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Home , Moment map

LECTURE 4: SYMPLECTIC GROUP ACTIONS

WEIMIN CHEN, UMASS, SPRING 07

1. Symplectic circle actions 1 We set S = R/2πZ throughout. 1 Let (M, ω) be a . A symplectic S -action on (M, ω) is a smooth 1 1 family ψt ∈ Symp(M, ω), t ∈ S , such that ψt+s = ψt ◦ ψs for any t, s ∈ S . One can d −1 easily check that the corresponding vector fields Xt ≡ dt ψt ◦ ψt is time-independent, i.e., Xt = X is constant in t. We call X the associated vector field of the given 1 symplectic S -action. Note that X is a symplectic vector field, i.e., ι(X)ω is a closed 1-form. When 1 ι(X)ω = dH is an exact 1-form, the corresponding symplectic S -action is called a 1 Hamiltonian S -action, and the function H : M → R is called a . Note that H is uniquely determined up to a constant. We point out that a symplectic 1 1 S -action on (M, ω) is automatically Hamiltonian if H (M; R) = 0. In general, ι(X)ω is only a closed form. In this case, one can perturb ω by adding a 1 sufficiently small S -invariant harmonic 2-form β so that ω + β is still symplectic and 1 2 S -invariant, and furthermore, the deRham cohomology class [ω +β] lies in H (M; Q). 0 1 A positive integral multiple of ω+β, ω ≡ N(ω+β), is a S -invariant symplectic form on 0 1 M such that the deRham cohomology class of ι(X)ω lies in H (M; Z). Consequently, 0 there exists a smooth function H : M → R/Z such that ι(X)ω = dH. Such a circle- valued smooth function H : M → R/Z is called a generalized moment map. As 1 far as the topology of the S -action is concerned, one may always assume that there is a generalized moment map. 1 Let H be a moment map or generalized moment map of a given symplectic S -action. We shall make the following two observations. −1 1 (1) Each level surface H (λ) is invariant under the S -action, because dH(X) = ω(X,X) = 0 (where X is the associated vector field of the given symplectic 1 S -action). 1 (2) A point p ∈ M is a fixed point of the S -action if and only if p is a critical point of H, i.e., dH = 0 at p. This is because p ∈ M is a fixed point of the 1 S -action iff X = 0 at p iff dH = ι(X)ω = 0 at p. Now consider a level surface H−1(λ) where λ is a regular value of H. Then H−1(λ) 1 is a hypersurface in M which does not contain any fixed points of the S -action. When 1 −1 −1 1 the S -action is free on H (λ), the quotient Bλ ≡ H (λ)/S is a smooth manifold. 1 −1 In general, the S -action may have finite isotropy on H (λ), and the quotient Bλ ≡ −1 1 H (λ)/S is a smooth orbifold. In any case, one observes that dim Bλ = dim M − 2. 1 2 WEIMIN CHEN, UMASS, SPRING 07

The next proposition shows that there exists a natural symplectic structure ωλ on Bλ. The symplectic manifold (or orbifold) (Bλ, ωλ) is called the symplectic quotient or the reduced space at λ. The process of going from (M, ω) to (Bλ, ωλ) is called symplectic reduction.

Proposition 1.1. There exists a canonically defined symplectic structure ωλ on Bλ ∗ −1 −1 1 such that π ωλ = ω|H−1(λ), where π : H (λ) → H (λ)/S ≡ Bλ is the projection. 1 −1 Proof. For simplicity, we assume the S -action on H (λ) is free, and consequently −1 Bλ is a smooth manifold. To simplify the notation, we set Q = H (λ). Let’s recall a basic result about differentiable actions on manifolds — the existence of local slice. In the present situation, the result amounts to say that for any point q ∈ Q, there exists a submanifold Oq of codimension 1 containing q, 1 1 such that S × Oq embedds into Q S -equivariantly. Oq is called a local slice at q, and the set {Oq|q ∈ Q} forms an atlas of charts for the differentiable structure on the 1 0 1 quotient Q/S . If q ∈ Q lies in S × Oq, then there is a local diffeomorphism φqq0 0 1 from a neighborhood U of q in the slice Oq0 into Oq and a function on U into S , fqq0 , such that U ⊂ Oq0 may be identified with the graph of fqq0 over the image of φqq0 1 1 in S × Oq. Note that with the differentiable structure on the quotient Q/S = Bλ 1 described above, the projection π : Q → Bλ becomes a principal S -bundle over Bλ, 1 with local trivializations of the bundle given by projections S × Oq → Oq, q ∈ Q. The symplectic structure ωλ is defined by pulling-back ω to each local slice Oq. This ∗ definition immediately gives the closedness of ωλ as well as the equation π ωλ = ω|Q. To see that ωλ is well-defined, i.e., the pull-back of ω to each local slice can be patched up, we note that the local slices are graphs over each other locally, and that the tangent 1 1 ω direction of S in S × Oq lies in TQ at each point. The nondegeneracy of ωλ follows ω ω from the fact that dim TqQ = dim TqM − dim TqQ = 1, so that TqQ is actually 1 generated by the tangent direction of S .  1 Example 1.2. (Product of S -actions). For j = 1, 2, let (Mj, ωj) be a symplectic 1 j 1 manifold with a symplectic S -action t 7→ ψt , t ∈ S . Then for any m1, m2 ∈ Z such 1 that gcd(m1, m2) = 1, there is a canonical S -action on the product (M1 ×M2, ω1 ×ω2), 1 1 2 t 7→ ψt, t ∈ S , where ψt = ψm1t × ψm2t. Moreover, if H1, H2 are moment maps of 1 1 2 the S -actions ψt , ψt respectively, then H = m1H1 + m2H2 is a moment map of the product ψt. To see this, note that if Xj, j = 1, 2, is the vector field on Mj 1 j which generates the S -action ψt , then X = hm1X1, m2X2i is the vector field on 1 M1 ×M2 which generates the S -action ψt. The claim about the moment maps follows immediately from ι(X)(ω1 × ω2) = m1ι(X1)ω1 + m2ι(X2)ω2. 1 Example 1.3. (Holomorphic S -actions on K¨ahler manifolds). For any holomorphic 1 S -action on a K¨ahler manifold, one can choose an invariant K¨ahlermetric, so that the 1 1 S -action becomes a symplectic S -action with respect to the invariant K¨ahler form. 2 1 Example 1.4. Consider (R , ω0) with symplectic S -action given by the complex it 1 2 multiplication z 7→ e z, t ∈ S . Here we identify R = C. To determine the moment 1 map, we note that the S -action is generated by the vector field X = −y∂x+x∂y. With 1 2 this we see the moment map is given by H(z) = − 2 |z| , because ι(X)ω0 = −ydy−xdx. LECTURE 4: SYMPLECTIC GROUP ACTIONS 3

