Proof: Assume the hypotheses of the Theorem. Let „¦0 be the pullback of „¦a to µ’1 (0) ‚

˜a

M. It is clear that each of the forms „¦0 is a closed, G-invariant 2-form on µ’1 (0).

˜

a

I ¬rst want to show that each of these can be written as a pullback of a 2-form on

M0 , i.e., that each is semi-basic for π0 . To do this, I need to characterize Tm µ’1 (0) in an

appropriate fashion. Now, the assumption that 0 be a clean value for µ implies µ’1 (0) is

a smooth submanifold of M and that for m ∈ µ’1 (0) any v ∈ Tm M lies in Tm µ’1 (0) if

and only if „¦a v, »— (x)(m) = 0 for all x ∈ g and all three values of a. Thus,

Tm µ’1 (0) = {v ∈ Tm M v, w i = v, w j = v, w k = 0, for all w ∈ Tm G·m}.

Also, by G-equivariance, G·m ‚ µ’1 (0) and hence Tm G·m ‚ Tm µ’1 (0). It follows that

v ∈ Tm G·m implies that v is in the null space of each of the forms „¦0 . Thus, each of the

˜a

˜

forms „¦0 is semi-basic for π0 , as we wished to show. This, combined with G-invariance,

a

— ˜

implies that there exist unique forms „¦0 on M0 that satisfy π0 („¦0 ) = „¦0 . Since π0 is a

a a a

0

submersion, the three 2-forms „¦a are closed.

To complete the proof, it su¬ces to show that the triple („¦0 , „¦0 , „¦0 ) actually de¬nes

1 2 3

an almost hyperK¨hler structure on M0 , for then we can apply Theorem 4.

a

We do this as follows: Use the associated metric , to de¬ne an orthogonal splitting

Tm µ’1 (0) = Tm G·m • Hm .

By the hypotheses of the theorem, the ¬bers of π0 are the G-orbits in µ’1 (0) and, for

each m ∈ µ’1 (0), the kernel of the di¬erential π0 (m) is Tm G·m. Thus, π0 (m) induces an

isomorphism from Hm to Tπ0 (m) M0 and, under this isomorphism, the restriction of the

˜a

form „¦0 to Hm is identi¬ed with „¦0 . a

˜˜˜

Thus, it su¬ces to show that the forms („¦0 , „¦0 , „¦0 ) de¬ne a hyperK¨hler structure

a

1 2 3

when restricted to Hm . By Proposition 2, to do this, it would su¬ce to show that Hm is

stable under the actions of Ri , Rj , and Rk . However, by de¬nition, Hm is the subspace

of Tm M that is orthogonal to the H-linear subspace Tm G·m · H ‚ Tm M. Since the

orthogonal complement of an H-linear subspace of Tm M is also an H-linear subspace, we

are done.

Note that the proof also shows that the dimension of the reduced space M0 is equal to

dim M ’4 dim(G/Gm ), since, at each point m ∈ µ’1 (0), the space Tm G·m is perpendicular

to Ri Tm G·m • Rj Tm G·m • Rk Tm G·m and this latter direct sum is orthogonal.

Unfortunately, it frequently happens that 0 is not a clean value of µ, in which case,

Theorem 5 cannot be applied to the action. Moreover, there does not appear to be any

simple way to perform hyperK¨hler reduction at the general clean value of µ in g— • g— • g—

a

L.8.17 146

(in marked contrast to the K¨hler case). In fact, for the general clean value ξ ∈ g— • g— • g—

a

of µ, the quotient space Mξ = Gξ \µ’1 (ξ) need not even have its dimension be divisible

by 4. (See the Exercises for a cautionary example.)

However, if [g, g]⊥ ‚ g— denotes the annihilator of [g, g] in g, then the points ξ ∈ [g, g]⊥

are the ¬xed points of the coadjoint action of G. It is then possible to perform hyperK¨hlera

⊥ ⊥ ⊥

reduction at any clean value ξ = (ξ1 , ξ2 , ξ3 ) ∈ [g, g] • [g, g] • [g, g] since, in this case,

we again have Gξ = G, and so the argument in the proof above that Hm ‚ Tm µ’1 (ξ)

is a quaternionic subspace for each m ∈ Tm µ’1 (ξ) is still valid. Of course, this is not

really much of a generalization, since reduction at such a ξ is simply reduction at 0 for the

modi¬ed (but still G-equivariant) momentum mapping µξ = µ ’ ξ.

