aa r X i v : . [ m a t h . AG ] M a y A survey of hypertoric geometry and topology
Nicholas Proudfoot Department of Mathematics, Columbia University, 10027
Abstract.
Hypertoric varieties are quaternionic analogues of toric varieties, importantfor their interaction with the combinatorics of matroids as well as for their prominentplace in the rapidly expanding field of algebraic symplectic and hyperk¨ahler 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.
Given a linear representation of a reductive complex algebraic group G , there are two naturalquotient constructions. First, one can take a geometric invariant theory (GIT) quotient,which may also be interpreted as a K¨ahler quotient by a maximal compact subgroup of G .Examples of this sort include toric varieties (when G is abelian), moduli spaces of spacialpolygons, and, more generally, moduli spaces of semistable representations of quivers. Asecond construction involves taking an algebraic symplectic quotient of the cotangent bundleof V , which may also be interpreted as a hyperk¨ahler quotient. The analogous examples ofthe second type are hypertoric varieties, hyperpolygon spaces, and Nakajima quiver varieties.The subject of this survey will be hypertoric varieties, which are by definition the vari-eties obtained from the second construction when G is abelian. Just as the geometry andtopology of toric varieties is deeply connected to the combinatorics of polytopes, hypertoricvarieties interact richly with the combinatorics of hyperplane arrangements and matroids.Furthermore, just as in the toric case, the flow of information goes in both directions.On one hand, Betti numbers of hypertoric varieties have a combinatorial interpretation,and the geometry of the varieties can be used to prove combinatorial results. Many purely al-gebraic constructions involving matroids acquire geometric meaning via hypertoric varieties,and this has led to geometric proofs of special cases of the g-theorem for matroids [HSt,7.4] and the Kook-Reiner-Stanton convolution formula [PW, 5.4]. Future plans include ageometric interpretation of the Tutte polynomial and of the phenomenon of Gale duality ofmatroids [BLP].On the other hand, hypertoric varieties are important to geometers with no interest incombinatorics simply because they are among the most explicitly understood examples ofalgebraic symplectic or hyperk¨ahler varieties, which are becoming increasingly prevalent inmany areas of mathematics. For example, Nakajima’s quiver varieties include resolutions ofSlodowy slices and Hilbert schemes of points on ALE spaces, both of which play major rolesin modern representation theory. Moduli spaces of Higgs bundles are currently receiving alot of attention in string theory, and character varieties of fundamental groups of surfacesand 3-manifolds have become an important tool in low-dimensional topology. Hypertoric Supported by a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship.
Acknowledgments.
The author is grateful to the organizers of the 2006 InternationalConference on Toric Topology at Osaka City University, out of which this survey grew.
Hypertoric varieties can be considered either as algebraic varieties or, in the smooth case,as hyperk¨ahler manifolds. In this section we give a constructive definition, with a strongbias toward the algebraic interpretation. Section 1.1 proceeds in greater generality thanis necessary for hypertoric varieties so as to unify the theory with that of other algebraicsymplectic quotients, most notably Nakajima quiver varieties.
Let G be a reductive algebraic group over the complex numbers acting linearly and effectivelyon a finite-dimensional complex vector space V . The cotangent bundle T ∗ V ∼ = V × V ∗ G on T ∗ V is hamilto-nian, with moment map µ : T ∗ V → g ∗ given by the equation µ ( z, w )( x ) = Ω( x · z, w ) for all z ∈ V , w ∈ V ∗ , x ∈ g .Suppose given an element λ ∈ Z ( g ∗ ) (the part of g ∗ fixed by the coadjoint action of G ), anda multiplicative character α : G → C × , which may be identified with an element of Z ( g ∗ Z ) bytaking its derivative at the identity element of G . The fact that λ lies in Z ( g ∗ ) implies that G acts on µ − ( λ ). Our main object of study in this survey will be the algebraic symplecticquotient M α,λ = T ∗ V//// α G = µ − ( λ ) // α G. Here the second quotient is a projective GIT quotient µ − ( λ ) // α G := Proj ∞ M m =0 n f ∈ Fun (cid:0) µ − ( λ ) (cid:1) (cid:12)(cid:12)(cid:12) ν ( g ) ∗ f = α ( g ) m f for all g ∈ G o , (1)where ν ( g ) is the automorphism of µ − ( λ ) defined by g .This quotient may be defined in a more geometric way as follows. A point ( z, w ) ∈ µ − ( λ )is called α -semistable if there exists a function f on µ − ( λ ) and a positive integer m suchthat ν ( g ) ∗ f = α ( g ) m f for all g ∈ G and f ( z, w ) = 0. It is called α -stable if it is α -semistableand its G -orbit in the α -semistable set is closed with finite stabilizers. Then the stable andsemistable sets µ − ( λ ) α − st ⊆ µ − ( λ ) α − ss ⊆ µ − ( λ )are nonempty and Zariski open, and there is a surjection µ − ( λ ) α − ss ։ M α,λ with ( z, w ) and ( z ′ , w ′ ) mapping to the same point if and only if the closures of their G -orbits intersect in µ − ( λ ) α − ss . In particular, the restriction of this map to the stable locus isnothing but the geometric quotient by G . For an introduction to geometric invariant theorythat explains the equivalence of these two perspectives, see [P2]. Remark 1.1.1
