On the Chow groups of certain cubic fourfolds

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold X in the family has an involution such that the induced involution on the Fano variety F of lines in X is symplectic and has a K3 surface S in the fixed locus. The main result establishes a relation between X and S on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family.     

Geometric invariant theory and Einstein–Weyl geometry

In this article, we give a survey of geometric invariant theory for Toric Varieties, and present an application to the Einstein-Weyl geometry. We compute the image of the Minitwistor space of the Honda metrics as a categorical quotient according to the most efficient linearization. The result is the complex weighted projective space CP_1,1,2. We also find and classify all possible quotients.     

On the symplectic structure of instanton moduli spaces

We study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of SU(2) instantons on both commutative and non-commutative R⁴. We identify global Darboux coordinates and quadratic Hamiltonians on classical phase spaces for which these quiver varieties are natural completions. We also show that the group of non-commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non-commutative instantons. This paper should be viewed as a step towards extending known results for Calogero–Moser spaces to the instanton moduli spaces.     

Chow’s Theorem for Semi-abelian Varieties and Bounds for Splitting Fields of Algebraic Tori

A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper, we generalize Chow's theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori.     

改进的QGA-ELM算法水稻叶面积指数反演模型

为了通过植被指数（VI）准确、可靠的获取不同施肥梯度、不同品种的水稻叶面积指数（LAI）,提出了一种基于改进的QGA-ELM算法应用于水稻LAI反演。首先通过8折交叉验证确定极限学习机(ELM)最佳的隐含层神经元个数与隐含层激活函数类型,再通过引入组合动态旋转角策略、单点混沌交叉操作、混沌变异操作、确定性选择策略、量子灾变操作对量子遗传算法（QGA）进行改进,最后使用改进后的QGA算法优化ELM神经网络输入层到隐含层的连接权值和隐含层的阈值。为了验证该模型普适性和有效性,依次建立多元线性回归、BP、ELM、QGA-ELM、改进的QGA-ELM算法5种模型,并在不同数据集上进行反演效果比较,结果表明：（1）对比QGA-ELM算法和改进的QGA-ELM算法进化过程,改进的算法能有效提升模型寻优能力,避免算法早熟,且能寻得更优结果。（2）对比五种算法在不同数据集上的反演效果,验证了NDVI,RVI与LAI之间主要为非线性关系,且ELM神经网络模型反演效果要优于BP神经网络模型和多元线性回归模型。（3）对比五种算法在不同数据集上的反演效果,改进的QGA-ELM算法绝大部分情况下拥有最高的反演精度和最低的误差,改进后的算法反演精度得到了明显提升,泛化性能也得到了增强。（4）改进的QGA-ELM算法在各种施肥梯度上均具有最高反演精度和最低误差,且精度较高,能为不同生长状况水稻LAI反演提供依据。（5）五种模型对庆和香LAI反演精度均要高于龙稻18,而改进的QGA-ELM算法在不同水稻品种上依然具有较高的反演精度,且在不同水稻品种上反演精度相差极小,远低于其他四种模型,能很好适应不同水稻品种LAI反演要求,极大提升模型的稳定性性,为不同水稻品种反演提供参考意义。     

Rook Poset Equivalence of Ferrers Boards

A natural construction due to K. Ding yields Schubert varieties from Ferrers boards. The poset structure of the Schubert cells in these varieties is equal to the poset of maximal rook placements on the Ferrers board under the Bruhat order. We determine when two Ferrers boards have isomorphic rook posets. Equivalently, we give an exact categorization of when two Ding Schubert varieties have identical Schubert cell structures. This also produces a complete classification of isomorphism types of lower intervals of 312-avoiding permutations in the Bruhat order.     

Discrete symmetry groups of vertex models in statistical mechanics

We analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations. They can be seen as generated by involutions corresponding respectively to two kinds of transformations onq×q matrices: the inversion of theq×q matrix and an (involutive) permutation of the entries of the matrix. We show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics. This approach enables one to analyze two-dimensionalq ⁴-state vertex models as simply as three-dimensional vertex models, or higher-dimensional vertex models. Various examples of birational symmetries of vertex models are analyzed. A particular emphasis is devoted to a three-dimensional vertex model, the 64-state cubic vertex model, which exhibits a polynomial growth of the complexity of the calculations. A subcase of this general model is seen to yield integrable recursion relations. We also concentrate on a specific two-dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes Yang-Baxter integrable. It is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces, or higher-dimensional algebraic varieties.     

A vanishing theorem for modular symbols on locally symmetric spaces

A modular symbol is the fundamental class of a totally geodesic submanifold embedded in a locally Riemannian symmetric space , which is defined by a subsymmetric space . In this paper, we consider the modular symbol defined by a semisimple symmetric pair (G,G'), and prove a vanishing theorem with respect to the -component in the Matsushima-Murakami formula based on the discretely decomposable theorem of the restriction . In particular, we determine explicitly the middle Hodge components of certain totally real modular symbols on the locally Hermitian symmetric spaces of type IV. Received: December 8, 1996     

Modular quintics in ℙ4

We will examine the arithmetic of some of the members of a pencil of symmetric quintics in projective 4‐space. We will give evidence for the modularity of some of the exceptional members (even the non‐rigid ones) and give a proof in one rigid case. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Residues in toric varieties

We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X.We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent X as a quotient (Y\{0})/C* such that the toric residue becomes the local residue at 0 in Y.