1. Symplectic review Let M 2n be a manifold with symplectic form ω ∈ Ω2(M), i.e. satisfying dω = 0 and ωn 6= 0. 2n n n ∗ n On R = Rq × Rp = T R we have the standard symplectic form n X ω0 = dqk ∧ dpk. k=1 By the following theorem, all symplectic are locally standard: Theorem 1.1 (Darboux’s theorem). For all x ∈ (M, ω) there exists a chart (U, φ), x ∈ U such n ∗ that φ(x) = 0 ∈ R and φ ω0 = ω. So there are no local invariants and all interesting questions are global.

2. Applications of pseudoholomorphic curves 2.1. Application 1: Exotic spaces. n = 1: Manifold with a volume form ω is symplectic. are area-preserving maps. 4 4 n = 2: M - lots of cool stuff happens in 4-dimensions: can we have exotic symplectic R ? Answer: Yes! But... 4 Theorem 2.1 (Gromov). Let (M , ω) have π2(M) = 0. Suppose that 4 ∼ 4 (M \KM , ω) = (R \K 4 , ω0) symp R 4 4 where KM ⊂⊂ M and KR4 ⊂⊂ R . Then 4 ∼ 4 (M , ω) = (R , ω0) symp One way to interpret this is through contact geometry. This can be proven with pseudoholo- morphic curves. 2.2. Application 2: Non-squeezing. Now suppose we have two symplectic manifolds of the same dimension. Can we embed symplectically? 2n 2n (M , ω1) ,→ (M , ω2) 1 symp 2 Certainly we need to have Z Z n 2n 2n n ω2 = Area(M2 ) ≥ Area(M1 ) = ω2 M2 M1 but this is not sufficient: 2n 2 2n−2 Theorem 2.2 (Gromov non-squeezing). (Br , ω0) ,→ (BR × R , ω0) if and only if r ≤ R. If you replace “symplectic form” with “volume form” then the volume being smaller is suffi- cient for an embedding to exist. This was the first time that people realised studying symplectic forms is very different to studying volume forms. 2 2.3. Application 3: Foliations of CP . An example of a symplectic 4-fold is 2 (CP , ωFS) 2 2 Can we characterise this manifold? CP is foliated by embedded S s which meet at one point: 2 • ωFS|S2 6= 0, i.e. symplectic S s. • [S2] · [S2] = 1, i.e. self-intersection number is 1. Theorem 2.3 (Gromov-McDuff). Let (M 4, ω) be compact, connected and 2 • there exists C ⊂ M symplectic s.t. C = S , ω|C 6= 0, [C] · [C] = 1, and • there does not exist symplectically embedded S2 in M s.t. [S2] · [S2] = −1. 4 ∼ 2 Then (M , ω) = (CP , cωFS) for some c > 0. symp 1 2

3. Strategy • Start with one pseudoholomorphic curve C • Look at M of C and show it’s nice. • This allows us to build a family of pseudoholomorphic curves filling out M. What do we mean by “look at the moduli space”? • Show M= 6 ∅. ¯−1 ¯ • Show M = ∂J (0)/G, where ∂J is the Cauchy-Riemann operator. ¯ • Local structure: The Fredholm theory for ∂J and transversality tells us what’s happening locally. ¯ • Global structure: Compactness theory for ∂J leads to M¯ , the compactification of M. 4. Pseudoholomorphic curves 2 2 2 Consider f : R → R , where f = (u, v) with u, v : R → R. Define the matrices  0 1   0 −1  j = ,J = . −1 0 1 0 Then we calculate that  0 −1   u u   0 1   −v v  J ◦ df ◦ j = x y = y x 1 0 vx vy −1 0 uy −ux and so if  −v + u v + u  df + J ◦ df ◦ j = y x x y = 0 uy + vx −ux + vy i.e. if J ◦ df = df ◦ j, the Cauchy-Riemann equations are satisfied and we conclude that f is holomorphic. The converse also holds. 2 2n n Now generalise the range R to R = C and define ¯ 1 ∂J f = 2 (df + J ◦ df ◦ j) (where we have extended J to a 2n-dimensional square matrix in the obvious way.) Then n df + J ◦ df ◦ j = 0 if and only if f : C → C is a holomorphic curve. So we have a condition for f to be holomorphic. Now more generally on a (M, ω) we need to find an almost complex structure, i.e. J ∈ End(TM) with J 2 = − Id. We say that J is • ω-compatible iff g(·, ·) = ω(·,J·) for some Riemannian metric g. • ω-tame iff ω(X,JX) > 0 for all X. (This is not ω-compatible since it isn’t necessarily symmetric, though it could be symmetrised.) Consider now a map f :Σ → (M, ω) where (Σ, j) is a complex curve. Denote by J the set of ω-compatible (or ω-tame) J. Then J is non-empty and contractible. Choose J ∈ J , and choose conformal coordinates s + it on (Σ, j). Then ¯ loc 1 1 ∂J f = 2 (∂sf + J∂tf)ds + 2 (∂tf − J∂sf)dt. These are the non-linear Cauchy-Riemann equations since J is not fixed but rather dependent on f. Now we say that f : (Σ, j) → (M, ω, J) ¯ is pseudoholomorphic (or J-holomorphic) if and only if ∂J f = 0. 5. Energy of a pseudoholomorphic curve The energy of f is defined as Z Z Z 1 2 ¯ 2 ∗ E(f) = |df| volΣ = |∂J f| volΣ + f ω. 2 Σ Σ Σ The first term is greater than or equal to 0, with equality if and only if f is is pseudoholomorphic. The second term is purely topological. Hence pseudoholomorphic curves minimise energy in their homology class. 3

