A hypertoric variety is a quaternionic analogue of a toric variety. Just as the topology of toric varieties is closely related to the combinatorics of polytopes, the topology of hypertoric varieties interacts richly with the combinatorics of hyperplane arrangements and matroids. Using finite field methods, we obtain combinatorial descriptions of the Betti numbers of hypertoric varieties, both for ordinary cohomology in the smooth case and intersection cohomology in the singular case. We also introduce a conjectural ring structure on the intersection cohomology of a hypertoric variety.
In this paper, we study and describe the universal Poisson deformation space of hypertoric varieties concretely. In the first application, we show that affine hypertoric varieties as conical symplectic varieties are classified by the associated regular matroids (this is a partial generalization of the result by Arbo and Proudfoot). As a corollary, we obtain a criterion when two quiver varieties whose dimension vector have all coordinates equal to one are isomorphic to each other. Then we describe all 4- and 6-dimensional affine hypertoric varieties as quiver varieties and give some examples of…
We establish a connection between the orbifold cohomology of hypertoric varieties and the Ehrhart theory of Lawrence polytopes. More specifically, we show that the dimensions of the orbifold cohomology groups of a hypertoric variety are equal to the coefficients of the Ehrhart $δ$-polynomial of the associated Lawrence polytope. As a consequence, we deduce a formula for the Ehrhart $δ$-polynomial of a Lawrence polytope and use the injective part of the Hard Lefschetz Theorem for hypertoric varieties to deduce some inequalities between the coefficients of the $δ$-polynomial.
The hypertoric variety $\mathfrak{M}_{\mathcal{A}}$ defined by an affine arrangement $\mathcal{A}$ admits a natural tropicalization, induced by its embedding in a Lawrence toric variety. We explicitly describe the polyhedral structure of this tropicalization. Using a recent result of Gubler, Rabinoff, and Werner, we prove that there is a continuous section of the tropicalization map.
We give an abstract definition of a hypertoric variety, generalizing the existing constructive definition. We construct a hypertoric variety associated with any zonotopal tiling, and we show that the previously known examples are exactly those varieties associated with regular tilings. In particular, the examples that we construct from irregular tilings have not appeared before. We conjecture that our construction gives a complete classification of hypertoric varieties, analogous to the classification of toric varieties by fans.
In this paper, we study the (singular) universal cover of an affine hypertoric variety. We show that it is given by another affine hypertoric variety, and taking the universal cover corresponds to taking the simplification of the associated hyperplane arrangement. Also, we describe the fundamental group of the regular locus of an affine hypertoric variety in general. In the latter part, we show that the hamiltonian torus action is block indecomposable if and only if $\mathbb{C}^*$-equivariant symplectic structures on the associated hypertoric variety are unique up to scalar. In particular, we …
We introduce elliptic hypertoric varieties, which is an elliptic analogue of hypertoric varieties and multiplicative hypertoric varieties. We also prove an elliptic version of Hikita conjecture, which relates elliptic (resp. additive and mulitplicative) hypertoric varieties to the equivariant elliptic cohomology (resp. ordinary cohomology and K-theory) of their 3d mirror hypertoric varieties.
Hypertoric varieties are quaternionic analogues of toric varieties, important for their interaction with the combinatorics of matroids as well as for their prominent place in the rapidly expanding field of algebraic symplectic and hyperkahler geometry. The aim of this survey is to give clear definitions and statements of known results, serving both as a reference and as a point of entry to this beautiful subject.
Recently McBreen and Webster constructed a tilting bundle on a smooth hypertoric variety (using reduction to finite characteristic) and showed that its endomorphism ring is Koszul. In this short note we present alternative proofs for these results. We simply observe that the tilting bundle constructed by Halpern-Leistner and Sam on a generic open GIT substack of the ambient linear space restricts to a tilting bundle on the hypertoric variety. The fact that the hypertoric variety is defined by a quadratic regular sequence then also yields an easy proof of Koszulity.
