LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY


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1 LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING Blowing up and symplectic cutting In complex geometry the blowingup 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. Consider the blowup of C n at the origin. This is the complex submanifold of C n CP n 1 C n {((z 1, z 2,, z n ), [w 1, w 2,, w n ]) (z 1, z 2,, z n ) [w 1, w 2,, w n ]}. Note that as a set C n = C n \ {0} CP n 1, with CP n 1 being the space of complex tangent lines through the origin. CP n 1 C n is called the exceptional divisor. The projection onto the first factor C n induces a biholomorphism between C n \ CP n 1 and C n \ {0}, and collapses the exceptional divisor CP n 1 onto the origin 0 C n. On the other hand, the projection onto the second factor CP n 1 defines C n as a holomorphic line bundle over CP n 1. Blowing up is often used in resolving singularities of complex subvarieties. We give some examples next to illustrate this. Example 1.1. (1) Consider the complex curve C {(z 1, z 2 ) C 2 z 1 z 2 = 0}. It is the union of lines C 1 = {z 1 = 0} and C 2 = {z 2 = 0} which intersect transversely at 0 C 2. Let us consider the preimage of the portion of C 1, C 2 in C 2 \ {0} in the blowup C 2 of C 2 at 0. It can be compactified into a complex curve C in C 2, which is a disjoint union of two smooth complex curves C 1 and C 2. The point here is that since C 1, C 2 intersect transversely at 0 C 2, the compactification C 1 and C 2 in C 2 are obtained by adding two distinct points in the exceptional divisor CP 1 C 2, which parametrizes the complex lines through 0 C 2. More concretely, note that and C 1 = {(0, z), [0, 1]) C 2 CP 1 z C}, C 2 = {(z, 0), [1, 0]) C 2 CP 1 z C}. We call C 1, C 2 the proper transform of C 1, C 2 under the blowing up C 2 C 2. (2) Consider the complex curve C = {(z 1, z 2 ) C 2 z1 2 = z3 2 }. It is called a cusp curve and is singular at 0 C 2. The proper transform C of C in the blowing up C 2 C 2 is identified with C = {(z 3, z 2 ), [z, 1]) C 2 CP 1 z C}. 1
2 2 WEIMIN CHEN, UMASS, SPRING 07 Note that C is a smooth curve in C 2 because the projection of C onto the exceptional curve CP 1 C 2 is nonsingular at z = 0. (3) Note that CP 2 can be regarded as C 2 with addition of a copy of CP 1 at the infinity, i.e., CP 2 = C CP 1. On the other hand, C 2 is the union of complex lines through the origin, which is parametrized by CP 1. Therefore the blow up of CP 2 at 0 C 2 is a (nontrivial) CP 1 bundle over CP 1. Topologically, it is diffeomorphic to CP 2 #CP 2, the connected sum of CP 2 with CP 2. Here CP 2 is CP 2 with the reversed orientation. More generally, we have Theorem 1.2. Let X be an ndimensional complex manifold. Then the blow up of X at one point is an ndimensional complex manifold X which is diffeomorphic to X#CP n. 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#CP n can be obtained by removing a ball centered at p and then collapsing the boundary S 2n 1 along the fibers of the Hopf fibration S 2n 1 CP n 1. 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 socalled symplectic cutting due to E. Lerman, which we shall describe next. Let (M, ω) be a symplectic manifold equipped with a Hamiltonian S 1 action, and let h : M R be a moment map of the S 1 action and let ɛ be a regular value of h. For simplicity we assume that the S 1 action on h 1 (ɛ) is free; this condition is unnecessary if one works with orbifolds. We introduce the following notations: we denote by M h>ɛ, M h ɛ the preimages of (ɛ, ) and [ɛ, ) under h : M R, and denote by M h ɛ the manifold which is obtained by collapsing the boundary h 1 (ɛ) of M h ɛ along the orbits of the S 1 action. Theorem 1.3. (E. Lerman). There is a natural symplectic structure ω ɛ on M h ɛ such that the restriction of ω ɛ to M h>ɛ M h ɛ equals ω. Proof. Consider the symplectic product (M C, ω ω 0 ) and the Hamiltonian S 1 action on it given by t (m, z) = (t m, e it z), m M, z C. The moment map is H(m, z) = h(m) 1 2 z 2. Observe the following identification H 1 (ɛ) = {(m, z) h(m) > ɛ, z = 2(h(m) ɛ)} {(m, 0) h(m) = ɛ} = M h>ɛ S 1 h 1 (ɛ). The theorem follows immediately from H 1 (ɛ)/s 1 = M h>ɛ h 1 (ɛ)/s 1 = M h ɛ, and that the symplectic structure ω ɛ on H 1 (ɛ)/s 1 equals ω when restricted to the open submanifold M h>ɛ.
