《微分几何：辛几何引论》课程教学资源（英文讲义）Lec10 THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM

LECTURE 1O:THE ATIYAH-GUILLEMIN-STERNBERGCONVEXITYTHEOREMCONTENTS11.The Atiyah-Guillemin-Sternberg Convexity Theorem32.Proof of the Atiyah-Guillemin-Sternberg Convexity theorem63.Morsetheory94.Studentpresentation1.THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM The statement of AGS convexity theorem.Let (M,w) be a compact connected symplectic manifold. In this and next lec-ture, we will study the special case where G = T is a compact connected abelianLiegroup,acting in Hamiltonian fashion on M.Since T is abelian, from standard Lie theory we know that T ~ Tk for some k.It follows that t Rk, and thus t* ~Rk. We have seen that in this case the momentmap μ : M → t* is a T-invariant map.ThemaintheoremwewanttoproveisTheorem 1.1 (Atiyah-Guillemin-Sternberg Convexity Theorem).Let (M,g,T,μ)be a compact connected Hamiltonian T-manifold.Then the image of μ is a converpolyhedron in t* whose vertices are the image of the T-fired points.Remark.In the general case of a Hamiltonian action of a compact Lie group G,the image of μ might be much more complicated. However, one can prove that theintersection of μ(M) with each Weyl chamber is a convex polyhedron.TExamples.Erample.Themomentmapfor thestandard rotational S'-action on S?isμ(r,y,z) = z.So the image of μ is the interval [-1, 1]. Observe that the pre-images of the points+l are north/south poles of S, which are exactly the fixed points of the Si-action-

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Contents 1. The Atiyah-Guillemin-Sternberg Convexity Theorem 1 2. Proof of the Atiyah-Guillemin-Sternberg Convexity theorem 3 3. Morse theory 6 4. Student presentation 9 1. The Atiyah-Guillemin-Sternberg Convexity Theorem ¶ The statement of AGS convexity theorem. Let (M, ω) be a compact connected symplectic manifold. In this and next lec￾ture, we will study the special case where G = T is a compact connected abelian Lie group, acting in Hamiltonian fashion on M. Since T is abelian, from standard Lie theory we know that T ' T k for some k. It follows that t ' R k , and thus t ∗ ' R k . We have seen that in this case the moment map µ : M → t ∗ is a T-invariant map. The main theorem we want to prove is Theorem 1.1 (Atiyah-Guillemin-Sternberg Convexity Theorem). Let (M, g, T, µ) be a compact connected Hamiltonian T-manifold. Then the image of µ is a convex polyhedron in t ∗ whose vertices are the image of the T-fixed points. Remark. In the general case of a Hamiltonian action of a compact Lie group G, the image of µ might be much more complicated. However, one can prove that the intersection of µ(M) with each Weyl chamber is a convex polyhedron. ¶ Examples. Example. The moment map for the standard rotational S 1 -action on S 2 is µ(x, y, z) = z. So the image of µ is the interval [−1, 1]. Observe that the pre-images of the points ±1 are north/south poles of S 2 , which are exactly the fixed points of the S 1 -action. 1