We construct a family of vertex algebras associated with a family of symplectic singularity/resolution, called hypertoric varieties. While the hypertoric varieties are constructed by a certain Hamiltonian reduction associated with a torus action, our vertex algebras are constructed by (semi-infinite) BRST reduction. The construction works algebro-geometrically and we construct sheaves of $\hbar$-adic vertex algebras over hypertoric varieties which localize the vertex algebras. We show when the vertex algebras are vertex operator algebras by giving explicit conformal vectors. We also show that …
We give a complete description of the equivariant quantum cohomology ring of any smooth hypertoric variety, and find a mirror formula for the quantum differential equation.
Motivated by the polynomial representation theory of the general linear group and the theory of symplectic singularities, we study a category of perverse sheaves with coefficients in a field $k$ on any affine unimodular hypertoric variety. Our main result is that this is a highest weight category whose Ringel dual is the corresponding category for the Gale dual hypertoric variety. On the way to proving our main result, we confirm a conjecture of Finkelberg-Kubrak in the case of hypertoric varieties. We also show that our category is equivalent to representations of a combinatorially-defined al…
We construct a Lagrangian torus fibration on a smooth hypertoric variety and a corresponding SYZ mirror variety using $T$-duality and generating functions of open Gromov-Witten invariants. The variety is singular in general. We construct a resolution using the wall and chamber structure of the SYZ base.
We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset.
For a smooth toric variety X over a field of positive characteristic, a T-equivariant étale cover Y \rightarrow T^*X^{(1)} trivializing the sheaf of crystalline differential operators on X is constructed. This trivialization is used to show that the sheaf of differential operators is a trivial Azumaya algebra along the fibers of the moment map. This result is then extended to certain Azumaya algebras on hypertoric varieties, whose global sections are central reductions of the hypertoric enveloping algebra in positive characteristic. A criteria for a derived Beilinson-Bernstein localization the…
We construct quantizations of multiplicative hypertoric varieties using an algebra of q-difference operators on affine space, where q is a root of unity in C. The quantization defines a matrix bundle (i.e. Azumaya algebra) over the multiplicative hypertoric variety and admits an explicit finite étale splitting. The global sections of this Azumaya algebra is a hypertoric quantum group, and we prove a localization theorem. We introduce a general framework of Frobenius quantum moment maps and their Hamiltonian reductions; our results shed light on an instance of this framework.
Following the idea of Aganagic--Okounkov \cite{AOelliptic}, we study vertex functions for hypertoric varieties, defined by $K$-theoretic counting of quasimaps from $\mathbb{P}^1$. We prove the 3d mirror symmetry statement that the two sets of $q$-difference equations of a 3d hypertoric mirror pairs are equivalent to each other, with Kähler and equivariant parameters exchanged, and the opposite choice of polarization. Vertex functions of a 3d mirror pair, as solutions to the $q$-difference equations, satisfying particular asymptotic conditions, are related by the elliptic stable envelopes. Vari…
We consider q-deformations of multiplicative hypertoric varieties, for q a non-zero element of an algebraically closed field of characteristic 0. We construct an algebra Dq of q-difference operators as a Heisenberg double in a braided monoidal category. We then focus on the case where q is specialized to a root of unity. In this setting, we use Dq to construct an Azumaya algebra on an l-twist of the multiplicative hypertoric variety, before showing that this algebra splits over the fibers of both the moment and resolution maps. Finally, we sketch a derived localization theorem for these Azumay…
We prove a conjecture of Etingof and the second author for hypertoric varieties, that the Poisson-de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we prove that this conjecture holds for an arbitrary conical variety admitting a symplectic resolution if and only if it holds in degree zero for all normal slices to symplectic leaves. The Poisson-de Rham homology of a Poisson cone inherits a second grading. In the hypertoric case, we compute the resulting 2-variable Poisson-de Rham-Poincare polynomial, and prove …
For any conical symplectic resolution, we give a conjecture relating the intersection cohomology of the singular cone to the quantum cohomology of its resolution. We prove this conjecture for hypertoric varieties, recovering the ring structure on hypertoric intersection cohomology that was originally constructed by Braden and the second author.