The algebraic symplectic quotient defined above may also be interpreted asa hyperk¨ahler quotient. The even dimensional complex vector space T ∗ V admits a completehyperk¨ahler metric, and the action of the maximal compact subgroup G R ⊆ G is hyper- Strictly speaking, an element of Z ( g ∗ Z ) only determines a character of the connected component of theidentity of G . It can be checked, however, that the notion of α -stability defined below depends only on therestriction of α to the identity component, therefore we will abusively think of α as sitting inside of Z ( g ∗ Z ). amiltonian , meaning that it is hamiltonian with respect to all three of the real symplecticforms on T ∗ V . Then M α,λ is naturally diffeomorphic to the hyperk¨ahler quotient of T ∗ V by G R , in the sense of [HKLR], at the value ( α, Re λ, Im λ ) ∈ g ∗ R ⊗ R . This was the originalperspective on both hypertoric varieties [BD] and Nakajima quiver varieties [N1]. For moreon this perspective in the hypertoric case, see Konno’s survey in this volume [K4, § α = 0 is the trivial character of G , then Equation (1) simplifies to M ,λ = Spec Fun (cid:0) µ − ( λ ) (cid:1) G . Furthermore, since M α,λ is defined as the projective spectrum of a graded ring whose degreezero part is the ring of invariant functions on µ − ( λ ), we always have a projective morphism M α,λ ։ M ,λ . (2)This morphism may also be induced from the inclusion of the inclusion µ − ( λ ) α − ss ⊆ µ − ( λ ) = µ − ( λ ) − ss . From this we may conclude that it is generically one-to-one, and therefore a partial resolution.When λ = 0, we have a distinguished point in M , , namely the image of 0 ∈ µ − (0) underthe map induced by the inclusion of the invariant functions into the coordinate ring of µ − (0).The preimage of this point under the morphism (2) is called the core of M α, , and will befurther studied (in the case where G is abelian) in Section 2.1.On the other extreme, if λ is a regular value of µ , then G will act locally freely on µ − ( λ ).In this case all points will be α -stable for any choice of α , and the GIT quotient M λ = µ − ( λ ) //G will simply be a geometric quotient. In particular, the morphism (2) becomes an isomor-phism. Both the case of regular λ and the case λ = 0 will be of interest to us.We call a pair ( α, λ ) generic if µ − ( λ ) α − st = µ − ( λ ) α − ss . In this case the moment mapcondition tells us that the stable set is smooth, and therefore that the quotient M α,λ bythe locally free G -action has at worst orbifold singularities. Using the hyperk¨ahler quotientperspective of Remark 1.1.1, one can prove the following Proposition. (See [K3, 2.6] or [HP1,2.1] in the hypertoric case, and [N1, 4.2] in the case of quiver varieties; the general case isno harder than these.) Proposition 1.1.2 If ( α, λ ) and ( α ′ , λ ′ ) are both generic, then the two symplectic quotients M α,λ and M α ′ ,λ ′ are diffeomorphic. Remark 1.1.3 If G is semisimple, then Z ( g ∗ ) = { } , and (unless G is finite) it will notbe possible to choose a regular value λ ∈ Z ( g ∗ ), nor a nontrivial character α . We will very4oon specialize, however, to the case where G is abelian. In this case Z ( g ∗ ) = g ∗ , the regularvalues form a dense open set, and the characters of G form a full integral lattice g ∗ Z ⊆ g ∗ . Let t n be the coordinate complex vector space of dimension n with basis { ε , . . . , ε n } , and let t d be a complex vector space of dimension d with a full lattice t d Z . Though t d Z is isomorphic tothe standard integer lattice Z d , we will not choose such an isomorphism. Let { a , . . . , a n } ⊂ t d Z be a collection of nonzero vectors such that the map t n → t d taking ε i to a i is surjective.Let k = n − d , and let t k be the kernel of this map. Then we have an exact sequence0 −→ t k ι −→ t n −→ t d −→ , (3)which exponentiates to an exact sequence of tori0 −→ T k −→ T n −→ T d −→ . (4)Here T n = ( C × ) n , T d is a quotient of T n , and T k = ker (cid:0) T n → T d (cid:1) is a subgroup with Liealgebra t k , which is connected if and only if the vectors { a i } span the lattice t d Z over theintegers. Note that every algebraic subgroup of T n arises in this way.The torus T n acts naturally via coordinatewise multiplication on the vector space C n ,thus so does the subtorus T k . For α ∈ ( t k ) ∗ Z a multiplicative character of T k and λ ∈ ( t k ) ∗ arbitrary, the algebraic symplectic quotient M α,λ = T ∗ C n //// α T k is called a hypertoric variety .The hypertoric variety M α,λ is a symplectic variety of dimension 2 d which admits acomplete hyperk¨ahler metric. The action of the quotient torus T d = T n /T k on M α,λ ishamiltonian with respect to the algebraic symplectic form, and the action of the maximalcompact subtorus T d R is hyperhamiltonian. In the original paper of Bielawski and Dancer[BD] the hyperk¨ahler perspective was stressed, and the spaces were referred to as “torichyperk¨ahler manifolds”. However, since we have worked frequently with singular reductionsas well as with fields of definition other than the complex numbers (see for example [HP1,P3, PW]), we prefer the term hypertoric varieties. Remark 1.2.1
