that [FA ] = 0. Then, in this case, µ’1 (0) is empty so the construction we made in the

example in the Lecture is vacuous. Here is how we can still get some information.

Fix any non-vanishing 2-form Ψ on Σ so that [Ψ] = [FA ]. Show that even though µ has

no regular values, Ψ is a non-trivial clean value of µ. Show also that, for any A ∈ µ’1 (Ψ),

the stabilizer GA ‚ Aut(P ) is a discrete (and hence ¬nite) subgroup of S 1 . Describe, as

fully as you can, the reduced space MΨ and its K¨hler structure. (Here, you will need to

a

keep in mind that Ψ is a ¬xed point of the coadjoint action of Aut(P ), so that reduction

away from 0 makes sense.)

8. Verify that Sp(n) is a connected Lie group of dimension 2n2 + n. (Hint: You will

¯

probably want to study the function f(A) = tAA = In .) Show that Sp(n) acts transitively

on the unit sphere S 4n’1 ‚ Hn de¬ned by the relation H(x, x) = 1. (Hint: First show

that, by acting by diagonal matrices in Sp(n), you can move any element of Hn into the

subspace Rn . Then note that Sp(n) contains SO(n).) By analysing the stabilizer subgroup

in Sp(n) of an element of S 4n’1 , show that there is a ¬bration

Sp(n’1) ’ Sp(n)

“

S 4n’1

and use this to conclude by induction that Sp(n) is connected and simply connected for

all n.

9. Prove Proposition 2. (Hint: First show how the maps Ri , Rj , and Rk de¬ne the

structure of a right H-module on V . Then show that V has a basis b1 , . . . , bn over H and

use this to construct an H-linear isomorphism of V with Hn . If you pick the basis ba

carefully, you will be done at this point. Warning: You must use the positive de¬niteness

of , !)

10. Show that if („¦1 , „¦2 , „¦3 ) is a hyperK¨hler structure on a real vector space V , with as-

a

sociated de¬ned maps Ri , Rj , and Rk and metric , , then („¦1 , Ri ) is a complex Hermitian

structure on V with associated metric , . Moreover, the C-valued 2-form Υ = „¦2 ’ ± „¦3 is

C-linear, i.e., Υ(Ri v, w) = ± Υ(v, w) for all v, w ∈ V . Show that Υ is non-degenerate on V

and hence that Υ de¬nes a (complex) symplectic structure on V (considered as a complex

vector space).

E.8.3 151

11. Let Sp(1) SU(2) act on Hn diagonally (i.e., as componentwise left multiplication n

copies of H). Compute the momentum mapping µ : Hn ’ sp(1) • sp(1) • sp(1) and show

that for all n ≥ 4, the map µ is surjective and that, for generic ξ ∈ sp(1) • sp(1) • sp(1),

we have Gξ = {±1}. In particular, for nearly all nonzero regular values of µ, the quotient

space Mξ = Gξ \µ’1 (ξ) has dimension 4n’9. Consequently, this quotient space is not even

K¨hler. (Hint: You may ¬nd it useful to recall that sp(1) = Im H R3 and that the

a

(co)adjoint action is identi¬able with SO(3) acting by rotations on R3 . In fact, you might

want to note that it is possible to identify sp(1) • sp(1) • sp(1) with R9 M3,3 (R) in such

a way that

µ(pq 1 u, · · · , pq n u) = R(p)µ(q 1 , · · · , q n )R(u)’1

¯ ¯

where R : Sp(1) ’ SO(3) is a covering homomorphism. Once this has been proved, you

can use facts about matrix multiplication to simplify your computations.) Now, again,

assuming that n ≥ 4, compute µ’1 (0), show that, once the origin in Hn is removed, 0 is

a regular value of µ and that Sp(1) acts freely on µ’1 (0). Can you describe the quotient

space? (You may ¬nd it helpful to note that SO(n) ‚ Sp(n) is the commuting subgroup

of Sp(1) embedded diagonally into Sp(n). What good is knowing this?)

