Victor Guillemin, Silvia Sabatini, Catalin Zara
The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray-Hirsch theorem. Then we apply this result to the equivariant cohomology theory of flag varieties.
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Borel-Hirzebruch type formula for the graph equivariant cohomology of a projective bundle over a GKM-graph
本文定义GKM理论中的腿丛及其射影化，证明射影化图等变上同调的Borel-Hirzebruch型公式，并说明其可实现Guillemin-Sabatini-Zara意义下的射影GKM纤维丛。
GKM graph locally modeled by $T^{n}\times S^{1}$-action on $T^{*}\mathbb{C}^{n}$ and its graph equivariant cohomology
该研究引入一类包含两种4n维带$T^{n}\times S^{1}$作用流形GKM图的带枝标号图，计算其图等变上同调环，借助剥壳结构给出模基并研究乘法结构。
This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem o…
On orientability, Poincaré duality, and connectivity of GKM graphs
该研究探讨抽象GKM图的组合可定向性，证明其可定向性等价于有理图上同调代数的庞加莱对偶性，并应用该结论得出可定向GKM图删去任意单个顶点后仍连通的结果。
Let $T$ be a torus of dimension $n>1$ and $M$ a compact $T-$manifold. $M$ is a GKM manifold if the set of zero dimensional orbits in the orbit space $M/T$ is zero dimensional and the set of one dimensional orbits in $M/T$ is one dimensional. For such a manifold these sets of orbits have the structure of a labelled graph and it is known that a lot of topological information about $M$ is encoded in this graph. In this paper we prove that every compact homogeneous space $M$ of non-zero Euler characteristic is of GKM type and show that the graph associated with $M$ encodes \emph{geometric} informa…
Suppose a compact torus $T$ acts on a closed smooth manifold $M$. Under certain conditions, Guillemin and Zara associate to $(M, T)$ a labeled graph $\mG_M$ where the labels lie in $H^2(BT)$. They also define the subring $H_T^*(\mG_M)$ of $\bigoplus_{v\in V(\mG_M)}H^*(BT)$, where $V(\mG_M)$ is the set of vertices of $\mG_M$ and we call $H_T^*(\mG_M)$ the "graph cohomology" ring of $\mG_M$. It is known that the equivariant cohomology ring of $M$ can be described by using combinatorial data of the labeled graph. The main result of this paper is to determine the ring structure of equivariant coho…
In this manuscript we develop the theory of poset pinball, a combinatorial game recently introduced by Harada and Tymoczko for the study of the equivariant cohomology rings of GKM-compatible subspaces of GKM spaces. Harada and Tymoczko also prove that in certain circumstances, a successful outcome of Betti poset pinball yields a module basis for the equivariant cohomology ring of the GKM-compatible subspace. Our main contributions are twofold. First we construct an algorithm (which we call the dimension pair algorithm) which yields the result of a successful outcome of Betti poset pinball for …
Equivariant cohomology of even-dimensional complex quadrics from a combinatorial point of view
本文从组合角度确定了偶数维复二次曲面诱导的GKM图等变上同调的环结构，建立了2n次生成元的乘法关系，并解释了x平方在n奇偶时的消失与非消失现象。
Modular law through GKM theory
本文通过 GKM 理论，为正则半单 Hessenberg 簇等几何对象的模律提供了初等且更简单的证明，进而支持了 Shareshian-Wachs 猜想等相关结论。
We show that the Frobenius character of the equivariant Borel-Moore homology of a certain positive $GL_n$-version of the unramified affine Springer fiber $Z_k$ studied by Goreski, Kottwitz and MacPherson is computed by the matrix coefficients of the $\nabla^k$-operator, which acts diagonally in the modified Macdonald basis. We do this by relating the combinatorial formula for the $\nabla^k$-operator we obtained in an earlier paper to the GKM paving of $Z_k$, and we give an algebraic presentation of the above homology as an explicit submodule of the Kostant-Kumar nil Hecke algebra. We then stud…
In this paper, we study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman-Weitsman's proof of the GKM theorem in this setting. The main example is the symplectic reduction X//S of a Hamiltonian T-manifold X by a subtorus S of T. This includes the class of symplectic toric orbifolds. We define the equivariant Chen-Ruan cohomology ring and use the above results to establish a combinatorial method of computing this equivariant Chen-Ruan cohomology in terms of orbifold fixed point data.