Now for any m = (m0, m1, ··· , mn) where each mi ∈ Z and gcd(m0, m1, ··· , mn) = 1 2n+2 1, consider more generally the symplectic S -action on (R , ω0), which is defined by im0t im1t imnt 1 (z0, z1, ··· , zn) 7→ (e z0, e z1, ··· , e zn), t ∈ S . 1 By Example 1.2, the moment map of the S -action is 1 H(z , z , ··· , z ) = − (m |z |2 + m |z |2 + ··· + m |z |2). 0 1 n 2 0 0 1 1 n n For the special case where m = (1, 1, ··· , 1), the level surface H−1(λ), λ < 0, is the 1 −1 (2n + 1)-dimensional sphere of radius −2λ, and the S -action on H (λ) is given by 1 n the Hopf fibration. The reduced space (Bλ, ωλ) at λ = − 2 is CP with ωλ being π n times the Fubini-Study form ω0 on CP , see Example 1.9 in Lecture 1.

Example 1.5. Note that for any m = (m0, m1, ··· , mn), the corresponding sym- 1 2n+2 2n+1 plectic S -action on (R , ω0) with weights m preserves the unit sphere S and 1 n commutes with the Hopf fibration. Hence there is an induced S -action on CP , which must be symplectic with respect to the Fubini-Study form and has the moment map 1 H(z , z , ··· , z ) = − (m |z |2 + m |z |2 + ··· + m |z |2) 0 1 n 2 0 0 1 1 n n 2 2 2 ∗ where |z0| + |z1| + ··· + |zn| = 1, due to the relation π ωλ = ω|H−1(λ) between the symplectic forms in the symplectic reduction given in Proposition 1.1, and the fact n 1 that the vector field on CP which generates the induced S -action is the push-down 2n+1 2n+1 n of the corresponding vector filed on S under the Hopf fibration π : S → CP . n 1 In terms of the homogeneous coordinates z0, z1, ··· , zn on CP , the S -action is given by im0t im1t imnt 1 [z0, z1, ··· , zn] 7→ [e z0, e z1, ··· , e zn], t ∈ S . The corresponding moment map is given by

1 2 2 2 H([z0, z1, ··· , zn]) = − Pn 2 (m0|z0| + m1|z1| + ··· + mn|zn| ). 2 j=0 |zj| 1 n Note that in order for the S -action on CP to be effective, one needs to impose additional conditions

gcd(m0 − mj, ··· , mn − mj) = 1, ∀j = 0, ··· , n.

On the other hand, for any m ∈ Z, the wights m−m ≡ (m0 −m, m1 −m, ··· , mn −m) 1 n defines the same S -action on CP as the weights m = (m0, m1, ··· , mn). Note that m the corresponding moment maps change by a constant 2 . Consider the case where n = 1. The above construction gives rise to symplectic 1 1 2 1 S -actions on CP = S , where different choices of weights m yield the same S -action. 1 1 If we let m = (0, −1) and identify CP = C ∪ {∞}, the S -action is simply given by the complex multiplication on C, which has moment map 1 1 H(z) = · . 2 1 + |z|2 The fixed points are {0, ∞}, and the corresponding critical values of the moment map 1 are H(0) = 2 , H(∞) = 0. The only difference between this example of a Hamiltonian 4 WEIMIN CHEN, UMASS, SPRING 07

1 2 S -action on S and the one given in Example 1.13 in Lecture 1 is that the former has 2 area π over S and the latter has area 4π. Definition 1.6. A smooth function f on a manifold M is called Morse-Bott if for n any critical point p ∈ M of f, there is a chart φ : R → M centered at p such that ∗ 2 2 φ f(x1, ··· , xn) = a1x1 + ··· + anxn + f(p), where each aj is −1, 0 or 1. The number of aj’s where aj < 0 is called the index of f at p. If none of the aj’s is zero for every critical point p, the function f is called a Morse function. Note that a Morse function has only isolated critical points, and for a Morse-Bott function in general, the set of critical points consists of a disjoint union of submanifolds of various co-dimensions, called the critical submanifolds. In the above standard chart, the critical submanifold is given by the equations xj = 0 for all j with aj 6= 0.

