Michel Brion
Consider a flag variety $Fl$ over an algebraically closed field, and a subvariety $V$ whose cycle class is a multiplicity-free sum of Schubert cycles. We show that $V$ is arithmetically normal and Cohen-Macaulay, in the projective embedding given by any ample invertible sheaf on $Fl$.
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F-rationality of multiplicity-free subvarieties
该研究在正特征域上证明了射影直线乘积中的无重数子簇具有F-有理奇点，强化了Brion关于此类子簇在特征0域上有有理奇点的结论。
We compute de Chow motive of certain subvarieties of the flags manifold and show that it is an Artin motive.
Cominuscule subvarieties of flag varieties
研究旗变种中自然定义的齐性余余子种，通过Dynkin图计算其子种的Dynkin图。
We consider Buch's rule for K-theory of the Grassmannian, in the Schur multiplicity-free cases classified by Stembridge. Using a result of Knutson, one sees that Buch's coefficients are related to Moebius inversion. We give a direct combinatorial proof of this by considering the product expansion for Grassmannian Grothendieck polynomials. We end with an extension to the multiplicity-free cases of Thomas and Yong.
On algebro-geometric quotients of torus-invariant subvarieties of the flag variety
本文研究了旗流形中torus不变子变种的代数几何商，通过matroid理论方法分解为薄施布特细胞，并证明了其存在性及拓扑结构。
Intersections of Schubert varieties and smooth $T$-stable subvarieties of flag varieties
本文研究了正则半单 Hessenberg 簇中 Bialynicki-Birula 胞腔闭包（即 Hessenberg Schubert 簇）的光滑性，通过考察 Schubert 簇与 Hessenberg 簇交集的不可约性与光滑性，得出了该簇光滑的充分条件。
Laurent polynomial mirrors for quiver flag zero loci
本文针对高维法诺簇分类问题，将Gelfand–Cetlin退化推广到Y形箭图旗簇，提出构造法诺箭图旗零轨迹洛朗多项式镜像的方法，并验证了99个四维情形的猜想镜像。
We compute the Newton--Okounkov bodies of line bundles on the complete flag variety of GL_n for a geometric valuation coming from a flag of translated Schubert subvarieties. The Schubert subvarieties correspond to the terminal subwords in the decomposition (s_1)(s_2s_1)(s_3s_2s_1)(...)(s_{n-1}...s_1) of the longest element in the Weyl group. The resulting Newton--Okounkov bodies coincide with the Feigin--Fourier--Littelmann--Vinberg polytopes in type A.
A subvariety of a complex projective space has a well-known dual variety, which is the set of its tangent hyperplanes. The purpose of this paper is to generalise this notion for a subvariety of a quite general partial flag variety. A similar biduality theorem is proved, and the dual varieties of Schubert varieties are described.
We define a variety of doubly indexed flags, this is a smooth, projective variety, and we describe it as an iterated over Grassmannian varieties. On the other hand, we consider the variety of partial flags which are stabilized by a given nilpotent endomorphism. We partition this variety into locally closed subvarieties which are vector bundles over varieties of the aforedmentioned type.
We systematically study the splitting of vector bundles on a smooth, projective variety, whose restriction to the zero locus of a regular section of an ample vector bundle splits. First, we find ampleness and genericity conditions which ensure that the splitting of the vector bundle along the subvariety implies its global splitting. Second, we obtain a simple splitting criterion for vector bundles on the Grassmannian and on partial flag varieties.
We consider smooth codimension two subcanonical subvarieties in $\mathbb{P}^n$ with $n \geq 5$, lying on a hypersurface of degree $s$ having a linear subspace of multiplicity $(s-2)$. We prove that such varieties are complete intersections. We also give a little improvement to some earlier results on the non existence of rank two vector bundles on $\mathbb{P}^4$ with small Chern classes.
