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LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. MILENA PABINIAK 0 1 0 2 Abstract. We consider a Hamiltonian Tn action on a compact symplectic man- g ifold (M,ω) with d isolated fixed points. For every fixed point p there exists a u class a H (M;Q) such that the collection a , for all fixed points, forms a A p ∈ T∗ { p} basis for H (M;Q) as an H (BT;Q) module. The map induced by inclusion, T∗ ∗ 5 ι : H (M;Q) H (MT;Q) = d Q[x ,...,x ] is injective. We will use such ∗ T∗ → T∗ ⊕j=1 1 n classes a to give necessary and sufficient conditions for f = (f ,...,f ) in ] { p} 1 d G d Q[x ,...,x ] to be in the image of ι , i.e. to represent an equiviariant coho- ⊕j=1 1 n ∗ S mology class on M. We may recover the GKM Theorem when the one skeleton is . h 2-dimensional. Moreover, our techniques give combinatorial description when we t a restrict to a smaller torus, even though we are then no longer in GKM case. m [ 1 v 0 0 9 0 Contents . 8 0 1. Introduction 2 0 1 2. Proof of the Main Theorem 7 : v i 3. Generating classes for Symplectic Toric Manifolds 10 X r 4. Examples 12 a 5. Working with integer coeficients 23 References 24 August 6, 2010. 1 2 MILENA PABINIAK 1. Introduction SupposethatacompactLiegroupactsonacompactmanifoldM. Theequivariant cohomology ring H∗(M;R) := H∗(M EG;R), with coefficients in a ring R, G ×G 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∗(M;R) tractable. The work of Goresky-Kottwitz-MacPherson [GKM] G 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 1.1 (Kirwan, [K]). Let torus T act on a symplectic manifold (M,ω) in a Hamiltonian fashion and let ι : MT M denote the natural inclusion of fixed points → into manifold. Then the induced map ι∗ : H∗(M;Q) H∗(MT;Q) is injective. T → T If MT are isolated points then also ι∗ : H∗(M;Z) H∗(MT;Z) is injective. T → T If there are d fixed points then H∗(MT;Q) = d Q[x ,...,x ], where n is the di- T ⊕j=1 1 n mensionofthetorus. Thereforewecanthinkaboutequivariantcohomologyclassesin H∗(MT;Q)asad-tupleofpolynomialsf = (f ,...,f ),witheachf inQ[x ,...,x ]. T 1 d j 1 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 S1 action. Theorem 1.2 (Chang, Skjelbred, [CS]). The image of ι∗ : H∗(M;Q) H∗(MT;Q) T → T is the set (cid:92) ι∗ (H∗(MH;Q)), MH T H where the intersection in H∗(MT;Q) is taken over all codimension-one subtori H of T T, and ι is the inclusion of MT into MH. MH Therefore we will consider a circle acting on a compact symplectic manifold (M,ω) in a Hamiltonian fashion with isolated fixed points and moment map µ : M R. It → turns out that with these assumptions we are in the Morse Theory setting. LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 3 Theorem 1.3 (Frankel [F], Kirwan [K]). Let a circle act on a symplectic manifold M with moment map µ : M R and isolated fixed points. Then 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 M. p The action of torus of higher dimension also carries a Morse function. For ξ t we ∈ define Φξ : M R, the component of moment map along ξ, by Φξ(p) = Φ,ξ . We → (cid:104) (cid:105) call ξ t generic if η,ξ = 0 for each weight η t∗ of T action on T M, for every p ∈ (cid:104) (cid:105) (cid:54) ∈ p in the fixed set MT. For such a generic ξ, Φξ is a Morse function with critical set MT. This map is a moment map for the action of a subcircle S (cid:44) 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 1.4 (Kirwan, [K]). Let a torus T act on a symplectic compact oriented manifold M with isolated fixed points, and let µ = Φξ : 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 be the negative weights of the T action on T M. Then there exists a class 1 k p a H2k(M;Z) such that p ∈ T a = ( 1)kΠk w ; • p|p − i=1 i a = 0 for all fixed points p(cid:48) MT p such that µ(p(cid:48)) µ(p). p p(cid:48) • | ∈ \{ } ≤ Moreover, taken together over all fixed points, these classes are a basis for the cohomology H∗(M;Z) as an H∗(BT;Z) module. T 