12. Apply the hyperK¨hler reduction procedure to H2 with G = S 1 acting by the rule

a

q1 eimθ q 1

e·

iθ

= ,

q2 einθ q 2

where m and n are relatively prime integers. Determine which values of µ are clean and

describe the resulting complex surfaces and their hyperK¨hler structures.

a

E.8.4 152

Lecture 9:

The Gromov School of Symplectic Geometry

In this lecture, I want to describe some of the remarkable new information we have

about symplectic manifolds owing to the in¬‚uence of the ideas of Mikhail Gromov. The

basic reference for much of this material is Gromov™s remarkable book Partial Di¬erential

Relations.

The fundamental idea of studying complex structures “tamed by” a given symplectic

structure was developed by Gromov in a remarkable paper Pseudo-holomorphic Curves on

Almost Complex Manifolds and has proved extraordinarily fruitful. In the latter part of

this lecture, I will try to introduce the reader to this theory.

Soft Techniques in Symplectic Manifolds

Symplectic Immersions and Embeddings. Before beginning on the topic of

symplectic immersions, let me recall how the theory of immersions in the ordinary sense

goes.

Recall that the Whitney Immersion Theorem (in the weak form) asserts that any

smooth n-manifold M has an immersion into R2n . This result is proved by ¬rst immersing

M into some RN for N 0 and then using Sard™s Theorem to show that if N > 2n,

one can ¬nd a vector u ∈ R so that u is not tangent to f(M) at any point. Then the

N

projection of f(M) onto a hyperplane orthogonal to u is still an immersion, but now into

RN ’1 .

This result is not the best possible. Whitney himself showed that one could always im-

merse M n into R2n’1 , although “general position” arguments are not su¬cient to do this.

This raises the question of determining what the best possible immersion or embedding

dimension is.

One topological obstruction to immersing M n into Rn+k can be described as follows:

If f: M ’ Rn+k is an immersion, then the trivial bundle f — (T Rn+k ) = M — Rn+k can

be split into a direct sum f — (T Rn+k ) = T M • ν f where ν f is the normal bundle of the

immersion f. Thus, if there is no bundle ν of rank k over M so that T M • ν is trivial,

then there can be no immersion of M into Rn+k .

The remarkable fact is that this topological necessary condition is almost su¬cient. In

fact, we have the following result of Hirsch and Smale for the general immersion problem.

Theorem 1: Let M and N be connected smooth manifolds and suppose either that M

is non-compact or else that dim(M) < dim(N ). Let f: M ’ N be a continuous map, and

suppose that there is a vector bundle ν over M so that f — (T N ) = T M • ν. Then f is

homotopic to an immersion of M into N

Theorem 1 can be interpreted as an example of what Gromov calls the h-principle,

which I now want to describe.

L.9.1 153

The h-Principle. Let π: V ’ X be a surjective submersion. A section of π is, by

de¬nition, a map σ: X ’ V which satis¬es π —¦ σ = idX . Let J k (X, V ) denote the space

of k-jets of sections of V , and let π k : J k (X, V ) ’ X denote the basepoint or “source”

projection. Given any section s of π, there is an associated section j k (s) of π k which is

de¬ned by letting j k (s)(x) be the k-jet of s at x ∈ X. A section σ of π k is said to be

holonomic if σ = j k (s) for some section s of π.

A partial di¬erential relation of order k for π is a subset R ‚ J k (X, V ). A section s

of π is said to satisfy R if j k (s)(X) ‚ R. We can now make the following de¬nition:

De¬nition 1: A partial di¬erential relation R ‚ J k (X, V ) satis¬es the h-principle if, for

every section σ of π k which satis¬es σ(X) ‚ R, there is a one-parameter family of sections

σt (0 ¤ t ¤ 1) of π k which satisfy the conditions that σt (X) ‚ R for all t, that σ0 = σ,

and that σ1 is holonomic.