Hypertoric varieties are determined by hyperplane arrangements. In this paper, we use stacky hyperplane arrangements to define the notion of hypertoric Deligne-Mumford stacks. Their orbifold Chow rings are computed. As an application, some examples related to crepant resolutions are discussed.
Vertex Superalgebras for Hypertoric Varieties and 3d Abelian Gauge Theories
Research Institute for Mathematical Sciences, Kyoto University（京都大学数学科学研究所）；Okinawa Institute of Science and Technology（冲绳科学技术大学院大学）；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器）；School of Mathematics, University of Edinburgh（爱丁堡大学数学学院）；DAMTP, University of Cambridge（剑桥大学应用数学与理论物理系）；Fachbereich Mathematik, Universität Hamburg（汉堡大学数学系）；Department of Mathematics and Statistics, Maynooth University（梅努斯大学数学与统计系）
We extend slightly the results of Evens-Mirković, and "compute" the characteristic cycles of Intersection Cohomology sheaves on the transversal slices in the double affine Grassmannian and on the hypertoric varieties. We propose a conjecture relating the hyperbolic stalks and the microlocalization at a torus-fixed point in a Poisson variety.
We show that the equivariant Chow cohomology ring of a toric variety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which localization holds in equivariant Chow cohomology with integer coefficients. We also compute the equivariant Chow cohomology of toric prevarieties and general complex hypertoric varieties in terms of piecewise polynomial functions.
We extend the methods developed in our earlier work to algorithmically compute the intersection cohomology Betti numbers of reductive varieties. These form a class of highly symmetric varieties that includes equivariant compactifications of reductive groups. Thereby, we extend a well-known algorithm for toric varieties.
In this thesis we compute motivic classes of hypertoric varieties, Nakajima quiver varieties and open de Rham spaces in a certain localization of the Grothendieck ring of varieties. Furthermore we study the $p$-adic pushforward of the Haar measure under a hypertoric moment map $μ$. This leads to an explicit formula for the Igusa zeta function $I_μ(s)$ of $μ$, and in particular to a small set of candidate poles for $I_μ(s)$. We also study various properties of the residue at the largest pole of $I_μ(s)$. Finally, if $μ$ is constructed out of a quiver $Γ$ we give a conjectural description of thi…
In previous work, we initiated the study of the cohomology of locally acyclic cluster varieties. In the present work, we show that the mixed Hodge structure and point counts of acyclic cluster varieties are essentially determined by the combinatorics of the independent sets of the quiver. We use this to show that the mixed Hodge numbers of acyclic cluster varieties of really full rank satisfy a strong vanishing condition.
We show that various genus zero Gromov-Witten invariants for flag varieties representing different homology classes are indeed the same. In particular, many of them are classical intersection numbers of Schubert cycles.
This paper introduces a two-parameter deformation of the cohomology of generalized flag varieties. One special case is the Belkale-Kumar deformation (used to study eigencones of Lie groups). Another picks out intersections of Schubert varieties that behave nicely under projections. Our construction yields a new proof that the Belkale-Kumar product is well-defined. This proof is shorter and more elementary than earlier proofs.
We show a canonical injective morphism from the quantum cohomology ring QH(G/P) to the associated graded algebra of QH(G/B), which is with respect to a nice filtration on QH(G/B) introduced by Leung and the author. This tells us the vanishing of a lot of genus zero, three-pointed Gromov-Witten invariants of flag varieties G/P.
