is given by ψt = z ’1 —¦ φt —¦ π. Since ψt identi¬es the symplectic structure „¦t with „¦0 on

ˆ

the annulus “overlap”, it follows that M is symplectic. This is Gromov™s symplectic blow

up procedure. Note that it can be e¬ected in such a way that the symplectic structure on

M \ {p} is not disturbed outside of an arbitrarily small ball around p. Note also that there

is a parameter involved, and that the symplectic structure is certainly not unique.

This is only describes a simple case. However, Gromov has shown how any compact

symplectic submanifold S 2k of M 2n can be blown up to become a symplectic “hypersurface”

ˆ ˆ

S in a new symplectic manifold M which has the property that M \ S is di¬eomorphic to

M \ S.

ˆˆ

L.9.4 156

The basic idea is the same as what we have already done: First, one mimics the

topological operations which would be performed if one were blowing up a complex sub-

manifold of a complex manifold. Thus, the submanifold S gets “replaced” by the complex

projectivization S = PNS of a complex normal bundle. Second, one shows how to de¬ne

ˆ

a symplectic structure on the resulting smooth manifold which can be made to agree with

the old structure outside of an arbitrarily small neighborhood of the blow up.

The details in the general case are somewhat more complicated than the case of

blowing up a single point, and Dusa McDu¬ ([McDu¬ 1984]) has written out a careful

construction. She has also used the method of blow ups to produce an example of a simply

connected compact symplectic manifold which has no K¨hler structure.

a

Hard Techniques in Symplectic Manifolds.

(Pseudo-) holomorphic curves. I begin with a fundamental de¬nition.

De¬nition 2: Let M 2n be a smooth manifold and let J : T M ’ T M be an almost

complex structure on M. For any Riemann surface Σ, we say that a map f: Σ ’ M is

J -holomorphic if f (± v) = J f (v) for all v ∈ T Σ.

Often, when J is clear from context, I will simply say “f is holomorphic”. Several

authors use the terminology “almost holomorphic” or “pseudo-holomorphic” for this con-

cept, reserving the word “holomorphic” for use only when the almost complex structure

on M is integrable to an actual complex structure. This distinction does not seem to be

particularly useful, so I will not maintain it.

It is instructive to see what this looks like in local coordinates. Let z = x+± y be a local

holomorphic coordinate on Σ and let w: U ’ R2n be a local coordinate on M. Then there

exists a matrix J(w) of functions on w(U) ‚ R2n which satis¬es w (J v) = J w(p) w (v) for

all v ∈ Tp U. This matrix of functions satis¬es the relation J2 = ’I2n . Now, if f: Σ ’ M

is holomorphic and carries the domain of the z-coordinate into U, then F = w —¦ f is easily

seen to satisfy the ¬rst order system of partial di¬erential equations

‚F ‚F

= J(F ) .

‚y ‚x

Since J2 = ’I2n , it follows that this is a ¬rst-order, elliptic, determined system of partial

di¬erential equations for F . In fact, the “principal symbol” of these equations is the same

as that for the Cauchy-Riemann equations. Assuming that J is su¬ciently regular (C ∞ is

su¬cient and we will always have this) there are plenty of local solutions. What is at issue

is the nature of the global solutions.

A parametrized holomorphic curve in M is a holomorphic map f: Σ ’ M. Sometimes

we will want to consider unparametrized holomorphic curves in M, namely equivalence

classes [Σ, f] of holomorphic curves in M where (Σ1 , f1 ) is equivalent to (Σ2 , f2 ) if there

exists a holomorphic map φ: Σ1 ’ Σ2 satisfying f1 = f2 —¦ φ.

L.9.5 157

We are going to be particularly interested in the space of holomorphic curves in M.

Here are some properties that hold in the case of holomorphic curves in actual complex

manifolds and it would be nice to know if they also hold for holomorphic curves in almost

complex manifolds.

Local Finite Dimensionality. If Σ is a compact Riemann surface, and f: Σ ’ M

is a holomorphic curve, it is reasonable to ask what the space of “nearby” holomorphic

curves looks like. Because the equations which determine these mappings are elliptic and

because Σ is compact, it follows without too much di¬culty that the space of nearby

holomorphic curves is ¬nite dimensional. (We do not, in general know that it is a smooth

manifold!)

Intersections. A pair of distinct complex curves in a complex surface always inter-

sect at isolated points and with positive “multiplicity.” This follows from complex analytic

geometry. This result is extremely useful because it allows us to derive information about

actual numbers of intersection points of holomorphic curves by applying topological inter-

section formulas. (Usually, these topological intersection formulas only count the number

of signed intersections, but if the surfaces can only intersect positively, then the topological

intersection numbers (counted with multiplicity) are the actual intersection numbers.)