2LECTURE1O:THEATIYAH-GUILLEMIN-STERNBERGCONVEXITYTHEOREMErample. Consider the standard T? on Cp?via(eiti,eit2) - [20 : 21 : 22] = [20 : eitz1 : eit2 2]The moment map is(o: - ()The image is a triangle in R2 with vertices (0, 0),(-, 0), (0, -). Observe that thesevertices are the image of [1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1] of Cp?, which are exactlythe fixed points of the T2-action. One can easily extend this example to arbitrarydimension, in which case the image of the moment map is a simplex.Erample. Hirzebruch surfaces. Application: Schur-Horn theorem.Let Abean Hermitian matrix whoseeigenvalues are≤2≤≤Arrange the diagonal entries of A in increasing order asai≤a2≤..≤anItwas shownbySchur thatfor each1≤k≤n,kkai≥a二1=1Horn prove the converse:For any sequences A and ak satisfying all the aboveinequalities, there exists an Hermitian matrix A whose diagonal entries are ax's andwhoseeigenvaluesareA's.We can give a symplectic geometric proof of Schur-Horn's theorem. For anyA =(i,...,An) e IRn, let H be the set of all n x n Hermitian matrices whoseeigenvalues are precisely Ai,...,An: As was explained in Yuguo's presentation, Hcan be identified with a U(n)-coadjoint orbit and thus is a symplectic manifold. Thecoadjoint action of U(n) on H is a Hamiltonian action. This action restricts to aHamiltonian action of the maximal torus Tn T C U(n) on Hx, whose momentmap isμ: H> -→Rn,A= (ai)nxn → (a11,a22,...,ann)=: d.The fixed points of the T action are the diagonal matrices, whose diagonal entrieshas to be 。 = (Ao(1),...,Ao(n), where is a permutation of (1,2,...,n). Soaccording to the Ativah-Guillemin-Sternberg convexity theorem. a vector a is thediagonal of a Hermitian matrix in Hx if and only if a lies in the convex hull of thepoints A.. This is equivalent to the Schur-Horn inequalities.Remark. Kostant extends Schur-Horn's theorem to more general coadjoint orbits

2 LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Example. Consider the standard T 2 on CP2 via (e it1 , eit2 ) · [z0 : z1 : z2] = [z0 : e it1 z1 : e it2 z2] The moment map is µ([z0 : z1 : z2]) = − 1 2  |z1| 2 |z| 2 , |z2| 2 |z| 2  The image is a triangle in R 2 with vertices (0, 0),(− 1 2 , 0),(0, − 1 2 ). Observe that these vertices are the image of [1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1] of CP2 , which are exactly the fixed points of the T 2 -action. One can easily extend this example to arbitrary dimension, in which case the image of the moment map is a simplex. Example. Hirzebruch surfaces. ¶ Application: Schur-Horn theorem. Let A be an Hermitian matrix whose eigenvalues are λ1 ≤ λ2 ≤ · · · ≤ λn. Arrange the diagonal entries of A in increasing order as a1 ≤ a2 ≤ · · · ≤ an. It was shown by Schur that for each 1 ≤ k ≤ n, X k i=1 ai ≥ X k i=1 λi . Horn prove the converse: For any sequences λk and ak satisfying all the above inequalities, there exists an Hermitian matrix A whose diagonal entries are ak’s and whose eigenvalues are λk’s. We can give a symplectic geometric proof of Schur-Horn’s theorem. For any λ = (λ1, · · · , λn) ∈ R n , let Hλ be the set of all n × n Hermitian matrices whose eigenvalues are precisely λ1, · · · , λn. As was explained in Yuguo’s presentation, Hλ can be identified with a U(n)-coadjoint orbit and thus is a symplectic manifold. The coadjoint action of U(n) on Hλ is a Hamiltonian action. This action restricts to a Hamiltonian action of the maximal torus T n ' T ⊂ U(n) on Hλ, whose moment map is µ : Hλ → R n , A = (aij )n×n 7→ (a11, a22, · · · , ann) =: ~a. The fixed points of the T action are the diagonal matrices, whose diagonal entries has to be λσ = (λσ(1), · · · , λσ(n)), where σ is a permutation of (1, 2, · · · , n). So according to the Atiyah-Guillemin-Sternberg convexity theorem, a vector ~a is the diagonal of a Hermitian matrix in Hλ if and only if ~a lies in the convex hull of the points λσ. This is equivalent to the Schur-Horn inequalities. Remark. Kostant extends Schur-Horn’s theorem to more general coadjoint orbits