We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of Hessenberg varieties over GL_n(C) and show that they have no odd-dimensional homology. We provide an explicit geometric construction which partitions each Hessenberg variety into pieces homeomorphic to affine space. We characterize these affine pieces by fillings of Young tableaux and show that the dimension of the af…
The characters of Kazhdan--Lusztig elements of the Hecke algebra over $S_n$ (and in particular, the chromatic symmetric function of indifference graphs) are completely encoded in the (intersection) cohomology of certain subvarieties of the flag variety. Considering the forgetful map to some partial flag variety, the decomposition theorem tells us that this cohomology splits as a sum of intersection cohomology groups with coefficients in some local systems of subvarieties of the partial flag variety. We prove that these local systems correspond to representations of subgroups of $S_n$. An expli…
We compute the Newton--Okounkov bodies of line bundles on a Bott--Samelson resolution of the complete flag variety of $GL_n$ for a geometric valuation coming from a flag of translated Schubert subvarieties. The Bott--Samelson resolution corresponds to the decomposition $(s_1)(s_2s_1)(s_3s_2s_1)(\ldots)(s_{n-1}\ldots s_1)$ of the longest element in the Weyl group, and the Schubert subvarieties correspond to the terminal subwords in this decomposition. We prove that the resulting Newton--Okounkov polytopes for semiample line bundles satisfy the additivity property with respect to the Minkowski s…
Regular nilpotent Hessenberg varieties form an important family of subvarieties of the flag variety, which are often singular and sometimes not normal varieties. Like Schubert varieties, they contain distinguished points called permutation flags. In this paper, we give a combinatorial characterization for a permutation flag of a regular nilpotent Hessenberg variety to be a singular point. We also apply this result to characterize regular nilpotent Hessenberg varieties which are normal algebraic varieties.
Let $\mathscr{L} \rightarrow X$ be an ample line bundle over a complex normal projective variety $X$. We construct a flag $X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n=X$ of subvarieties for which the associated Okounkov body for $\mathscr{L}$ is a rational polytope. In the case when $X$ is a homogeneous surface, and the pseudoeffective cone of $X$ is rational polyhedral, we also show that the global Okounkov body is a rational polyhedral cone if the flag of subvarieties is suitably chosen. Finally, we provide an application to the asymptotic study of group representations.
We suggest a possible programme to associate geometric "flag-like" data to an arbitrary simple quantum group, in the spirit of the noncommutative algebraic geometry developed by Artin, Tate, and Van den Bergh. We then carry out this programme for the standard quantum SL(n) of Drinfel'd and Jimbo, where the varieties involved are certain T-stable subvarieties of the (ordinary) flag variety.
In this paper we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We give a connectedness criterion for semisimple Hessenberg varieties generalizing a criterion given by Anderson and Tymoczko. We show that nilpotent Hessenberg varieties are rationally connected.
We study certain nearby cycles sheaves on an affine flag manifold which arise naturally in the Beilinson-Gaitsgory deformation of the affine flag manifold to the affine Grassmannian. We study the multiplicity functions we introduced in an earlier paper, which encode the data of the Jordan-Hoelder series. We prove the multiplicity functions are polynomials in q, and we give a sharp bound for their degrees. Our results apply as well to the nearby cycles in the p-adic deformation of Laumon-Haines-Ngo, and also to Wakimoto sheaves.
We study the $\bG_a^M$ degenerations $\Fl^a_\la$ of the type $A$ flag varieties $\Fl_\la$. We describe these degenerations explicitly as subvarieties in the products of Grassmanians. We construct cell decompositions of $\Fl^a_\la$ and show that for complete flags the number of cells is equal to the normalized median Genocchi numbers $h_n$. This leads to a new combinatorial definition of the numbers $h_n$. We also compute the Poincar\' e polynomials of the complete degenerate flag varieties via a natural statistics on the set of Dellac's configurations, similar to the length statistics on the s…
We describe a stratification on the double flag variety $G/B^+\times G/B^-$ of a complex semisimple algebraic group $G$ analogous to the Deodhar stratification on the flag variety $G/B^+$, which is a refinement of the stratification into orbits both for $B^+\times B^-$ and for the diagonal action of $G$, just as Deodhar's stratification refines the orbits of $B^+$ and $B^-$. We give a coordinate system on each stratum, and show that all strata are coisotropic subvarieties. Also, we discuss possible connections to the positive and cluster geometry of $G/B^+\times G/B^-$, which would generalize …
In this paper we construct free resolutions of certain class of closed subvarieties of affine spaces (the so-called "opposite big cells" of Grassmannians). Our class covers the determinantal varieties, whose resolutions were first constructed by A. Lascoux (Adv. Math., 1978). Our approach uses the geometry of Schubert varieties. An interesting aspect of our work is its connection to the computation of the cohomology of homogeneous bundles (that are not necessarily completely reducible) on partial flag varieties.
We consider tangent cones to Schubert subvarieties of the flag variety $G/B$, where $B$ is a Borel subgroup of a reductive complex algebraic group $G$ of type $E_6$, $E_7$ or $E_8$. We prove that if $w_1$ and $w_2$ form a good pair of involutions in the Weyl group $W$ of $G$ then the tangent cones $C_{w_1}$ and $C_{w_2}$ to the corresponding Schubert subvarieties of $G/B$ do not coincide as subschemes of the tangent space to $G/B$ at the neutral point.