Kahler Area Bounds. If M happens to be a K¨hler manifold, with K¨hler form

a a

¨

„¦, then the area of the image of a holomorphic curve f: Σ ’ M is given by the formula

f — („¦).

Area f(Σ) =

Σ

In particular, since „¦ is closed, the right hand side of this equation depends only on the

homotopy class of f as a map into M. Thus, if (Σt , ft ) is a continuous one-parameter

family of closed holomorphic curves in a K¨hler manifold, then they all have the same

a

area. This is a powerful constraint on how the images can behave, as we shall see.

Now the ¬rst two of these properties go through without change in the case of almost

complex manifolds.

In the case of local ¬niteness, this is purely an elliptic theory result. Studying the

linearization of the equations at a solution will even allow one to predict, using the Atiyah-

Singer Index Theorem, an upper bound for the local dimension of the moduli space and,

in some cases, will allow us to conclude that the moduli space near a given closed curve is

actually a smooth manifold (see below).

As for pairs of complex curves in an almost complex surface, Gromov has shown that

they do indeed only intersect in isolated points and with positive multiplicity (unless they

have a common component, of course). Both Gromov and Dusa McDu¬ have used this

fact to study the geometry of symplectic 4-manifolds.

The third property is only valid for K¨hler manifolds, but it is highly desirable. The

a

behaviour of holomorphic curves in compact K¨hler manifolds is well understood in a large

a

part because of this area bound. This motivated Gromov to investigate ways of generalizing

this property.

L.9.6 158

Symplectic Tamings. Following Gromov, we make the following de¬nition.

De¬nition 3: A symplectic form „¦ on M tames an almost complex structure J if it is

J -positive, i.e., if it satis¬es „¦(v, J v) > 0 for all non-zero tangent vectors v ∈ T M.

The reader should be thinking of K¨hler geometry. In that case, the symplectic form

a

„¦ and the complex structure J satisfy „¦(v, J v) = v, v > 0. Of course, this generalizes to

the case of an arbitrary almost K¨hler structure.

a

Now, if M is compact and „¦ tames J , then for any Riemannian metric g on M (not

necessarily compatible with either J or „¦) there is a constant C > 0 so that

|v § J v| ¤ C „¦(v, J v)

where |v§J v| represents the area in Tp M of the parallelogram spanned by v and J v in Tp M

(see the Exercises). In particular, it follows that, for any holomorphic curve f: Σ ’ M,

we have the inequality

f — („¦).

Area f(Σ) ¤ C

Σ

Just as in the K¨hler case, the integral on the right hand side depends only on the homotopy

a

class of f. Thus, if an almost complex structure can be tamed, it follows that, in any metric

on M, there is a uniform upper bound on the areas of the curves in any continuous family

of compact holomorphic curves in M.

C C

Example: Let N3 denote the complex Heisenberg group. Thus, N3 is the complex Lie

group of matrices of the form «

1xz

g = 0 1 y.

001

C

Let “ ‚ N3 be the subgroup all of whose entries belong to the ring of Gaussian integers

Z[±].

C

Let M = N3 /“. Then M is a compact complex 3-manifold. I claim that the complex

structure on M cannot be tamed by any symplectic form.

To see this, consider the right-invariant 1-form

«

0 ω1 ω3

= 0 ω2 .

dg g ’1 0

0 0 0

Since they are right-invariant, it follows that the complex 1-forms ω1 , ω2 , ω3 are also well-

de¬ned on M. De¬ne the metric G on M to be the quadratic form

G = ω1 —¦ ω1 + ω2 —¦ ω2 + ω3 —¦ ω3 .

L.9.7 159

C

Now consider the holomorphic curve Y : C ’ N3 de¬ned by

«

1 00

Y (y) = 0 1 y.

0 01

Let ψ: C ’ M be the composition. It is clear that ψ is doubly periodic and hence de¬nes

an embedding of a complex torus into M. It is clear that the G-area of this torus is 1.

C

Now N3 acts holomorphically on M on the left (not by G-isometries, of course). We

can consider what happens to the area of the torus ψ(C) under the action of this group.

Speci¬cally, for x ∈ C, let ψx denote ψ acted on by left multiplication by the matrix

«

1 x0

0 1 0.

0 01

Then, as the reader can easily check, we have

2

—

ψx (G) = 1 + |x|2 ) dy .

