Klaus Kuennemann, Harry Tamvakis
Let X=G/P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact type.
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。
继续探索
We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS `span' the space of all infinitesimal VHS; and (3) show that the cohomology classes dual the Schubert VHS form a basis of the invariant characteristic cohomology associated to the infinitesimal period relation (a.k.a. Griffiths transversality).
Hodge rigidity of Chern classes
本文研究Schubert胞的Chern-Schwartz-MacPherson类的齐次分量，证明在合适条件下每个分量由不可约子簇表示，扩展Huh的结果并适用于更多情况。
A normal variety $X$ is called $H$-spherical for the action of the complex reductive group $H$ if it contains a dense orbit of some Borel subgroup of $H$. We resolve a conjecture of Hodges--Yong by showing that their spherical permutations are characterized by permutation pattern avoidance. Together with results of Gao--Hodges--Yong this implies that the sphericality of a Schubert variety $X_w$ with respect to the largest possible Levi subgroup is characterized by this same pattern avoidance condition.
Local cohomology with support in Schubert varieties
本文研究广义旗形簇上局部上同调sheaves在闭Schubert簇的支持结构，利用Kazhdan-Lusztig理论和Saito的混合Hodge模理论，通过Kempf引入的Grothendieck-Cousin复形计算简单分解因子、Hodge滤和权重滤。
Local cohomology with Schubert support on compact Hermitian symmetric spaces
本文为紧致埃尔米特对称空间上舒伯特簇支撑的局部上同调模提供了统一的根系构造，给出权滤过公式，并应用于对称行列式与Pfaffian簇，同时计算了霍奇同调与上同调亏格。
For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are spherical for the action of a Levi subgroup. We evidence the conjecture, employing the combinatorics of Demazure modules, and work of R. Avdeev-A. Petukhov, M. Can-R. Hodges, R. Hodges-V. Lakshmibai, P. Karuppuchamy, P. Magyar-J. Weyman-A. Zelevinsky, N. Perrin, J. Stembridge, and B. Tenner. In type A, we establish connections with the key polynomials of A. …
Hodge structures of the surface of planes in a cubic 5-fold
本文研究一般三次五重簇中平面模空间的几何与形变，证明其为典范丛丰沛的光滑射影曲面，且特定族的1次Hodge结构变分极大，核心技术为hyper-Kähler几何与Hodge数精细计算。
Around a class version of the Hodge index theorem for singular varieties
本文概述了奇异复代数簇霍奇指标定理特征类版本的最新进展，澄清了文献中L类概念的关系，并给出了该猜想成立的多种新情况，包括紧 toric 簇、Schubert 簇等。
We give several new formulas which are useful for Schubert Calculus associated with the orthogonal groups and related orthogonal degeneracy loci.
We obtain an algorithm computing the Chern-Schwartz-MacPherson (CSM) classes of Schubert cells in a generalized flag manifold G/B. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class is obtained by applying certain Demazure-Lusztig type operators to the CSM class of a cell of dimension one less. These operators define a representation of the Weyl group on the homology of G/B. By functoriality, we deduce algorithmic expressions for CSM classes of Schubert cells in any flag manifold G/P. We conjecture that the CSM classes of Sch…
Applications of Homogeneous Fiber Bundles to the Schubert Varieties
本文利用旗簇上齐次纤维丛框架研究 Schubert 簇与等变嵌入的关系，证明了相关簇的环面性质，识别了奇妙簇，并给出了光滑球面 Schubert 簇的刻画及其局部刚性等应用。
We treat two quite different problems related to changes of complex structures on Kähler manifolds by using global geometric method. First, by using operators from Hodge theory on compact Kähler manifold, we present a closed explicit extension formula for holomorphic canonical forms in different complex structures. As applications, we give a closed explicit formula for certain canonical sections of Hodge bundles on marked and polarized moduli spaces of projective manifolds, and provide a closed explicit extension formula for holomorphic pluricanonical forms under certain natural conditions. Se…
We prove a formula for the structure constants of multiplication of equivariant Schubert classes in both equivariant cohomology and equivariant K-theory of Kac-Moody flag manifolds G/B. We introduce new operators whose coefficients compute these (in a manifestly polynomial, but not positive, way), resulting in a formula much like and generalizing the positive Andersen-Jantzen-Soergel/Billey and Graham/Willems formulae for the restriction of classes to fixed points. Our proof involves Bott-Samelson manifolds, and in particular, the (K-)cohomology basis dual to the (K-)homology basis consisting …
