LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. MILENA PABINIAK Abstract. We consider a Hamiltonian action of n-dimensional torus, T n, on a compact symplectic manifold (M, ω) with d isolated fixed points. For every fixed point p there exists (though not unique) a class a p H T (M; Q) such that the collection {a p}, over all fixed points, forms a basis for H T (M; Q) as an H (BT; Q) module. The map induced by inclusion, ι : H T (M; Q) H T (M T ; Q) = d j=q[x,..., x n] is injective. We will use such classes {a p} to give necessary and sufficient conditions for f = (f,..., f d ) in d j=q[x,..., x n] to be in the image of ι, i.e. to represent an equiviariant cohomology class on M. When the one skeleton is 2-dimensional, we recover the GKM Theorem. Moreover, our techniques give combinatorial description of H K(M; Q), for a subgroup K T, even though we are then no longer in GKM case. Contents. Introduction 2 2. Proof of the Main Theorem 8 3. Generating classes for Symplectic Toric Manifolds 4. Examples 4 5. Working with integer coeficients 25 References 26 March, 2.
2 MILENA PABINIAK. Introduction Suppose that a compact Lie group acts on a compact, closed, connected and oriented manifold M. The equivariant cohomology ring H G (M; R) := H (M G EG; R), with coefficients in a ring R, encodes topological information about the manifold and the action. In the case of a Hamiltonian action on a symplectic manifold, a variety of techniques has made computing H G (M; R) tractable. The work of Goresky-Kottwitz-MacPherson [GKM] describes this ring combinatorially when G is a torus, R a field, and the action has very specific form. We give a more general description that has a similar flavor. A theorem of Kirwan [K] states that the inclusion of the fixed points induces an injective map in equivariant cohomology. Theorem. (Kirwan, [K]). Let a torus T act on a symplectic compact connected manifold (M, ω) in a Hamiltonian fashion and let ι : M T M denote the natural inclusion of fixed points into manifold. Then the induced map ι : H T (M; Q) H T (MT ; Q) is injective. If M T consists of isolated points then also ι : H T (M; Z) H T (MT ; Z) is injective. If there are d fixed points then H T (MT ; Q) = d j= Q[x,..., x n ], where n is the dimension of the torus. Therefore we can think about an equivariant cohomology class in H T (MT ; Q) as a d-tuple of polynomials f = (f,..., f d ), with each f j in Q[x,..., x n ]. The goal of this paper is to give necessary and sufficient conditions for a d-tuple of polynomials to be in the image of ι, that is to represent an equiviariant cohomology class on M. The following result of Chang and Skjelbred [CS] guarantees that we only need to consider the case of an S action. Theorem.2 (Chang, Skjelbred, [CS]). The image of ι : H T (M; Q) H T (MT ; Q) is the set ι M (H H T (MH ; Q)), H where the intersection in H T (MT ; Q) is taken over all codimension-one subtori H of T, and ι M H is the inclusion of M T into M H. In fact the only nontrivial contributions to this intersection are those codimension subtori H which appear as isotropy groups of some elements
LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 3 of M (that is M H M T 2 ). Therefore we will consider a circle acting on a compact, connected and closed symplectic manifold (M, ω) in a Hamiltonian fashion with isolated fixed points and moment map µ : M R. Unless otherwise stated, all the manifold considered in this paper are assumed to be compact, closed and connected. It turns out that with these assumptions we are in the Morse Theory setting. Theorem.3 (Frankel [F], Kirwan [K]). In the above setting, the moment map µ is a perfect Morse function on M (for both ordinary and equivariant cohomology). The critical points of µ are the fixed points of M, and the index of a critical point p is precisely twice the number of negative weights of the circle action on T p M. The Morse function is called perfect if the number of critical points of index k is equal to the dimension of k-th cohomology group. The action of a torus of higher dimension also carries a Morse function. For ξ t we define Φ ξ : M R, the component of moment map along ξ, by Φ ξ (p) = Φ, ξ. We call ξ t generic if η, ξ for each weight η t of T action on T p M, for every p in the fixed set M T. For a generic, rational ξ, Φ ξ is a Morse function with critical set M T. This map is a moment map for the action of a subcircle S T generated by ξ t. Using Morse Theory, Kirwan constructed equivariant cohomology classes that form a basis for integral equivariant cohomology ring of M. Then the existence of a basis for rational equivariant cohomology ring of M follows. We quote this theorem with the integral coeficients, and action of T, although in this paper we work mostly with rational coefficients and circle actions. In Section 5 we describe possible generalizations of our work to integer coefficients. Theorem.4 (Kirwan, [K]). Let a torus T act on a symplectic compact manifold M with isolated fixed points, and let µ = Φ ξ : M R be a component of moment map Φ along generic ξ t. Let p be any fixed point of index 2k and let w,..., w k be the negative weights of the T action on T p M. Then there exists a class a p H 2k T (M; Z) such that a p p = Π k i= w i; a p p = for all fixed points p M T \ {p} such that µ(p ) µ(p).