We show that in a cominuscule partial flag variety G/P, the multiplicity of an arbitrary point on a Richardson variety X_w^v = X_w \cap X^v is the product of its multiplicities on the Schubert varieties X_w and X^v.
Regular semisimple Hessenberg varieties are smooth subvarieties of the flag variety, and their examples contain the flag variety itself and the permutohedral variety which is a toric variety. We give a complete classification of Fano and weak Fano regular semisimple Hessenberg varieties in type A in terms of combinatorics of Hessenberg functions. In particular, we show that if the anti-canonical bundle of a regular semisimple Hessenberg variety is nef, then it is in fact nef and big.
Gorenstein toric Schubert varieties in Grassmannians
本文研究格拉斯曼流形中的环面Schubert簇，给出了Gorenstein环面Schubert簇的扇的显式描述，并证明其均为Fano簇。
This paper aims to focus on Richardson varieties on symplectic groups, especially their combinatorial characterization and defining equations. Schubert varieties and opposite Schubert varieties have profound significance in the study of generalized flag varieties which are not only research objects in algebraic geometry but also ones in representation theory. A more general research object is Richardson variety, which is obtained by the intersection of a Schubert variety and an opposite Schubert variety. The structure of Richardson variety on Grassmannian and its combinatorial characterization…
Notes on the cohomology of partial Hessenberg varieties
本文研究部分旗簇中Hessenberg簇的上同调，通过将其与全旗簇中Hessenberg簇的上同调关联展开分析，该簇家族包含多种知名子簇，近年研究活跃且与超平面配置等领域有意外关联。
Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of maximal multiplicity vanishing in Frobenius splitting. In this paper we define the algebraic analogue of this concept and construct a Frobenius splitting vanishing with maximal multiplicity on the diagonal of the full flag variety. Our splitting induces a diagonal Frobenius splitting of maximal multiplicity for a special class of smooth Schubert varieties first considered by Kempf. Consequences are Frobenius splitting of tangent bundles, of blow-ups along the diagonal in flag varieties along with the LMP and Wahl conjectures in …
We use the theory of Mori dream spaces to prove that the global Okounkov body of a Bott-Samelson variety with respect to a natural flag of subvarieties is rational polyhedral. In fact, we prove more generally that this holds for any Mori dream space which admits a flag of Mori dream spaces satisfying a certain regularity condition. As a corollary, Okounkov bodies of effective line bundles over Schubert varieties are shown to be rational polyhedral. In particular, it follows that the global Okounkov body of a flag variety $G/B$ is rational polyhedral. As an application we show that the asymptot…
Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator $X$ and a nondecreasing function $h$. The family of Hessenberg varieties for regular $X$ is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and $K$-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends o…
Restriction varieties in the orthogonal Grassmannian are subvarieties of $OG(k, n)$ defined by rank conditions given by a flag that is not necessarily isotropic with respect to the relevant symmetric bilinear form. In particular, Schubert varieties of Type B and D are examples of restriction varieties. In this paper, we introduce a resolution of singularities for restriction varieties in $OG(k, n)$, and give a description of their singular locus by studying components of the exceptional locus of the resolution.