In the hypertoric case, the diffeomorphism of Proposition 1.1.2 can be made T d R -equivariant [HP1, 2.1]. The case in which λ = 0 will be of particular importance, and it is convenient to encode thedata that were used to construct the hypertoric variety M α, in terms of an arrangement of5ffine hyperplanes with some additional structure in the real vector space ( t d ) ∗ R = ( t d ) ∗ Z ⊗ Z R .A weighted, cooriented, affine hyperplane H ⊆ ( t d ) ∗ R is an affine hyperplane along witha choice of nonzero integer normal vector a ∈ t d Z . Here “affine” means that H need notpass through the origin, and “weighted” means that a is not required to be primitive. Let r = ( r , . . . , r n ) ∈ ( t n ) ∗ be a lift of α along ι ∗ , and let H i = { x ∈ ( t d ) ∗ R | x · a i + r i = 0 } be the weighted, cooriented, affine hyperplane with normal vector a i ∈ ( t d ) ∗ Z . (Choosing adifferent r corresponds to simultaneously translating all of the hyperplanes by a vector in( t d ) ∗ Z .) We will denote the collection { H , . . . , H n } by A , and write M ( A ) = M α, for the corresponding hypertoric variety. We will refer to A simply as an arrangement ,always assuming that the weighted coorientations are part of the data. Remark 1.3.1
We note that we allow repetitions of hyperplanes in our arrangement ( A may be a multi-set), and that a repeated occurrence of a particular hyperplane is not thesame as a single occurrence of that hyperplane with weight 2. On the other hand, little islost by restricting one’s attention to arrangements of distinct hyperplanes of weight one.Since each hyperplane H i comes with a normal vector, it seems at first that it wouldmake the most sense to talk about an arrangement of half-spaces, where the i th half-spaceconsists of the set of points that lie on the positive side of H i with respect to a i . The reasonthat we talk about hyperplanes rather than half-spaces is the following proposition, provenin [HP1, 2.2]. Proposition 1.3.2
The T d -variety M ( A ) does not depend on the signs of the vectors a i . In other words, if we make a new hypertoric variety with the same arrangement ofweighted hyperplanes but with some of the coorientations flipped, it will be T d -equivariantlyisomorphic to the hypertoric variety with which we started. We call the arrangement A simple if every subset of m hyperplanes with nonemptyintersection intersects in codimension m . We call A unimodular if every collection of d linearly independent vectors { a i , . . . , a i d } spans t d over the integers. An arrangement whichis both simple and unimodular is called smooth . The following proposition is proven in[BD, 3.2 & 3.3]. Proposition 1.3.3
The hypertoric variety M ( A ) has at worst orbifold (finite quotient) sin-gularities if and only if A is simple, and is smooth if and only if A is smooth. In [HP1] we consider an extra C × action on M ( A ) that does depend on the coorientations. A = { H , . . . , H n } be a central arrangement, mean-ing that r i = 0 for all i , so that all of the hyperplanes pass through the origin. Then M ( A ) isthe singular affine variety M , . Let ˜ A = { ˜ H , . . . , ˜ H n } be a simplification of A , by whichwe mean an arrangement defined by the same vectors { a i } ⊂ t d , but with a different choiceof r ∈ ( t n ) ∗ , such that ˜ A is simple. This corresponds to translating each of the hyperplanesin A away from the origin by some generic amount. Then M ( ˜ A ) maps T -equivariantly to M ( A ) by Equation (2), and Proposition 1.3.3 tell us that it is in fact an “orbifold resolu-tion”, meaning a projective morphism, generically one-to-one, in which the source has atworst orbifold singularities. The structure of this map is studied extensively in [PW]. The definition of a hypertoric variety in Section 1.2 is constructive, modeled on the definitionof toric varieties as GIT quotients of the form C n // α T k , or equivalently as symplectic quotientsby compact tori. In the case of toric varieties, there are also abstract definitions. In thesymplectic world, one defines a toric orbifold to be a symplectic orbifold of dimension 2 d alongwith an effective Hamiltonian action of a compact d -torus, and proves that any connected,compact toric orbifold arises from the symplectic quotient construction [De, LT]. In thealgebraic world, one defines a toric variety to be a normal variety admitting a torus actionwith a dense orbit, and then proves that any semiprojective toric variety with at worstorbifold singularities arises from the GIT construction. This idea goes back to [Co], and canbe found in this language in [HSt, 2.6].It is natural to ask for such an abstract definition and classification theorem for hypertoricvarieties, either from the standpoint of symplectic algebraic geometry or that of hyperk¨ahlergeometry. In the hyperk¨ahler setting, such a theorem was proven in [Bi, 3,4]. Theorem 1.4.1
Any complete, connected, hyperk¨ahler manifold of real dimension d whichadmits an effective, hyperhamiltonian action of the compact torus T d R is T d R -equivariantlydiffeomorphic, and Taub-NUT deformation equivalent, to a hypertoric variety. Any suchmanifold with Euclidean volume growth is T d R -equivariantly isometric to a hypertoric variety. An analogous algebraic theorem has not been proven, but it should look something likethe following.