1 Proposition 1.7. A moment map (or generalized moment map) of a symplectic S - action is Morse-Bott with the following additional properties: (1) the critical subman- ifolds are symplectic submanifolds, (2) the index at each critical point is always an even number. Proof. The proposition follows from an equivariant version of the Darboux theorem, which also gives a model for the moment map near a critical point. Since the problem is local, there is no difference for the case of generalized moment map. 1 1 Let (M, ω) be a symplectic manifold with a symplectic S -action ψt, t ∈ S . The equivariant version of Darboux theorem states as follows: for any fixed point p ∈ M, 2n ∗ there is a chart φ : R → M centered at p, such that φ ω = ω0 the standard 2n 1 ∗ −1 2n symplectic form on R , and the pull-back S -action φ ψt ≡ φ ◦ ψt ◦ φ on R is 2n n linear and is given under the standard identification R = C by

im1t im2t imnt 1 (z1, z2, ··· , zn) 7→ (e z1, e z2, ··· , e zn), t ∈ S , for some m1, m2, ··· , mn ∈ Z with gcd(m1, m2, ··· , mn) = 1. 1 Let H be a moment map of the symplectic S -action on M. Then the equivariant Darboux theorem together with Example 1.4 implies that there exists a chart φ : 2n R → M centered at p such that 1 φ∗H(x , ··· , x , y , ··· , y ) = − [m (x2 + y2) + ··· + m (x2 + y2)] + H(p) 1 n 1 n 2 1 1 1 n n n for some m1, m2, ··· , mn ∈ Z with gcd(m1, m2, ··· , mn) = 1. With a further change of coordinates it follows immediately that H is a Morse-Bott function. The index at p is twice of the number of positive mj’s, hence is always an even number. The critical submanifold is locally defined near p by the equations xj = yj = 0 for all j with mj 6= 0, hence is symplectic. 1 Next we sketch a proof of the equivariant Darboux theorem. We pick an S -invariant metric on M, and by Theorem 1.15 of Lecture 2 (parametric version) it gives rise to 1 an S -invariant J ∈ J (M, ω). Note that the Hermitian metric on (M,J), gJ (·, ·) ≡ 1 ω(·,J·), is also S -invariant. LECTURE 4: SYMPLECTIC GROUP ACTIONS 5

1 Let p ∈ M be a fixed point. Then the induced S -action on TpM, which is naturally a Hermitian vector space with J and gJ , has the form n n X X imj t 1 zjuj 7→ e zjuj, t ∈ S , j=1 j=1 with respect to a unitary basis u1, u2, ··· , un, where mj ∈ Z, j = 1, 2, ··· , n, obeys gcd(m1, m2, ··· , mn) = 1. 2n We define a chart ψ : R → M centered at p by n X ψ :(x1, x2, ··· , xn, y1, y2, ··· , yn) 7→ expp( (xj + iyj)uj), j=1 where the exponential map expp is with respect to the metric gJ . Note that with 1 2n respect to the linear S -action on R (identified with TpM via the unitary basis 1 u1, u2, ··· , un), the local diffeomorphism ψ is S -equivariant because the metric gJ is. 2n ∗ ∗ Now consider the pull-back symplectic form on R , ψ ω. Both ψ ω and ω0 are 1 1 2n ∗ S -invariant with respect to the linear S -action on R , and ψ ω = ω0 at the origin 2n 0 ∈ R . The proof of Lemma 1.1 in Lecture 3 works perfectly in an equivariant context, from which the equivariant Darboux theorem follows. 

Note that with respect to the metric gJ (for any J ∈ J (M, ω)) the gradient vector field for a moment map H is given by grad H = JX, where X is the vector field which 1 generates the S -action. 1 The fact that a moment map of a symplectic S -action is Morse-Bott with even index at each critical point has the following corollary by a standard argument in Morse theory. 1 Corollary 1.8. Let H : M → R be a moment map of a Hamiltonian S -action on a compact, connected manifold. Then each level surface H−1(λ) is connected. In particular, the critical submanifolds at the maximal and minimal values of H are connected. The standard model for a moment map H near a critical point as we obtained in the proof of Proposition 1.7 allows one to explicitly analyzing the change of the topology of the reduced spaces passing a critical value of the moment map in terms of the weights mj at each critical point in the pre-image of that critical value. On the other hand, for any interval I of regular values, Morse theory allows one to identify H−1(I) −1 diffeomorphically with the product H (λ0)×I for any λ0 ∈ I (e.g. using the gradient 1 flow of H), which can be made S -equivariantly. The next proposition describes the −1 relation between the symplectic form ω on H (I) and the reduced spaces (Bλ, ωλ), 1 −1 λ ∈ I, and the first Chern class of the S -principal bunble π : H (λ0) → Bλ0 . For 1 −1 simplicity we assume that the S -action on H (I) is free, so that each reduced space −1 Bλ, λ ∈ I, is a smooth manifold. Note that all H (λ), Bλ, λ ∈ I, are diffeomorphic; 1 we denote the underlying manifolds by P , B respectively. (We warn that the S -action on each H−1(λ), λ ∈ I, is assumed to be on the left, so when H−1(λ) is regarded as a 1 S -principal bundle, where the action is always assumed to be on the right, we mean 6 WEIMIN CHEN, UMASS, SPRING 07