3 LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY 3 Let δ > 0 and let M = {(z 1, z 2,, z n ) C n n j=1 z j 2 < δ} be the ball of radius δ in C n centered at the origin. Given with the standard symplectic structure ω 0, M admits a Hamiltonian S 1 action t (z 1, z 2,, z n ) = (e it z 1, e it z 2,, e it z n ), which has a moment map h(z 1, z 2,, z n ) = 1 n 2 j=1 z j 2. For any 0 < ɛ < δ 2, ɛ is a regular value of h such that the S 1 action on h 1 (ɛ) is free. As we explained earlier, M h ɛ is a symplectic blowup 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 ɛ on the blowup of the standard symplectic ball of radius δ equals the standard symplectic structure ω 0 when restricted to the shell region {(z 1, z 2,, z n ) C n 2ɛ < n j=1 z j 2 < 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 torus action. Let (M, ω) be a symplectic manifold with a Hamiltonian T n action, and let µ : M (t n ) be the moment map. Suppose ξ 0 t n generates a circle T 1 T n. Then the induced S 1 action on M has the moment map h = µ, ξ 0. Because the induced S 1 action commutes with the T n action, and both M h>ɛ and h 1 (ɛ) are T n  invariant, there is an induced T n action on M h>ɛ h 1 (ɛ)/s 1 = M h ɛ, which is also Hamiltonian. The corresponding moment map µ ɛ : M h ɛ (t n ) is induced from the restriction of µ on M h ɛ. Hence the image of µ ɛ in (t n ) is µ ɛ (M h ɛ ) = µ(m) {ξ (t n ) ξ, ξ 0 ɛ}. Example 1.4. (Equivariant blowingup at a fixed point). Hamiltonian T 2 action on CP 2 given by (t 1, t 2 ) [z 0, z 1, z 2 ] = [z 0, e it 1 z 1, e it 2 z 2 ] Consider the standard which is considered in Example 2.4 (1) of Lecture 4. Here we shall double the symplectic form used in Example 2.4 (1) of Lecture 4, and consequently the moment map changes to z 1 2 µ([z 0, z 1, z 2 ]) = ( z z z 2 2, z 2 2 z z z 2 2 ), and the image of µ is {(x 1, x 2 ) R 2 x 1 + x 2 1, x 1, x 2 0}. Now consider the S 1 action t [z 0, z 1, z 2 ] = [z 0, z 1, e it z 2 ], which has moment map h([z 0, z 1, z 2 ]) = z z z 2. The symplectic cut CP 2, 2 h 1 2 which is a symplectic blowingup of CP 2 at the fixed point [0, 0, 1], has an induced z 2 2
4 4 WEIMIN CHEN, UMASS, SPRING 07 Hamiltonian T 2 action. The image of the corresponding moment map is {(x 1, x 2 ) R 2 x 1 + x 2 1, x 1 0, 0 x }. Note that this is the same as the image of the moment map in Example 2.4 (4) in Lecture 4 (a T 2 action on a Hirzebruch surface). By Delzant s classification theorem, these two T 2 actions are equivalent. 2. Thurston s construction Recall that If M 1, M 2 are symplectic manifolds with symplectic structures ω 1, ω 2, then the product M 1 M 2 is also a symplectic manifold with a natural symplectic structure ω 1 ω 2. In this section we shall consider more generally constructing symplectic structures on a manifold which is a fiber bundle whose fiber and base are both symplectic manifolds. The construction is due to Thurston. Recall that a smooth map π : M B between smooth manifolds is said to be a fiber bundle (or locally trivial fibration) with fiber F (also a smooth manifold) if there is an open cover {U α } of B and a collection of diffeomorphisms φ α : π 1 (U α ) U α F such that π = pr 1 φ α, where pr 1 : U α F U α is the projection. The maps φ α are called local trivializations. We denote by F b = π 1 (b) the fiber over b B and by φ α (b) : F b F the restriction of φ α to F b followed by the projection onto F. The maps φ βα : U α U β Diff(F ) defined