Conjecture 1.4.2
Any connected, symplectic, algebraic variety which is projective over itsaffinization and admits an effective, hamiltonian action of the algebraic torus T d is equiv-ariantly isomorphic to a Zariski open subset of a hypertoric variety. Hausel and Sturmfels call a toric variety semiprojective if it is projective over its affinization and has atleast one torus fixed point. Homotopy models
In this section we fix the vector configuration { a , . . . a n } ⊆ t d Z , consider three spaces thatare T d -equivariantly homotopy equivalent to the hypertoric variety M α,λ for generic choiceof ( α, λ ). Each space is essentially toric rather than hypertoric in nature, and therefore mayprovide a way to think about hypertoric varieties in terms of more familiar objects. Recallthat if λ = 0 then M α,λ = M ( ˜ A ) for a simple hyperplane arrangement ˜ A , in which thepositions of the hyperplanes (up to simultaneous translation) are determined by α . If, onthe other hand, λ is a regular value, then M α,λ = M λ is independent of α . Recall from Section 1.3 that we have an equivariant orbifold resolution M ( ˜ A ) → M ( A ) , and from Section 1.1 that the fiber L ( ˜ A ) ⊆ M ( ˜ A ) over the most singular point of M ( A )is called the core of M ( ˜ A ). The primary interest in the core comes from the followingproposition, originally proven in [BD, 6.5] from the perspective of Proposition 2.1.4. Proposition 2.1.1
The core L ( ˜ A ) is a T d R -equivariant deformation retract of M ( ˜ A ) . Remark 2.1.2
In fact, Proposition 2.1.1 holds in the greater generality of Section 1.1, foralgebraic symplectic quotients M α, by arbitrary reductive groups [P1, 2.8]. The cores ofNakajima’s quiver varieties play an important role in representation theory, because thefundamental classes of the irreducible components form a natural basis for the top nonvan-ishing homology group of M α, , which may be interpreted as a weight space of an irreduciblerepresentation of a Kac-Moody algebra [N2, 10.2].We now give a toric interpretation of L ( ˜ A ). For any subset U ⊆ { , . . . , n } , let P U = { x ∈ ( t d ) ∗ R | x · a i + r i ≥ i ∈ U and x · a i + r i ≤ i / ∈ U } . (5)Thus P U is the polyhedron “cut out” by the cooriented hyperplanes of ˜ A after reversing thecoorientations of the hyperplanes with indices in U . Since ˜ A is a weighted arrangement, P U is a labeled polytope in the sense of [LT]. Let E U = { ( z, w ) ∈ T ∗ C n | w i = 0 if i ∈ U and z i = 0 if i / ∈ U } and X U = E U // α T k . E U ⊆ µ − (0), and therefore X U = E U // α T k ⊆ µ − (0) // α T k = M ( ˜ A ) . The following proposition is proven in [BD, 6.5], but is stated more explicitly in this languagein [P1, 3.8].