3LECTURE10:THE ATIYAH-GUILLEMIN-STERNBERGCONVEXITYTHEOREM Application: number of T-fixed points.As an application of theAGS convexitytheorem, we canproveProposition 1.2.Let (M,g)be a compact connected symplectic manifold admittinga Hamiltonian T-action. Suppose there ecists m E M such that T acts locally freeatm.Thentheremustbeatleastk+1firedpoints,wherek=dimTProof.Since the action is locally free at m, the stabilizerTm is a finite subgroup ofT, and thus tm=[o}.It follows thatIm(dμm) = tm = t*.So dμm is surjective, i.e. μ is a submersion near m. It follows that μ : M → t* is anopen map near m. In particular, μ(p) is an interior point of μ(M). So μ(M) is anon-degenerate convex polytope in t* ~ Rk, which has at least k +1 vertices. Sinceeach vertex is the image of a T-fixed point, the T action has at least k + 1 fixed口points.2. PROOF OF THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREMWewill followthe Guillemin-Sternberg approach toprovethe convexity theorem.The Atiyah approach is sketched in Ana Canas de Silver's book. The equivariant Darboux theorems.Suppose G is a compact connected Lie group acting smoothly on M. Supposemo is a fixed point of the G-action.We can endow with M a G-invariant Riemannianmetric.Then sufficiently small geodesic balls around mo is a contractibleinvariantdomain, and using the ordinary Poincare lemma it is easy to prove the followinginariant Poincar'e lemma: Any invariant closed k-form in a neighborhood of mois the differential of an invariantk -1-form.As a consequence, we can proveTheorem 2.1 (Equivariant Darboux theorem).Let (M,w;), i=1,2 be symplecticG-spaces. Let m be a fired point of G so that wi(m) = w2(m). Then there is aninvariant neighborhood U of m and an equivariant diffeomorphism f of U into Msothatf(m)=mand f*w2=w1.口Proof. Apply Moser's trick as before.Details left as an exercise.Now suppose (M,w)bea symplectic G-manifold, and let m be a G-fixed point.Then the isotropy action of G on TmM,g · w= dgm(o),is a linear G-action, i.e. a representation, of G on TmM.Moreover, if we fix anG-invariant Riemannian metric on M, then the exponential map exp : TmM -→ Mis equivariant.Itfollowsfrom theprevious theoremthat

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM 3 ¶ Application: number of T-fixed points. As an application of the AGS convexity theorem, we can prove Proposition 1.2. Let (M, g) be a compact connected symplectic manifold admitting a Hamiltonian T-action. Suppose there exists m ∈ M such that T acts locally free at m. Then there must be at least k + 1 fixed points, where k = dim T. Proof. Since the action is locally free at m, the stabilizer Tm is a finite subgroup of T, and thus tm = {0}. It follows that Im(dµm) = t 0 m = t ∗ . So dµm is surjective, i.e. µ is a submersion near m. It follows that µ : M → t ∗ is an open map near m. In particular, µ(p) is an interior point of µ(M). So µ(M) is a non-degenerate convex polytope in t ∗ ' R k , which has at least k + 1 vertices. Since each vertex is the image of a T-fixed point, the T action has at least k + 1 fixed points. 2. Proof of the Atiyah-Guillemin-Sternberg Convexity theorem We will follow the Guillemin-Sternberg approach to prove the convexity theorem. The Atiyah approach is sketched in Ana Canas de Silver’s book. ¶ The equivariant Darboux theorems. Suppose G is a compact connected Lie group acting smoothly on M. Suppose m0 is a fixed point of the G-action. We can endow with M a G-invariant Riemannian metric. Then sufficiently small geodesic balls around m0 is a contractible invariant domain, and using the ordinary Poincare lemma it is easy to prove the following invariant Poincar’e lemma: Any invariant closed k-form in a neighborhood of m0 is the differential of an invariant k − 1-form. As a consequence, we can prove Theorem 2.1 (Equivariant Darboux theorem). Let (M, ωi), i = 1, 2 be symplectic G-spaces. Let m be a fixed point of G so that ω1(m) = ω2(m). Then there is an invariant neighborhood U of m and an equivariant diffeomorphism f of U into M so that f(m) = m and f ∗ω2 = ω1. Proof. Apply Moser’s trick as before. Details left as an exercise. Now suppose (M, ω) be a symplectic G-manifold, and let m be a G-fixed point. Then the isotropy action of G on TmM, g · v = dgm(v), is a linear G-action, i.e. a representation, of G on TmM. Moreover, if we fix an G-invariant Riemannian metric on M, then the exponential map exp : TmM → M is equivariant. It follows from the previous theorem that

