局截面空间。
Let $(X,\,D)$ be an $m$-pointed compact Riemann surface of genus at least $2$. For each $x \,\in\, D$, fix full flag and concentrated weight system $α$. Let $P \mathcal{M}_ξ$ denote the moduli space of semi-stable parabolic vector bundles of rank $r$ and determinant $ξ$ over $X$ with weight system $α$, where $r$ is a prime number and $ξ$ is a holomorphic line bundle over $X$ of degree $d$ which is not a multiple of $r$. We compute the Chen-Ruan cohomology of the orbifold for the action on $P \mathcal{M}_ξ$ of the group of $r$-torsion points in ${\rm Pic}^0(X)$.
BPS cohomology for rank 2 degree 0 Higgs bundles (and more)
本研究给出秩2次数0半稳定希格斯丛模叠的E-级数与粗模空间相交E-多项式的比较公式，该公式适用于多种2-Calabi-Yau情形，为Davison的BPS上同调猜想提供证据，并用于上同调χ-无关性检验。
In this paper, we provide two different resolutions of structural sheaves of projectivized tangent bundles of smooth complete intersections. These resolutions allow in particular to obtain convenient (and completely explicit) descriptions of cohomology of twisted symmetric powers of cotangent bundles of complete intersections, which are easily implemented on computer. We then provide several applications. First, we recover the known vanishing theorems on the subject, and show that they are optimal via some non-vanishing theorems. Then, we study the symmetric algebra of global sections of symme…
Parabolic bundles and the intersection cohomology of moduli spaces of vector bundles on curves
本文提出纯几何框架，利用分解定理分析抛物模空间遗忘映射，几何证明递归公式，将0次模空间交点Betti数计算归约到1次情形。
Cohomology of vector bundles on the moduli space of parabolic connections on $\mathbb{P}^1$ minus $5$ points
本文研究了复射影直线去掉五个点上秩为二的抛物连接的模空间，并计算其上同调，推广了Arinkin关于几何兰格朗日对应的结果，同时给出了该模空间的紧化描述。
Computing the cohomology of the tensor product of two vector bundles is central in the study of their moduli spaces and in applications to representation theory, combinatorics and physics. These computations play a fundamental role in the construction of Brill-Noether loci, birational geometry and $S$-duality. Using recent advances in the Minimal Model Program for moduli spaces of sheaves on $\mathbb{P}^2$, we compute the cohomology of the tensor product of general semistable bundles on $\mathbb{P}^2$. This solves a natural higher rank generalization of the polynomial interpolation problem. Mo…
Let $L_1,\dots,L_s$ be line bundles on a smooth variety $X\subset \mathbb{P}^r$ and let $D_1,\dots,D_s$ be divisors on $X$ such that $D_i$ represents $L_i$. We give a probabilistic algorithm for computing the degree of intersections of polar classes which are in turn used for computing the Euler characteristic of linear combinations of $L_1,\dots,L_s$. The input consists of generators for the homogeneous ideals $I_X, I_{D_i} \subset \mathbb{C}[x_0,\ldots,x_r]$ defining $X$ and $D_i$.
Quantum cohomology of projective bundles over $\Pee^n$
初步版本的结果得到了加强。此外，还部分地验证了Batyrev关于$\Pee^n$上与线丛直和相关联的投射丛的量子上同调公式.
In this paper, we study the cohomology of vector bundles on projective space defined as kernels or cokernels of general maps $V_1 \to V_2$, where the $V_i$ are direct sums of line bundles or certain exceptional bundles. We prove an asymptotic cohomology vanishing theorem. We characterize the stability of general Steiner bundles on projective space. We also give a criterion for a Steiner bundle to be ample.