Proposition 2.1.3
The variety X U is isomorphic to the toric orbifold classified by theweighted polytope P U . It is not hard to see that the subvariety E U // α T k ⊆ M ( ˜ A ) lies inside the core L ( ˜ A ) of M ( ˜ A ). In fact, these subvarieties make up the entire core, as can be deduced from [BD, § Proposition 2.1.4 L ( ˜ A ) = [ P U bounded E U // α T k ⊆ M ( ˜ A ) . Thus L ( ˜ A ) is a union of compact toric varieties sitting inside the hypertoric M ( ˜ A ), gluedtogether along toric subvarieties as prescribed by the combinatorics of the polytopes P U andtheir intersections in ( t d ) ∗ R . Example 2.1.5
Consider the two hyperplane arrangement pictured below, with all hyper-planes having primitive normal vectors. Note that there are two primitive vectors to choosefrom for each hyperplane (one must choose a direction), but the corresponding hypertoricvarieties and their cores will be independent of these choices by Proposition 1.3.2. In the
41 3 1 22 34 first picture, the core consists of a C P (the toric variety associated to a triangle) and a C P blown up at a point (the toric variety associated to a trapezoid) glued together along a C P (the toric variety associated to an interval). In the second picture, it consists of two copiesof C P glued together at a point. Remark 2.1.6
Each of the core components E U is a lagrangian subvariety of M ( ˜ A ), there-fore its normal bundle in M ( ˜ A ) is isomorphic to its cotangent bundle. Furthermore, each E U has a T d -invariant algebraic tubular neighborhood in M ( ˜ A ) (necessarily isomorphic tothe total space of T ∗ X U ), and these neighborhoods cover M ( ˜ A ). Thus M ( ˜ A ) is a unionof cotangent bundles of toric varieties, glued together equivariantly and symplectically in amanner prescribed by the combinatorics of the bounded chambers of ˜ A . It is possible to takeProposition 2.1.3 and Equation (5) as a definition of X U , and this remark as a definition of M ( ˜ A ). The affine variety M ( A ) may then be defined as the spectrum of the ring of globalfunctions on M ( ˜ A ). 9 emark 2.1.7 Though Propositions 2.1.1, 2.1.3, and 2.1.4 appear in the literature only for˜ A simple, this hypothesis should not be necessary. Let B ( ˜ A ) = T ∗ C n // α T k . This variety is a GIT quotient of a vector space by the linear action of a torus, and istherefore a toric variety. Toric varieties that arise in this way are called
Lawrence toricvarieties . The following proposition is proven in [HSt, § Proposition 2.2.1
The inclusion M ( ˜ A ) = µ − (0) // α T k ֒ → T ∗ C n // α T k = B ( ˜ A ) is a T d R -equivariant homotopy equivalence. This Proposition is proven by showing that any toric variety retracts equivariantly ontothe union of those T d -orbits whose closures are compact. In the case of the Lawrence toricvariety, this is nothing but the core L ( ˜ A ). Given α ∈ ( t k ) ∗ Z , we may define stable and semistable sets( C n ) α − st ⊆ ( C n ) α − ss ⊆ C n as in Section 1.1, and the toric variety X α = C n // α T k may be defined as the categoricalquotient of ( C n ) α − st by T k . In analogy with Section 1.1, we will call α generic if the α -stable and α -semistable sets of C n coincide. In this case the categorical quotient will besimply a geometric quotient, and X α will be the toric orbifold corresponding to the polytope P ∅ of Section 2.1. We consider two characters to be equivalent if their stable sets are thesame, and note that there are only finitely many equivalence classes of characters, given bythe various combinatorial types of P ∅ for different simplifications ˜ A of A . Let α , . . . , α m bea complete list of representatives of equivalence classes for which ∅ 6 = ( C n ) α − st = ( C n ) α − ss .Let ( C n ) ℓf be the set of vectors in C n on which T k acts locally freely, meaning with finitestabilizers. For any character α of T k , the stable set ( C n ) α − st is, by definition, contained in( C n ) ℓf . Conversely, every element of ( C n ) ℓf is stable for some generic α [P4, 1.1], therefore( C n ) ℓf = m [ i =1 ( C n ) α i − st . Though µ − ( λ ) α − st is never empty, ( C n ) α − st sometimes is.
10e define the nonhausdorff space X ℓf = ( C n ) ℓf / T k = m [ i =1 ( C n ) α i − st /T k = m [ i =1 X α i to be the union of the toric varieties X α i along the open loci of commonly stable points.For an arbitrary λ ∈ ( t k ) ∗ , consider the projection π λ : µ − ( λ ) ֒ → T ∗ C n → C n . The following proposition is proven in [P2, 1.3].