Thus, the G-area of the torus ψx (C) goes o¬ to in¬nity as x tends to in¬nity. Obviously,

there can be no taming of the complex manifold M. (In particular, M cannot carry a

K¨hler structure compatible with its complex structure.)

a

Gromov™s Compactness Theorem. In this section, I want to discuss Gromov™s

approach to compactifying the connected components of the space of unparametrized holo-

morphic curves in M.

Example: Before looking at the general case, let us look at what happens in a very

familiar case: The case of algebraic curves in CP2 with its standard Fubini-Study metric

and symplectic form „¦ (normalized so as to give the lines in CP2 an area of 1).

Since this is a K¨hler metric, we know that the area of a connected one-parameter

a

family of holomorphic curves (Σt , ft ) in CP2 is constant and is equal to an integer d =

—

Σt ft („¦), called the degree. To make matters as simple as possible, let me consider the

curves degree by degree.

d = 0. In this case, the “curves” are just the constant maps and (in the unparametrized

case) clearly constitute a copy of CP2 itself. Note that this is already compact.

d = 1. In this case, the only possibility is that each Σt is just CP1 and the holo-

morphic map ft must be just a biholomorphism onto a line in CP2 . Of course, the space

of lines in CP2 is compact, just being a copy of the dual CP2 . Thus, the space M1 of

unparametrized holomorphic curves in CP2 is compact. Note, however, that the space H1

of holomorphic maps f: CP1 ’ CP2 of degree 1 is not compact. In fact, the ¬bers of the

natural map H1 ’ M1 are copies of Aut(CP1 ) = PSL(2, C).

L.9.8 160

d = 2. This is the ¬rst really interesting case. Here again, degree 2 (connected,

parametrized) curves in CP2 consist of rational curves, and the images f: CP1 ’ CP2

are of two kinds: the smooth conics and the double covers of lines. However, not only is

this space not compact, the corresponding space of unparametrized curves is not compact

either, for it is fairly clear that one can approach a pair of intersecting lines as closely as

one wishes.

In fact, the reader may want to contemplate the one-parameter family of hyperbolas

xy = »2 as » ’ 0. If we choose the parametrization

f» (t) = [t, »t2 , »] = [1, x, y],

—

then the pullback ¦» = f» („¦) is an area form on CP1 whose total integral is 2, but (and the

reader should check this), as » ’ 0, the form ¦» accumulates equally at the points t = 0

and t = ∞ and goes to zero everywhere else. (See the Exercises for a further discussion.)

Now, if we go ahead and add in the pairs of lines, then this “completed” moduli

space is indeed compact. It is just the space of non-zero quadratic forms in three variables

(irreducible or not) up to constant multiples. It is well known that this forms a CP5 .

In fact, a further analysis of low degree mappings indicates that the following phe-

nomena are typical: If one takes a sequence (Σk , fk ) of smooth holomorphic curves in CP2 ,

then after reparametrizing and passing to a subsequence, one can arrange that the holo-

morphic curves have the property that, at a ¬nite number of points p± ∈ Σ, the induced

k

—

metric fk („¦) on the surface goes to in¬nity and the integral of the induced area form on

a neighborhood of each of these points approaches an integer while along a ¬nite number

of loops γi , the induced metric goes to zero.

The ¬rst type of phenomenon is called “bubbling”, for what is happening is that

a small 2-sphere is in¬‚ating and “breaking o¬” from the surface and covering a line in

CP2 . The second type of phenomenon is called “vanishing cycles”, a loop in the surface

is literally contracting to a point. It turns out that the limiting object in CP2 is a union

of algebraic curves whose total degree is the same as that of the members of the varying

family.

Thus, for CP2 , the moduli space Md of unparametrized curves of degree d has a

compacti¬cation Md where the extra points represent decomposable or degenerate curves

with “cusps”.

Other instances of this “bubbling” phenomena have been discovered. Sacks and Uh-

lenbeck showed that when one wants to study the question of representing elements of

π2 (X) (where X is a Riemannian manifold) by harmonic or minimal surfaces, one has

to deal with the possibility of pieces of the surface “bubbling o¬” in exactly the fashion

described above.

More recently, this sort of phenomenon has been used in “reverse” by Taubes to

construct solutions to the (anti-)self dual Yang-Mills equations over compact 4-manifolds.

With all of this evidence of good compacti¬cations of moduli spaces in other problems,

Gromov had the idea of trying to compactify the connected components of the “moduli

L.9.9 161

space” M of holomorphic curves in a general almost complex manifold M. Since one would

certainly expect the area function to be continuous on each compacti¬ed component, it