4LECTURE1O:THEATIYAH-GUILLEMIN-STERNBERGCONVEXITYTHEOREMTheorem 2.2 (Equivariant Darboux theorem, version 2).Let (M,w) be a symplec-tic G-manifold and m a fired point of the G-action.Then there is an invariantneighborhood U of m inM and an equivariant diffeomorphism from (U,w)into(TmM,wm) so that p(m) = 0 and p*wm = w.Onecan identifythe symplecticvector space(TmM,wm)withthe complexspaceCn. SoTheorem 2.3 (Equivariant Darboux theorem, version 3).Let (M,w) be a symplec-tic G-manifold and m a fired point of the G-action.Then there is an invariantneighborhood U of m in M and local compler coordinates zi,., zn so that on U,the symplectic form can be written asEdzAdkW2iand theG-actionbecomes a linear symplecticG-actionon Cn. Darboux theorem for the moment mapNow let's turn back to the case G -T is a torus of dimension n.Let ai,...,Qn Et* be the weights of the isotropy representation of T on TmM. In other words,eit. (21,.., zn) = (eia1()z1,..,eian(t) n).Theorem 2.4.Let pEU be a point whose coordinate is z.Then() = (m) + [PXk2k口Proof.Exercise.Remark.Locally a symplectic action is always Hamiltonian since the first cohomol-ogy vanishes.TThelocal convexityNow let's go back tothetheorem.Let U be the invariant neighborhood given by the equivariant Darboux theoremabove.Then the image of U under the moment map μ,near the point μ(m), isu(m)+siak0≤k=1In otherwords,weprovedProposition 2.5. Let (M,w, T, μ) be a Hamiltonian T-space and m a T-fired pointThen there erists a neighborhood U of m so that μ(U) is a cone with verter μ(m)

4 LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Theorem 2.2 (Equivariant Darboux theorem, version 2). Let (M, ω) be a symplec￾tic G-manifold and m a fixed point of the G-action. Then there is an invariant neighborhood U of m in M and an equivariant diffeomorphism ϕ from (U, ω) into (TmM, ωm) so that ϕ(m) = 0 and ϕ ∗ωm = ω. One can identify the symplectic vector space (TmM, ωm) with the complex space C n . So Theorem 2.3 (Equivariant Darboux theorem, version 3). Let (M, ω) be a symplec￾tic G-manifold and m a fixed point of the G-action. Then there is an invariant neighborhood U of m in M and local complex coordinates z1, · · · , zn so that on U, the symplectic form can be written as ω = 1 2i X k dzk ∧ dz¯k, and the G-action becomes a linear symplectic G-action on C n . ¶ Darboux theorem for the moment map. Now let’s turn back to the case G = T is a torus of dimension n. Let α1, · · · , αn ∈ t ∗ be the weights of the isotropy representation of T on TmM. In other words, e it · (z1, · · · , zn) = (e iα1(t) z1, · · · , eiαn(t) zn). Theorem 2.4. Let p ∈ U be a point whose coordinate is z. Then µ(p) = µ(m) +X k |zk| 2 2 αk. Proof. Exercise. Remark. Locally a symplectic action is always Hamiltonian since the first cohomol￾ogy vanishes. ¶ The local convexity. Now let’s go back to the theorem. Let U be the invariant neighborhood given by the equivariant Darboux theorem above. Then the image of U under the moment map µ, near the point µ(m), is ( µ(m) + Xn k=1 skαk      0 ≤ sk ≤ ε ) . In other words, we proved Proposition 2.5. Let (M, ω, T, µ) be a Hamiltonian T-space and m a T-fixed point. Then there exists a neighborhood U of m so that µ(U) is a cone with vertex µ(m).