Proposition 2.3.1 If λ is a regular value of µ , then π λ has image ( C n ) ℓf , and the fibers of π λ are affine spaces of dimension d . Corollary 2.3.2
The variety M λ = µ − ( λ ) /T k is an affine bundle over X ℓf = ( C n ) ℓf /T k . It follows from Corollary 2.3.2 that the natural projection M λ → X ℓf is a weak homotopyequivalence, meaning that it induces isomorphisms on all homotopy and homology groups.It is not a homotopy equivalence in the ordinary sense because it does not have a homotopyinverse–in particular, it does not admit a section. Example 2.3.3
Consider the action of C × on C by the formula t · ( z , z ) = ( tz , t − z ). Amultiplicative character of C × is given by an integer α , and that character will be generic ifand only if that integer is nonzero. The equivalence class of generic characters will be givenby the sign of that integer, so we let α = − α = 1. The corresponding stable setswill be ( C ) α − st = C r { z = 0 } and ( C ) α − st = C r { z = 0 } . The corresponding toric varieties X α and X α will both be isomorphic to C , and X ℓf = X α ∪ X α will be the (nonhausdorff) union of two copies of C glued together away from theorigin.The moment map µ : C × ( C ) ∨ → ( t k ) ∗ ∼ = C is given in coordinates by µ ( z, w ) = z w − z w . The hypertoric variety M α = µ − (0) //T k at a generic character is isomorphic to T ∗ C P , and its core is the zero section C P . It isdiffeomorphic to M λ = µ − ( λ ) / C × , which is, by Corollary 2.3.2, an affine bundle over X ℓf .If we trivialize this affine bundle over the two copies of C , we may write down a family ofaffine linear maps ρ z : C → C such that, over a point 0 = z ∈ C , the fibers of the two trivialbundles are glued together using ρ z . Doing this calculation, we find that ρ z ( w ) = w + z − . Remark 2.3.4
Both Proposition 2.1.1 and Corollary 2.3.2 show that a hypertoric varietyis equivariantly (weakly) homotopy equivalent to a union of toric orbifolds. In the case of11roposition 2.1.1 those toric orbifolds are always compact, and glued together along closedtoric subvarieties. In the case of Corollary 2.3.2 those toric orbifolds may or may not becompact, and are glued together along Zariski open subsets to create something that has atworst orbifold singularities, but is not Hausdorff. In general, there is no relationship betweenthe collection of toric varieties that appear in Proposition 2.1.1 and those that appear inCorollary 2.3.2.
Remark 2.3.5
Corollary 2.3.2 generalizes to abelian quotients of cotangent bundles of ar-bitrary varieties, rather than just vector spaces [P4, 1.4]. A more complicated statementfor nonabelian groups was used by Crawley-Boevey and Van den Bergh [CBVdB] to provea conjecture of Kac about counting quiver representations over finite fields.
In this Section we discuss the cohomology of the orbifold M ( ˜ A ) and the intersection coho-mology of the singular variety M ( A ), focusing on the connection to the combinatorics ofmatroids. In Section 3.4 we explain how hypertoric varieties can be used to compute coho-mology rings of nonabelian algebraic symplectic quotients, as defined in Section 1.1. Thereare a number of results on the cohomology of hypertoric varieties that we won’t discuss,including computations of the intersection form on the L -cohomology of M ( ˜ A ) [HSw] andthe Chen-Ruan orbifold cohomology ring of M ( ˜ A ) [GH, JT]. A simplicial complex ∆ on the set { , . . . , n } is a collection of subsets of { , . . . , n } , calledfaces, such that a subset of a face is always a face. Let f i (∆) denote the number of faces of∆ of order i , and define the h-polynomial h ∆ ( q ) := d X i =0 f i q i (1 − q ) d − i , where d is the order of the largest face of ∆. Although the numbers f i (∆) are themselvesvery natural to consider, it is unclear from the definition above why we want to encode themin this convoluted way. The following equivalent construction of the h -polynomial is lesselementary but better motivated.To any simplicial complex one associates a natural graded algebra, called the Stanley-Reisner ring , defined as follows: SR (∆) := C [ e , . . . , e n ] . *Y i ∈ S e i (cid:12)(cid:12)(cid:12) S / ∈ ∆ + .
12n order to agree with the cohomological interpretation that we will give to this ring inTheorem 3.2.2, we let the generators e i have degree 2. Consider the Hilbert seriesHilb( SR (∆) , q ) := ∞ X i =0 dim SR i (∆) q i , which may be expressed as a rational function in q . The following proposition (see [St, § II.2])says that the h -polynomial is the numerator of that rational function. Proposition 3.1.1
Hilb( SR (∆) , q ) = h ∆ ( q ) / (1 − q ) d . M ( ˜ A ) Let ∆ A be the simplicial complex consisting of all sets S ⊆ { , . . . , n } such that the normalvectors { a i | i ∈ S } are linearly independent. This simplicial complex is known as the matroid complex associated to A . The Betti numbers of M ( ˜ A ) were computed in [BD,6.7], but the following combinatorial interpretation was first observed by [HSt, 1.2]. LetPoin M ( ˜ A ) ( q ) = d X i =0 dim H i ( M ( ˜ A )) q i be the even degree Poincar´e polynomial of M ( ˜ A ). Theorem 3.2.1
The cohomology of M ( ˜ A ) vanishes in odd degrees, and Poin M ( ˜ A ) ( q ) = h ∆ A ( q ) . Theorem 3.2.1 is a consequence of the following stronger result.