1 the conjugate S -action. For example, in this way the first Chern class of the Hopf 3 1 1 fibration S → CP evaluates positively on the fundamental class of CP .) 2 1 Proposition 1.9. (1) Let c ∈ H (B; Z) be the first Chern class of the S -principal bundle π : P → B and let {ωλ|λ ∈ I} be a smooth family of symplectic forms on B such that their deRham cohomology classes satisfy

[ωλ] = [ωµ] − 2π(λ − µ) · c. 1 There is an S -invariant symplectic form ω on P × I with a moment map H equal to the projection P × I → I and with reduced spaces (B, ωλ), λ ∈ I. 1 (2) Conversely, every S -invariant symplectic form ω arises in the above way. More- 1 1 over, up to S -equivariant such a S -invariant symplectic form on P × I is uniquely determined by the family of symplectic forms {ωλ|λ ∈ I} on B. d Proof. (1) Since the deRham cohomology class of dλ ωλ represents −2πc, there exists a smooth family of imaginary valued 1-forms Aλ on P , i.e., the connection 1-forms, such i 1 ∗ d 1 that 2π dAλ = − 2π π dλ ωλ. Let X be the vector field which generates the S -action. 1 Then Aλ(X) = −i because as we remarked before P is regarded as an S -principal bundle on B with the conjugate action. Set αλ = iAλ. Then αλ(X) = 1, and ∗ d π dλ ωλ + dαλ = 0. With these understood, ∗ ω = π ωλ + αλ ∧ dλ 1 is an S -invariant symplectic form on P × I with a moment map H equal to the projection P × I → I and with reduced spaces (B, ωλ), λ ∈ I. (2) Note that as a 2-form on P × I, ω may be written as

ω = βλ + αλ ∧ dλ 1 2 for some αλ ∈ Ω (P ) and βλ ∈ Ω (P ). Let X be the vector field which generates the 1 1 S -action. Since the moment map of the S -action is the projection P × I → I, we see that ι(X)βλ = 0 and αλ(X) = 1. The former implies that βλ descents to a smooth ∗ family of 2-forms ωλ on B, so that π ωλ = βλ. The nondegeneracy of ω implies that each ωλ is nondegenerate, and the closedness of ω implies d dω = 0, β + dα = 0. λ dλ λ λ Note that −iαλ are connection 1-forms on P , so that the first Chern class c is repre- i 1 sented by 2π d(−iαλ) = 2π dαλ. This gives the relation [ωλ] = [ωµ] − 2π(λ − µ) · c. For the uniqueness, consider two different such symplectic forms ∗ 0 ∗ 0 ω = π ωλ + αλ ∧ dλ, ω = π ωλ + αλ ∧ dλ. One can form a smooth family of such symplectic forms ∗ 0 ωt = π ωλ + ((1 − t)αλ + tαλ) ∧ dλ, t ∈ [0, 1]. The uniqueness follows from an equivariant version of Moser’s stability theorem applied to ωt. We leave it as an exercise.  LECTURE 4: SYMPLECTIC GROUP ACTIONS 7

1 1 Example 1.10. (1) Consider the S -action on CP = C ∪ {∞} at the end of Example 1.5, which is given by the complex multiplication of eit and has moment map H(z) = 2 1/2(1 + |z| ). Since in this case the reduced spaces are a single point, ωλ = 0, and therefore the symplectic form

ω = αλ ∧ dλ.

In the polar coordinates (r, θ) on C, αλ = dθ, so that 1 rdr ∧ dθ dx ∧ dy ω = dθ ∧ d( ) = = . 2(1 + r2) (1 + r2)2 (1 + x2 + y2)2 Direct calculation shows Z Z ∞ Z ∞ dxdy ω = 2 2 2 = π 1 (1 + x + y ) CP −∞ −∞ as we claimed. 1 2n+2 (2) Consider the S -action on (R , ω0) in Example 1.4 with weights m = (1, 1, ··· , 1). The moment map is 1 H(z , z , ··· , z ) = − (|z |2 + |z |2 + ··· + |z |2). 0 1 n 2 0 1 n −1 1 For any λ < 0 the level surface H (λ) is the sphere of radius −2λ, and the S -action −1 n on H (λ) is given by the Hopf fibration, with the quotient being CP . We have 1 claimed in Example 1.4 that the symplectic form ωλ at λ = − 2 is π times the Fubini- n Study form on CP , which is normalized so that the integral of its n-th power over n R n n equals 1. Note that this implies that n ω = π . We shall next give an CP CP −1/2 independent verification of this fact using Proposition 1.9. First note that as λ → 0 the form ωλ converges to 0. This gives, by Proposition 1.9, 1 [ω− 1 ] = 0 − 2π(− − 0) · c = π · c, 2 2 2 n where c is the first Chern class of the Hopf fibration. It is known that c ∈ H (CP ; Z) = n n Z is the positive generator, so that c [CP ] = 1. This implies that Z n n ω 1 = π n − 2 CP as we claimed. 1 2 (3) Consider the S -action on CP in Example 1.5 with weights m = (0, −1, −2). There are three fixed points [1, 0, 0], [0, 1, 0] and [0, 0, 1], where the moment map has 1 values 0, 2 and 1 respectively. Using the standard model for the moment map near a critical point as in the proof of Proposition 1.7, it is easy to check that for any 1 regular value λ, the reduced space Bλ is the weighted projective space CP (1, 2), 2 2 which is the quotient space (C \{(0, 0)})/ ∼, where (z1, z2) ∼ (zz1, z z2). (Note that 1 CP (1, 2) is a 2-dimensional orbiford, with one singular point of order 2.) However, 1 1 −1 for λ ∈ (0, 2 ), the first Chern class of the (orbifold) S -pricipal bundle H (λ) → Bλ 1 2 1 1 1 2 1 equals − 2 ∈ H (CP (1, 2); Q) and for λ ∈ ( 2 , 1), it equals 2 ∈ H (CP (1, 2); Q). Note 1 that the first Chern class changes by 1 when passing the critical value λ = 2 . 8 WEIMIN CHEN, UMASS, SPRING 07

