Klaus Altmann, Amelie Flatt, Lutz Hille
For any two nef line bundles F and G on a toric variety X represented by lattice polyhedra P respectively Q, we present the universal equivariant extension of G by F under use of the connected components of the set theoretic difference of Q and P.
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We give several mild conditions on a toric bundle on a nonsingular toric variety under which the projectivization of the toric bundle is Frobenius split.
We show that the moduli space of rank three toric vector bundles on smooth projective toric varieties satisfies Murphy's law, that is, every singularity type of finite type arises somewhere on this space.
We give a criterion for a projectivized toric vector bundle to be a Mori dream space and describe its Cox ring using generators and relations. Both of these results are in terms of the matroids of all symmetric powers of the bundle. We also give a criterion for a toric vector bundle to be big and describe several interesting examples of toric vector bundles which highlights how positivity properties for toric vector bundles are more complicated than for toric line bundles.
In this paper, using Klyachko's classification theorem we study positivity and semi-stability of toric vector bundles on a class of nonsingular projective toric varieties, known as Bott towers. In particular, we give a criterion of $s$-jet ampleness of line bundles and characterize nef and big line bundles on Bott towers using Bott numbers. We obtain a criterion for the ampleness of discriminant zero semi-stable toric vector bundles on nonsingular projective varieties. We also describe toric subbundles of toric vector bundles on nonsingular toric varieties.
We prove a weak version of a bigness criterion for equivariant vector bundles on toric varieties.
Simplicial toric stack bundles are smooth Deligne-Mumford stacks over smooth varieties with fibre a toric Deligne-Mumford stack. We compute the Grothendieck $K$-theory of simplicial toric stack bundles and study the Chern character homomorphism.
In this note we prove that any toric Fano manifold with nef tangent bundle is a product of projective spaces. In particular, it implies that Campana-Peternell conjecture hold for toric manifolds.
The automorphism group of a projective bundle P(E) over a simplicial toric variety is described when the bundle E is a direct sum of line bundles. Applications to study of moduli of complete intersections on toric varieties, including Calabi Yau m-folds and Del Pezzo surfaces, are indicated.
In this paper, we investigate when there exists a wild hypersurface bundle over a smooth proper toric variety in positive characteristic. In particular, we determine the possibilities for toric varieties with Picard number at most three or toric Fano varieties of dimension at most four. Moreover, we can construct wild hypersurface bundles over them.
We introduce extended toric Deligne-Mumford stacks. We use an extended toric Deligne-Mumford stack to get the toric stack bundle and compute its orbifold Chow ring. Finally we generalize one result of Borisov, Chen and Smith so that the orbifold Chow ring of the toric stack bundle and the Chow ring of its crepant resolution are fibres of a flat family.
Moduli of toric principal bundles
本文基于Kaveh-Manon对典范主丛的分类，构建了具有给定总等变特征类的典范主丛的模空间，作为部分旗变种的积空间的局部闭子流形。
Equivariant vector bundles on toric schemes over semirings
该研究定义了半环上概形的等变向量丛，证明幂等半域上环面概形的环面向量丛可等变分裂为线丛直和，还研究了等变皮卡群并给出对应Klyachko分类定理版本。
For any $n\geq 3$, we explicitly construct smooth projective toric $n$-folds of Picard number $\geq 5$, where any nontrivial nef line bundles are big.
We give a simple combinatorial proof of the toric version of Mori's theorem that the only $n$-dimensional smooth projective varieties with ample tangent bundle are the projective spaces $\mathbb{P}^n$.
We classify holomorphic as well as algebraic torus equivariant principal $G$-bundles over a nonsingular toric variety $X$, where $G$ is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results.
We study line bundles on toric DM stacks $\mathbb{P}_{\mathbfΣ}$ of dimension two. We give a combinatorial criterion of when infinitely many line bundles on $\mathbb{P}_{\mathbfΣ}$ have trivial cohomology. We further discuss the structure of the set of such line bundles.
We give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope. Furthermore, we show that for a smooth toric surface and a smooth toric variety of Picard rank 2, there exists an ample line bundle with respect to which the tangent bundle is stable if and only if it is an iterated blow-up of projective space.
Using the notion of a valuation into the semifield of piecewise linear functions, we give a classification of torus equivariant flat families of finite type over a toric variety base, by certain piecewise linear maps between fans. As a consequence we derive a classification of toric vector bundles phrased in terms of tropicalized linear spaces. We use these tools to give a characterization of the Mori dream space property for a projectivized toric vector bundle.