Theorem 3.2.2
There is a natural isomorphism of graded rings H ∗ T d ( M ( ˜ A )) ∼ = SR (∆ A ) . The action of T d on M ( ˜ A ) is equivariantly formal [K1, 2.5], therefore the Hilbert seriesof H ∗ T d ( M ( ˜ A )) is equal to Poin M ( ˜ A ) ( q ) / (1 − q ) d , and Theorem 3.2.1 follows immediately fromProposition 3.1.1. Theorem 3.2.2 was proven for ˜ A smooth in [K1, 2.4] from the perspectiveof Section 2.1, and in the general case in [HSt, 1.1] from the perspective of Section 2.2. Herewe give a new, very short proof, from the perspective of Section 2.3. Proof of 3.2.2:
By Proposition 1.1.2, Remark 1.2.1, and Corollary 2.3.2, H ∗ T d ( M ( ˜ A )) ∼ = H ∗ T d ( M λ ) ∼ = H ∗ T d ( X ℓf ) ∼ = H ∗ T d (( C n ) ℓf /T k ) ∼ = H ∗ T n (( C n ) ℓf ) . Given a simplicial complex ∆ on { , . . . , n } , Buchstaber and Panov build a T n -space Z ∆ called the moment angle complex with the property that H ∗ T n ( Z ∆ ) ∼ = SR (∆) [BP, 7.12].13n the case of the matroid complex ∆ A , there is a T n -equivariant homotopy equivalence Z ∆ A ≃ ( C n ) ℓf [BP, 8.9], which completes the proof. M ( A ) The singular hypertoric variety M ( A ) = M , is contractible, hence its ordinary cohomologyis trivial. Instead, we consider intersection cohomology, a variant of cohomology introducedby Goresky and MacPherson which is better at probing the topology of singular varieties[GM1, GM2]. Let Poin M ( A ) ( q ) = d − X i =0 dim IH i ( M ( A )) q i be the even degree intersection cohomology Poincar´e polynomial of M ( A ). We will interpretthis polynomial combinatorially with a theorem analogous to Theorem 3.2.1.A minimal nonface of ∆ A is called a circuit . Given an ordering σ of { , . . . , n } , definea σ - broken circuit to be a circuit minus its smallest element with respect to the ordering σ . The σ -broken circuit complex bc σ ∆ A is defined to be the collection of subsets of { , . . . , n } that do not contain a σ -broken circuit. Though the simplicial complex bc σ ∆ A depends on the choice of σ , its h -polynomial does not. The following theorem was provedby arithmetic methods in [PW, § Theorem 3.3.1
The intersection cohomology of M ( A ) vanishes in odd degrees, and Poin M ( A ) ( q ) = h bc σ ∆ A ( q ) . Given the formal similarity of Theorems 3.2.1 and 3.3.1, it is natural to ask if there is ananalogue of Theorem 3.2.2 in the central case. The most naive guess is that the equivariantcohomology IH ∗ T d ( M ( A )) is naturally isomorphic to the Stanley-Reisner ring SR (bc σ ∆ A ),but this guess is problematic for two reasons. The first is that intersection cohomologygenerally does not admit a ring structure, and therefore such an isomorphism would besurprising. The second and more important problem is that the ring SR (bc σ ∆ A ) dependson σ , while the vector space IH ∗ T d ( M ( A )) does not. Since the various rings SR (bc σ ∆ A ) fordifferent choices of σ are not naturally isomorphic to each other, they cannot all be naturallyisomorphic to IH ∗ T d ( M ( A )), even as vector spaces. These problems can be addressed andresolved by the following construction.Let R ( A ) = C [ a − , . . . , a − n ] be the subring of the ring of all rational functions on C n generated by the inverses of the linear forms that define the hyperplanes of A . There is asurjective map ϕ from C [ e , . . . , e n ] to R ( A ) taking e i to a − i . Given a set S ⊆ { , . . . , n } and a linear relation of the form P i ∈ S c i a i = 0, the element k S = P i ∈ S c i Q j ∈ S r { i } e j liesin the kernel I ( A ) of φ , and in fact I ( A ) is generated by such elements. Since k S is clearlyhomogeneous, R ( A ) is a graded ring, with the usual convention of deg e i = 2 for all i .14he following proposition, proven in [PS, 4], states that the ring R ( A ) is a simultaneousdeformation of the various Stanley-Reisner rings SR (bc σ ∆ A ). Proposition 3.3.2
The set { k S | S a circuit } is a universal Gr¨obner basis for I ( A ) , andthe choice of an ordering σ of { , . . . , n } defines a flat degeneration of R ( A ) to the Stanley-Reisner ring SR (bc σ ∆ A ) . Example 3.3.3