Chow theory of toric variety bundles
本文研究了托尔奇簇包的Chow理论，通过扇形位移规则和等价乘法性，建立了普通和等价上同调与Minkowski权重及分段多项式函数的对应关系，并应用于对数枚举几何中的平衡条件。
On the cohomology groups of local systems over Hilbert modular varieties via Higgs bundles
设$X$是Hilbert模簇，$\mathbb{V}$是$X$上具有无穷单值的非平凡局部系.本文利用Higgs丛的方法研究了上同调群$H^k(X,\mathbb{V})$上的Saito混合Hodge结构。在其它结果中，我们证明了Eichler-Shimura同构，给出了Hodge数的维数公式，并证明了混合Hodge结构在$\mathbb{R}$上是分裂的。这些结果类似于Matsushima-Shimura [Annals of Mathematics 78，1963]中的余紧情形，并补充了Freitag [Book：Hilbert模形式，Springer-Verlag，Berlin，1990]中的常系数结果。
On the intersection cohomology of the moduli of $\mathrm{SL}_n$-Higgs bundles on a curve
本文利用Ngô型支撑不等式研究曲线上任意次数SL_n-希格斯丛模空间的交截上同调，证明了奇异情形下的支撑定理及Hausel-Thaddeus拓扑镜像对称猜想的一个版本，并推广了Harder-Narasimhan定理。
Partial positivity: geometry and cohomology of q-ample line bundles
主要从上同调的角度对线丛的部分正性条件进行了综述。虽然目前的工作是在很大程度上的瞬态性质，我们提出了一些小的改进，现有的文献和一个新的结果：Kodaira型消灭定理有效的q-样本杜波依斯因子和日志典型对。
We prove the Hodge symmetry type result on the Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in the direct sum of holomorphic line bundles. Consequently, we show the vanishing and non-vanishing of Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in holomorphic line bundles.
Positivity on subvarieties and vanishing of higher cohomology
研究了限制于完全交子簇的线丛的正性与高上同调群消失之间的关系。基于这种联系，我们证明了推广的消灭定理的Kawamata-Viehweg和藤田是有效的不一定大线丛。
Variation of the holomorphic determinant bundle
本文证明了Deligne上同调中计算抽象复流形之间真浸入纤维上的全纯向量丛的上同调行列式的Grothendieck-Riemann-Roch公式不随丛的变形而变化.
Cohomology of Coherent Sheaves and Series of Supernatural Bundles
我们表明，上同调表的任何相干层的射影空间是一个收敛的-但可能是无限的-总和的正实倍数的上同调表，我们称之为超自然层。
Cohomology of line bundles on Schubert varieties in the Kac-Moody setting
本文利用外尔群组合学给出了Kac Moody集下线丛的非零上同调模的顶最大和顶最小指标.我们还证明了一个满射定理。
The Determinant of Cohomology and Moduli of $λ$-Connections
本文研究了通过余子式余子式线丛的Atiyah藻酸盐对偶来理解稳定丛的Hodge-Deligne模空间。
Stability of representation cohomology sheaves on moduli spaces of bundles
本文证明了向量丛模空间上切层及更一般的表示上同调层的斜率稳定性，适用于亏格至少为2的曲线和连通约化群，并处理了奇异模空间情形。
Generalised Kostka-Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles
设$G$是半单代数群，$B$是博雷尔子群.我们考虑推广Lusztig的q-类似物的重量多重性，其中的正根的集合被替换为多集的重量的$B$ -子模块的任意有限维$G$ -模块。
We conjecture and prove closed-form index expressions for the cohomology dimensions of line bundles on del Pezzo and Hirzebruch surfaces. Further, for all compact toric surfaces we provide a simple algorithm which allows expression of any line bundle cohomology in terms of an index. These formulae follow from general theorems we prove for a wider class of surfaces. In particular, we construct a map that takes any effective line bundle to a nef line bundle while preserving the zeroth cohomology dimension. For complex surfaces, these results explain the appearance of piecewise polynomial equatio…
Vector bundles without intermediate cohomology and the trichotomy result
本文回顾了无中间上同调向量丛的研究历程，从 Horrocks 的经典分解定理到 Faenzi 和 Pons-Llopis 于 2021 年针对 ACM 簇获得的三分法结果，揭示了其与导出范畴等领域的深刻联系。
We show that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can always be computed in terms of a topological index.
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