1 For a Hamiltonian S -action with at most isolated fixed points, the moment map is a Morse function. Propositions 1.7 and 1.9 show that the weights of the induced action on the tangent space of each fixed point contain vital information about the equivari- ant of the manifold. There are certain constraints amongst the weights of the fixed points, as shown in the following beautiful theorem of Duistermaat and Heckman. 1 Theorem 1.11. (Duistermaat-Heckman). Assume a Hamiltonian S -action on a compact 2n-dimensional symplectic manifold (M, ω) has only isolated fixed points. Let H : M → R be a moment map, and let e(p) denote the product of the weights at a fixed point p. Then Z ωn X e−2π~H(p) e−2π~H = n! ne(p) M p ~ for every ~ ∈ C, where the sum on the right-hand side runs over all fixed points of the 1 S -action. Example 1.12. If one expands both sides of the Duistermaat-Heckman formula as power series in ~ and then compares the coefficients, the following set of constraints are obtained: X H(p)k = 0, for k = 0, 1, ··· , n − 1, e(p) p and Z X H(p)n ωn = (−2π)n . e(p) M p 1 2 We shall check this out on an example of S -action on CP as discussed in Example 1.5, with weights m = (0, −2, −5). It is easy to check that there are three isolated fixed points p1 = [1, 0, 0], p2 = [0, 1, 0], p3 = [0, 0, 1], which have weights (−2, −5), (2, −3), and (5, 3) respectively. Let H be the standard moment map as given in Example 1.5. 5 Then H(p1) = 0, H(p2) = 1, and H(p3) = 2 . We also know (see Example 1.10 (2)) R 2 2 that 2 ω = π . With the preceding understood, the set of constraints obtained CP from the Duistermaat-Heckman formula are the following for this example: 1 1 1 0 1 5 + + = 0, + + 2 = 0 (−2) · (−5) 2 · (−3) 5 · 3 (−2) · (−5) 2 · (−3) 5 · 3 and 02 12 ( 5 )2 (−2π)2( + + 2 ) = π2. (−2) · (−5) 2 · (−3) 5 · 3 1 We end with a discussion on the question as to which symplectic S -actions on a compact closed manifold are Hamiltonian. Note that a necessary condition is that the 1 S -action must have a fixed point, because the fixed points correspond to the critical points of a moment map, and a smooth (R-valued) function on a compact manifold must have a critical point. We will give a sufficient condition utilizing the following lemma. LECTURE 4: SYMPLECTIC GROUP ACTIONS 9

Lemma 1.13. Let (M, ω) be a compact closed symplectic manifold equipped with a 1 R n symplectic S -action. We assume without loss of generality that M ω = 1. Denote 1 by α the homology class of an orbit of the S -action in H1(M; R). Then α is Poincar´e n 2n−1 1 dual to 2π · [ι(X)ω ] ∈ H (M; R), where X is the vector field generating the S - action.

1 1 Proof. Let ψt, t ∈ S , be the S -action and let γ(t) = ψt(q), q ∈ M, be a non-constant orbit in M. We choose a volume form σ ∈ Ω2n(M) which is supported in a small R 1 neighborhood of γ and satisfies M σ = 1. By averaging over S we may assume that ∗ R n n ψt σ = σ. Since by assumption M ω = 1, we see that σ and ω are cohomologous, n 2n−1 1 and σ − ω = dβ for some β ∈ Ω (M), which, by averaging over S , may be ∗ assumed to satisfy ψt β = β also. Now we have n ι(X)σ − ι(X)ω = ι(X)dβ = LX β − d(ι(X)β) = d(−ι(X)β), n ∗ so that ι(X)σ and ι(X)ω are cohomologous. (Note that LX β = 0 because ψt β = β.) On the other hand, ι(X)σ is Poincar´edual to α, because if we let D be a fiber of the normal bundle of γ in M, then Z 1 Z Z 2π · ι(X)σ = 2π · σ = σ = 1. D 2π γ×D M The lemma follows immediately.  A compact, closed symplectic manifold (M, ω) of dimension 2n is said of Lefschetz type if n−1 1 2n−1 n−1 1 ∧ω : H (M; R) → H (M; R): α 7→ α ∪ [ω] , ∀α ∈ H (M; R) is an isomorphism. It is known that compact K¨ahler manifolds are of Lefschetz type. Proposition 1.14. Suppose (M, ω) is a compact, closed symplectic manifold of Lef- 1 schetz type. Then a symplectic S -action on (M, ω) is Hamiltonian if and only if there is a fixed point.