4 MILENA PABINIAK Moreover, taken together over all fixed points, these classes are a basis for the cohomology H T (M; Z) as an H (BT; Z) module. In the above theorem we use the convention that empty product is equal to. We will call the above classes Kirwan classes. These classes may be not unique. Goldin and Tolman give a different basis for the cohomology ring H T (M; Z) in [GT]. They require a p p = for all fixed points p p of index less then or equal 2k (where 2k is index of p). Goldin and Tolman s classes, if they exist, are unique. Therefore they are called canonical classes. For our purposes, it is enough to have some basis for the rational equivariant cohomology ring with respect to circle action, and with the following property ( ) elements of the basis are in such a bijection with the fixed points that a class corresponding to a fixed point of index 2k evaluated at any fixed point is or a homogeneous polynomial of degree k. We will call elements of a basis satisfying condition ( ) generating classes. Kirwan classes and Goldin-Tolman canonical classes satisfy the above condition. Another important ingredient of our proof is the Atiyah-Bott, Berline- Vergne (ABBV) localization theorem. For a fixed point p let e(p) be the equivariant Euler class of tangent bundle T p M, which in this case is equal to the product of weights of the torus action. Theorem.5 (ABBV Localization, [AB][BV]). Let M be a compact oriented manifold equipped with an S action with isolated fixed points, and let α H S (M; Q). Then as elements of H (BS ; Q) = Q[x], M α = p α p e(p), where the sum is taken over all fixed points. In this paper, we show how to obtain relations describing the image of ι (H T (M)) H T (MT ) by applying the Localization Theorem. Our Main Theorem is:
LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 5 Theorem.6. Let a circle act on a closed compact connected symplectic manifold M in a Hamiltonian fashion, with isolated fixed points p,..., p d. Let f = (f,..., f d ) d j= Q[x] = H (M S S ; Q). Suppose we are given a basis {a p } of H T (M; Q), satisfying condition ( ). Then f is an equivariant cohomology class on M if and only if for every fixed point p of index 2k, k < n we have d f j a p (p j ) Q[x], e(p j ) j= where a p (p j ) denotes ι p j (a p ), with ι pj : p j M the inclusion of the fixed point p j into M. Note that if p is a fixed point of index 2n, this condition is automatically satisfied. This is because a p is nonzero only at p, and there its value is the Euler class e(p). Therefore it is sufficient to check the above condition only for points of index strongly less then 2n = dimm. Remark.7. Using the Localization Theorem, we easily see that if f is cohomology class, then these conditions must be satisfied. The interesting part of the theorem is that they are sufficient to describe H T (M) as a subring of H T (MT ). Example.8. Recovering the GKM Theorem. Consider the standard Hamiltonian S action on S 2 by rotation with weight ax. The isolated fixed points are south and north poles which we will denote by p and p 2 respectively. The Goldin-Tolman class associated to p is. It exists due to Theorem.6 in [GT] as the moment map is index increasing. The Theorem.6 then says that f = (f, f 2 ) represents equivariant cohomology class if and only if f a (p ) e(p ) + f 2 a (p 2 ) e(p 2 ) = f ax + f 2 ax = f f 2 ax Q[x]. The above condition is exactly the same as the condition () in [GH]. Using the solution for this special case, together with the Chang-Skjelbred Lemma, Goldin and Holm recover the GKM Theorem in Section and 2 of [GH]. Let M be a compact, connected, symplectic manifold with a Hamiltonian, effective action of a torus T and with finitely many fixed points. Let N M be the set of points whose orbits under the G action are -dimensional. The
6 MILENA PABINIAK one-skeleton of M is the closure N. The manifold M is called a GKM manifold if N has finitely many connected components N α. Theorem.9 ([GKM] and [GH],[TW2]). Let M be a GKM manifold with a Hamiltonian torus action by G. Let M G be the fixed point set, and N be the one-skeleton. Let r : M G M be the inclusion of the fixed point set to M and j : M G N be the inclusion to N. The induced maps r : H G (M) H G (MG ) and j : H G (N) H G (MG ) on equivariant cohomology have the same image. Theorem.6 is useful only if we know the restrictions to the fixed points of a set of generating classes (whose existence is guaranteed by Theorem.4). It is not surprising that there is a translation from the values of generating classes at fixed points to relations defining H T (M) H T (MT ). Our translation provides a particularly combinatorial description that is easy to apply in examples. Although we cannot compute these classes in general, there are algorithms that work for a wide class of spaces, for example GKM spaces, including symplectic toric manifolds and flag manifolds (see[t]). For the sake of completeness we will describe an algorithm for obtaining Kirwan classes for symplectic toric manifolds in Section 3. The choice of a p assigned to fixed point p may be not unique, even for symplectic toric manifolds. In the case when moment map is so called index increasing and the manifold is a GKM manifold; this also includes symplectic toric manifolds), uniqueness was proved by Goldin and Tolman in [GT]. A particularly interesting application of our theorem is when we want to restrict the action of T to an action of a subtorus S T such that M S = M T, and compute ι (H S (M)) H S (MS ) = H S (MT ). We call this process specialization of the T action to the action of subtorus S. Having generating classes for T action we can easily compute generating classes for S action using the projection t s. Theorem.6 gives relations that cut out ι (H S (M)) H S (MT ). In particular we can use this method to restrict the torus action on a symplectic toric manifold to a generic circle, i.e. such a circle S for which M S = M T (see Examples 4. and 4.2). A priori we only require that M S is finite, as we still want to describe H S (M) by analyzing the relations on polynomials defining the image
LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 7 ι (H S (M)) H S (MS ) = Q[x,... x k ]. However it turns out that this requirement implies M T = M S. We can explain this fact using Morse theory. If Φ : M t is a moment map for T action and ξ t is generic, then Φ ξ, a component of Φ along ξ, is a perfect Morse function with critical set M T. Therefore dim H i (M) = M T. Similarly, taking µ = pr s Φ for the moment map for S action, and any generic η s, we obtain µ η which is also a perfect Morse function for M. Thus M S = dim H i (M) = M T. As obviously M T M S, the sets must actually be equal. Consider restriction of the GKM action of T to a generic subcircle S: H T (M) H T (M T ) GKM relations H S (M) H S (M S ) GKM relations not enough GKM relations are sufficient to describe the image of H T (M) in H T (MT ), but their projections are not sufficient to describe the image of H S (M) in H S (MT ). However projecting generating classes and using the Main Theorem to construct relations from such a basis will give all the relations we need. The GKM Theorem is a very powerful tool that allows us to compute the image under ι of H T (M) H T (MT ). However this theorem cannot be applied if for some codimension subtorus H T we have dim M H > 2. Goldin and Holm in [GH] provide a generalization of this result to the case where dim M H 4 for all codimension subtori H T. An important corollary is that, in the case of Hamiltonian circle actions, with isolated fixed points, on manifolds of dimension 2 or 4, the rational equivariant cohomology ring can be computed solely from the weights of the circle action at the fixed points. In dimension 2 this is given for example by the GKM Theorem. In dimension 4 one can apply the algorithm presented by Goldin and Holm in [GH] or use the fact that any such S action is actually a specialization of a toric T 2 action (see [K2]). If one wishes to compute integral equivariant cohomology ring, one will need additional piece of information, so called isotropy skeleton ([GO]). Godinho in [GO] presents such an algorithm. Information encoded in the isotropy skeleton is essential. There cannot exist an algorithm computing the integral equivariant cohomology
8 MILENA PABINIAK only from the fixed points data. Karshon in [K](Example ), constructs two 4-dimensional S spaces with the same weights at the fixed points but different integral equivariant cohomology ring. This suggests that we should not hope for an algorithm computing the rational equivariant cohomology ring from the weights at the fixed points for manifolds of dimension greater then 4. More information is needed. Tolman and Weitsman used generating classes to compute the equivariant cohomology ring in case of semifree action in [TW]. Their work gave us the idea for constructing necessary relations described in the present paper using information from generating classes. Our proof was also motivated by the work of Goldin and Holm [GH] where the Localization Theorem and dimensional reasoning were used. Organization. In Section 2, we prove our main result. In Section 3, we describe the case of toric symplectic manifolds and construct generating classes for their equivariant cohomology. We present several examples in Section 4. We finish our work with possible generalization to integer coefficients in Section 5. Acknowledgments. The author is grateful to Tara Holm for suggesting this problem and for helpful conversations, and to the referees for their comments and suggestions. 2. Proof of the Main Theorem Let a circle act on a manifold M in a Hamiltonian fashion with isolated fixed points p,..., p d. Let {a p } be a basis of H T (M; Q), satisfying condition ( ). We want to show that if f = (f,..., f d ) d j= Q[x] = H (M S S ) satisfies that d f j a p (p j ) Q[x], e pj j= for every fixed point p, then f is an equivariant cohomology class on all of M. Proof. The moment map is a Morse function. Therefore the idex of a fixed point is well defined. We first compare the equivariant Poincaré polynomials of M and M S to determine the minimum number of relations we will need to describe the image of H S (M) in H S (M S ). Let b k be the number of
LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 9 fixed points of index 2k. Then d = n k= b k is the total number of fixed points. By Theorem.3 of Frankel and Kirwan, we know that b k is also the 2k-th Betti number of M. Hamiltonian S -spaces are always equivariantly formal, that is H S (M; Q) = H (M; Q) H (BS ; Q) as modules. Thus the equivariant Poincaré polynomial for M is P S M (t) = P M(t)P S pt (t) = (b + b t 2 +... + b n t 2n )( + t 2 + t 4 +...) = = b + (b + b )t 2 +... + (b + b +... + b k )t 2k +... + dt 2n + dt 2(n+) +.... For the M S -set of fixed points, we have P S M S (t) = d k= Therefore we need precisely P S pt (t) = d + d t 2 +... + dt 2n + dt 2(n+) +... d (b + b +... + b k ) = b k+ +... + b n relations of degree k, for each k < n. By Poincaré duality, this number is equal to b + b +... + b n k. For any f = (f,..., f d ) d j= Q[x] = H S (M S ) we introduce the notation K j f j (x) = a jk x k, k= with a jk Q. Then a jk are independent variables, and we want to find conditions on these variables that will guarantee f H (M). Relations of S type d s j a jk = j= for some constants s j s are called relations of degree k, as they involve the coefficients of x k. For any fixed point p of index 2(k ), its generating class a p assigns to each fixed point p j either or a homogeneous polynomial of degree (k ). Denote by c p j the rational number satisfying a p (p j ) e(p j ) = cp j xk n.