We introduce a collection of convex polytopes associated to a torus-equivariant vector bundle on a smooth complete toric variety. We show that the lattice points in these polytopes correspond to generators for the space of global sections and we relate edges to jets. Using the polytopes, we also exhibit toric vector bundles that are ample but not globally generated, and toric vector bundles that are ample and globally generated but not very ample.
We explain a method for calculating the cohomology of line bundles on a toric variety in terms of the cohomology of certain constructible sheaves on the polytope. We show its effective use by means of some examples.
On the instability of syzygy bundles on toric surfaces
本文研究环面上合冲束的斜率稳定性，证明除两类特殊环面外，总可选取极化使高次幂线丛对应的合冲束失稳。
We define the notion of a piecewise linear map from a fan $Σ$ to $\tilde{\mathfrak{B}}(G)$, the cone over the Tits building of a linear algebraic group $G$. Let $X_Σ$ be a toric variety with fan $Σ$. We show that when $G$ is reductive the set of integral piecewise linear maps from $Σ$ to $\tilde{\mathfrak{B}}(G)$ classifies the isomorphism classes of (framed) toric principal $G$-bundles on $X_Σ$. This in particular recovers Klyachko's classification of toric vector bundles, and gives new classification results for the orthogonal and symplectic toric principal bundles.
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.
This paper constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth $4$-dimensional toric Fano varieties. The tilting bundles lead to a large class of explicit Calabi-Yau-$5$ algebras, obtained as the corresponding rolled-up helix algebra. A database of the full strong exceptional collections can be found in the package QuiversToricVarieties for the computer algebra system Macaulay2.
Cox rings of projectivized toric vector bundles and toric flag Bundles
本文研究射影化环面向量丛在直和运算下何时保持Mori dream空间性质，利用全旗丛关系给出判定条件，并计算了射影空间切丛的全旗丛Cox环表示。
We give a complete classification of equivariant vector bundles of rank two over smooth complete toric surfaces and construct moduli spaces of such bundles. This note is a direct continuation of an earlier note where we developed a general description of equivariant sheaves on toric varieties. Here we give a first application of that description.
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
Toric vector bundles with trivial Chern class and flag decorations
本文用热带方法研究完备环面簇上Chern类平凡的环面向量丛，证明其可经特征标扭曲后由低维簇拉回，推广了Payne定理，并构造了无此类非平凡丛的例子，引入旗装饰作为关键工具。
Exceptional Collections of Line Bundles for Smooth Toric Fano Surfaces and Threefolds
该研究利用Bayer-Popescu-Sturmfels的对角胞腔分解，得到单模射影环面曲面上满的强例外线丛集合，又通过Hanlon-Hicks-Lazarev的对角分解，为18个光滑环面法诺三维簇中的16个构造了此类线丛集合。
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。
We give a presentation of the moduli stack of toric vector bundles with fixed equivariant total Chern class as a quotient of a fine moduli scheme of framed bundles by a linear group action. This fine moduli scheme is described explicitly as a locally closed subscheme of a product of partial flag varieties cut out by combinatorially specified rank conditions. We use this description to show that the moduli of rank three toric vector bundles satisfy Murphy's Law, in the sense of Vakil. The preliminary sections of the paper give a self-contained introduction to Klyachko's classification of toric …
We give a complete answer to the question of (semi)stability of tangent bundle of any nonsingular projective complex toric variety with Picard number 2 by using combinatorial crietrion of (semi)stability of an equivariant sheaf. We also give a complete answer to the question of (semi)stability of tangent bundle of all toric Fano 4-folds with Picard number (\leq) 3 which are classified by Batyrev \cite{batyrev}. We have constructed a collection of equivariant indecomposable rank 2 vector bundles on Bott tower and pseudo-symmetric toric Fano varieties. Further in case of Bott tower, we have show…
An affine local criterion for toric projective space bundles
该研究探讨等维环态射成为射影空间丛的条件，证明仿射刚性定理并将其应用于长极射线相关的环收缩，得到无需ℚ-因子性的射影空间丛定理。
In this note we study properties of partially ample line bundles on simplicial projective toric varieties. We prove that the cone of q-ample line bundles is a union of rational polyhedral cones, and calculate these cones in examples. We prove a restriction theorem for big q-ample line bundles, and deduce that q-ampleness of the anticanonical bundle is not invariant under flips. Finally we prove a Kodaira-type vanishing theorem for q-ample line bundles.
