We obtain a formula for structure constants of certain variant form of Bott-Samelson classes for equivariant oriented cohomology of flag varieties. Specializing to singular cohomology/K-theory, we recover formulas of structure constants of Schubert classes of Goldin-Knutson, and that of structure constants of Segre-Schwartz-MacPherson classes of Su. We also obtain a formula for K-theoretic stable basis. Our method comes from the study of formal affine Demazure algebra, so is purely algebraic, while the above mentioned results are geometric.
We obtain an explicit presentation of the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation of the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational cobordism rings of wonderful symmetric varieties of minimal rank.
Our concern in this paper is the dimension and inclusion relations of Schubert varieties in twisted partial affine flag varieties. In the end we apply our results to some local models of certain Schubert varieties.
We compute the Hilbert series of general weighted flag varieties and discuss a computer-aided method to determine their defining equations. We apply our results to weighted flag varieties coming from the Lie groups of type G_2 and GL(6), to construct some families of polarised projective varieties in codimensions 8 and 6, respectively.
Let G be a semisimple affine algebraic group and P a parabolic subgroup of G. We classify all flag varieties G/P which admit an action of the commutative unipotent group G_a^n with an open orbit.
We prove the existence of tilting objects on generalized Brauer--Severi varieties, some relative flags and some twisted forms of relative flags. As an application we obtain tilting objects on certain homogeneous varieties of classical type and on certain twisted forms of homogeneous varieties of type $A_n$ and $C_n$.
Based on the Brieskorn-Slodowy-Grothendieck diagram, we write the holomorphic structures (or filtrations) of the ADE Lie algebra bundles over the corresponding type ADE flag varieties, over the cotangent bundles of these flag varieties, and over the corresponding type $ADE$ singular surfaces. The main tool is the cohomology of line bundles over flag varieties and their cotangent bundles.
The rim-hook rule for quantum cohomology of the Grassmannian allows one to reduce quantum calculations to classical calculations in the cohomology of the Grassmannian. We use the Abelian/non-Abelian correspondence for cohomology to prove a rim-hook removal rule for the cohomology of quiver flag varieties. Quiver flag varieties are generalisations of type A flag varieties; this result is new even in the flag case. This gives an effective way of computing products in their cohomology, reducing computations to that in the cohomology ring of the Grassmannian. We then prove a quantum rim-hook rule …
We compute the automorphism groups of finite and cofinite ind-grassmannians, as well as of the ind-variety of maximal flags indexed by Z_{>0}. We pay special attention to differences with the case of ordinary flag varieties.
A classification of double flag varieties of complexity 0 and 1 is obtained. An application of this problem to decomposing tensor products of irreducible representations of semisimple Lie groups is considered.
We define the class of admissible linear embeddings of flag varieties. The definition is given in the general language of algebraic geometry. We then prove that an admissible linear embedding of flag varieties has a certain explicit form in terms of linear algebra. This result enables us to show that any direct limit of admissible embeddings of flag varieties is isomorphic to an ind-variety of generalized flags as defined in [DP]. These latter ind-varieties have been introduced in terms of the ind-group SL(\infty) (respectively, O(\infty) or Sp(\infty) for isotropic generalized flags), and the…
We obtain A- and B- series of flag manifolds for T-equivariant tangent bundles and equivariant Chern polynomial following the arguments of Lian, Liu, Yau.
The aim of this paper is to construct certain closed embeddings of Grassmannian varieties, using tensor operations on vector bundles. These embeddings generalize Segre and Plücker morphisms.
Consider the natural torus action on a partial flag manifold $Fl$. Let $Ω_I\subset Fl$ be an open Schubert variety, and let $c^{sm}(Ω_I)\in H_T^*(Fl)$ be its torus equivariant Chern-Schwartz-MacPherson class. We show a set of interpolation properties that uniquely determine $c^{sm}(Ω_I)$, as well as a formula, of `localization type', for $c^{sm}(Ω_I)$. In fact, we proved similar results for a class $κ_I\in H_T^*(Fl)$ --- in the context of quantum group actions on the equivariant cohomology groups of partial flag varieties. In this note we show that $c^{SM}(Ω_I)=κ_I$.
In this paper we prove some vanishing theorems for the twisted Dolbeault cohomology of the complete flag varieties associated to a simple, simply connected algebraic group.
We classify all products of flag varieties with finitely many orbits under the diagonal action of the general linear group. We also classify the orbits in each case and construct explicit representatives. This generalizes the classical Schubert decompostion, which states that the GL(n)-orbits on a product of two flag varieties correspond to permutations. Our main tool is the theory of quiver representations.
We introduce two new general methods to compute the Chow motives of twisted flag varieties and settle a 20-year-old conjecture of Markus Rost about the Rost invariant for groups of type E7.
Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on its Langlands dual. This new symmetry is only revealed if Schubert calculus is elevated from cohomology or K theory to the elliptic level.