For a projective algebraic variety $V$ with isolated singularities, endowed with a metric induced from an embedding, we consider the analysis of the natural partial differential operators on the regular part of $V$. We show that, in the complex case, the Laplacians of the de Rham and Dolbeault complexes are discrete operators except possibly in degrees $n,n\pm 1$, where $n$ is the complex dimension of $V$. We also prove a Hodge theorem on the operator level and the $L^2$--Stokes theorem outside the degrees $n-1,n$. We show that the $L^2$-Stokes theorem may fail to hold in the case of real alge…
On Two Algebraic Realizations of Schubert Calculus
研究复格拉斯曼流形上舒伯特演算的两种代数实现，即通过作用于舒尔多项式的微分算子和作用于外代数的顶点算子，利用玻色子 - 费米子对应建立统一框架，关联舒伯特演算与多种数学理论。
Consider the holomorphic bundle with connection on $\mathbb P^1-\{0,1,\infty\}$ corresponding to the regular hypergeometric differential operator \[ \prod_{j=1}^h(D-α_j)-z\prod_{j=1}^h(D-β_j), \qquad D=z\frac{d}{dz}. \] If the numbers $α_i$ and $β_j$ are real and for all $i$ and $j$ the number $α_i-β_j$ is not integer, then the bundle with connection is known to underlie a complex polarizable variation of Hodge structures. We calculate some Hodge invariants for this variation, in particular, the Hodge numbers. From this we derive a conjecture of Corti and Golyshev. We also use non-abelian Hodg…
In this paper, we show that the generating function for linear Hodge integrals over moduli spaces of stable maps to a nonsingular projective variety $X$ can be connected to the generating function for Gromov-Witten invariants of $X$ by a series of differential operators $\{ L_m \mid m \geq 1 \}$ after a suitable change of variables. These operators satisfy the Virasoro bracket relation and can be seen as a generalization of the Virasoro operators appeared in the Virasoro constraints for Kontsevich-Witten tau-function in the point case. This result is an extension of the work in \cite{LW} for t…
Hodge structures and differential operators
本文是算术驱动的非技术性极限混合Hodge结构定义导论，以常微分算子经典理论的自然表述提出若干论断，并用初等方法证明。
Topological Gysin Coherence for Algebraic Characteristic Classes of Singular Spaces
本文证明了Gysin凝聚示性类框架适用于Hodge理论型示性类$IT_{1,\ast}$，从而证明了关于Grassmannian中Schubert簇的Brasselet等人猜想，并研究了Chern类和Todd类。
A classical result of Schubert calculus is an inductive description of Schubert cycles using divided difference (or push-pull) operators in Chow rings. We define convex geometric analogs of push-pull operators and describe their applications to the theory of Newton-Okounkov convex bodies. Convex geometric push-pull operators yield an inductive construction of Newton-Okounkov polytopes of Bott-Samelson varieties. In particular, we construct a Minkowski sum of Feigin-Fourier-Littelmann-Vinberg polytopes using convex geometric push-pull operators in type A.
Mixed Hodge structures and Siegel operators
本文研究局部对称簇上同调的混合霍奇结构，证明其权滤过与西格尔算子定义的典范权模形式空间上的余秩滤过一致，并由此推导出总西格尔算子的满射性及障碍空间。
In this paper we prove a new generic vanishing theorem for $X$ a complete homogeneous variety with respect to an action of a connected algebraic group. Let $A, B_0\subset X$ be locally closed affine subvarieties, and assume that $B_0$ is smooth and pure dimensional. Let $\mathcal{P}$ be a perverse sheaf on $A$ and let $B=g B_0$ be a generic translate of $B_0$. Then our theorem implies $(-1)^{\operatorname{codim} B}χ(A\cap B, \mathcal{P}|_{A\cap B})\geq 0$. As an application, we prove in full generality a positivity conjecture about the signed Euler characteristic of generic triple intersection…
We introduce a version of the Cartier isomorphism for de Rham cohomology valid for associative, not necessarily commutative algebras over a field of positive characteristic. Using this, we imitate the well-known argument of P. Deligne and L. Iluusie and prove, in some cases, a conjecture of M. Kontsevich which claims that the Hodge-to-de Rham, a.k.a. Hochschild-to-cyclic spectral sequence degenerates.