Toric principal bundles, Tits buildings and reduction of structure group
本文基于环面主丛的分片线性映射分类，刻画其等变自同构群并给出结构群约化判据，同时引入 Tits 厦的 Helly 数并提出锐上界问题。
Exceptional sequences of line bundles on a smooth projective toric surface are automatically full when they can be constructed via augmentation. By using spherical twists, we give examples that there are also exceptional sequences which can not be constructed this way but are nevertheless full.
We show that a nef and big line bundle whose adjoint bundle has non-zero global sections on a nonsingular toric weak Fano 3-fold is normally generated. As a consequence, we see that all ample line bundles on a nonsingular toric weak Fano 3-fold are normally generated. We also show that an ample line bundle whose adjoint bundle has non-zero global sections on a Gorensein toric Fano 3-fold is normally generated. In our proof we do not use full classifications of Fano polytopes.
Positivity properties of divisors on Toric Vector Bundles
本文结合 Cox 环表示与拟阵理论，计算射影化环面向量丛的 Newton-Okounkov 体及有效、nef 锥，并据此验证其 Fano 性质与 Fujita 猜想。
Ample line bundles on certain toric fibered 3-folds
本文研究具有到射影直线的满射环面等变态射的射影非奇异环面3维流形，证明了其上的丰富线丛总是正规生成。
We study the positivity of exterior powers of the tangent sheaf on toric varieties in order to generalize results by Campana and Peternell about 3-folds with nef second exterior power of the tangent bundle. Using the theory of equivariant vector bundles and the toric MMP, we establish in the smooth case a criterion for the positivity of $Λ^m\mathcal T_X$ in terms of wall relations. As an application, we classify smooth toric varieties of arbitrary dimension $n\ge3$ with $Λ^2\mathcal T_X$ nef and those of dimension $n\ge 4$ with $Λ^3\mathcal T_X$ ample.
Note on Euler characteristic of a toric vector bundle
本文基于凸链与托尔等变向量束的关系，证明凸链在格点上的求和等于向量束的欧拉特征。
We study projectivizations of a special class of toric vector bundles that includes cotangent bundles, whose associated Klyachko filtrations are particularly simple. For these projectivized bundles, we give generators for the cone of effective divisors and a presentation of the Cox ring as a polynomial algebra over the Cox ring of a blowup of a projective space along a sequence of linear subspaces. As applications, we show that the projectivized cotangent bundles of some toric varieties are not Mori dream spaces and give examples of projectivized toric vector bundles whose Cox rings are isomor…
Asymptotics of immaculate line bundles on smooth toric Deligne-Mumford stacks
本文研究了光滑环面 Deligne-Mumford 栈上的无瑕线丛，给出了存在无穷多无瑕线丛的判据，肯定回答了 Borisov 和 Wang 的问题，并描述了其渐近行为。
Equivariant Chern Classes of Toric Vector Bundles over a DVR and Bruhat--Tits Buildings
本文定义 DVR 上正常环面格式中环面向量丛的等变陈类，借助多面体复形上的分段多项式函数及扩展 Bruhat–Tits 建筑给出组合刻画，并联系非阿基米德 Arakelov 理论。
We prove that every toric quiver flag variety $Y$ is isomorphic to a fine moduli space of cyclic modules over the algebra $\text{End}(T)$ for some tilting bundle $T$ on $Y$. This generalises the well known fact that $\mathbb{P}^n$ can be recovered from the endomorphism algebra of $\bigoplus_{0\leq i\leq n} \mathcal{O}_{\mathbb{P}^n}(i)$.
We show that an equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a nonvanishing global section at every point, and deduce that the underlying vector bundle is trivial if and only if its restriction to every invariant curve is trivial. We apply our methods and results to study, in particular, the vector bundles M_L that arise as the kernel of the evaluation map on sections of L, when L is an ample line bundle. We give examples of twi…
We study strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks $\mathbb{P}_{\mathbfΣ}$ with rank of Picard group at most two. We prove that any strong exceptional collection of line bundles generates the derived category of $\mathbb{P}_{\mathbfΣ}$, as long as the number of elements in the collection equals the rank of the (Grothendieck) $K$-theory group of $\mathbb{P}_{\mathbfΣ}$.