We give the construction of weighted Lagranngiann Grassmannians$wLGr(3,6)$ and weighted partial $A_3$ flag variety $wFl_{1,3}$ coming from the symplectic Lie group $Sp(6,\mathbb C)$ and the general linear group $GL(4,\mathbb C)$ respectively. We give general formulas for their Hilbert series in terms of Lie theoretic data. We use them as key varieties (Format) to construct some families of polarized 3-folds in codimension 7 and 9. At the end, we list all the distinct weighted flag varieties in codimension $4\le c\le 10$.
For a complete toric variety, we obtain an explicit formula for the localized equivariant Todd class in terms of the combinatorial data -- the fan. This is based on the equivariant Riemann-Roch theorem and the computation of the equivariant cohomology and equivariant homology of toric varieties.
These notes are the written version of my lectures at the Banach Center mini-school "Schubert Varieties" in Warsaw, May 18-22, 2003. Their aim is to give a self-contained exposition of some geometric aspects of Schubert calculus.
The main goal of this paper is to give a unified description for the structure of the small quantum cohomology rings for all homogeneous spaces of SL_n(C).
We study the equivariant cobordism rings for the action of a torus $T$ on smooth varieties over an algebraically closed field of characteristic zero. We prove a theorem describing the rational $T$-equivariant cobordism rings of smooth projective $G$-spherical varieties with the action of a maximal torus $T$ of $G$. As an application, we obtain explicit presentations for the rational equivariant cobordism rings of smooth projective horospherical varieties of Picard number one.
These are extended notes of a talk given at Maurice Auslander Distinguished Lectures and International Conference (Woods Hole, MA) in April 2013. Their aim is to give an introduction into Schubert calculus on Grassmannians and flag varieties. We discuss various aspects of Schubert calculus, such as applications to enumerative geometry, structure of the cohomology rings of Grassmannians and flag varieties, Schur and Schubert polynomials. We conclude with a survey of results of V.Kiritchenko, V.Timorin and the author on a new approach to Schubert calculus on full flag varieties via combinatorics…
We say that a group G acts infinitely transitively on a set X if for every integer m the induced diagonal action of G is transitive on the cartesian mth power of X with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of affine cones over flag varieties, the second of non-degenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups of a reinforced type.
Let $G$ be a split semisimple algebraic group over a field and let $A^*$ be an oriented cohomology theory in the sense of Levine--Morel. We provide a uniform approach to the $A^*$-motives of geometrically cellular smooth projective $G$-varieties based on the Hopf algebra structure of $A^*(G)$. Using this approach we provide various applications to the structure of motives of twisted flag varieties.
We give an effective infinitesimal Torelli theorem for cyclic covers of G/P, where G is a simple algebraic group and P is a maximal parabolic subgroup.
In this article we obtain many results on the multiplicative structure constants of $T$-equivariant Grothendieck ring of the flag variety $G/B$. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_{T}(G/B)$ to $R(T)\otimes R(T)$, where $R(T)$ denotes the representation ring of the torus $T$. We further apply our results to describe the multiplicative structure constants of $K(X)_{\mathbb Q}$ where $X$ is the wonderful compactification of the adjoint group of $G$, in terms of the structure constants of Schubert varieties in the Grothendieck ring of $G/B$.
A Richardson variety in a flag variety is an intersection of two Schubert varieties defined by transverse flags. We define and study relative Richardson varieties, which are defined over a base scheme with a vector bundle and two flags. To do so, we generalize transversality of flags to a relative notion, versality, that allows the flags to be non-transverse over some fibers. Relative Richardson varieties share many of the geometric properties of Richardson varieties. We generalize several geometric and cohomological facts about Richardson varieties to relative Richardson varieties. We also pr…
We found an explicit description of all $GL(n,\RR)$-Whittaker functions as oscillatory integrals and thus constructed equivariant mirrors of flag manifolds. As a consequence we proved the Virasoro conjecture for flag manifolds.
We determine generators for the codimension 1 Chow group of the moduli spaces of genus zero stable maps to flag varieties G/P. In the case of SL flags, we find all relations between our generators, showing that they essentially come from $\bar M_{0,n}$. In addition, we analyze the codimension 2 classes on the moduli spaces of stable maps to Grassmannians and prove a new codimension 2 relation. This will lead to a partial reconstruction theorem for the Grassmannian of 2 planes.
This expository note surveys some results on equivariant K-theory of varieties with a torus action, focusing on recent work with Sam Payne and Richard Gonzales. It is based on my contribution to the Clifford Lectures at Tulane University in March 2015.
We characterize embeddability of algebraic varieties into smooth toric varieties and prevarieties. Our embedding results hold also in an equivariant context and thus generalize a well known embedding theorem of Sumihiro on quasiprojective G-varieties. The main idea is to reduce the embedding problem to the affine case. This is done by constructing equivariant affine conoids, a tool which extends the concept of an equivariant affine cone over a projective G-variety to a more general framework.