Let X be an orthogonal Grassmannian parametrizing isotropic subspaces in an even dimensional vector space equipped with a nondegenerate symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in the singular and quantum cohomology ring of X as a polynomial in certain special Schubert classes. Our analysis reveals a surprising relation between the Schubert calculus on even and odd orthogonal Grassmannians. We also study eta polynomials, a family of polynomials defined using raising operators whose algebra agrees with the Schubert calculus on X.
A $σ$-operator on a complexification $V_{\C}$ of an $\R$-vector space $V_{\R}$ is an operator $A \in \rm{End}_{\C} (V_{\C})$ such that $σ(A) = 0$ where $σ(z)$ denotes the Weierstrass $σ$-function. In this paper we define the notion of the strongly pseudo-real $σ$-operator and prove that there is one to one correspondence between real mixed Hodge structures and strongly pseudo-real $σ$-operators.
These notes were written to serve as an easy reference for \cite{Wang-AF}. All the results in this presentation are well-known (or quasi-well-known) theorems in Hodge theory. Our main purpose was to give a unified approach based on a variation formula of the Lefschetz star operator, following \cite{Wang-k}. It fits quite well with Timorin's $T$-Hodge theory, i.e. the Hodge theory on the space of differential forms divided by $T$ (i.e. forms like $T\wedge u$), where $T$ is a finite wedge product of Kähler forms.
Let X be a symplectic or odd orthogonal Grassmannian parametrizing isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in H^*(X,Z) as a polynomial in certain special Schubert classes. We study theta polynomials, a family of polynomials defined using raising operators whose algebra agrees with the Schubert calculus on X. Furthermore, we prove that theta polynomials are special cases of Billey-Haiman Schubert polynomials and use this connection to express the former as positive linear …
This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over C and over C[t_1, t_2,...,t_n]. We show these group actions are the same as an action of simple transpositions studied geometrically by M. Brion, and give topological meaning to the divided difference operators studied by Berstein-Gelfand-Gelfand, Demazure, Kostant-Kumar, and others. We analyze these representations using the combinatorial approach to equivariant cohomology introduced by Goresky-Kottwitz-MacPherson. We find that each permutation represent…
In this paper we introduce new definition of Hodge structures and show that $\R$-Hodge structures are determined by $\R$-linear operators that are annihilated by the Weierstrass $σ$-function
Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K theory
该研究探讨部分旗流形上左乘诱导的Demazure-Lusztig算子，证明其生成Schubert胞腔的相关特征类，并构造出满足量子积相容Leibniz法则的等变量子左除差算子。
The purpose of this paper is to develop a new theory of gauges in mixed characteristic. Namely, let $k$ be a perfect field of characteristic $p>0$ and $W(k)$ the $p$-typical Witt vectors. Making use of Berthelot's arithmetic differential operators, we define for a smooth formal scheme $\mathfrak{X}$ over $W(k)$, a new sheaf of algebras $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ which can be considered a higher dimensional analogue of the (commutative) Dieudonne ring. Modules over this sheaf of algebras can be considered the analogue (over $\mathfrak{X}$) of the gauges of Ekedahl and Fontai…
Theta operators on Hodge type Shimura varieties
本文提出了一种新的模p重量移位微分算子，用于Hodge类型Shimura品种，并展示了其在Serre猜想中的应用。
Defect Triangles and Intersection-Space Hodge Atom Shadows for Calabi--Yau Conifolds
本文通过投影三角形陈述，组织了交集空间Hodge原子阴影包，研究了交集空间原子阴影与交集同调包的比较，并在混合Hodge实现Banagl中 exact 序列下，识别了IIB消失原子与实现核。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。