Levi subgroup actions on Schubert varieties are studied. In the case of partial flag varieties, the horospherical actions are determined. This leads to a characterization of the toroidal and horospherical partial flag varieties with Picard number 1. In the more general case, we provide a set of necessary conditions for the action of a Levi subgroup on a Schubert variety to be toroidal. The singular locus of a (co)minuscule Schubert variety is shown to contain all the $L_{max}$-stable Schubert subvarieties, where $L_{max}$ is the standard Levi subgroup of the maximal parabolic which acts on the…
Global positroid varieties
本文定义了A型全局仿射Grassmannian中的全局正型簇，证明其族平坦，并描述特殊纤维结构及其不可约分支与仿射Richardson簇的联系。
We study Hilbert-Samuel multiplicity for points of Schubert varieties in the complete flag variety, by Groebner degenerations of the Kazhdan-Lusztig ideal. In the covexillary case, we give a positive combinatorial rule for multiplicity by establishing (with a Groebner basis) a reduced and equidimensional limit whose Stanley-Reisner simplicial complex is homeomorphic to a shellable ball or sphere. We show that multiplicity counts the number of facets of this complex. We also obtain a formula for the Hilbert series of the local ring. In particular, our work gives a multiplicity rule for Grassman…
We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension. We also prove a similar result for intersections inside an ambient subvariety of the torus, when the tropicalizations meet inside a facet of multiplicity 1. The proofs require not only the geometry of compactified tropicalizations of subvarieties of toric varieties, but also new results about the geometry of finite type schemes over non-noetherian valuation rings of …
Semi-toric degenerations of Richardson varieties arising from cluster structures on flag varieties
本文证明旗簇上由簇结构产生的环面退化可诱导 Richardson 簇的半环面退化，并将其作为 Morier-Genoud 与 Kogan-Miller 相关构造的推广。
Hessenberg varieties $\mathcal{H}(X,H)$ form a class of subvarieties of the flag variety $G/B$, parameterized by an operator $X$ and certain subspaces $H$ of the Lie algebra of $G$. We identify several families of Hessenberg varieties in type $A_{n-1}$ that are $T$-stable subvarieties of $G/B$, as well as families that are invariant under a subtorus $K$ of $T$. In particular, these varieties are candidates for the use of equivariant methods to study their geometry. Indeed, we are able to show that some of these varieties are unions of Schubert varieties, while others cannot be such unions. Amo…
Regular Hessenberg varieties for the minimal indecomposable Hessenberg space
本文研究复约化群旗簇中极小不可分解Hessenberg空间的正则Hessenberg簇几何，计算仿射胞腔闭包并建立对应关系，分析奇点轨迹推广结果，给出Hessenberg-Schubert簇相关刻画与公式，分类A型奇异置换旗并推广至所有李型。
Recursive Formula for the Equations of Hessenberg Varieties
研究如何确定旗簇中黑森伯格簇的闭子簇结构，通过对黑森伯格函数施加偏序并分析相关簇关系，给出在舒伯特胞腔中确定切出给定黑森伯格簇方程的递归公式，还为泰莫茨科的相关结果提供几何证明。
Hessenberg varieties are a family of subvarieties of the flag variety, including the Springer fibers, the Peterson variety, and the entire flag variety itself. The seminal example arises from a problem in numerical analysis and consists for a fixed linear operator M of the full flags V_1 \subsetneq V_2 >... \subsetneq V_n in GL_n with M V_i contained in V_{i+1} for all i. In this paper I show that all Hessenberg varieties in type A_n and semisimple and regular nilpotent Hessenberg varieties in types B_n,C_n, and D_n can be paved by affine spaces. Moreover, this paving is the intersection of a …
Introduction to the Cohomology of the Flag Variety
本章基于Monk的构造性代数方法，介绍了Grassmannians与旗流形的上同调环，推导了代表Schubert簇上同调类的Schur和Schubert多项式公式，总结了Schubert calculus的严格发展及计算方法，并指出了相关领域的开放问题。
Tangent cones to Schubert varieties for Kac--Moody groups
针对$\tilde{A}_{n-1}$型仿射Kac–Moody群的旗簇，证明不同Weyl群对合对应的Schubert子簇在单位陪集点处的切锥不重合，推广了有限维情形的相关结果，所用技术工具涉及Weyl群嵌入组合学与幂么根基余伴随轨道。
Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a "Giambelli formula" expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert ba…
In a classical-type flag variety, we consider a Schubert variety associated to a vexillary (signed) permutation, and establish a combinatorial formula for the Hilbert-Samuel multiplicity of a point on such a Schubert variety. The formula is expressed in terms of excited Young diagrams, and extends results for Grassmannians due to Krattenthaler, Lakshmibai-Raghavan-Sankaran, and for the maximal isotropic (symplectic and orthogonal) Grassmannians to Ghorpade-Raghavan, Raghavan-Upadhyay, Kreiman, and Ikeda-Naruse. We also provide a new proof of a theorem of Li-Yong in the type A vexillary case. T…
In this paper we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular nilpotent and arbitrary elements of \mathfrak{gl}_n(\C) are paved by affines. For example, our results prove that any Hessenbe…
Semi-infinite orbits in affine flag varieties and homology of affine Springer fibers