MILENA PABINIAK If f is an equivariant cohomology class of M then f a p is also. The Localization Theorem gives the relation d f j a p (p j ) a p f = Q[x]. e pj M j= We may rewrite this in the following form: d f j a p (p j ) a p f = e ( p j ) M = = = j= d f j c p j xk n j= d j= d j= c p j c p j K j a jl x l x k n l= K j a jl x k n+l Q[x]. l= Using the convention a jl = for l > K j, we can write ( d n k ) d a p f = a jl x k n+l + M = j= n k l= c p j l= d c p j a jl x k n+l + j= j= d j= c p j c p j K j l=n k+ K j l=n k+ a jl x k n+l a jl x k n+l. The second component is an element of Q[x] as all the exponents of x are nonnegative. Thus M a pf is in Q[x] if and only if all the coefficients of x in the first component (that is coefficients of negative powers of x) are. Therefore for any fixed point p and any l =,..., n k, where 2(k ) is the index of p, we get the following relation of degree l: d c p j a jl =. j= Note that these relations are independent. We will show this by explicit computation. It is enough to show that for any l all the relations of degree l are independent, as relations of different degree involve different subset of
LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. variables {a jk }. Suppose that in some degree l these relations in a jl s are not independent. That is, there are rational numbers s p, not all zero, such that ajl = ( ) d d s p c p j a jl = s p c p j a jl p p j= As a jl are independent variables, we have p s pc p j =, for all j =,..., d. Multiplying both sides by e(p j )x k n we obtain s p e(p j )c p j xk n =. Recall the definition of c p j p j= to notice that the above equation is equivalent to s p a p (p j ) =. p That means p s pa p vanishes on every fixed point and therefore is the class, although it is a nontrivial combination of classes a p. This contradicts the independence of the generating classes a p s. Now we will count the relations just constructed. As noted above, a fixed point of index 2(k ) gives relations of degrees,..., n k. Therefore a relation of degree n k is obtained from each fixed point of index 2(k ) or less. That means we get a relation of degree k for each fixed point of index 2(n k ) or less. Counting relations of degree k over contributions from all fixed points, we have found b + b +... + b n k of them, exactly as many as we need. This completes the proof. 3. Generating classes for Symplectic Toric Manifolds Let M 2n be a symplectic toric manifold with a Hamiltonian action of T = T n and moment map image a Delzant polytope P t = (R n ). In particular P is simple, rational and smooth. Denote by M the one-skeleton of M, that is, the union of all T-orbits of dimension. Closures of connected components of M are spheres, called the isotropy spheres. Denote by V the vertices of P, and by E the -dimensional faces of P, also called edges. Vertices correspond to the fixed points of the torus action, while edges correspond to the isotropy spheres. Fix a generic ξ R n, so that for any p, q V we have p, ξ q, ξ. Orient the edges so that i(e), ξ < t(e), ξ for
2 MILENA PABINIAK any edge e, where i(e), t(e) are initial and terminal points of e. For any v (Q n ) (R n ) denote by prim(v) (Z n ) the primitive integral vector in direction of v. Note that prim(t(e) i(e)) is the weight of T action on the isotropy sphere corresponding to the edge e. For any p V let G p denote the smallest face containing p and all points q V with p, ξ < q, ξ which are connected with p by an edge. We will call G p the flow up face for p. We define the class a p H (M S S ) by for q V \ G p a p (q) = prim(r q) for q G p r where the product is taken over all r V \G p such that r and q are connected by an edge of P. We use convention that empty product is. If k edges terminate at p then the n k edges starting from p belong to the face G p (as polytope is simple, exactly n edges meet at each vertex). Similarly, for any q G p, there are n k edges meeting q that are contained in the face G p and k edges connecting q to vertices outside the face G p. Therefore the class a p assigns to each fixed point or a homogeneous polynomial of degree k. Such classes satisfy the GKM conditions and thus are in the image of the equivariant cohomolgy of M. The class a p constructed this way is the canonical equivariant extension (see [LS], Corollary 3.5) of the cohomology class Poincare dual to the submanifold of M mapping to the face G p. These two facts can be proved using the notion of the axial function introduced in [GZ]. The classes we have just defined are also linearly independent, which follows easily from the fact that a p can be nonzero only at vertices q greater or equal to p in the partial order given by the orientation of edges. Our first example is a set of generating classes for CP 2 presented in Figure y y(y x) x Figure. Generating classes for CP 2. Next we give an example where the generating classes are not unique. The
LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 3 above algorithm gives the basis presented on Figure 2. However classes in Figure 3 also form a basis. y y(y x) y x y x y Figure 2. The basis of the equivariant cohomology ring given by the above algorithm. Notice that this algorithm can be used to compute equivariant cohomology ny y y(y x) (n + )y x y ( x y) Figure 3. Different basis of the equivariant cohomology ring. even if we do not have the weights for the full T n action. It is enough to know the weights of S action, the fact that this action is a specialization of some toric action and positions of isotropy spheres for that toric action. These weights are just projections of T weights under π : t s. That is the S weight on edge e is π (prim(t(e) i(e)) ). With this information, we can find the flow up face G p for any fixed point p. The above algorithm gives that for q V \ G p a p (q) = π (prim(r q) ) for q G p r where the product is taken over all r V \G p such that r and q are connected by an isotropy sphere. Having generating classes for S action, we may apply the Main Theorem to obtain all relations needed to describe ι (H S (M)). This gives us a method for computing equivariant cohomology for a cirlce
4 MILENA PABINIAK action that happens to be part of a toric action. We expect that also for Hamiltonian circle actions with isolated fixed points that are part of a GKM action, the equivariant cohomology ring can be computed from the weights at fixed points and positions of isotropy spheres for this GKM action. 4. Examples Example 4.. Consider the product of CP 2 blown up at a point and CP CP 2 CP = {([x : x 2 ][y : y : y 2 ][z : z ]) x y 2 x 2 y = }, and the following T 3 action on this space: (e iu, e iv, e iw ) ([x : x 2 ][y : y : y 2 ][z : z ]) = ([e iu x : x 2 ][e iv y : e iu y : y 2 ][e iw z : z ]). This is a symplectic toric manifold with moment map ( x 2 µ([x : x 2 ][y : y : y 2 ][z : z ]) = x 2 + y 2 y 2, y 2 y 2, z 2 ) z 2 where x 2 = x 2 + x 2 2, and similarly for y 2 and z 2. The moment polytope is shown in Figure 4.. Using the algorithm from Section 3 we can 6 8 2 5 4 7 3 Figure 4. Moment polytope for CP 2 CP. compute generating classes for the equivariant cohomology with respect to T action. They are presented in the table below.
LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 5 class v v 2 v 3 v 4 v 5 v 6 v 7 v 8 A y y x y y x A 2 y y x y y x x x x A 3 x x x x z z x z z A 4 z z z z x(x y) x(x y) A 5 x(x y) x(x y) yz (y x)z A 6 yz (y x)z xz xz A 7 xz xz xz(y x) A 8 xz(y x) We want to compute equivariant cohomology with respect to the action of S T 3 given by u (u, 2u, u). More precisely, our action is: e iu ([x : x 2 ], [y : y : y 2 ], [z : z ]) = ([e iu x : x 2 ], [e i2u y : e iu y : y 2 ], [e iu z : z ]). Note that we still have the same eight fixed points, namely: v = ([ : ], [ : : ], [ : ]), v 2 = ([ : ], [ : : ], [ : ]),
6 MILENA PABINIAK v 3 = ([ : ], [ : : ], [ : ]), v 4 = ([ : ], [ : : ], [ : ]), v 5 = ([ : ], [ : : ], [ : ]), v 6 = ([ : ], [ : : ], [ : ]), v 7 = ([ : ], [ : : ], [ : ]), and v 8 = ([ : ], [ : : ], [ : ]). The weights of this circle actions are: fixed point weights index v u, 2u, u v 2 u, 2u, u 2 v 3 u, u, u 2 v 4 u, 2u, u 2 v 5 u, u, u 4 v 6 u, 2u, u 4 v 7 u, u, u 4 v 8 u, u, u 6 We compute generating classes for the S action from the classes for the T action using the projection map x u, y 2u, z u. They are presented in the table below, together with a row with 2u3 e(v i ) that is useful for further computations. v v 2 v 3 v 4 v 5 v 6 v 7 v 8 2u 3 e(v i ) - -2-2 2-2 A A 2 2u u 2u u A 3 u u u u A 4 u u u u A 5 u 2 u 2 A 6 2u 2 u 2 A 7 u 2 u 2 A 8 u 3
LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 7 We keep denoting by f j the restriction of f to a fixed point v j. condition that 8 f j A e(v j ) = f A Q[u], M implies that: j= The Thus f u 3 + f 2 u 3 + 2f 3 u 3 + f 4 u 3 + 2f 5 u 3 + f 6 u 3 + 2f 7 u 3 + 2f 8 u 3 Q[u]. f f 2 2f 3 f 4 + 2f 5 + f 6 + 2f 7 2f 8 (u 3 ) Q[u], Similarly, using class the A 2 we get f 2 + f 5 + f 6 f 8 (u 2 ) Q[u], Other classes give: f 3 + f 5 + f 7 f 8 (u 2 ) Q[u], f 4 + f 6 + 2f 7 2f 8 (u 2 ) Q[u], f 5 f 8 (u) Q[u], 2f 6 2f 8 (u) Q[u], and f 7 f 8 (u) Q[u]. Therefore f = (f,..., f d ) represents equivariant cohomology class if and only if it satisfies: the degree relations: (f i f j ) (u)q[u], for every i and j, the degree relations: f 3 + f 5 + f 7 f 8 (u 2 ) Q[u] f 2 + f 5 + f 6 f 8 (u 2 ) Q[u] f 4 + f 6 + 2f 7 2f 8 (u 2 ) Q[u] f f 2 2f 3 + 2f 5 (u 2 ) Q[u] the degree 2 relation: f f 2 2f 3 f 4 + 2f 5 + f 6 + 2f 7 2f 8 (u 3 )Q[u]. Example 4.2. In the case of the specialization for a T n action on M 2n (i.e. a symplectic toric manifold) to the action of some generic S (i.e. with M S = M T ), we can proceed using this simple algorithm. The weights of T n action are easy to read from moment polytope - they are just primitive integer vectors in the directions of the edges. To get the weights for our chosen S -action, we just need to use the appropriate
8 MILENA PABINIAK projection π : t (s ). To compute the basis of generating classes we use the method from Section 3 with ξ a generator of our S to get a T-basis, and then we project with π. If the fixed points are p,..., p d, we denote by a,..., a d the generating classes assigned to them and by G,..., G d the faces of moment polytope that are the flow up faces of the corresponding fixed point. Recall that for any v (Q n ) (R n ) we denote by prim(v) (Z n ) the primitive integral vector in direction of v. Using this notation, and the construction from Section 3, Theorem.6 states that f = (f,..., f d ) d j= Q[x] is an equivariant cohomology class of M if and only if for any fixed point p l we have d j= f j a l (p j ) e(p j ) = {j p j G l } f j r π(prim(r p j)) e(p j ) Q[x], where the product is taken over all vertices r not in G l such that r and p j are connected by an edge. The equivariant Euler class e(p j ) is a product of all weights at p j, therefore, up to a multiplication by a rational constant, it is equal to π(prim(r p j )), r where the product is taken over all vertices r connected to p j. above condition is equivalent to {j p j G l } f j r π(prim(r p j)) Q[x], Thus the where product is taken over all fixed points r G l that are connected with p j by an edge in G l. Consider, for example, vertex v 3 in the Example 4. above. The face G 3 is the face spanned by v 3, v 5, v 7, v 8. The weights at v 3 corresponding to edges that are in G 3 are u, u, for v 5 : u, u, for v 7 : u, u and for v 8 : u, u. Therefore relation we get is: f 3 u 2 + f 5 u 2 + f 7 u 2 + f 8 u 2 Q[u]. After clearing denominators, we obtain relation f 3 f 5 f 7 + f 8 (u 2 )Q[u].
LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 9 Example 4.3. Consider the following T 2 action on M = CP 4 : (e it, e is ) [z : z : z 2 : z 3 : z 4 ] = [z : e it z : e i2t z 2 : e i3t z 3 : e i(t+s) z 4 ]. This action has 5 fixed points with the following weights: fixed point weight p = [ : : : : ] x, 2x, 3x, x + y p 2 = [ : : : : ] x, x, 2x, y p 3 = [ : : : : ] 2x, x, x, y x p 4 = [ : : : : ] 3x, 2x, x, y 2x p 5 = [ : : : : ] x y, y, x y, 2x y We want to find relations among f j s so that f = (f,..., f 5 ) 5 j=q[x, y] is in the image of H T (M) H 2 T (M T 2 ) = 5 2 j=q[x, y]. By the topological Schur Lemma, Theorem.2, this image is r M (H H T (M H )), 2 H where intersection is taken over all codimension subtori H which appear as isotropy groups of some elements of M (that is M H M T 2 ). We have chosen an identification T 2 = S S with the first circle factor corresponding to x, and the second to y variable in H (BT) = Q[x, y]. In this example there are two relevant subgroups of T 2 : H = S {} T 2 and H 2 = {} S T 2. In the first case, M H = {[ : z : : : z 4 ]} = CP and S = T 2 /H acts on M H by e is [ : z : : : z 4 ] = [z : z : : : e is z 4 ]. There are two fixed points: p 2 and p 5. 5 j=q[x, y] (see Example.8): We get the following relation in f 2 f 5 (y) Q[x, y]. In the case of H 2 have that M 2 := M H 2 = {[z : z : z 2 : z 3 : ]} = CP 3. Fixed points of this action are
2 MILENA PABINIAK fixed point weight index p = [ : : : : ] x, 2x, 3x p 2 = [ : : : : ] x, x, 2x 2 p 3 = [ : : : : ] 2x, x, x 4 p 4 = [ : : : : ] 3x, 2x, x 6 The moment map is [z : z : z 2 : z 3 : ] 2 ( z 2 3 i= z i 2 + 2 z 2 2 3 i= z i 2 + 3 z 3 2 3 i= z i 2 ). To find the relations we first need to compute generating classes. We easily get that: class p p 2 p 3 p 4 A A 2 x ax bx A 3 2x 2 cx 2 A 4 6x 3 where a, b, c are some parameters. Dimension reasons give = M 2 A 3 = 2x 2 2x 3 + cx2 6x 3 thus c = 6. We would like to apply the Goldin-Tolman formula (Theorem.6 in [GT]) to compute the values of other generating classes. Goldin and Tolman worked with a very special collection of generating classes, called the canonical classes. The canonical class assigned to a fixed point p needs to vanish at all other points of index less than or equal to the index p (see comments below Theorem.4). In this particular example, all our fixed points are of different index and therefore the above classes are canonical classes in the sense of Goldin and Tolman. This allows us to apply Theorem.6 from [GT] and compute that A 2 (p 4 ) = 6a x 4. Substituting this result into = M 2 A 2 gives that a = 2 is the unique solution. Therefore generating classes are class p p 2 p 3 p 4 A A 2 x 2x 3x A 3 2x 2 6x 2 A 4 6x 3 The relations we obtain in this way are
LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 2 A 3 : f 3 f 4 (x) Q[x, y], A 2 : f 2 2f 3 + f 4 (x 2 ) Q[x, y], A : f 3f 2 + 3f 3 f 4 (x 3 ) Q[x, y]. Simplifying the relations and putting all the results together we get that f = (f,..., f 5 ) 5 j= Q[x, y] is in the image of H (M) 5 T 2 j=q[x, y] if and only if it satisfies f i f j (x) Q[x, y], f 2 2f 3 + f 4 (x 2 ) Q[x, y], f 2f 2 + f 3 (x 2 ) Q[x, y], f 3f 2 + 3f 3 f 4 (x 3 ) Q[x, y]. Example 4.4. Let T SO(5) be the maximal 2-torus in SO(5) and let O λ be the coadjoint orbit of SO(5) through a generic point λ t, the dual of the Lie algebra t of T. The torus T acts on O λ in a Hamiltonian fashion. We compute the T/H equivariant cohomology of M = O λ //H, the symplectic reduction of O λ by a circle H T fixing an S 2 in O λ and chosen so that the reduced space is a manifold. The inclusion h t induces the projection Π H : t h. To obtain the moment map for action of H, Φ H : O λ h, we need to compose the moment map Φ T for the T action with this projection. We choose a regular value, µ, of Φ H and define M = O λ //H := Φ H (µ)/h. The residual action of G := T/H = S on M is Hamiltonian and the moment map image can be identified with a slice of the moment polytope of O λ presented in Figure 5. We will compute equivariant cohomology of M with respect to this T/H action. This action has 8 fixed points which we denote p,..., p 8. For each i there is a splitting of torus T = H H i such that H i fixes a sphere S 2 i O λ, and there is q i S 2 i such that Φ T (q i ) = p i. The residual G = T/H action on T pi M is isomorphic to the H i action on N qi S 2 i, the normal bundle to S 2 i in M. To obtain the weights of G action on T p i M, take the T weights at the north or the south pole of S 2 i and compute their images under the projection t h i. This projection is a map Q[α, β] Q[x] that sends α to x, and the weight assigned to S i to. One of the T weights will go to under this map. The three remaining weights are the G weights. Note that either pole will give the same result, as the T weights differ by a