by φ βα (b) = φ β (b) φ α (b) 1 are called the transition functions. The following construction is due to Thurston where M is assumed to be compact. Lemma 2.1. Let π : M B be a fiber bundle with fiber F. Suppose there is a closed 2form τ on M such that the restriction of τ to each fiber F b is a symplectic form on F b, and that the base B is a symplectic manifold with a symplectic form β. Then for large enough N > 0, the 2form ω N τ + Nπ β is a symplectic structure on M. Proof. Let 2m = dim M and 2n = dim B. Then dim F = 2m 2n. Note that (π β) k = 0 for any k > n. Hence n 1 ωn m = N n (τ m n (π β) n + N l n τ m l (π β) l ). Since M is compact, the lemma follows easily from the fact that τ m n (π β) n is a volume form on M. To see that τ m n (π β) n is a volume form, let p M be any point and let b = π(p). Pick a basis v 1,, v 2m 2n of T p F b and pick a basis u 1,, u 2n of T b B. Let u i T pm be a lift of u i, i.e., π (u i ) = u i, i = 1,, 2n. Then v 1,, v 2m 2n, u 1,, u 2n form a basis of T pm. With this understood, we have l=0 τ m n (π β) n (v 1,, v 2m 2n, u 1,, u 2n) = ±τ m n (v 1,, v 2m 2n ) (π β) n (u 1,, u 2n) = ±τ m n (v 1,, v 2m 2n ) β n (u 1,, u 2n ) because π β(v j, ) = β(π (v j ), ) = 0 for all j. Since the restriction of τ to F b is a symplectic form, τ m n (v 1,, v 2m 2n ) 0, and since β is a symplectic form on B,
5 LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY 5 β n (u 1,, u 2n ) 0 also. Hence τ m n (π β) n (v 1,, v 2m 2n, u 1,, u 2n) 0 and τ m n (π β) n is a volume form on M. In what follows, we shall give some criteria which give existence of the closed 2form τ in the above lemma. Recall that a fiber bundle π : M B with fiber F is called oriented if F is oriented and the transition functions φ βα (b), b B, all lie in the subgroup Diff + (F ) Diff(F ) of orientationpreserving diffeomorphsims of F. Note that each fiber F b has an induced orientation through the diffeomorphisms φ α (b) : F b F, which is independent of the choice of α. We call π : M B a symplectic fibration if F is a symplectic manifold with a symplectic form σ, and the transition functions φ βα (b) all lie in Symp(F, σ) Diff + (F ). In this case, each fiber F b has an induced symplectic form σ b = φ α (b) (σ), which is independent of the choice of α. For symplectic fibrations, the existence of the closed 2form τ can be reduced to a cohomological condition. Lemma 2.2. Suppose π : M B is a symplectic fibration with fiber (F, σ). If there exists a cohomology class a H 2 (M) such that i b a = [σ b], where i b : F b M is the inclusion, then there exists a closed 2form τ Ω 2 (M) such that i b τ = σ b, b B, [τ] = a H 2 (M). Proof. Pick a closed 2form τ 0 on M which represents a. Let {φ α : π 1 (U α ) U α F } be a set of local trivializations where {U α } is an open cover of B by balls. Let σ α Ω 2 (U α F ) be the pullback of σ Ω 2 (F ) by the projection to F. Then note that for any b B, i b (φ ασ α ) = σ b. Since i b a = [σ b], [τ 0 ] = a, and each U α is contractible, we see that τ 0 and φ ασ α are cohomologous on π 1 (U α ), therefore, there exists a 1form λ α such that φ ασ α τ 0 = dλ α, α. Pick a partition of unity {ρ α } subordinate to {U α }, i.e., α ρ α = 1 and suppρ α U α. We define τ = τ 0 + d((π ρ α )λ α ). α Note that d((π ρ α )λ α ) = d(π ρ α ) λ α + (π ρ α )dλ α and i b (d(π ρ α)) = 0. Hence i b τ = i b τ 0 + α ρ α (b)i b (dλ α) = α ρ α (b)i b (τ 0 + dλ α ) = α ρ α (b)i b φ ασ α = α ρ α (b)σ b = σ b.