Forgetting a subspace from a partial flag yields another partial flag composed of fewer subspaces. This induces a forgetful map $π: X \to X'$ between the corresponding flag varieties. We prove here that, for a degree large enough, the variety associated with degree d stable maps sending their marked points within Schubert varieties $X_i$ of $X$ is a rationally connected fibration over its image, which parametrizes degree $π_* d$ stable maps sending their marked points within the Schubert varieties $π(X_i)$ of $X'$. The Euler characteristic of these varieties are quantum $K$-invariants. Our res…
We give an alternative proof of a recent result by Pasquier stating that for a generalized flag variety $X=G/P$ and an effective $\mathbb{Q}$-divisor $D$ stable with respect to a Borel subgroup the pair $(X,D)$ is Kawamata log terminal if and only if $\lfloor D\rfloor=0$.
Following Skorobogatov and Serganova we construct the embeddings of universal torsors over del Pezzo surfaces in the cones over flag varieties, considered as the closed orbits in the projectivization of quasiminiscule representations. We give the approach that allows us to construct the embeddings of universal torsors over del Pezzo surfaces of the degree one. This approach uses the Mumford stability and slightly differs from the approach of the authors named above.
We introduce a superpotential for partial flag varieties of type $A$. This is a map $W: Y^\circ \to \mathbb{C}$, where $Y^\circ$ is the complement of an anticanonical divisor on a product of Grassmannians. The map $W$ is expressed in terms of Plücker coordinates of the Grassmannian factors. This construction generalizes the Marsh--Rietsch Plücker coordinate mirror for Grassmannians. We show that in a distinguished cluster chart for $Y$, our superpotential agrees with earlier mirrors constructed by Eguchi--Hori--Xiong and Batyrev--Ciocan-Fontanine--Kim--van Straten. Our main tool is quantum Sch…
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 $\bigoplus_{0\leq i\leq n} \mathcal{O}_{\mathbb{P}^n}(i)$.
Let $P$ be a parabolic subgroup in $G=SL_n(\mathbf k)$, for $\mathbf k$ an algebraically closed field. We show that there is a $G$-stable closed subvariety of an affine Schubert variety in an affine partial flag variety which is a natural compactification of the cotangent bundle $T^*G/P$. Restricting this identification to the conormal variety $N^*X(w)$ of a Schubert divisor $X(w)$ in $G/P$, we show that there is a compactification of $N^*X(w)$ as an affine Schubert variety. It follows that $N^*X(w)$ is normal, Cohen-Macaulay, and Frobenius split.
We construct a basis of the equivariant $K$-theory of Bott towers, and we describe precisely the multiplicative structure of these algebras. We deduce similar results for Bott-Samelson varieties. Thanks to the link between flag varieties and Bott-Samelson varieties, we give a method to compute the structure constants of the equivariant $K$-theory of flag varieties in the basis constructed by Kostant and Kumar.
We use geometry to give a short proof of an equivariant Pieri rule in the classical flag manifold. This rule is due to Robinson, who gave an algebraic proof.
We prove that the invariant Hilbert scheme parametrising the equivariant deformations of the affine multicone over a flag variety is, under certain hypotheses, an affine space. The proof is based on the construction of a wonderful variety in a fixed multiprojective space.
We prove an explicit combinatorial formula for certain structure constants of the T-equivariant cohomology of the flag manifold SLn/B. Our result generalizes the Pieri-type formula in ordinary cohomology proved by Sottile in 1996. Our result also gives a Pieri-type formula for the double Schubert polynomials introduced by Lascoux and Schutzenberger.
We investigate the secant dimensions and the identifiablity of flag varieties parametrizing flag of sub vector spaces of a fixed vector space. We give numerical conditions ensuring that secant varieties of flag varieties have the expected dimension, and that a general point on these secant varieties is identifiable.
We study the equivariant K-group of the affine flag manifold with respect to the Borel group action. We prove that the structure sheaf of the (infinite-dimensional) Schubert variety in the K-group is represented by a unique polynomial, which we call the affine Grothendieck polynomial.
Using Quot schemes and a localization theorem we study Gromov-Witten invariants for partial flag varieties. The strategy is to extend A. Bertram's result of Gromov-Witten invariants for special Schubert varieties of Grassmannians to the case of partial flag varieties. To do so a Grothendieck's Quot scheme is generalized for flags and proven to be an irreducible, rational, smooth, projective variety following Strømme \cite{St}.
The Witt ring of a complex flag variety describes the interesting -- i.e. torsion -- part of its topological KO-theory. We show that for a large class of flag varieties, these Witt rings are exterior algebras, and that the degrees of the generators can be determined by Dynkin diagram combinatorics. Besides full flag varieties, projective spaces, and other varieties whose Witt rings were previously known, this class contains many flag varieties of exceptional types. Complete results are obtained for flag varieties of types G2 and F4. The results also extend to flag varieties over other algebrai…
Explicit combinatorial cancellation-free rules are given for the product of an equivariant line bundle class with a Schubert class in the torus-equivariant K-theory of a Kac-Moody flag manifold. The weight of the line bundle may be dominant or antidominant, and the coefficients may be described either by Lakshmibai-Seshadri paths or by the alcove model of the first author and Postnikov. For Lakshmibai-Seshadri paths, our formulas are the Kac-Moody generalizations of results of Griffeth-Ram and Pittie-Ram for finite dimensional flag manifolds. A gap in the proofs of the mentioned results is add…