Let d = 2, identify t d R with R , and let a = (cid:18) (cid:19) , a = a = (cid:18) (cid:19) , and a = (cid:18) − − (cid:19) . The two arrangements pictured in Example 2.1.5 are two different simplifications of theresulting central arrangement A . We then have R ( A ) ∼ = C [ e , . . . , e ] (cid:14) h e − e , e e + e e + e e , e e + e e + e e i . By taking the initial ideal with respect to some term order, we get the Stanley-Reisner ringof the corresponding broken circuit complex.In Theorem 3.3.4, proven in [BrP], we show that R ( A ) replaces the Stanley-Reisner ringin the “correct” analogue of Theorem 3.2.2. Theorem 3.3.4
Suppose that A is unimodular. The equivariant intersection cohomologysheaf IC T d ( M ( A )) admits canonically the structure of a ring object in the bounded equivariantderived category of M ( A ) . This induces a ring structure on IH ∗ T d ( M ( A )) , which is naturallyisomorphic to R ( A ) . The problems of classifying h -polynomials of matroid complexes and their broken circuitcomplexes remain completely open. Hausel and Sturmfels explore the restrictions on h ∆ A ( q )imposed by Theorem 3.2.2 in [HSt, § M ( ˜ A ) → M ( A ) [PW, § As in Section 1.1, let G be a reductive complex algebraic group acting linearly on a complexvector space V , and let T ⊆ G be a maximal torus. We need the further technical assumptionthat V has no nonconstant T -invariant functions, which is equivalent to asking that any GITquotient of V by T is projective. The inclusion of T into G induces a surjection g ∗ ։ t ∗ , whichrestricts to an inclusion of Z ( g ∗ ) into t ∗ . Thus a pair of parameters ( α, λ ) ∈ Z ( g ∗ Z ) × Z ( g ∗ )may be interpreted as parameters for T as well as for G . Suppose given α ∈ Z ( g ∗ Z ) such that15 α,
0) is generic for both G and T , so that the symplectic quotients M α, ( G ) and M α, ( T )are both orbifolds. Our first goal for this section is to describe the cohomology of M α, ( G )in terms of that of M α, ( T ).Both M α, ( G ) and M α, ( T ) inherit actions of the group C × induced by scalar multipli-cation on the fibers of the cotangent bundle of V . LetΦ( G ) : H ∗ G × C × ( T ∗ V ) → H ∗ C × ( M α, ( G ))and Φ( T ) : H ∗ T × C × ( T ∗ V ) → H ∗ C × ( M α, ( T ))be the equivariant Kirwan maps , induced by the C × -equivariant inclusions of µ − G (0) α − st and µ − T (0) α − st into T ∗ V . The map Φ( T ) is known to be surjective [HP1, 4.5], and Φ( G )is conjectured to be so, as well. The abelian Kirwan map Φ( T ) makes the equivariantcohomology ring H ∗ C × ( M α, ( T )) into a module over H ∗ T × C × ( T ∗ V ). The Weyl group W = N ( T ) /T acts both on the source and the target of Φ( T ), and the map is W -equivariant.Let ∆ ⊆ t ∗ be the set of roots of G (not to be confused with the simplicial complexes ∆that we discussed earlier), and consider the W -invariant class e = Y β ∈ ∆ β ( x − β ) ∈ Sym t ∗ ⊗ C [ x ] ∼ = H ∗ T × C × ( T ∗ V ) . The following theorem was proven in [HP, 2.4].
Theorem 3.4.1 If Φ( G ) is surjective, then there is a natural isomorphism H ∗ C × ( M α, ( G )) ∼ = H ∗ C × ( M α, ( T )) W (cid:14) Ann( e ) , where Ann( e ) is the ideal of classes annihilated by e . We note that the abelian quotient M α, ( T ) is a hypertoric variety, and the C × -equivariantring H ∗ C × ( M α, ( T )) was explicitly described in [HP1, 4.5] and [HH, 3.5]. Thus, modulosurjectivity of the Kirwan map, Theorem 3.4.1 tells us how to compute the cohomology ringof arbitrary symplectic quotients constructed in the manner of Section 1.1. In [HP, § C × -equivariant cohomology rings of hyperpolygonspaces, a result which originally appeared in [HP2, 3.2] as an extension of the nonequivariantcomputation in [K2, 7.1].Although the proof of Theorem 3.4.1 uses the C × -action in a crucial way, Hausel hasconjectured a simpler, nonequivariant version. Let Φ ( G ) be the map obtained from Φ( G )16y setting the equivariant parameter x to zero, and let e = Y β ∈ ∆ β ∈ Sym t ∗ ∼ = H ∗ T ( T ∗ V ) . Note that e is not the class obtained from e by setting x to zero, rather it is a square rootof that class. Conjecture 3.4.2 If Φ is surjective, then there is a natural isomorphism H ∗ ( M α, ( G )) ∼ = H ∗ ( M α, ( T )) W (cid:14) Ann( e ) . We end by combining Conjecture 3.4.2 with Theorem 3.3.4 to produce a conjecture thatwould put a ring structure on the intersection cohomology groups of M , ( G ). The hypothesisthat A be unimodular in Theorem 3.3.4 is equivalent to requiring that the orbifold resolution M ( ˜ A ) of M ( A ) is actually smooth. The analogous assumption in this context is that M α, ( G )and M α, ( T ) are smooth for generic choice of α . Conjecture 3.4.3
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