Let $Γ$ be a finite d-valent graph and G an n-dimensional torus. An ``action'' of G on $Γ$ is defined by a map, $α$, which assigns to each oriented edge e of $Γ$ a one-dimensional representation of G (or, alternatively, a weight, $α_e$, in the weight lattice of G). For the assignment, $e \to α_e$, to be a schematic description of a ``G-action'', these weights have to satisfy certain compatibility conditions: the GKM axioms. We attach to $(Γ, α)$ an equivariant cohomology ring, $H_G(Γ)=H(Γ,α)$. By definition this ring contains the equivariant cohomology ring of a point, $\SS(\fg^*) = H_G(pt)$, …
Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this paper, we analyze the equivariant cohomology of Springer fibers for $GL_n(\mathbb{C})$ using results of Kumar and Procesi that describe this equivariant cohomology as a quotient ring. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equ…
We construct a divided difference operator using GKM theory. This generalizes the classical divided difference operator for the cohomology of the complete flag variety. This construction proves a special case of a recent conjecture of Shareshian and Wachs. Our methods are entirely combinatorial and algebraic, and rely heavily on the combinatorics of root systems and Bruhat order.
Torus Equivariant Cohomology for the $Δ$-Springer Fiber
本文定义了一个环面U作用于Δ-斯普林纤维Y_{n,λ,s}，并给出其等变上同调环的博雷尔式呈现，方法基于轨道谐波变形技术及Chou-Matsumura-Rhoades和Abe-Horiguchi的方法。
The local homological structure of generalized splines
本文研究了广义样条的局部同调结构，通过两个精确序列揭示其与局部环同调机器的协调性，并证明了特定条件下广义样条模块的自由性。
We prove that the center of a regular block of parabolic category O for the general linear Lie algebra is isomorphic to the cohomology algebra of a corresponding Springer fiber. This was conjectured by Khovanov. We also find presentations for the centers of singular blocks, which are cohomology algebras of Spaltenstein varieties.
We give a vanishing theorem for the monodromy eigenspaces of the Milnor fibers of complex line arrangements. By applying the modular bound of the local system cohomology groups given by Papadima-Suciu, the result is deduced from the vanishing of the cohomology of certain Aomoto complex over finite fields. In order to prove this, we introduce degeneration homomorphisms of Orlik-Solomon algebras.
We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classi…
Peterson varieties are a special class of Hessenberg varieties that have been extensively studied e.g. by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley-Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety $Y$ in type $A_{n-1}$, with respect to a natural $S^1$-action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculu…
For several important classes of manifolds acted on by the torus, the information about the action can be encoded combinatorially by a regular n-valent graph with vector labels on its edges, which we refer to as the torus graph. By analogy with the GKM-graphs, we introduce the notion of equivariant cohomology of a torus graph, and show that it is isomorphic to the face ring of the associated simplicial poset. This extends a series of previous results on the equivariant cohomology of torus manifolds. As a primary combinatorial application, we show that a simplicial poset is Cohen-Macaulay if it…
We study the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of odd degree cohomology, so that $X_h$ is an equivariantly formal manifold. The equivariant and ordinary cohomology of $X_h$ are described using GKM-theory. The main goal of this paper is to show the connection between the manifolds $X_h$ and the semisimple Hessenberg v…
Springer fibers and the Delta Conjecture at $t=0$
本研究引入推广A型Springer纤维的Δ-Springer簇，给出其上同调环的显式表示与对称群作用，为t=0时的Delta猜想提供几何实现，并推广了对角幂零矩阵概形的相关结果。
The classical BKK theorem computes the intersection number of divisors on toric variety in terms of volumes of corresponding polytopes. It was observed by Pukhlikov and the first author that the BKK theorem leads to a presentation of the cohomology ring of toric variety as a quotient of the ring of differential operators with constant coefficients by the annihilator of an explicit polynomial. In this paper we generalize this construction to the case of quasitoric bundles. These are fiber bundles with generalized quasitoric manifolds as fibers. First we obtain a generalization of the BKK theore…
We study the geometry and topology of exotic Springer fibers for orbits corresponding to one-row bipartitions from an explicit, combinatorial point of view. This includes a detailed analysis of the structure of the irreducible components and their intersections as well as the construction of an explicit affine paving. Moreover, we compute the ring structure of cohomology by constructing a CW-complex homotopy equivalent to the exotic Springer fiber. This homotopy equivalent space admits an action of the type C Weyl group inducing Kato's original exotic Springer representation on cohomology. Our…
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