6 6 WEIMIN CHEN, UMASS, SPRING 07 Finally, [τ] = [τ 0 ] = a H 2 (M), and the lemma is proved. According to the general theory of fiber bundles, given an oriented fiber bundle π : M B with fiber F, where F is a symplectic manifold with an orientationcompatible symplectic form σ, the question as whether π : M B is a symplectic fibration with fiber (F, σ) boils down to the understanding of the homotopy type of the space Diff + (F )/Symp(F, σ). Lemma 2.3. Let F be a compact, closed, oriented surface. Then Diff + (F )/Symp(F, σ) is contractible. Consequently, any oriented surface bundle over a smooth manifold is a symplectic fibration. Proof. Let T be the space of orientationcompatible symplectic structures on F which has the same total area of σ. Then T is contractible, with (σ, t) (1 t)σ + tσ, where σ T and 0 t 1, being the contraction of T to the point σ T. We will show that Diff + (F )/Symp(F, σ) is homeomorphic to T, from which the lemma follows. To see this, we consider the action of Diff + (F ) on T by g σ = g (σ ), g Diff + (F ), σ T. The action is obviously continuous with respect to appropriate topology on the two spaces. We claim it is transitive. To see this, let α 0, α 1 T be any two elements. Then α t = (1 t)α 0 + tα 1 T for 0 t 1, and [α t ] = [α 0 ] for all t. By Moser s stability theorem, there exists a smooth family of g t Diff + (F ) with g 0 = id, such that g t α t = α 0. Particular, g 1 α 1 = α 0, and the action of Diff + (F ) on T is transitive. This implies that Diff + (F )/Symp(F, σ) is homeomorphic to T as Symp(F, σ) is the isotropy subgroup at σ T. The next lemma gives a simple criterior for the hypothesis in Lemma 2.2. Lemma 2.4. Suppose π : M B is a symplectic fibration with fiber (F, σ). If c 1 (T F ) is a nonzero multiple of [σ], then there exists a cohomology class a H 2 (M) such that i b a = [σ b], where i b : F b M is the inclusion. Proof. Let E be the subbundle of T M where at each point the fiber consists of vectors which are tangent to the fiber of π : M B. Then the assumption that π : M B is a symplectic fibration implies that E is a symplectic vector bundle with the symplectic bilinear form which at p M equals σ b (p) where b = π(p). Note that i b E = T F b, therefore i b c 1(E) = c 1 (T F b ) = λ[σ b ] for some λ 0. We can take a = 1 λ c 1(E) H 2 (M). We obtain the following corollary. Corollary 2.5. Let π : M B be an oriented surface bundle over a symplectic manifold, where the genus of the fiber is not 1. Then M admits a symplectic structure such that each fiber is a symplectic submanifold.