Toric vector bundles and the number of zeros of vertical systems
本文针对垂直参数化多项式方程组，通过关联环面向量丛，给出其达到复非零解最大数量的充要条件，并推导了该最大零点数量的新公式。
Counting aCM Toric Bundles of Rank Two on the Veronese Surface
本文通过 Klyachko 滤过方法，对 Veronese 曲面上环面等变秩二 aCM 向量丛的同构类（模超平面类扭转）进行计数，并给出代表性例子。
On the derived category of a toric stack bundle
本文研究了分裂环带叠片丛上环面等变凝聚层导出范畴的性质，并给出了其半正交分解。
Numerical tropical line bundles and toric b-divisors
研究非常紧致的非常代数簇Y的热带紧缩中的线丛与关联热带变种上的典范除子关系，构建数值等价类的自然单射映射，并揭示热带 nef 锥与典范除子的双射关系，拓展Baker对曲线的专精理论。
Toric vector bundles over a discrete valuation ring and Bruhat-Tits buildings
本文利用从环面概形扇形到$GL(r)$的分段线性映射，对离散赋值环上的环面等变向量丛进行了分类，并给出了等变分裂判据，为算术几何研究奠定基础。
Morelli's computation of the K-theory of a toric variety X associates a polyhedrally constructible function on a real vector space to every equivariant vector bundle E on X. The coherent-constructible correspondence lifts Morelli's constructible function to a complex of constructible sheaves kappa(E). We show that certain filtrations of the cohomology of kappa(E) coming from Morse theory coincide with the Klyachko filtrations of the generic stalk of E. We give Morse-theoretic (i.e. microlocal) conditions for a complex of constructible sheaves to correspond to a vector bundle, and to a nef vect…
We describe all of the smooth complete toric threefolds of Picard number 5 with no nontrivial nef line bundles, and show that no such examples exist with Picard number less than 5.
Let $\sE$ be an ample rank $r$ bundle on a smooth toric projective surface, $S$, whose topological Euler characteristic is $e(S)$. In this article, we prove a number of surprisingly strong lower bounds for $c_1(\sE)^2$ and $c_2(\sE)$. We also enumerate the exceptions to either the inequality $c_1(\sE)^2\ge 4e(S)$ or the inequality $c_2(\sE)\ge e(S)$ holding.
CORRECTION. One of the main results in this paper contains a fatal error. We cannot conclude the existence of nontrivial vector bundles on X from the nontriviality of its K-group. The K-group that is computed here is the Grothendieck group of perfect complexes and not vector bundles. Since the varieties are not quasi-projective, existence of nontrivial perfect complexes says nothing about the existence of nontrivial vector bundles. We thank Sam Payne for drawing our attention to the error and Christian Haesemeyer for explanations about the K-theory. Abstract: Following Sam Payne's work, we stu…
Let $X$ be a nonsingular complex projective toric variety. We address the question of semi-stability as well as stability for the tangent bundle $T{X}$. In particular, a complete answer is given when $X$ is a Fano toric variety of dimension four with Picard number at most two, complementing earlier work of Nakagawa. We also give an infinite set of examples of Fano toric varieties for which $TX$ is unstable; the dimensions of this collection of varieties are unbounded. Our method is based on the equivariant approach initiated by Klyachko and developed further by Perling and Kool.
Vector bundles on proper toric 3-folds and certain other schemes
本研究证明满足特定修改条件的n维正常代数概形（含非射影情形）存在秩为n且顶陈数任意大的非平凡向量丛，应用于环面簇得出三维正常环面簇均有秩3的这类向量丛，并刻画了适用该结果的高维环面簇类。
Let $X$ be a toric Del-Pezzo surface and let $Crit(W) \subset (\mathbb{C}^{\ast})^n$ be the solution scheme of the Landau-Ginzburg system of equations. Denote by $X^{\circ}$ the polar variety of $X$. Our aim in this work is to describe a map $L : Crit(W) \rightarrow Fuk_{trop}(X^{\circ})$ whose image under homological mirror symmetry corresponds to a full strongly exceptional collection of line bundles.
Given a complex projective manifold $X$ and a divisor $D$ with normal crossings, we say that the logarithmic tangent bundle $T_X(-\log D)$ is R-flat if its pull-back to the normalization of any rational curve contained in $X$ is the trivial vector bundle. If moreover $-(K_X+D)$ is nef, then the log canonical divisor $K_X+D$ is torsion and the maximally rationally chain connected fibration turns out to be a smooth locally trivial fibration with typical fiber $F$ being a toric variety with boundary divisor $D_{|F}$.
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