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Lecture 5: Some Basic Constructions in Symplectic Topology

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Lecture 5: Some Basic Constructions in Symplectic Topology LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in a space by the space of complex tangent lines through that point. It is a local operation which can be explicitly written down as follows. n Consider the blow-up of C at the origin. This is the complex submanifold of n n−1 C × CP n Ce ≡ {((z1, z2, ··· , zn), [w1, w2, ··· , wn])|(z1, z2, ··· , zn) ∈ [w1, w2, ··· , wn]}. n n n−1 n−1 Note that as a set Ce = C \{0} t CP , with CP being the space of complex n−1 n tangent lines through the origin. CP ⊂ Ce is called the exceptional divisor. The n n n−1 projection onto the first factor C induces a biholomorphism between Ce \ CP and n n−1 n C \{0}, and collapses the exceptional divisor CP onto the origin 0 ∈ C . On the n−1 n other hand, the projection onto the second factor CP defines Ce as a holomorphic n−1 line bundle over CP . Blowing up is often used in resolving singularities of complex subvarieties. We give some examples next to illustrate this. 2 Example 1.1. (1) Consider the complex curve C ≡ {(z1, z2) ∈ C |z1z2 = 0}. It is the union of lines C1 = {z1 = 0} and C2 = {z2 = 0} which intersect transversely 2 2 at 0 ∈ C . Let us consider the pre-image of the portion of C1,C2 in C \{0} in the 2 2 2 blowup Ce of C at 0. It can be compactified into a complex curve Ce in Ce , which is a disjoint union of two smooth complex curves Ce1 and Ce2. The point here is that 2 2 since C1,C2 intersect transversely at 0 ∈ C , the compactification Ce1 and Ce2 in Ce 1 2 are obtained by adding two distinct points in the exceptional divisor CP ⊂ Ce , which 2 parametrizes the complex lines through 0 ∈ C . More concretely, note that 2 1 Ce1 = {(0, z), [0, 1]) ∈ C × CP |z ∈ C}, and 2 1 Ce2 = {(z, 0), [1, 0]) ∈ C × CP |z ∈ C}. 2 2 We call Ce1, Ce2 the proper transform of C1, C2 under the blowing up Ce → C . 2 2 3 (2) Consider the complex curve C = {(z1, z2) ∈ C |z1 = z2}. It is called a cusp 2 curve and is singular at 0 ∈ C . The proper transform Ce of C in the blowing up 2 2 Ce → C is identified with 3 2 2 1 Ce = {(z , z ), [z, 1]) ∈ C × CP |z ∈ C}. 1 2 WEIMIN CHEN, UMASS, SPRING 07 2 Note that Ce is a smooth curve in Ce because the projection of Ce onto the exceptional 1 2 curve CP ⊂ Ce is non-singular at z = 0. 2 2 1 (3) Note that CP can be regarded as C with addition of a copy of CP at the 2 1 2 infinity, i.e., CP = C t CP . On the other hand, C is the union of complex lines 1 2 through the origin, which is parametrized by CP . Therefore the blow up of CP at 2 1 1 0 ∈ C is a (nontrivial) CP -bundle over CP . Topologically, it is diffeomorphic to 2 2 2 CP #CP2, the connected sum of CP with CP2. Here CP2 is CP with the reversed orientation. More generally, we have Theorem 1.2. Let X be an n-dimensional complex manifold. Then the blow up of X at one point is an n-dimensional complex manifold Xe which is diffeomorphic to X#CPn. Let (M, ω) be a symplectic manifold of dimension 2n, and let p ∈ M be a point. We would like to define a symplectic analog of blowing up of M at p. To this end, note that topologically M#CPn can be obtained by removing a ball centered at p and then 2n−1 2n−1 n−1 collapsing the boundary S along the fibers of the Hopf fibration S → CP . We will show that there is a canonical symplectic structure on the resulting manifold (depending on the “symplectic size” of the ball removed); this is a special case of the so-called symplectic cutting due to E. Lerman, which we shall describe next. 1 Let (M, ω) be a symplectic manifold equipped with a Hamiltonian S -action, and 1 let h : M → R be a moment map of the S -action and let be a regular value of 1 −1 h. For simplicity we assume that the S -action on h () is free; this condition is unnecessary if one works with orbifolds. We introduce the following notations: we denote by Mh>, Mh≥ the pre-images of (, ∞) and [, ∞) under h : M → R, and −1 denote