7 LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY 7 Proof. For an oriented surface F, c 1 (T F )[F ] = 2 2g where g is the genus of F. Hence when g 1, c 1 (T F ) is a nonzero multiple of [σ] for any symplectic form σ on F. The corollary follows from Lemmas When the fiber F = T 2, we have Proposition 2.6. Suppose M is an oriented surface bundle over a symplectic manifold such that the fiber class [F b ] H 2 (M) is nonzero. Then M admits a symplectic structure such that each fiber is a symplectic submanifold. Proof. By Lemma 2.3, M is a symplectic fibration with fiber (F, σ). By Lemma 2.2 and then Lemma 2.1, it suffices to show that there exists a cohomology class a H 2 (M) such that a[f b ] 0, because H 2 (F ) = R, and this would imply that i b a = λ[σ b] for some λ 0. Since H 2 (M; R) = Hom(H 2 (M, R), R), and [F b ] H 2 (M) is nonzero, there exists an a H 2 (M) such that a[f b ] 0. Example 2.7. Let H : S 3 CP 1 be the Hopf fibration. Then π : S 3 S 1 CP 1 defined by (x, t) H(x) is a T 2 bundle over CP 1. Note that H 2 (S 3 S 1 ) = H 2 (S 3 ) H 0 (S 1 ) H 1 (S 3 ) H 1 (S 1 ) = 0 by the Kunneth formula, so that S 3 S 1 can not be symplectic. This shows that the conditions related to the existence of τ in Thurston s construction (cf. Lemma 2.1) are necessary. Example 2.8. (The KodairaThurston Manifold, cf. Example 1.16 in Lecture 1). Consider the 4manifold M = S 1 N where N is the nontrivial T 2 bundle over S 1 defined by N = [0, 1] T 2 /, where (0, x, y) (1, x + y, y). Naturally M is a T 2  bundle over T 2. We claim that the fiber class is nonzero in H 2 (M). This is equivalent to say that the fiber class is nonzero in H 2 (N). But this follows from the fact that N S 1 has a section [(t, 0, 0)], t [0, 1], which has a nonzero intersection product with the fiber. By Proposition 2.6, M is a symplectic manifold. M can not be Kähler, because H 1 (M) = R 3, which has an odd dimension. This is the first example of symplectic, nonkähler manifold. 3. Symplectic fiber sums Recall the symplectic neighborhood theorem that the symplectic structure on a regular neighborhood of a compact symplectic submanifold is determined by the induced symplectic structure on the symplectic submanifold and the isomorphism class of the normal bundle as a symplectic vector bundle, or equivalently as a complex vector bundle (cf. Lecture 3, Theorem 1.3). This theorem is the basis of a connected sum construction in symplectic category, called the symplectic fibre sum. We note that such a connected sum construction is not available in the holomorphic category. In fact, the symplectic fibre sum is the major technique of investigating the difference between the category of symplectic manifolds and that of Kähler manifolds. The following theorem, due to R. Gompf, is a simple, but an important, example.
8 8 WEIMIN CHEN, UMASS, SPRING 07 Theorem 3.1. (Gompf, 1995) Every finitely presentable group is the fundamental group of a compact symplectic 4manifold. It is known that every finitely presentable group is the fundamental group of a compact 4manifold. On the other hand, it was proved that there exist no algorithms which can be used to classify all the finitely presentable groups. As a consequence, we obtain the following complexity result about compact 4manifolds: there exist no algorithms which can be used to classify all the compact 4manifolds (topological or smooth). The above theorem of Gompf shows that the same holds for symplectic 4manifolds. On the other hand, it is known that there are severe constraints on the fundamental group of a Kähler surface. Gompf s theorem shows that the set of symplectic 4manifolds is significantly larger than that of Kähler surfaces. Now we describe the symplectic fiber sum. For j = 1, 2 let (M j, ω j ) be a symplectic manifold of dimension 2n