1. Symplectic geometry review

Let M2n be a manifold with symplectic form ω ∈ Ω2(M), i.e. satisfying dω = 0 and ωn 6= 0.On R2n = Rnq × Rnp = T ∗Rn we have the standard symplectic form

ω0 =n∑k=1

dqk ∧ dpk.

By the following theorem, all symplectic geometries are locally standard:

Theorem 1.1 (Darboux’s theorem). For all x ∈ (M,ω) there exists a chart (U, φ), x ∈ U suchthat φ(x) = 0 ∈ Rn 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.Symplectomorphisms are area-preserving maps.n = 2: M4 - lots of cool stuff happens in 4-dimensions: can we have exotic symplectic R4?

Answer: Yes! But...

Theorem 2.1 (Gromov). Let (M4, ω) have π2(M) = 0. Suppose that

(M4\KM , ω) ∼=symp

(R4\KR4 , ω0)

where KM ⊂⊂M4 and KR4 ⊂⊂ R4. Then

(M4, ω) ∼=symp

(R4, ω0)

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 thesame dimension. Can we embed symplectically?

(M2n1 , ω1) →

symp(M2n

2 , ω2)

Certainly we need to have∫M2

ωn2 = Area(M2n2 ) ≥ Area(M2n

1 ) =

∫M1

ωn2

but this is not sufficient:

Theorem 2.2 (Gromov non-squeezing). (B2nr , ω0) → (B2

R × R2n−2, 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 symplecticforms is very different to studying volume forms.

2.3. Application 3: Foliations of CP2. An example of a symplectic 4-fold is

(CP2, ωFS)

Can we characterise this manifold? CP2 is foliated by embedded S2s which meet at one point:

• ωFS |S2 6= 0, i.e. symplectic S2s.• [S2] · [S2] = 1, i.e. self-intersection number is 1.

Theorem 2.3 (Gromov-McDuff). Let (M4, ω) be compact, connected and

• there exists C ⊂M symplectic s.t. C = S2, ω|C 6= 0, [C] · [C] = 1, and• there does not exist symplectically embedded S2 in M s.t. [S2] · [S2] = −1.

Then (M4, ω) ∼=symp

(CP2, cωFS) for some c > 0.

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3. Strategy

• Start with one pseudoholomorphic curve C• Look at moduli space 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= ∅.• Show M = ∂−1

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

Consider f : R2 → R2, where f = (u, v) with u, v : R2 → R. Define the matrices

j =

(0 1−1 0

), J =

(0 −11 0

).

Then we calculate that

J df j =

(0 −11 0

)(ux uyvx vy

)(0 1−1 0

)=

(−vy vxuy −ux

)and so if

df + J df j =

(−vy + ux vx + uyuy + vx −ux + vy

)= 0

i.e. if J df = df j, the Cauchy-Riemann equations are satisfied and we conclude that f isholomorphic. The converse also holds.

Now generalise the range R2 to R2n = Cn and define

∂Jf = 12(df + J df j)

(where we have extended J to a 2n-dimensional square matrix in the obvious way.) Thendf + J df j = 0 if and only if f : C→ Cn is a holomorphic curve. So we have a condition forf to be holomorphic.

Now more generally on a symplectic manifold (M,ω) we need to find an almost complexstructure, i.e. J ∈ End(TM) with J2 = − 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 setof ω-compatible (or ω-tame) J . Then J is non-empty and contractible. Choose J ∈ J , andchoose conformal coordinates s+ it on (Σ, j). Then

∂Jfloc= 1

2(∂sf + J∂tf)ds+ 12(∂tf − J∂sf)dt.

These are the non-linear Cauchy-Riemann equations since J is not fixed but rather dependenton f . Now we say that

f : (Σ, j)→ (M,ω, J)

is pseudoholomorphic (or J-holomorphic) if and only if ∂Jf = 0.

5. Energy of a pseudoholomorphic curve

The energy of f is defined as

E(f) =1

2

∫Σ|df |2 volΣ =

∫Σ|∂Jf |2 volΣ +

∫Σf∗ω.

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 intheir homology class.

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6. Moduli space of a pseudoholomorphic curve

Fix A ∈ H2(M). The moduli space of a pseudoholomorphic curve is then

M = f : (S2, j)→ (M,ω, J) | ∂Jf = 0, f∗[S2] = A/G

where G = PSL(2,C) is the space of Mobius transformations acting on S2. Note that in generalM is non-compact:

Example 6.1. Let M = CP2 and, considering S2 = CP1, define the map fm : CP1 → CP2 by

fm([z1 : z2]) = [z21 : z2

2 : mz1z2].

Then

fm(CP1) =

[z1 : z2 : z3] : z1z2 = z23/m

2

→[z1 : z2 : z3] : z1z2 = 0 as m→∞,

which is the union of two CP1s but is not a map from CP1.Now take some Mobius transformation φm ∈ G such that

φm([z1 : z2]) = [z1 : mz2].

Reparametrise by this Mobius transformation:

fm φm([z1 : z2]) = fm([z1 : mz2]) = [z21/m

2 : z22 : z1z2]

Now we find that

fm φm(CP1)→ [0 : z1 : z2]which is just a single CP1! (The same procedure with φm−1 would give us the other CP1.)

Theorem 6.2 (Gromov compactness theorem for pseudoholomorphic curves). Let Jn ∈ J (forω-tame) with Jn → J ∈ J in C∞. Let fn : (S2, j) → (M,ω, Jn) be Jn-holomorphic with auniform bound on the energy, i.e. supn(En(fn)) < ∞. Then there exists a subsequence of fnconverging to a J-stable map (f, z).

What is (f, z)? It is a bubble tree. In the above example, we got a pair of spheres. Letus briefly (and without any diagrams since I cannot be bothered to texify them) describe thenotation we will use for bubble trees. Consider the bubble tree to be a tree T comprised ofvertices α (the centres of each sphere) connected by edges αEβ (representing the nodal pointszαβ where two spheres with centres α and β touch). Define the subtree Tαβ to be the verticesand edges connected to β which do not contain the edge αEβ.

So (f, z) is a collection

(fαα∈T , zαβαEβ)

where fα : (S2, j)→ (M,ω, J) are J-holomorphic and zαβ ∈ S2 are nodal points, such that

(a) αEβ =⇒ fα(zαβ) = fβ(zβα),

(b) α ∈ T =⇒ zαβ 6= zαβ′

if β 6= β′,(c) fα(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 anequality.

Question: What does fn → (f, z) really mean? It’s Gromov convergence. Take E(f) =∑α∈T E(fα) and Mαβ(f) =

∑γ∈Tαβ E(fγ) (so that Mαβ gives the energy of the subtree Tαβ.

Then we have the following definition:

Definition 6.3. fn → (f, z) means that there exists φαnα∈T , φαn ∈ G such that

• (Map) For all α ∈ T , fn φαnC∞−→ fα on S2\Zα. (i.e. up to reparametrisation, fn → f

away from nodal points.)

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• (Energy) There is no energy lost in the limit at a nodal point zαβ:

Mαβ(f) = limε→0

limn→∞

E(fαn |Bε(zαβ)

)• (Rescaling) Let αEβ be an edge. Then

(φαn)−1 φβnC∞−→ zαβ

on S2\zβα.

Department of Mathematics, University College London, Gower Street, London, WC1E 6BT,United Kingdom

E-mail address: christopher.evans.13@ucl.ac.uk