by Mh≥ the manifold which is obtained by collapsing the boundary h () of 1 Mh≥ along the orbits of the S -action. Theorem 1.3. (E. Lerman). There is a natural symplectic structure ω on Mh≥ such that the restriction of ω to Mh> ⊂ Mh≥ equals ω. 1 Proof. Consider the symplectic product (M ×C, ω⊕ω0) and the Hamiltonian S -action on it given by it t · (m, z) = (t · m, e z), m ∈ M, z ∈ C. 1 2 The moment map is H(m, z) = h(m) − 2 |z| . Observe the following identification H−1() = {(m, z)|h(m) > , |z| = p2(h(m) − )} t {(m, 0)|h(m) = } 1 −1 = Mh> × S t h (). −1 1 −1 1 The theorem follows immediately from H ()/S = Mh> t h ()/S = Mh≥, and −1 1 that the symplectic structure ω on H ()/S equals ω when restricted to the open submanifold Mh>. LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY 3 Let δ > 0 and let M = {(z , z , ··· , z ) ∈ n| Pn |z |2 < δ} be the ball of radius √ 1 2 n C j=1 j n δ in C centered at the origin. Given with the standard symplectic structure ω0, M 1 admits a Hamiltonian S -action −it −it −it t · (z1, z2, ··· , zn) = (e z1, e z2, ··· , e zn), 1 Pn 2 δ which has a moment map h(z1, z2, ··· , zn) = 2 j=1 |zj| . For any 0 < < 2 , is a 1 −1 regular value of h such that the S -action on h () is free. As we explained√ earlier, Mh≥ is a symplectic blow-up of M, the standard symplectic ball of radius δ, at the origin. Symplectic blowing up of a general symplectic manifold can be defined by grafting the above construction into the manifold. More precisely, let (M, ω) be a symplectic manifold and let p ∈ M be a point. By Darboux theorem, there exists a δ > 0 such that a neighborhood√ of p in M is symplectomorphic to the standard symplectic ball of radius δ. By Theorem 1.3, since√ the symplectic structure ω on the blow-up of the standard symplectic ball of radius δ equals the standard symplectic structure ω0 n Pn 2 when restricted to the shell region {(z1, z2, ··· , zn) ∈ C |2 < j=1 |zj| < δ}, one can graft it into the symplectic manifold (M, ω). We end with a brief discussion of symplectic cutting in the presence of a Hamiltonian n torus action. Let (M, ω) be a symplectic manifold with a Hamiltonian T -action, and n ∗ n 1 n let µ : M → (t ) be the moment map. Suppose ξ0 ∈ t generates a circle T ⊂ T . 1 Then the induced S -action on M has the moment map h = hµ, ξ0i. Because the 1 n −1 n induced S -action commutes with the T -action, and both Mh> and h () are T - n −1 1 invariant, there is an induced T -action on Mh> t h ()/S = Mh≥, which is also n ∗ Hamiltonian. The corresponding moment map µ : Mh≥ → (t ) is induced from the n ∗ restriction of µ on Mh≥. Hence the image of µ in (t ) is ∗ n ∗ ∗ µ(Mh≥) = µ(M) ∩ {ξ ∈ (t ) |hξ , ξ0i ≥ }. Example 1.4. (Equivariant blowing-up at a fixed point). Consider the standard 2 2 Hamiltonian T -action on CP given by −it1 −it2 (t1, t2) · [z0, z1, z2] = [z0, e z1, e z2] which is considered in Example 2.4 (1) of Lecture 4. Here we shall double the sym- plectic form used in Example 2.4 (1) of Lecture 4, and consequently the moment map changes to 2 2 |z1| |z2| µ([z0, z1, z2]) = ( 2 2 2 , 2 2 2 ), |z0| + |z1| + |z2| |z0| + |z1| + |z2| 2 and the image of µ is {(x1, x2) ∈ R |x1 + x2 ≤ 1, x1, x2 ≥ 0}. 1 Now consider the S -action it t · [z0, z1, z2] = [z0, z1, e z2], 2 |z2| 2 which has moment map h([z0, z1, z2]) = − 2 2 2 . The symplectic cut 1 , |z0| +|z1| +|z2| CP h≥− 2 2 which is a symplectic blowing-up of CP at the fixed point [0, 0, 1], has an induced 4 WEIMIN CHEN, UMASS, SPRING 07 2 Hamiltonian T -action. The image of the corresponding moment map is 1 {(x , x ) ∈ 2|x + x ≤ 1, x ≥ 0, 0 ≤ x ≤ }. 1 2 R 1 2 1 2 2 Note that this is the same as the image of the moment map in Example 2.4 (4) in 2 Lecture 4 (a T -action on a Hirzebruch surface). By Delzant’s classification theorem, 2 these two T -actions are equivalent. 2. Thurston’s construction Recall that If M1, M2 are symplectic manifolds with symplectic structures ω1, ω2, then the product M1 × M2 is also a symplectic manifold with a natural symplectic structure ω1 ⊕ ω2.
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