and let Q j M j be a compact symplectic submanifold of dimension 2n 2 which has a trivial normal bundle. By the symplectic neighborhood theorem, a regular neighborhood of Q j in M j is symplectomorphic to (Q j B 2 (r 0 ), ω j dx dy) for some r 0 > 0, where B 2 (r 0 ) = {(x, y) R 2 x 2 + y 2 < r 2 0 }. Now suppose there exists a symplectomorphism φ : (Q 1, ω 1 ) (Q 2, ω 2 ). Then for any 0 < r 1 < r 0, there is a symplectomorphism Φ : (Q 1 A(r 1, r 0 ), ω 1 dx dy) (Q 2 A(r 1, r 0 ), ω 2 dx dy) lifting φ, where A(r 1, r 0 ) = {(x, y) R 2 r1 2 < x2 + y 2 < r0 2 }, which interchanges the inner boundary and the outer boundary of A(r 1, r 0 ). To define Φ, we let r, θ be the polar coordinates on B 2 (r 0 ). Then dx dy = rdr dθ = du dθ where u = 1 2 r2. With this understood, we define Φ : (q, u, θ) (φ(q), 1 2 (r2 0 + r 2 1) u, θ). We can construct a new symplectic manifold by taking out a regular neighborhood of Q 1, Q 2 and gluing the complements via Φ: (M 1 \ Q 1 B 2 (r 1 )) (M 2 \ Q 2 B 2 (r 1 ))/ where (q, u, θ) Φ(q, u, θ), (q, u, θ) Q 1 A(r 1, r 0 ). We denote it by M 1 # Q1 =Q 2 M 2. Remark 3.2. (1) The most useful case of this construction is when dim M j = 4. In this case, Q j is an embedded symplectic surface with selfintersection Q 2 j = 0. Note that by Moser s argument, there exists a symplectomorphism φ : (Q 1, ω 1 ) (Q 2, ω 2 ) if and only if Q 1, Q 2 have the same genus, and the total areas Q 1 ω 1 = Q 2 ω 2. However, the second condition is not essential because it can always be arranged by replacing of one of ω 1, ω 2 with an appropriate multiple. (2) The diffeomorphism type of the resulting manifold M 1 # Q1 =Q 2 M 2 depends on a number of things. First, it depends on the identification of a regular neighborhood of Q j to Q j B 2 (r 0 ). Such identifications are parametrized by the socalled framings, i.e., the set of trivializations of the trivial bundle Q j R 2 over Q j, which may be identified with H 1 (Q j ; Z) (i.e., the set of homotopy classes of maps from Q j to S 1 ).
9 LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY 9 Second, it depends on the isotopy class of φ : Q 1 Q 2. For example, when Q j = T 2 is a torus, the gluing data may be summarized into a 3 3 matrix a b 0 c d 0, m n 1 ( ) a b where SL(2, Z) parametrizes the isotopy classes of symplectomorphisms c d φ : T 2 T 2 and (m, n) H 1 (T 2 ; Z) = Z Z are the framings. (3) Note that the symplectic fibre sum construction requires the existence of a symplectomorphism A(r 1, r 0 ) A(r 1, r 0 ) of the annulus which interchanges the inner and outer boundaries. There exist no higher dimensional analogs because such a symplectomorphism could be used to glue two higher dimensional balls to obtain a symplectic S 2k for some k > 1, which we know does not exist. Thus the symplectic fiber sum construction can only be performed along a symplectic submanifold of codimension 2. (4) Note that in the gluing region Q 1 A(r 1, r 0 ) in a symplectic fiber sum M 1 # Q1 =Q 2 M 2, there is a free Hamiltonian S 1 action which acts trivially on the Q 1 factor and acts as complex multiplication on the A(r 1, r 0 ) factor. Lerman s symplectic cutting, when applied in this setting, allows one to undo the symplectic fiber sum, i.e., producing M 1, M 2 from M 1 # Q1 =Q 2 M 2. This is where the name symplectic cutting was coming from, i.e., it gives an inverse of symplectic gluing. In the remaining of this section we shall explain the basic ideas of the proof of Gompf s theorem (i.e. Theorem 3.1) with a simple example. A key ingredient is the following fact. Lemma 3.3. There exists a compact symplectic 4manifold V with an embedded symplectic torus T of selfintersection T 2 = 0 such that the complement of a regular neighborhood of T in V is simplyconnected. Proof. The manifold V will be the Kähler surface