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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


= ,
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

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