Very roughly speaking, a partial di¬erential relation satis¬es the h-principle if, when-

ever the “topological” conditions for a solution to exist are satis¬ed, then a solution exists.

For example, if X = M and V = M — N , where dim(N ) ≥ dim(M), then there is

an (open) subset R = Imm(M, N ) ‚ J 1 (M, M — N ) which consists of the 1-jets of graphs

of (local) immersions of M into N . What the Hirsch-Smale immersion theory says is that

Imm(M, N ) satis¬es the h-principle if either dim M = dim N and M has no compact

component or else dim M < dim N .

Of course, the h-principle does not hold for every relation R. The real question

is how to determine when the h-principle holds for a given R. Gromov has developed

several extremely general methods for proving that the h-principle holds for various partial

di¬erential relations R which arise in geometry. These methods include his theory of

topological sheaves and techniques like his method of convex integration. They generally

work in situations where the local solutions of a given partial di¬erential relation R are

easy to come by and it is mainly a question of “patching together” local solutions which

are fairly “¬‚exible”.

Gromov calls this collection of techniques “soft” to distinguish them from the “hard”

techniques, such as elliptic theory, which come from analysis and deal with situations where

the local solutions are somewhat “rigid”.

Here is a sample of some of the results which Gromov obtains by these methods:

Theorem 2: Let X 2n be a smooth manifold and let V ‚ Λ2 T — (M) denote the open

subbundle consisting of non-degenerate 2-forms ω ∈ Tx X. Let Z 1 (X, V ) ‚ J 1 (X, V )

denote the space of 1-jets of closed non-degenerate 2-forms on X. Then, if X has no

compact component, Z 1 (X, V ) satis¬es the h-principle.

In particular, Theorem 2 implies that a non-compact, connected X has a symplectic

structure if and only if it has an almost symplectic structure.

Note that this result is de¬nitely not true for compact manifolds. We have already seen

several examples, e.g., S 1 — S 3, which have almost symplectic structures but no symplectic

structures because they do not satisfy the cohomology ring obstruction. Gromov has asked

the following:

L.9.2 154

Question : If X 2n is compact and connected and satis¬es the condition that there exists

an element u ∈ HdR (X, R) which satis¬es un = 0, does Z 1 (X, V ) satisfy the h-principle?

2

The next result I want to describe is Gromov™s theorem on symplectic immersions.

This theorem is an example of a sort of “restricted h-principle” in that it is only required

to apply to sections σ which satisfy speci¬ed cohomological conditions.

First, let me make a few de¬nitions: Let (X, Ξ) and (Y, Ψ) be two connected symplectic

manifolds. Let S(X, Y ) ‚ J 1(X, X — Y ) denote the space of 1-jets of graphs of (local)

symplectic maps f: X ’ Y . i.e., (local) maps f: X ’ Y which satisfy f — (Ψ) = Ξ. Let

„ : S(X, Y ) ’ Y be the obvious “target projection”.

Theorem 3: If either X is non-compact, or dim(X) < dim(Y ), then any section σ of

S(X, Y ) for which the induced map s = „ —¦ σ: X ’ Y satis¬es the cohomological condition

s— ([Ψ]) = [Ξ] is homotopic to a holonomic section of S(X, Y ).

This result can be also stated as follows: Suppose that either X is non-compact or else

that dim(X) < dim(Y ). Let φ: X ’ Y be a smooth map which satis¬es the cohomological

condition φ— [Ψ] = [Ξ]. Suppose that there exists a bundle map f: T X ’ φ— (T Y ) which

is symplectic in the obvious sense. Then φ is homotopic to a symplectic immersion.

As an application of Theorem 3, we can now prove the following result of Narasimham

and Ramanan.

Corollary : Any compact symplectic manifold (M, „¦) for which the cohomology class

[„¦] is integral admits a symplectic immersion into (CPN , „¦N ) for some N n.