This work discusses combinatorial and arithmetic aspects of cohomology vanishing for divisorial sheaves on toric varieties. We obtain a refined variant of the Kawamata-Viehweg theorem which is slightly stronger. Moreover, we prove a new vanishing theorem related to divisors whose inverse is nef and has small Kodaira dimension. Finally, we give a new criterion for divisorial sheaves for being maximal Cohen-Macaulay.
We give sufficient conditions for the semisimplicity of quantum cohomology of Fano varieties of Picard rank 1. We apply these techniques to prove new semisimplicity results for some Fano varieties of Picard rank 1 and large index. We also give examples of Fano varieties having a non semisimple small quantum cohomology but a semisimple big quantum cohomology.
A general vanishing result for the first cohomology group of affine smooth complex varieties with values in rank one local systems is established. This is applied to the determination of the monodromy action on the first cohomology group of the Milnor fiber of some line arrangements, including the monomial arrangement and the exceptional reflection arrangement of type $G_{31}$.
We use the toric degeneration of Bott-Samelson varieties and the description of cohomolgy of line bundles on toric varieties to deduce vanishings results for the cohomology of lines bundles on Bott-Samelson varieties.
In this paper we study a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials $f_{i,j}$ were introduced by Abe-Harada-Horiguchi-Masuda. We show that every polynomial $f_{i,j}$ is an alternating sum of certain Schubert polynomials.
We use a theorem of Tolman and Weitsman to find explicit formulæfor the rational cohomology rings of the symplectic reduction of flag varieties in C^n, or generic coadjoint orbits of SU(n), by (maximal) torus actions. We also calculate the cohomology ring of the moduli space of $n$ points in CP^k, which is isomorphic to the Grassmannian of k-planes in C^n, by realizing it as a degenerate coadjoint orbit.
The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this variety's cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action. Sixteen years later, Garsia and Procesi made Springer's work more transparent and accessible by presenting the cohomology ring as a graded quotient of a polynomial ring. They combinatorially describe an explicit basis for this quotient. The goal of this paper is to generalize their work. Our main result deepens their analysis of Springer varieties and extend…
This is an expository lecture, for the Abel bicentennial (Oslo, 2002), describing some recent work on the (small) quantum cohomology ring of Grassmannians and other homogeneous varieties.
We study the Fulton-Macpherson Chow cohomology of affine toric varieties. In particular, we prove that the Chow cohomology vanishes in positive degree. We prove an analogous result for the operational $K$-theory defined by Anderson and Payne.
We study intersection cohomology of character varieties for punctured Riemann surfaces with prescribed monodromies around the punctures. Relying on previous result from Mellit for semisimple monodromies we compute the intersection cohomology of character varieties with monodromies of any Jordan type. This proves the Poincaré polynomial specialization of a conjecture from Letellier.
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).
In this paper we study the étale cohomology groups associated to abelian varieties. We obtain necessary and sufficient conditions for an abelian variety to have semistable reduction (or purely additive reduction which becomes semistable over a quadratic extension) in terms of the action of the absolute inertia group on the étale cohomology groups with finite coefficients.
A horospherical variety is a normal algebraic variety where a connected reductive algebraic group acts with an open orbit isomorphic to a torus bundle over a flag variety. In this article we study the cohomology of line bundles on complete horospherical varieties.
By using vector field techniques, we compute the ordinary and equivariant cohomology rings of Hilbert scheme of points in the projective plane in relation with that of a Grassmann variety.
In this paper we construct a spectral sequence computing a modified version of morphic cohomology of a toric variety (even when it is singular) in terms of combinatorial data coming from the fan of the toric variety.
We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks.
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 …
In this short note we prove that if X is a separably rationally connected variety over an algebraically closed field of positive characteristic, then H^1(X, O_X)=0.
We describe the weight filtration in the cohomology of toric varieties. We present a role of the Frobenius automorphism in an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We obtain results concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complex $IH_T^*(X)\otimes H^*(T)$. We also describe the weight filtration in $IH^*(X)$.