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∗(M;Z) in [GT]. T Theyrequirea = 0forallfixedpointsp(cid:48) = pofindexlessthenorequal2k. Goldin p p(cid:48) | (cid:54) and Tolman’s classes, if they exist, are unique. Therefore they are called canonical classes. Forourpurposes, itisenoughtohavesomebasisfortherationalequivariant cohomology ring with respect to circle action, and with the following property 4 MILENA PABINIAK (*) 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 0 or a homogeneous polynomial of degree k. We will call elements of a basis satisfying condition (*) generating classes. Kirwan classes and Goldin-Tolman cannonical 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 M, which in this case is equal to the product of weights of p the torus action. Theorem1.5(ABBVLocalization,[AB][BV]). LetM beacompactmanifoldequipped with an S1 action with isolated fixed points, and let α H∗ (M;Q). Then as ele- ∈ S1 ments of H∗(BS1;Q) = Q[x], (cid:90) (cid:88) α p α = | , e(p) M p where the sum is taken over all fixed points. Inthispaper,weshowhowtoobtainrelationsdescribingtheimageofι∗(H∗(M)) T ⊂ H∗(MT) by applying the Localization Theorem. Our Main Theorem is: T Theorem 1.6. Let a circle act on a manifold M in a Hamiltonian fashion with isolated fixed points p ,...,p , and let f = (f ,...,f ) d Q[x] = H∗ (MS1;Q). 1 d 1 d ∈ ⊕j=1 S1 Suppose we are given a -a basis of (H∗(M);Q) satisfying condition (*). Then { p} T f is an equivariant cohomology class on M if and only if for every fixed point p of index 2k, 0 k < n have ≤ d (cid:88) f a (p ) j p j Q[x], e(p ) ∈ j j=1 where by a (p ) we mean ι∗ (a ), with ι : p (cid:44) M inclusion of the fixed point p p j pj p pj j → j into M. Note that if p is a fixed point of index n, this condition is automatically satisfied. This is because a is nonzero only at p, and there its value is the Euler class e(p). p Therefore it is sufficient to check the above condition only for points of index 2k, 0 k < n. ≤ LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 5 Remark 1.7. 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∗(M) as a subring of H∗(MT). T T Example 1.8. Consider the standard Hamiltonian S1 action on S2 by rotation with weight ax. The isolated fixed points are south and north poles which we will denote by p and p respectively. Kirwan class associated to p is 1. The Theorem then says 1 2 1 that f = (f ,f ) represents equivariant cohomology class if and only if 1 2 f a (p ) f a (p ) f f f f 1 1 1 + 2 1 2 = 1 + 2 = 1 − 2 Q[x]. e(p ) e(p ) ax ax ax ∈ 1 2 − This observation, together with the Chang-Skjelbred Lemma, recovers the GKM theorem. Theorem 1.6 is useful only if we know the restrictions to the fixed points of a set of generating classes (whose existence is guaranteed by Theorem 1.4). It is not surprising that there is a translation from the values of generating classes at fixed points to relations defining H∗(M) H∗(MT). Our translation provides a T ⊂ T 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 assigned to fixed point p may be not unique, even for toric varieties. Choosing p a generic subcircle of T, we get moment map for this circle action that is a Morse function and therefore we can talk about flow up from the fixed point p. If for every fixed point p(cid:48) in the flow up at p the index of p(cid:48) is strictly greater then the index of p, then the choice of a is unique (see [GZ]). We then say that the moment map is p index increasing. A particularly interesting application of our theorem is when we want to restrict the action of T to an action of a subtorus S (cid:44) T such that MS = MT, and compute → ι∗(H∗(M)) H∗(MS) = H∗(MT). We call this process specialization to the S ⊆ S S 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 1.6 gives → relations that cut out ι∗(H∗(M)) H∗(MT). In particular we can use this method S ⊆ S 6 MILENA PABINIAK to restrict the torus action on a symplectic toric manifold to a generic circle, i.e. such a circle S for which MS = MT (see Example 1a and b). A priori we only require that MS is finite, as we still want to describe H∗(M) by analyzing the relations on S polynomials defining the image ι∗(H∗(M)) H∗(MS) = Q[x ,...x ]. However it S ⊆ S ⊕ 1 k turns out that this requirement implies MT = MS. 