1 Proof. It suffices to show that the S -action is Hamiltonian if it has a fixed point. Suppose the action has a fixed point. Then the homology class of an orbit must be zero because it shrinks to a fixed point. This implies by Lemma 1.13 that its Poincar´e n n−1 1 dual 2π · [ι(X)ω ], which is the image of 2π · [ι(X)ω] under ∧ω : H (M; R) → 2n−1 n−1 H (M; R), is also zero. But (M, ω) is of Lefschetz type so that ∧ω is isomorphic. 1 This shows that ι(X)ω is exact, and the S -action is Hamiltonian.  Let (M, ω) be a compact, closed symplectic manifold. If dim M ≤ 4, then every 1 symplectic S -action which has a fixed point is Hamiltonian. McDuff found the first 1 example of symplectic S -actions on a 6-dimensional manifold which has a fixed point but is not Hamiltonian. In her example the fixed points are not isolated. One has not been able to find an example which has only isolated fixed points but is not Hamiltonian. 10 WEIMIN CHEN, UMASS, SPRING 07

Conjecture 1.15. Let (M, ω) be a 6-dimensional compact, closed symplectic manifold. 1 Then a symplectic S -action on (M, ω) is Hamiltonian if there is a fixed point and all fixed points are isolated.

2. Hamiltonian torus actions n 1 n We denote by T = (S ) the n-torus. The corresponding and its dual n n ∗ n are denoted by t and (t ) respectively. Since T is abelian, the Lie bracket is trivial, n n ∗ n n and t and (t ) can be canonically identified with R , with the pairing between t n ∗ n and (t ) given by the inner product on R . n Let (M, ω) be a symplectic manifold, and let T act (effectively) on M via sym- plectomorphisms. Then for any ξ ∈ tn, one has a 1-parameter group of symplecto- morphisms exp(tξ). We denote by Xξ the vector field on M which generates the flow n n exp(tξ). Note that for any ξ, η ∈ t ,[Xξ,Xη] = X[ξ,η] = 0 since T is abelian. On the other hand, each Xξ is a symplectic vector field, i.e., ι(Xξ)ω is closed. We say n n that a symplectic T -action on (M, ω) is weakly Hamiltonian if for any ξ ∈ t , ι(Xξ)ω = dHξ for some smooth function Hξ on M (note that Hξ is uniquely deter- mined up to a constant). In order to define Hamiltonian actions, we recall the concept of . Let F,H be smooth functions on M. We denote by XF ,XH the corresponding Hamil- tonian vector fields, i.e., ι(XF )ω = dF , ι(XH )ω = dH. Then the Poisson bracket of F,H is defined and denoted by

{F,H} ≡ ω(XF ,XH ) = dF (XH ) = −dH(XF ).

In particular, {F,H} = 0 means that the Hamiltonian function F is constant under the flow generated by XH (and vice versa). The set of smooth functions on (M, ω) becomes a Lie algebra under the Poisson bracket. n With the above understood, a weakly Hamiltonian T -action is called Hamilton- n ian if for any ξ, η ∈ t , the Poisson bracket {Hξ,Hη} = 0. (In general, a weakly Hamiltonian is called Hamiltonian if ξ 7→ Hξ can be chosen to be a Lie algebra homomorphism.) n The next lemma shows that in many cases a weakly Hamiltonian T -action is au- tomatically Hamiltonian.

n Lemma 2.1. For any weakly Hamiltonian T -action on (M, ω), the Poisson bracket n {Hξ,Hη} is a constant function on M for any ξ, η ∈ t . In particular, a weakly n n Hamiltonian T -action is Hamiltonian if exp(tξ) has a fixed point for any ξ ∈ t (e.g., when M is compact, closed).

Proof. It follows from the following straightforward calculation. LECTURE 4: SYMPLECTIC GROUP ACTIONS 11

d(ω(Xξ,Xη)) = d(ι(Xη)ι(Xξ)ω)

= (d ◦ ι(Xη) + ι(Xη) ◦ d)(ι(Xξ)ω)

= LXη (ι(Xξ)ω)

= ι(LXη Xξ)ω = ι([Xη,Xξ])ω = 0.

n Here we use the fact that ι(Xξ)ω, ι(Xη)ω are closed, and the assumption that T is abelian so that [Xη,Xξ] = 0.  n The moment map of a Hamiltonian T -action on (M, ω) is a smooth map n ∗ n µ : M → (t ) = R , n n such that for any ξ ∈ t = R ,

Hξ(p) = hµ(p), ξi, ∀p ∈ M, is a Hamiltonian function for exp(tξ), i.e., ι(Xξ)ω = dHξ. Note that the assignment ξ 7→ Hξ is linear. n Remark 2.2. (1) The moment map always exists. For example, let ξ1, ··· , ξn ∈ t ∗ ∗ n ∗ be a basis, and let ξ1, ··· , ξn ∈ (t ) be the corresponding dual basis. Then ∗ ∗ µ(p) = Hξ1 (p)ξ1 + ··· + Hξn (p)ξn, ∀p ∈ M, is a moment map. (2) The moment map is uniquely defined up to a constant vector in (tn)∗. n n (3) Because of the condition {Hξ,Hη} = 0 for any ξ, η ∈ t and the fact that T is n n connected, the moment map µ : M → R is T -invariant, i.e., µ(g · p) = µ(p) for any n g ∈ T .