Example: One of the simplest things to do is take M = Hn and let G ‚ Sp(n)

be a closed subgroup. It is not di¬cult to show (see the Exercises) that the standard

hyperK¨hler structure on Hn has its three 2-forms given by

a

1t

i „¦1 + j „¦2 + k „¦3 = d¯ § dq

q

2

where q : Hn ’ Hn is the identity, thought of as a Hn -valued function on Hn . Using this

formula, it is easy to show that the standard left action of Sp(n) on Hn is Poisson, with

momentum mapping µ : Hn ’ sp(n) • sp(n) • sp(n) given by the formula*

1

q i t q, q j t q, q k t q .

µ(q) = ¯ ¯ ¯

2

Note that 0 is not a clean value of µ with respect to the full action of Sp(n). However, the

situation can be very di¬erent for a closed subgroup G ‚ Sp(n): Let πg : sp(n) ’ g be the

orthogonal projection relative to the Ad-invariant inner product on sp(n). The momentum

mapping for the action of G on Hn is then given by

1

πg (q i t q ), πg (q j t q ), πg (q k t q ) .

µG (q) = ¯ ¯ ¯

2

Since G is compact, there is an orthogonal direct sum g = z • [g, g], where z is the tangent

algebra to the center of G. Thus, there will be a hyperK¨hler reduction for each clean

a

value of µG that lies in z•z•z.

Let us now consider a very simple example: Let S 1 ‚ Sp(n) act diagonally on Hn by

« 1 « iθ 1

the action

q eq

¬.· ¬ . ·

eiθ · . = . .

. .

qn eiθ q n

* Here, I am identifying sp(n) with sp(n)— via the positive de¬nite, Ad-invariant sym-

metric bilinear pairing de¬ned by a, b = ’ 2 tr(ab + ba). (Because of the noncommuta-

1

tivity of H, we do not have tr(ab) = tr(ba) for all a, b ∈ sp(n).)

L.8.18 147

Then it is not di¬cult to see that the momentum mappping can be identi¬ed with the

map

µ(q) = tq i q.

¯

The reduced space Mp for any p = 0 is easily seen to be complex analytically equivalent to

T — CPn’1 , and the induced hyperK¨hler structure is the one found by Calabi. In particular,

a

for n = 2, we recover the Eguchi-Hansen metric.

In the Exercises, there are other examples for you to try.

The method of hyperK¨hler reduction has a wide variety of applications. Many of the

a

interesting moduli spaces for Yang-Mills theory turn out to have hyperK¨hler structures

a

because of this reduction procedure. For example, as Atiyah and Hitchin [AH] show, the

space of magnetic monopoles of “charge” k on R3 turns out to have a natural hyperK¨hler

a

structure that is derived by methods extremely similar to the example presented earlier of

a K¨hler structure on the moduli space of ¬‚at connections over a Riemann surface.

a

Peter Kronheimer [Kr] has used the method of hyperK¨hler reduction to construct,

a

3

for each quotient manifold Σ of S , an asymptotically locally Euclidean (ALE) Ricci-¬‚at

self-dual Einstein metric on a 4-manifold MΣ whose boundary at in¬nity is Σ. He then

went on to prove that all such metrics on 4-manifolds arise in this way.

Finally, it should also be mentioned that the case of metrics on manifolds M 4n with

holonomy Sp(n) · Sp(1) can also be treated by the method of reduction. I don™t have time

to go into this here, but the reader can ¬nd a complete account in [GL].

L.8.19 148

Exercise Set 8:

Recent Applications of Reduction

1. Show that the following two de¬nitions of compatibility between an almost complex

structure J and a metric g on M 2n are equivalent

(i) (g, J ) are compatible if g(v) = g(J v) for all v ∈ T M.

(ii) (g, J ) are compatible if „¦(v, w) = J v, w de¬nes a (skew-symmetric) 2-form on M.

2. A Non-Integrable Almost Complex Structure. Let J be an almost complex

structure on M. Let A1,0 ‚ C — A1 (M) denote the space of C-valued 1-forms on M that

satisfy ±(J v) = ± ±(v) for all v ∈ T M.

(i) Show that if we de¬ne A0,1 (M) ‚ C — A1 (M) to be the space of C-valued 1-forms on

M that satisfy ±(J v) = ’± ±(v) for all v ∈ T M, then A0,1 (M) = A1,0 (M) and that

A1,0(M) © A0,1 (M) = {0}.