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 Φξ - component of Φ along ξ, is a perfect Morse function with critical set MT. Therefore (cid:80) dimHi(M) = MT . Similarly, taking µ = pr Φ - the moment map s∗ | | ◦ for S action, and any generic η s, we obtainn µη which is also a perfect Morse ∈ function for M. Thus MS = (cid:80) dimHi(M) = MT . As obviously MT MS, the | | | | ⊂ sets must actually be equal. Consider restriction of the GKM action of T to a generic subcircle S: H (M) H (MT) GKMrelations T T H (M) H (MS) GKMrelationsnotenough S S GKM relations are sufficient to describe the image of H∗(M) in H∗(MT), but their T T “projections”arenotsufficienttodescribetheimageofH∗(M)inH∗(MT). However S S projecting generating classes and using 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 of ι∗(H∗(M) (cid:44) H∗(MT)). However this theorem cannot be applied if for some codi- T → T mension 1 subtorus H (cid:44) T we have dim MH > 2. Goldin and Holm in [GH] provide → a generalization of this result to the case where dim MH 4 for all codimension 1 ≤ subtori H (cid:44) T. An important corollary is that, in the case of Hamiltonian circle → actions, with isolated fixed points, on manifolds of dimension 2 or 4, 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 action is actually a specialization of a toric 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 LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 7 [GO] presents such an algorithm. Information encoded in the isotropy skeleton is essential. There cannot exist an algorithm computing the integral equivariant cohomology only from the fixed points data. Karshon in [K1](Example 1), constructs two S1 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 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 Lo- calization 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 prob- lem and for helpful conversations. 2. Proof of the Main Theorem Let a circle act on a manifold M in a Hamiltonian way with isolated fixed points p ,...,p and let f = (f ,...,f ) d Q[x] = H∗ (MS1). We want to show that 1 d 1 d ∈ ⊕j=1 S1 f is an equivariant cohomology class of M if and only if for every fixed point p have d (cid:88) f a (p ) j p j Q[x], e ∈ j=1 pj where a denotes generating class assigned to p. p Proof. The moment map is a Morse function. Therefore idex of a fixed point is well defined. We first compare the equivariant Poincaré polynomials of M and MS1 to determine the minimum number of relations we will need. Let b be the number of k fixedpointsofindex2k. Thend = (cid:80)n b isthetotalnumberoffixedpoints. Using k=0 k Theorem 1.3 of Frankel and Kirwan, we know that b is also the 2k-th Betti number k 8 MILENA PABINIAK of M. Hamiltonian S1 spaces are always equivariantly formal, that is H∗ (M;Q) = S1 ∼ H∗(M;Q) H∗(BS1;Q) as modules. Thus the equivariant Poincaré polynomial for ⊗ M is PS1(t) = P (t)PS1(t) = (b +b t2 +...+b t2n)(1+t2 +t4 +...) = M M pt 0 1 n = b +(b +b )t2 +...+(b +b +...+b )t2k +...+dt2n +dt2(n+1) +.... 0 0 1 0 1 k For the MS1-set of fixed points, we have d (cid:88) PS1 (t) = PS1(t) = d +dt2 +...+dt2n +dt2(n+1) +... MS1 pt k=1 Therefore we need precisely d (b +b +...+b ) = b +...b 0 1 k k+1 n − relations of degree k, for each 0 k < n. Using Poincaré duality we see that the ≤ number above is equal to b +b +...+b . 