Let p be a point in M. We next give a description of the image of dµp : TpM → n ∗ n n n (t ) = R . Let us consider the subspace of t = R which annihilates the im- n n age, i.e., the set of ξ ∈ t = R such that hdµp(Y ), ξi = 0 for all Y ∈ TpM. Ob- serve the identity hdµp(Y ), ξi = (dHξ)p(Y ) = ωp(Xξ,Y ). Since ω is nondegenerate, we see immediately that the set of ξ which annihilates the image of dµp : TpM → n ∗ n n (t ) = R is the subspace {ξ ∈ t |Xξ = 0 at p}, or equivalently, the subspace {ξ ∈ tn|p is a fixed point of the subgroup exp(tξ)}. In particular, since the princi- pal orbit, i.e., the set of points in M which has trivial isotropy, is open and dense for an effective action, we see that the set of regular values of the moment map n ∗ n µ : M → (t ) = R is open and dense in the image µ(M). n ∗ n n Let λ ∈ (t ) = R be a regular value of µ. Since µ is T -invariant, we see that the −1 n −1 n level surface µ (λ) is T -invariant. The quotient space Bλ ≡ µ (λ)/T , which is an orbifold in general of dimension dim M − 2n, has a natual symplectic structure ωλ. The space (Bλ, ωλ) is called the reduced space at λ (its proof is similar to the case of 12 WEIMIN CHEN, UMASS, SPRING 07

1 S -action, cf. Proposition. 1.1). Note that dim M − 2n ≥ 0, namely, the dimension of the torus is at most half of the dimension of the symplectic manifold which the torus acts on. When the dimension of the torus equals half of the dimension of the symplectic manifold, the reduced spaces consist of single points, and the preimages µ−1(λ) are orbits of the torus action, which are easily seen to be embedded Lagrangian tori (they are Lagrangian because of the conditon {Hξ,Hη} = ω(Xξ,Xη) = 0 for any ξ, η ∈ tn). The fundamental result concerning Hamiltonian torus actions is the following con- vexity theorem, due to Atiyah and Guillemin-Sternberg independently. Theorem 2.3. (Atiyah-Guillemin-Sternberg). Let (M, ω) be a compact, connected n symplectic manifold which is equipped with a Hamiltonian T -action of moment map n n µ : M → R . Then the fixed points of the T -action form a finite union of connected symplectic submanifolds Q1, ··· ,QN , such that on each Qj, the moment map µ has a n constant value λj ∈ R , and the image of µ is the convex hull of λj, i.e., N N X X n µ(M) = { xjλj| xj = 1, xj ≥ 0} ⊂ R . j=1 j=1 n n n Example 2.4. (1) T -action on CP . Consider the following Hamiltonian T -action n on CP

−it1 −it2 −itn (t1, t2, ··· , tn) · [z0, z1, z2, ··· , zn] = [z0, e z1, e z2, ··· , e zn], which has moment map 2 2 1 |z1| |zn| µ([z0, z1, ··· , zn]) = ( 2 2 , ··· , 2 2 ). 2 |z0| + ··· + |zn| |z0| + ··· + |zn| n n Pn 1 Clearly µ(CP ) = {(x1, x2, ··· , xn) ∈ R | i=1 xi ≤ 2 , xi ≥ 0}. The fixed points are p0 = [1, 0, ··· , 0], p1 = [0, 1, ··· , 0], ··· , pn = [0, 0, ··· , 1], which are mapped under µ 1 1 n to λ0 = (0, 0, ··· , 0), λ1 = ( 2 , 0, ··· , 0), ··· , λn = (0, 0, ··· , 2 ) respectively. µ(CP ) is the n-simplex with vertices λ0, λ1, ··· , λn. 2 2 (2) A non-effective T -action on CP . Consider the following non-effective action

−it1 −2it2 (t1, t2) · [z0, z1, z2] = [z0, e z1, e z2], which has moment map 2 2 1 |z1| 2|z2| µ([z0, z1, z2]) = ( 2 2 2 , 2 2 2 ). 2 |z0| + |z1| + |z2| |z0| + |z1| + |z2| 2 1 The image µ(CP ) is the triangle with vertices (0, 0), ( 2 , 0) and (0, 1). 2 1 1 (3) T -actions on CP × CP . Consider the following action

