《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.pdf_大学文库

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

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

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

6 LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Finally a point µ(m) is a vertex if and only if n1 = n for the point m, i.e. the stabilizer of m is T itself. So m is a fixed point of the T-action. 3. Morse theory It remains to prove the Guillemin-Sternberg lemma. ¶ Morse-Bott functions. Let M be a compact manifold and f ∈ C ∞(M) a smooth function. Recall that a point m ∈ M is called a critical point of f if dfm = 0. The set of all critical points of f is denoted by Crit(f). Now let m ∈ Crit(f) be a critical point of f. Consider the Hessian map Hessm(f) : TmM × TmM → R defined by (1) Hessm(f)(Xm, Ym) = X(Y f)(m), where X and Y are any vector fields whose value at m are Xm and Ym respectively. Lemma 3.1. For m ∈ Crit(f), Hessm(f) is well-defined, symmetric and bilinear. Proof. Symmetry follows from the fact X(Y f)(m) − Y (Xf)(m) = [X, Y ]f(m) = hdfm, [X, Y ]mi = 0, Since X(Y f)(m) = Xm(Y f), the right hand side of (1) is independent on the choice of X and is bilinear on Xm. By symmetry, X(Y f)(m) = Y (Xf)(m), the right hand side of (1) is also independent of the choice of Y and is bilinear on Ym. Recall that a critical point m ∈ Crit(f) is called non-degenerate if the bilinear form Hessm(f) is non-degenerate. A function f ∈ C ∞(M) is called a Morse function if Crit(f) is discrete, and each m ∈ Crit(f) is non-degenerate. Obviously any Morse function on a compact manifold has only finitely many critical points. Definition 3.2. A function f ∈ C ∞(M) is called a Morse-Bott function if Crit(f) is a manifold (with different components, Ci , which may have different dimensions), and for each m ∈ M the Hessian Hessm(f) is non-degenerate in the direction transverse to Crit(f) (equivalently, the kernel of Hessm(f) is TmCrit(f)). ¶ The index. Now suppose f is a Morse-Bott function. Let n ±(m) be the number of positive/negative eigenvalues of Hessm(f) at m ∈ Crit(f). They are of course constant along each connected component Ci of Crit(f). We shall denote them by n ± j and call them the index/coindex of Cj . Now we equip M with a Riemannian metric. Note that the gradient vector field ∇f vanishes exactly on Crit(f). Using the Riemannian metric g one can identify

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM 7 Hessf with ∇2 f. Let φt be the negative gradient flow of f. Then for any m ∈ Cj , one has the decomposition TmM = TmCj ⊕ E + m ⊕ E − m, where E ± are spanned by the eigenspace of positive/negative eigenvalues of ∇2 f respectively. Observe that as t → ∞, every point moves to point in Crit(f) under the flow φt . For each j the consider the set of points that flow to Cj as t → ∞, Ws (Cj ) = {m ∈ M | φt(m) → Cj as t → +∞}. Fact: Ws (Cj ) is a submanifold of M of dimension n + j + dim(Cj ). Moreover, at each m ∈ Cj , TmWs (Cj ) = TmCj ⊕ E + m. We will call Ws (Cj ) the stable submanifold of Cj . Similarly by studying the negative gradient flow as t → −∞, one gets the unstable submanifold of Cj , Wu (Cj ) = {m ∈ M | φt(m) → Cj as t → −∞}, which is a submanifold of dimension n − j + dim(Cj ), whose tangent space at m ∈ Cj is TmWu (Cj ) = TmCj ⊕ E − m. We shall use the following topological observation: If N is a submanifold of M of codimension at least 2, then M \ N is connected. We have seen n + j = codim(Wu (Cj )), n− j = codim(Ws (Cj )). Note that n − j means each point in Cj is a local minimum. Proposition 3.3. Let f be a Morse-Bott function. If none of these n − j ’s equals to 1, then there exists a unique j such that n − j = 0. In other words, there is a unique Cj on which f is local minimum, and thus an absolute minimum. Proof. By the assumption, if n − j ≥ 0, then n − j ≥ 2. The union of all these stable submanifolds has codimension at least 2. It follows that their complement, Mc , is a nonempty connected subset in M. Since M is the union of all stable submanifolds, there exists at least one critical component Cj with index n − j = 0. Note that Ws (Cj ) is an open subset of M. Moreover, since Mc is connected, it cannot be a union of more than one disjoint open subset. It follows that such Cj is unique. So f attains its absolute minimum on Cj . Remark. By the same way one can show that if none of the n + j ’s equals to 1, then there exists a unique Ck on which f is local maximum, and thus an absolute maximum.

8 LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM ¶ Proof of the Guillemin-Sternberg lemma. Obviously the Guillemin-Sternberg lemma follows from Theorem 3.4. Let (M, ω, T, µ) be a compact connected Hamiltonian T-space. Then for any X ∈ g, µ X is a Morse-Bott function, and all of its indeces n ± j ’s are even. Proof. Let X ∈ g and m ∈ Crit(µ X). Let T1 be the closure of the 1-parameter subgroup {exp(tX) |t ∈ R}. Then m is a fixed point for the action of T1. Let α1, · · · , αn be the weights for the isotropy representation of T1 on TmM. Then as we have seen, the moment map of this T1 action is locally of the form µ1(p) = µ1(m) +X |zk| 2 2 αk. On the other hand, µ1 = π ◦ µ, where π is induced by the inclusion T1 ⊂ T. So µ X(p) = µ X 1 (p) = µ X 1 (m) +X |zk| 2 2 αk(X) = µ X 1 (m) +X αk(X) 2 (x 2 k + y 2 k ). It follows that the critical points of µ X near m are defined locally by the equations zj = zj+1 = · · · = zd = 0, where j is chosen so that αj (X) = · · · = αn(X) = 0 and α1(X) 6= 0, · · · , αj−1(X) 6= 0, and d is half the dimension of M. In other words, the critical set Crit(µ X) is a smooth submanifolds near each point m ∈ Crit(µ X). So µ X is a Morse-Bott function. Moreover, the Hessian of µ X at m has eigenvalues α1(X), α1(X), · · · , αn(X), αn(X), 0, · · · , 0. In particular, all indeces are even. ¶ Connectedness of level sets. One can say more from the Morse-Bott theory. Lemma 3.5. Let f be a Morse-Bott function such that none of the n ± j ’s equals 1. Then for any regular value r of f, f −1 (r) is a connected submanifold of M. Proof. According to the previous proposition, we see that there exists a unique Cj on which f attains its absolute minimum, and there exists a unique Ck one which f attains its local maximum. Let M0 = (M \ Crit(f)) ∪ Cj ∪ Ck be the complement of the union of all other critical levels. It is an open dense connected subset of M. Consider the negative gradient flow φt on M0. Since r is a regular value, f −1 (r) is a smooth submanifold of M, and min f < r < max f. So each orbit intersects with f −1 (r) at a unique point. So the map f −1 (r) × R → M0, (m, t) 7→ φt(m) is a diffeomorphism. Since M0 is connected, f −1 (r) must be connected.