0 1 n−k−1 If we introduce the notation f (x) = (cid:80)Kj a xk, then relations of type j k=0 jk d (cid:88) s a = 0 j jk j=0 for some constants s ’s are called relations of degree k. For any fixed point p of index j 2(k 1), itscanonicalclassassignstoeachfixedpoint0orahomogeneouspolynomial − of degree (k 1). Denote by cp rational number satisfying ap(pj) = cpxk−1−n. If f is − j e(pj) j an equivariant cohomology class of M then f a is also. The Localization Theorem p · gives the relation d (cid:88) f a (p ) j p j Q[x]. e ∈ j=1 pj LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. 9 We may rewrite it in the following form: (cid:90) d (cid:88) f a (p ) j p j a f = = p e p ) M ( j j=1 d (cid:88) = f cpxk−1−n = j j j=1   d Kj (cid:88) (cid:88) = cpj  ajlxl xk−1−n = j=1 l=0   d Kj (cid:88) (cid:88) = cpj  ajlxk−1−n+l ∈ Q[x] j=1 l=0 The coefficients of negative powers of x need to be 0. Therefore for any fixed point p and any l = 0,...,n k we get the following relation of degree l: − d (cid:88) cpa = 0. j jl j=1 Note that in any degree, these relations are independent. We will show this by explicit computation. Suppose that in some degree l these relations in a ’s are not jl independent. That is, there are rational numbers b , not all zero, such that p (cid:32) (cid:33) (cid:32) (cid:33) d d (cid:88) (cid:88) (cid:88) (cid:88) 0 = b cpa = b cp a p j jl p j jl p j=1 j=1 p for all possible a . Thus for all j = 1,...,d, we have (cid:80) b cp = 0. Multiplying both jl p p j sides by e(p )xk−1−n we obtain j (cid:88) b e(p )cpxk−1−n = 0. p j j p Recall the definition of cp to notice that the above equation is equivalent to j (cid:88) b a (p ) = 0. p p j p (cid:80) That means b a vanishes on every fixed point and therefore is the 0 class, al- p p p though it is a nontrivial combination of classes a . Here we get contradiction with p the independence of the a ’s. p 10 MILENA PABINIAK To count the relations we have just constructed, notice that a relation of degree n k is obtained from each fixed point of index 2(k 1) or less. Dually we get a rela- − − tion of degree k for each fixed point of index 2(n k 1) or less. Counting relations of − − degree k over contributions from all fixed points, we have found b +b +...+b 0 1 n−k−1 of them, exactly as many as we need. This completes the proof. (cid:3) 3. Generating classes for Symplectic Toric Manifolds Let M2n be a symplectic toric manifold with a Hamiltonian action of T = Tn and moment polytope P (Rn)∗. Denote by M the one-skeleton of M, that is, 1 ⊂ the union of all T-orbits of dimension 1. Closures of connected components of M 1 are spheres, called the isotropy spheres. Denote by V the vertices of P, and by E the 1-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 Rn, so that for any p,q V we have p,ξ = q,ξ . Orient the edges ∈ ∈ (cid:104) (cid:105) (cid:54) (cid:104) (cid:105) so that i(e),ξ < t(e),ξ for any edge e, where i(e), t(e) are initial and terminal (cid:104) (cid:105) (cid:104) (cid:105) points of e. For any v (Zn)∗ (Rn)∗ denote by prim(v) the primitive integral ∈ ⊂ vector in direction of v. Label each edge with the primitive integral weight in the direction of the edge in (Rn)∗. This is the weight of T action on the corresponding isotropy sphere. For any p V let G denote the smallest face of positive dimension p ∈ containing p and only such points q V for which p,ξ < q,ξ . We will call G p ∈ (cid:104) (cid:105) (cid:104) (cid:105) the flow up face for p. We define the class a H∗ (MS1) by p ∈ S1  0 for q V G p a (q) = ∈ \ p (cid:81)  prim(r q) for q G r − ∈ p where the product is taken over all r V G such that r and q are connected by p ∈ \ an edge of P. We use convention that empty product is 1. Such classes satisfy the GKM conditions and thus are in the image of the equivariant cohomolgy of M. They are Poincaré duals to the homology classes defined by the flow up submanifolds from fixed points (in Morse theoretic sense). All fixed points are assigned a value which is 0 or a homogeneous polynomial of degree equal to number of edges terminating at this vertex (as polytope is smooth, exactly n edges meet at each vertex). These two facts can be proved using the notion of the axial function introduced in [GZ].