22 MILENA PABINIAK β α + β 2α + β v 7 v 8 h α v 5 v 6 Π H p p 2 p 3 p 4 p 5 p 6 p 7 p 8 v 3 v 4 µ v v 2 Figure 5. Moment Polytope for T 2 action on coadjoint orbit of SO(5) through a generic point. multiple of the weight assigned to the edge representing S i, and this weight vanishes on h i. For our example we have fixed point weight index p x, x, x p 2 x, x, 2x 2 p 3 x, x, 2x 2 p 4 x, x, x 2 p 5 x, x, x 4 p 6 x, 2x, x 4 p 7 x, 2x, x 4 p 8 x, x, x 6 We will use generating classes for T action on the whole coadjoint orbit to obtain generating classes for G-equivariant cohomology of M. Note that there will be 8 generating classes for H G (M). Let A j denote a canonical class for the T action associated to a fixed point v j, for j =,..., 8, with α, β as in Figure 5. Figure 6 presents their values in a chart, while Figure 7
LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 23 presents them graphically. There is a surjective map κ : H T (O λ) H G (M), and therefore generating class v v 2 v 3 v 4 v 5 v 6 v 7 v 8 A A 2 α α (α + β) (2α + β) (α + β) (2α + β) A 3 β (2α + β) β (2α + β) 2(α + β) 2(α + β) A 4 α(2α + β) α(2α + β) (α + β)(2α + β) (α + β)(2α + β) A 5 (α + β)β (α + β)(2α + β) (α + β)β (α + β)(2α + β) A 6 α(α + β)(2α + β) α(α + β)(2α + β) A 7 β(α + β)(2α + β) β(α + β)(2α + β) A 8 αβ(α + β)(2α + β) Figure 6. The values of generating classes for T 2 action on coadjoint orbit of SO(5). classes for G action on M are images of some Q[α, β]-linear combinations of A j s. To compute the value of κ(a) at p i for A H T (O λ), take the value of A at north or south pole of S i and compute its image under the map Q[α, β] Q[x] that sends α to x, and the weight assigned to S i to. The kernel of map κ was described by Tolman and Weitsman in [TW3]. Their result implies that for i = 5,... 8, κ(a i ) =. Therefore the image of κ is generated over Q[x] by κ(a ),..., κ(a 4 ), κ(βa ),..., κ(βa 4 ). Moreover, these 8 classes are Q[x]-linearly independent. They are not Kirwan classes as described in Theorem.4. However they satisfy all the requirements needed to apply our Theorem, that is condition ( ). They are in bijection with the fixed points and a class corresponding to a fixed point of index 2k evaluated at any fixed point is or a homogenous polynomial of degree k. We present them, together with the Euler class, in the following table: class p p 2 p 3 p 4 p 5 p 6 p 7 p 8 a = κ(a ) a 2 = κ(a 2 ) x x x x a 3 = κ(a 3 ) x 2x x 2x x a 4 = κ(βa ) x 2x x x 2x x a 5 = κ(a 4 ) 2x 2 x 2 a 6 = κ(βa 2 ) x 2 x 2 a 7 = κ(βa 3 ) x 2 4x 2 x 2 a 8 = κ(βa 4 ) x 3 Euler class x 3 2x 3 2x 3 x 3 x 3 2x 3 2x 3 x 3
24 MILENA PABINIAK A A 2 (α + β) (2α + β) (α + β) (2α + β) α α 2(α + β) 2(α + β) A 3 A 4 (α + β)(2α + β) (α + β)(2α + β) β (2α + β) α(2α + β) β (2α + β) α(2α + β) β(α + β) (α + β)(2α + β) A 5 A 6 α(α + β)(2α + β) β(α + β) (α + β)(2α + β) α(α + β)(2α + β) β(α + β)(2α + β) β(α + β)(2α + β) A 7 A 8 αβ(α + β)(2α + β) Figure 7. The generating classes for T 2 action on coadjoint orbit of SO(5). Therefore we get the following relations on f = (f,..., f 8 ): a : 2f f 2 f 3 2f 4 + 2f 5 + f 6 + f 7 2f 8 (x 3 ) Q[x] a 2 : f 3 2f 5 f 7 + 2f 8 (x 2 ) Q[x] a 3 : 2f 2f 2 2f 7 + 2f 8 (x 2 ) Q[x] a 4 : 2f + 2f 2 + 2f 4 2f 5 2f 6 + 2f 8 (x 2 ) Q[x] a 5 : 2f 7 2f 8 (x) Q[x] a 6 : 2f 5 2f 8 (x) Q[x] a 7 : 2f + 4f 2 2f 8 (x) Q[x]
LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 25 Simplifying these relations we can put them in the following form: the degree relations: All (f i f j ) (x)q[x], the degree relations: f 3 2f 5 f 7 + 2f 8 (x 2 ) Q[x] f f 2 f 7 + f 8 (x 2 ) Q[x] f 3 + 3f 4 f 5 f 6 (x 2 ) Q[x] f 2 f 3 f 6 + f 7 (x 2 ) Q[x] the degree 2 relation: 2f f 2 f 3 2f 4 + 2f 5 + f 6 + f 7 2f 8 (x 3 )Q[x]. 5. Working with integer coeficients In this section we analyze which parts of the paper would work over integer coefficients. In the case of isolated fixed points, injectivity theorem of Kirwan, Theorem., holds over Z. The Kirwan basis constructed in Theorem.4 is a basis for integral equivariant cohomology ring. The main tool in our proof, that is the Localization Theorem, also holds for integral coefficients ([AP]). Our method of transferring the information about the basis into explicit relations works over any ring. Therefore the Main Theorem holds also for integral coefficients. There we were considering only circle actions. The analysis of the action of torus of higher dimension could be simplified to analyzing only subcirles actions thanks to Chang Skjelbred Lemma. This Lemma does not hold over Z in general. Tolman and Weitsman in [TW2], Section 4, construct an explicit example where this lemma fails. In this example S S acts on S 2 S 2, rotating each sphere with speed 2. However, under special assumptions, Franz and Puppe in [FP] proved integral Chang Skjelbred Lemma. Proposition 5. (Franz, Puppe, [FP]). Let X be a T-space such that H T (X) is free over H (BT), where the cohomology is with integral coefficients. Denote by X i the i-skeleton of X, that is the union of all orbits of dimension i. Then H T (X) H T (X )
26 MILENA PABINIAK is exact. If in addition the isotropy group of each x X is contained in a proper subtorus of T then the following sequence is exact: H T (X) H T (X ) H + T (X, X ). The injectivity of the map H T (X) H T (X ) follows from the Localization Theorem. In the Tolman-Weitsman example mentioned above the proper subtorus assumption was not satisfied, as there all orbits of dimension 2 have isotropy group Z 2 Z 2 S S. References [AB] M. Atiyah and R. Bott. The moment map and equivariant cohomology. Topology, 23 (984), 28. [AP] C. Allday and V. Puppe, Cohomological Methods in Transformation Groups, Cambridge University Press, 993 [BV] N. Berline and M. Vergne. Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante. C.R. Acad. Sci. Paris Sér. I Math., 295 (982), 539 54. [CS] T. Chang and T. Skjelbred, The topological Schur lemma and related results, Annals of Mathematics (974), 37-32. [F] T. Frankel, Fixed points on Kahler manifolds, Annals of Mathematics, Vol. 7, No., Jul., 959, pages -8. [FP] Matthias Franz and Volker Puppe, Exact sequences for equivariantly formal spaces, http://arxiv.org/abs/math/372 [GH] R. Goldin and T. Holm, The equivariant cohomology of Hamiltonian G-spaces from residual S actions, Mathematical Research Letters 8, 67-77 (2). [GKM] M. Goresky, R. Kottwitz and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorems, Invent. math 3 (998)25-83. [GO] L.Godinho Equivariant cohomology of S -actions on 4-manifolds, Canad. Math. Bull. 5 (27), 365-376. [GT] R. Goldin and S. Tolman Towards generalizing Schubert calculus in the symplectic category, Journal of Symplectic Geometry Volume 7, Number 4 (29), 449-473. [GZ] V. Guillemin and C. Zara The existence of generating families for the cohomology ring of a graph, Advances in Mathematics 74 (23) 5-53. [K] F.C.Kirwan, The cohomology of quotients in symplectic and algebraic geometry, Princeton University Press, 984. [K] Y. Karshon, Hamiltonian torus actions, Geometry and Physics, Lecture notes in Pure and Applied Mathematics Series 84, Marcel Dekker, 996, p.22-23
LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 27 [K2] Y. Karshon Periodic Hamiltonian flows on four dimensional manifolds, Memoirs Amer. Math. Soc. 672, 999 [LS] Y. Lin, R. Sjamaar Equivariant symplectic Hodge theory and the dg-lemma,j. Symplectic Geom. Volume 2, Number 2 (24), 267-278. [T] J. Tymoszko, An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson, Snowbird lectures in algebraic geometry, 69-88, Contemp. Math. 388, Amer. Math. Soc., Providence, RI, 25. Available at arxiv:math/53369. [TW] S. Tolman and J. Weitsman, On the semifree symplectic circle actions with isolated fixed points, Topology 39 (2) 299-39. [TW2] S. Tolman and J. Weitsman, On the cohomology rings of Hamiltonian T-spaces, pp.25-258 in: Northen California symplectic geometry seminar, Transl., Ser 2 96 (45), Am. Math. Soc., Providence, RI 999. [TW3] S. Tolman and J. Weitsman, The cohomology rings of abelian symplectic quotients, Communications in Analysis and Geometry, Volume, Number 4, 75-773. NY Milena Pabiniak, Department of Mathematics, Cornell University, Ithaca E-mail address: milena@math.cornell.edu