6. Moduli space of a pseudoholomorphic curve

Fix A ∈ H2(M). The moduli space of a pseudoholomorphic curve is then 2 ¯ 2 M = {f :(S , j) → (M, ω, J) | ∂J f = 0, f∗[S ] = A}/G 2 where G = PSL(2, C) is the space of M¨obiustransformations acting on S . Note that in general M is non-compact: 2 2 1 1 2 Example 6.1. Let M = CP and, considering S = CP , define the map fm : CP → CP by 2 2 fm([z1 : z2]) = [z1 : z2 : mz1z2]. Then 1  2 2 fm(CP ) = [z1 : z2 : z3]: z1z2 = z3/m → {[z1 : z2 : z3]: z1z2 = 0} as m → ∞, 1 1 which is the union of two CP s but is not a map from CP . Now take some M¨obius transformation φm ∈ G such that

φm([z1 : z2]) = [z1 : mz2]. Reparametrise by this M¨obiustransformation: 2 2 2 fm ◦ φm([z1 : z2]) = fm([z1 : mz2]) = [z1/m : z2 : z1z2] Now we find that 1 fm ◦ φm(CP ) → {[0 : z1 : z2]} 1 1 which is just a single CP ! (The same procedure with φm−1 would give us the other CP .)

Theorem 6.2 (Gromov compactness theorem for pseudoholomorphic curves). Let Jn ∈ J (for ∞ 2 ω-tame) with Jn → J ∈ J in C . Let fn :(S , j) → (M, ω, Jn) be Jn-holomorphic with a uniform bound on the energy, i.e. supn(En(fn)) < ∞. Then there exists a subsequence of fn converging to a J- (f, z). What is (f, z)? It is a bubble tree. In the above example, we got a pair of spheres. Let us briefly (and without any diagrams since I cannot be bothered to texify them) describe the notation we will use for bubble trees. Consider the bubble tree to be a tree T comprised of vertices α (the centres of each sphere) connected by edges αEβ (representing the nodal points αβ z where two spheres with centres α and β touch). Define the subtree Tαβ to be the vertices and edges connected to β which do not contain the edge αEβ. So (f, z) is a collection α αβ ({f }α∈T , {z }αEβ) where f α :(S2, j) → (M, ω, J) are J-holomorphic and zαβ ∈ S2 are nodal points, such that (a) αEβ =⇒ f α(zαβ) = f β(zβα), (b) α ∈ T =⇒ zαβ 6= zαβ0 if β 6= β0, (c) f α(S2) = {∗} =⇒ #Zα ≥ 3, where Zα = {zαβ : αEβ} is the set of nodal points on the α-sphere. This final condition is stability. Then (f, z) is called a “stable J-holomorphic map of genus 0, modelled on the tree T .” In Sacks-Uhlenbeck, we found an inequality for the energy. Here however, we’ll obtain an equality. Question: What does fn → (f, z) really mean? It’s Gromov convergence. Take E(f) = P E(f α) and M (f) = P E(f γ) (so that M gives the energy of the subtree T . α∈T αβ γ∈Tαβ αβ αβ Then we have the following definition: α α Definition 6.3. fn → (f, z) means that there exists {φn}α∈T , φn ∈ G such that ∞ α C α 2 α • (Map) For all α ∈ T , fn ◦ φn −→ f on S \Z . (i.e. up to reparametrisation, fn → f away from nodal points.) 4

• (Energy) There is no energy lost in the limit at a nodal point zαβ:  α  Mαβ(f) = lim lim E fn |B (zαβ ) ε→0 n→∞ ε • (Rescaling) Let αEβ be an edge. Then α −1 β C∞ αβ (φn) ◦ φn −→ z on S2\{zβα}.

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