−it1 −it2 (t1, t2) · ([z0, z1], [w0, w1]) = ([z0, e z1], [w0, e w1]). The moment map is 2 2 1 |z1| |w0| µ([z0, z1], [w0, w1]) = ( 2 2 , 2 2 ), 2 |z0| + |z1| |w0| + |w1| LECTURE 4: SYMPLECTIC GROUP ACTIONS 13 and the fixed points are ([1, 0], [1, 0]), ([1, 0], [0, 1]), ([0, 1], [1, 0]) and ([0, 1], [0, 1]), with 1 1 1 1 values under µ being (0, 0), (0, 2 ), ( 2 , 0) and ( 2 , 2 ) respectively. The image of the moment map is 1 µ( 1 × 1) = {(x , x ) ∈ 2|0 ≤ x , x ≤ }. CP CP 1 2 R 1 2 2 2 1 1 Consider the following (effective) T -action on CP × CP −it1 −it1−it2 (t1, t2) · ([z0, z1], [w0, w1]) = ([z0, e z1], [w0, e w1]). The moment map is 2 2 2 1 |z1| |w0| |w0| µ([z0, z1], [w0, w1]) = ( 2 2 + 2 2 , 2 2 ), 2 |z0| + |z1| |w0| + |w1| |w0| + |w1| and the fixed points are ([1, 0], [1, 0]), ([1, 0], [0, 1]), ([0, 1], [1, 0]) and ([0, 1], [0, 1]), with 1 1 1 1 values under µ being (0, 0), ( 2 , 2 ), ( 2 , 0) and (1, 2 ) respectively. The image of µ is the 1 1 1 1 parallelagram with vertices (0, 0), ( 2 , 2 ), ( 2 , 0) and (1, 2 ). 2 1 1 Consider the following (effective) T -action on CP × CP −it1 −2it1−it2 (t1, t2) · ([z0, z1], [w0, w1]) = ([z0, e z1], [w0, e w1]). The moment map is 2 2 2 1 |z1| 2|w0| |w0| µ([z0, z1], [w0, w1]) = ( 2 2 + 2 2 , 2 2 ), 2 |z0| + |z1| |w0| + |w1| |w0| + |w1| and the fixed points are ([1, 0], [1, 0]), ([1, 0], [0, 1]), ([0, 1], [1, 0]) and ([0, 1], [0, 1]), with 1 1 3 1 values under µ being (0, 0), (1, 2 ), ( 2 , 0) and ( 2 , 2 ) respectively. The image of µ is the 1 1 3 1 parallelagram with vertices (0, 0), (1, 2 ), ( 2 , 0) and ( 2 , 2 ). 2 2 (4) A T -action on Hirzebruch surface CP #CP2. Here the Hirzebruch surface is given as the complex surface 1 2 M ≡ {([a, b], [x, y, z]) ∈ CP × CP |ay = bx}. 2 2 1 2 The T -action on M is the restriction of the following T -action on CP × CP −it1 −it1 −it2 (t1, t2) · ([a, b], [x, y, z]) = ([e a, b], [e x, y, e z]), which leaves M invariant. The moment map is 1 |a|2 |x|2 |z|2 µ([a, b], [x, y, z]) = ( + , ), 2 |a|2 + |b|2 |x|2 + |y|2 + |z|2 |x|2 + |y|2 + |z|2 and there are four fixed points on M, which are ([1, 0], [1, 0, 0]), ([1, 0], [0, 0, 1]), ([0, 1], [0, 1, 0]), ([0, 1], [0, 0, 1]). 1 1 1 The corresponding values under the moment map are (1, 0), ( 2 , 2 ), (0, 0) and (0, 2 ). 2 1 So the image of µ is {(x1, x2) ∈ R |x1 + x2 ≤ 1, x1 ≥ 0, 0 ≤ x2 ≤ 2 }. n Let (M, ω) be a symplectic manifold with a Hamiltonian T -action and let µ : M → n ∗ k n (t ) be the corresponding moment map. For any k < n let T ⊂ T be a sub-torus. k Then there is naturally an induced Hamiltonian T -action on M. The moment map of the induced action is µ : M → (tn)∗ composed with the projection (tn)∗ → (tk)∗. 14 WEIMIN CHEN, UMASS, SPRING 07

1 2 Example 2.5. Consider the S -action on CP which is induced from the standard 2 2 1 2 T -action on CP considered in Example 2.4 (1) by the embedding S ⊂ T given by 1 t 7→ (t, 2t). This is the same S -action we considered in Example 1.10 (3). 2 2 Note that the image of the moment map of the T -action on CP is the triangle 1 1 2 with vertices (0, 0), ( 2 , 0) and (0, 2 ). Its projection onto the line Rh1, 2i ⊂ R is the 1 2 line segment between the points (0, 0) and ( 5 , 5 ), which are the images of the vertices 1 1 (0, 0) and (0, 2 ) under the projection. Note that the image of the vertex ( 2 , 0) is the middle point ( 1 , 1 ) of the line segment. Compare with the moment map in Example 10 5 √ 1.10 (3) and notice that the length of the vector h1, 2i is 5. In general the set of regular values of the moment map is divided into several 2 3 chambers. We illustrate this with the following example of a T -action on CP . 2 3 Example 2.6. Consider the Hamiltonian T -action on CP −it1 −2it1 −it2 (t1, t2) · [z0, z1, z2, z3] = [z0, e z1, e z2, e z3], which has moment map 2 2 2 1 |z1| + 2|z2| |z3| µ([z0, z1, z2, z3]) = ( , ). 2 P3 2 P3 2 j=0 |zj| j=0 |zj| 1 The image of µ is the triangle with vertices (0, 0), (1, 0) and (0, 2 ). The set of regular values of µ is the interior of the triangle with the line segment between the points 1 1 ( 2 , 0), (0, 2 ) removed. So it is divided into two chambers by the line segment. Notice 3 that the “wall” that divides the two chambers is the image of {[0, z, 0, w] ∈ CP } under 2 µ, which is fixed by the diagonal sub-torus {(t, t)} ⊂ T . A compact, connected symplectic manifold of dimension 2n is called toric if it n admits an effective Hamiltonian T -action. Delzant showed that such a space together n with the T -action is uniquely determined by the image of the moment map, and n moreover, there exists a T -invariant complex structure with respect to which the symplectic form is K¨ahler. 1 Compact, connected symplectic 4-manifolds with a Hamiltonian S -action have been 1 classified, from which it is known that such spaces are all K¨ahler and the S -actions 2 are holomorphic. However, S. Tolman constructed an example of a Hamiltonian T - action on a compact, connected 6-dimensional manifold which does not admit any 1 S -invariant holomorphic K¨ahlerstructure.

References [1] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Mono- graphs, 2nd edition, Oxford Univ. Press, 1998.