which is CP 2 blown up at 9 points, and T will be the proper transform of a smooth cubic curve in CP 2. More precisely, take two generic cubic polynomials P 1, P 2 such that the zeroes {P 1 = 0}, {P 2 = 0} CP 2 are smooth curves which intersect transversely at 9 distinct points. For any λ = [a, b] CP 1, the cubic curve {P λ ap 1 + bp 2 = 0} CP 2 contains all of the 9 points where P 1, P 2 intersect. Since for distinct λ, λ the cubic curves {P λ = 0}, {P λ = 0} are distinct, and since their intersection product is 9, it follows that {P λ = 0} and {P λ = 0} intersect only at these 9 points, and furthermore, the intersection is transversal. It follows that CP 2 is the union of these cubic curves, and the complex surface obtained by blowing up at these 9 points is a disjoint union of the proper transform of these cubic curves in CP 2, which is parametrized by CP 1. A generic member is a smoothly embedded torus of selfintersection 0. Since the blow up of an algebraic surface is still an algebraic surface, we see in particular that V is Kähler. To see that the complement of a regular neighborhood of T in V is simplyconnected, we use the VanKampen theorem. To this end, we denote by ν(t ) a regular neighborhood of T in V. Then V is the union of the complement V \ ν(t ) and ν(t )
10 10 WEIMIN CHEN, UMASS, SPRING 07 along a 3torus ν(t ). We pick a base point x 0 ν(t ). First, we observe that the class of the meridian of T in π 1 (V \ ν(t ), x 0 ) is zero because the meridian bounds an embedded disc in V \ ν(t ). To see this, recall that V is CP 2 blown up at 9 points where the family of cubic curves intersect transversely, and that T is the proper transform of a fixed smooth cubic. In particular, the exceptional curve at any of the blown up point intersects T transversely, so that the part of the exceptional curve in V \ ν(t ) is an embedded disc bounded by the meridian of T. With this understood, now observe that there is a homomorphism π 1 (ν(t ), x 0 ) π 1 (V \ ν(t ), x 0 ) such that the natural homomorphism induced by inclusion π 1 ( ν(t ), x 0 ) π 1 (ν(t ), x 0 ) followed by this homomorphism equals the natural homomorphism induced by inclusion π 1 ( ν(t ), x 0 ) π 1 (V \ ν(t ), x 0 ). It follows from the VanKampen theorem that there exists a homomorphism π 1 (V, x 0 ) π 1 (V \ ν(t ), x 0 ), such that the natural homomorphism induced by inclusion π 1 (V \ ν(t ), x 0 ) π 1 (V, x 0 ) followed by this homomorphism equals the identity on π 1 (V \ ν(t ), x 0 ). This implies that V \ ν(t ) is simplyconnected because V is simplyconnected. Now suppose X is a symplectic 4manifold with an embedded symplectic torus T of selfintersection 0. Let Y V # T =T X be the symplectic fiber sum. Then by VanKampen theorem, π 1 (Y, x 0 ) is obtained from π 1 (X, x 0 ) by setting the free loops contained in T nullhomotopic, for any x 0 X. Example 3.4. In this example we will illustrate how to construct a compact symplectic 4manifold with fundamental group Z, using the symplectic fiber sum construction. To this end, we consider the symplectic 4manifold X, where X = S 1 S 1 S 1 S 1. The symplectic structure on X is ω = dθ 1 dθ 2 + dθ 3 dθ 4 + dθ 2 dθ 3, where θ j, j = 1, 2, 3, 4, is the angular coordinate on the jth copy of S 1 in X. There are two disjoint, embedded symplectic tori in X: T 1 = {1} {1} S 1 S 1, and T 2 = { 1} S 1 S 1 {1}. The symplectic fiber sum Y = V # T T1 X# T2 =T V has a fundamental group which is obtained from π 1 (X) by setting the jth copy of S 1 in X nullhomotopic, for j = 2, 3, 4. Clearly π 1 (Y ) = Z. We remark that Y can not be Kähler because the first Betti number b 1 (Y ) = 1 which is odd. References [1] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs, 2nd edition, Oxford Univ. Press, 1998.
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