本文研究仿射 Springer 纤维由仿射 Schubert 子簇给出的有限并的同调，证明在足够正则条件下自然包含映射在所有次数上单射，并将其解释为加权轨道积分的范畴化，同时刻画仿射 Schubert 簇为半无限轨道闭包的交。
Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety arising in the study of quantum cohomology, geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent Hessenberg varieties for all classical types, generalizing results of de Concini-Lusztig-Procesi and Kostant. This paving is in fact the intersection of a particular Bruhat decomposition with the Hessenberg variety. The nonempty cells of the paving and their dimensions are identified by combinatorial conditions on roots. We use the paving t…
Intersection multiplicity one for the Belkale-Kumar product in G/B
本文证明复半单代数群完全旗簇上 Belkale-Kumar 乘积的结构系数均为 0 或 1，采用几何与组合方法，给出统一证明并推导推论。
We study a family of subvarieties of the flag variety defined by certain linear conditions, called Hessenberg varieties. We compare them to Schubert varieties. We prove that some Schubert varieties can be realized as Hessenberg varieties and vice versa. Our proof explicitly identifies these Schubert varieties by their permutation and computes their dimension. We use this to answer an open question by proving that Hessenberg varieties are not always pure dimensional. We give examples that neither semisimple nor nilpotent Hessenberg varieties need be pure; the latter are connected, non-pure-dime…
Let $G$ be a connected semisimple algebraic group and let $H \subset G$ be a connected reductive subgroup. Given a flag variety $X$ of $G$, a result of Vinberg and Kimelfeld asserts that $H$ acts spherically on $X$ if and only if for every irreducible representation $R$ of $G$ realized in the space of sections of a homogeneous line bundle on $X$ the restriction of $R$ to $H$ is multiplicity free. In this case, the information on restrictions to $H$ of all such irreducible representations of $G$ is encoded in a monoid, which we call the restricted branching monoid. In this paper, we review the …
Let X=G/B be a complete flag variety, and L' and L" two line bundles on X. Consider the cup product map H^{d'}(X,L') x H^{d"}(X, L") --> H^{d}(X,L), where L=L' x L" and d=d'+d". We answer two natural questions about the map above: When is it a nonzero map of irreducible G-representations? Conversely, given generic irreducible representations V' and V" of G, which irreducible components of V' x V" may appear in the right hand side of the map above? We also give bounds on the multiplicities appearing in a tensor product, and relate these considerations to the boundary of the Littlewood-Richardso…
We introduce the notion of a cominuscule point in a Schubert variety in a generalized flag variety for a semisimple group. We derive formulas expressing the Hilbert series and multiplicity of a Schubert variety at a cominuscule point in terms of the restrictions of classes in torus-equivariant K-theory and cohomology to that point, generalizing previously known formulas for flag varieties of cominuscule type. Thus, we can calculate Hilbert series and multiplicities in cases where these were previously unknown. The formulas for Schubert varieties are special cases of more general formulas valid…
Multiplicity-free products of Schubert divisors
研究任意域上旗簇中舒伯特除子类无重数乘积在 Chow 环中的次数，能将其分解为舒伯特簇类线性组合，通过研究组合系数给出判别准则，确定系数情况。
In this expository paper we describe a powerful combinatorial formula and its implications in geometry, topology, and algebra. This formula first appeared in the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey discovered it independently five years later, and it played a prominent role in her work to evaluate certain polynomials closely related to Schubert polynomials. Billey's formula relates many pieces of Schubert calculus: the geometry of Schubert varieties, the action of the torus on the flag variety, combinatorial data about permutations, the cohomology of the flag vari…
Regular semisimple Hessenberg varieties are subvarieties of the flag variety $\mathrm{Flag}(\mathbb{C}^n)$ arising naturally in the intersection of geometry, representation theory, and combinatorics. Recent results of Abe-Horiguchi-Masuda-Murai-Sato and Abe-DeDieu-Galetto-Harada relate the volume polynomials of regular semisimple Hessenberg varieties to the volume polynomial of the Gelfand-Zetlin polytope $\mathrm{GZ}(λ)$ for $λ=(λ_1,λ_2,\ldots,λ_n)$. The main results of this manuscript use and generalize tools developed by Anderson-Tymoczko, Kiritchenko-Smirnov-Timorin, and Postnikov, in orde…
On Positivity for the Peterson Variety
本文探讨旗流形 $G/B$ 及其子变形上同调环中的几何正性，阐述了 Schubert 变形基的非负结构常数，并解释了该正性现象如何扩展至特定的幂零 Hessenberg 变形（Peterson 变形），同时提出了向其他 Hessenberg 变形扩展的开放问题。
Dimensions of type $A$ Hessenberg varieties over a fixed sheet
本文将极小层上Hessenberg簇维数相同的结论推广到任意层，证明了固定Hessenberg函数时，A型旗簇中同一层算子定义的Hessenberg簇维数一致。
Morita equivalences, moduli spaces and flag varieties
本文通过将双 Bruhat 细胞视为装饰模空间，研究了其作为辛双群oid 的积分，并展示了与莫rita 等价和 meromorphic 连接的关系。
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