LECTURE 10:THE ATIYAH-GUILLEMIN-STERNBERGCONVEXITY THEOREM5This local convexity theorem has a relative version: Let Ti CT be a subgroup,and mi E M is a fixed point under the induced Ti action.As we have seen inlecture 8, the moment map μi of this Ti-action is μi = diT o μ. Note that the mapdiT :t*→t is nothing else but the projection :t*→t*if we identify t as asubspace of t. Applying the previous arguments, we can find a neighborhood Ui ofmi so that μi(Ui) is the cone with vertex μi(mi) described as above. It follows thatitspreimageisProposition 2.6. Let (M,w,T,μ) be a Hamiltonian T-space, T a subgroup of T,and mi a Ti-fired point. Let ai, .. ,an, be the weights of the isotropy representationof Ti on TmM. Then there erists a neighborhood Ui of mi so that μ(Ui), nearμ(mi),isu(m)+0≤Sk The global convexity.In the next section we shall proveLemma 2.7 (Guillemin-Sternberg lemma). For any X e g, the function μX : M -→R has a unique local minimum/marimum.Proof of the Atiyah-Guillemin-Sternberg converity theorem. Since M is compact, theimage μ(M) is compact in g*. Let E g* be a point in the boundary of μ(M). Takeapoint mEM sothat μ(m)=.LetTi=Tmbethe stabilizer of m and letai,...,an E ti be theweights of the isotropy representation of Ti on TmM. Thenthere exists a neighborhood U of m in M so thatμ(U)<SkSkWedenote S=π-1((szak| Sk≥O)). LetSi be a boundary component ofS. Since Si has codimension at least 1, one can choose X e g so that(n,X)=ofornESkand(n,X)<for n in the interior of S.Nowsuppose(s,X)=a.Thenforanym'eU,μ(m) = (μ(m),X)= (E +n,X)≤a,i.e. a is a local maximum of μX. According to the Guillemin-Sternberg lemma, it isan absolute maximum. So <μ(M),X)=μx(M) ≤ a.Applying this argument to all boundary components S of S, we conclude thatμ(M) sits in the cone + S. It follows that μ(M) is a convex polyhedron

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM 5 This local convexity theorem has a relative version: Let T1 ⊂ T be a subgroup, and m1 ∈ M is a fixed point under the induced T1 action. As we have seen in lecture 8, the moment map µ1 of this T1-action is µ1 = dιT ◦ µ. Note that the map dιT : t ∗ → t ∗ 1 is nothing else but the projection π : t ∗ → t ∗ 1 if we identify t ∗ 1 as a subspace of t. Applying the previous arguments, we can find a neighborhood U1 of m1 so that µ1(U1) is the cone with vertex µ1(m1) described as above. It follows that its preimage is Proposition 2.6. Let (M, ω, T, µ) be a Hamiltonian T-space, T1 a subgroup of T, and m1 a T1-fixed point. Let α 1 1 , · · · , α1 n1 be the weights of the isotropy representation of T1 on Tm1M. Then there exists a neighborhood U1 of m1 so that µ(U1), near µ(m1), is ( µ(m) + π −1 Xn1 k=1 skαk !      0 ≤ sk ≤ ε ) . ¶ The global convexity. In the next section we shall prove Lemma 2.7 (Guillemin-Sternberg lemma). For any X ∈ g, the function µ X : M → R has a unique local minimum/maximum. Proof of the Atiyah-Guillemin-Sternberg convexity theorem. Since M is compact, the image µ(M) is compact in g ∗ . Let ξ ∈ g ∗ be a point in the boundary of µ(M). Take a point m ∈ M so that µ(m) = ξ. Let T1 = Tm be the stabilizer of m and let α1, · · · , αn ∈ t1 be the weights of the isotropy representation of T1 on TmM. Then there exists a neighborhood U of m in M so that µ(U) = ( ξ + π −1 Xn1 k=1 skαk !      0 ≤ sk ≤ ε ) . We denote S = π −1 ({ Pskαk | sk ≥ 0}). Let S1 be a boundary component of S. Since S1 has codimension at least 1, one can choose X ∈ g so that hη, Xi = 0 for η ∈ Sk and hη, Xi < 0 for η in the interior of S. Now suppose hξ, Xi = a. Then for any m0 ∈ U, µ X(m0 ) = hµ(m0 ), Xi = hξ + η, Xi ≤ a, i.e. a is a local maximum of µ X. According to the Guillemin-Sternberg lemma, it is an absolute maximum. So hµ(M), Xi = µ X(M) ≤ a. Applying this argument to all boundary components Sk of S, we conclude that µ(M) sits in the cone ξ + S. It follows that µ(M) is a convex polyhedron