Proof: Since the cohomology class [„¦] is integral, there exists a smooth map φ: M ’ CPN

for some N su¬ciently large so that φ— [„¦N ] = [„¦]. Then, choosing N n, we can

arrange that there also exists a symplectic bundle map f: T M ’ f — (T CPN ) (see the

Exercises). Now apply Theorem 3.

As a ¬nal example along these lines, let me state Gromov™s embedding result. Here,

the reader should be thinking of the di¬erence between the Whitney Immersion Theorems

and the Whitney Embedding Theorems: One needs slightly more room to embed than to

immerse.

Theorem 4: Suppose that (X, Ξ) and (Y, Ψ) are connected symplectic manifolds and

that either X is non-compact and dim(X) < dim(Y ) or else that dim(X) < dim(Y ) ’ 2.

Suppose that there exists a smooth embedding φ: X ’ Y and that the induced map on

bundles φ : T X ’ φ— (T Y ) is homotopic through a 1-parameter family of injective bundle

maps •t : T X ’ φ— (T Y ) (with •0 = φ ) to a symplectic bundle map •1 : T X ’ φ— (T Y ).

Then φ is isotopic to a symplectic embedding •: X ’ Y .

This result is actually the best possible, for, as Gromov has shown using “hard” tech-

niques (see below), there are counterexamples if one leaves out the dimensional restrictions.

Note by the way that, because Theorem 4 deals with embeddings rather than immersions,

it not straightforward to place it in the framework of the h-principle.

L.9.3 155

Blowing Up in the Symplectic Category. We have already seen in Lecture 6 that

certain operations on smooth manifolds cannot be carried out in the symplectic category.

For example, one cannot form connected sums in the symplectic category.

However, certain of the operations from the geometry of complex manifolds can be car-

ried out. Gromov has shown how to de¬ne the operation of “blowing up” in the symplectic

category.

Recall how one “blows up” the origin in Cn . To avoid triviality, let me assume that

n > 1. Consider the subvariety

X = {(v, [w]) ∈ Cn — CPn’1 | v ∈ [w]} ‚ Cn — CPn’1 .

It is easy to see that X is a smooth embedded submanifold of the product and that the

projection π: X ’ Cn is a biholomorphism away from the “exceptional point” 0 ∈ Cn .

Moreover, if „¦0 and ¦ are the standard K¨hler 2-forms on Cn and CPn’1 respectively,

a

then, for each > 0, the 2-form „¦ = „¦0 + ¦ is a K¨hler 2-form on X.

a

Now, Gromov realized that this can be generalized to a “blow up” construction for

any point p on any symplectic manifold (M 2n , „¦). Here is how this goes:

First, choose a neighborhood U of p on which there exists a local chart z: U ’ Cn

which is symplectic, i.e., satis¬es z — („¦0 ) = „¦, and satis¬es z(p) = 0. Suppose that the ball

B2δ (0) in Cn of radius 2δ centered on 0 lies inside z(U). Since π: π ’1 B2δ (0) ’ B2δ (0)

— —

is a di¬eomorphism, there exists a closed 2-form ¦ on B2δ (0) so that π — (¦) = ¦. Since

—

˜ ˜

— —

2

HdR B2δ (0) = 0, there exists a 1-form • on B2δ (0) so that d• = ¦.

—

Now consider the family of symplectic forms „¦0 + d• on B2δ (0). By using a homotopy

argument exactly like the one used to Prove Theorem 1 in Lecture 6, it easily follows that

for all t > 0 su¬ciently small, there exists an open annulus A(δ ’ µ, δ + µ) and a one-

—

parameter family of di¬eomorphisms φt : A(δ ’ µ, δ + µ) ’ B2δ (0) so that

φ— („¦0 ) = „¦0 + t d•.

t

It follows that we can set

M = π ’1 Bδ+µ (0) ∪ψt M \ z ’1 φt Bδ’µ (0)

— —

ˆ

where