(ii) Show that if J is an integrable almost complex structure, then, for any ± ∈ A1,0 (M),

the 2-form d± is (at least locally) in the ideal generated by A1,0(M). (Hint: Show

that, if z : U ’ Cn is a holomorphic coordinate chart, then, on U, the space A1,0 (U)

is spanned by the forms dz 1 , . . . , dz n . Now consider the exterior derivative of any

linear combination of the dz i .)

It is a celebrated result of Newlander and Nirenberg that this condition is su¬cient

for J to be integrable.

(iii) Show that there is an almost complex structure on C2 for which A1,0 (C2 ) is spanned

by the 1-forms

ω 1 = dz 1 ’ z 1 d¯2

¯z

ω 2 = dz 2

and that this almost complex structure is not integrable.

3. Let U(2) act diagonally on C2n , thought of as n > 2 copies of C2 (the action on

each factor is the standard one and the K¨hler structure on each factor is the standard

a

one). Regarding C2n as the space of 2-by-n matrices with complex entries, show that

the momentum mapping in this case is (up to a constant factor) given by µ(z) = ± z t z .

¯

(As usual, identify u(2) with u(2)— by using the nondegenerate bilinear pairing x, y =

’tr(xy).) Note that ξ0 = ± I2 ∈ u(2) is a ¬xed point of the coadjoint action and describe

K¨hler reduction at ξ0 . On the other hand, if ξ ∈ u(2) has eigenvalues ±»1 and ±»2

a

where »1 > »2 > 0, show that, even though ξ is a clean value of µ and Gξ ‚ U(2) acts

freely on µ’1 (ξ), the metric gξ de¬ned on the quotient Mξ so that πξ : µ’1 (ξ) ’ Mξ is a

Riemannian submersion is not compatible with „¦ξ .

E.8.1 149

4. Examine the coadjoint orbits of G = SL(2, R) and show that G·ξ carries a G-invariant

Riemannian metric if and only if Gξ is compact. Classify the coadjoint orbits of G =

SL(3, R) and show that none of them carry a G-invariant Riemannian metric. In particular,

none of them carry a G-invariant K¨hler metric.

a

5. The point of this problem is to examine the coadjoint orbits of the compact group U(n)

and to construct, on each one, the K¨hler metric compatible with the canonical symplectic

a

structure. As usual, we identitfy u(n) with u(n)— via the Ad-invariant positive de¬nite

symmetric bilinear form x, y = ’tr(xy). Thus, ξ ∈ u(n) is to be regarded as the linear

functional x ’ ξ, x . This allows us to identify the adjoint and coadjoint representa-

tions. Of course, since U(n) is a matrix group Ad(a)(x) = axa’1 = ax t ¯ for a ∈ U(n)

a

and x ∈ u(n). Recall that every skew-Hermitian matrix ξ can be diagonalized by a unitary

transformation. Consequently, each (co)adjoint orbit is the orbit of a unique matrix of the

form «

± ξ1 0 · · · 0

¬ 0 ± ξ2 · · · 0 ·

ξ=¬ . . ·, ξ1 ≥ ξ2 ≥ · · · ≥ ξn .

.

. .

..

. .

. . .

0 · · · ± ξn

0

Fix ξ and let n1 , . . . , nd ≥ 1 be the multiplicities of the eigenvalues (i.e., n1 + · · · + nd = n

and ξj = ξk if and only if, for some r, we have n1 + · · · + nr ¤ j ¤ k < n1 + · · · + nr+1 ).

Then U(n)ξ = U(n1 ) — U(n2 ) — · · · — U(nd ) (i.e., the obvious block diagonal subgroup).

Let ω = g ’1 dg = (ωj k ) be the canonical left-invariant form on U(n) and let πξ :

¯

U(n) ’ U(n)/U(n)ξ = U(n)·ξ be the canonical projection. Show that the formulae

±

— —

2(ξj ’ξk ) ωk¯ § ωk¯ 2(ξj ’ξk ) ωk¯—¦ωk¯

πξ („¦ξ ) = and πξ (hξ ) =

2

k>j k>j

de¬ne the symplectic form „¦ξ and a compatible K¨hler metric hξ on the orbit U(n)·ξ. (In

a

fact, this hξ is the only U(n)-invariant, „¦ξ -compatible metric on U(n)·ξ.)

Remark: If you know about roots and weights, it is not hard to generalize this

construction so that it works for the coadjoint orbits of any compact Lie group. The same

uniqueness result is true as well: If G is compact and ξ ∈ g— is any element, there is a

unique G-invariant K¨hler metric hξ on G·ξ that is compatible with „¦ξ .

a

6. Let M = Cn1 • Cn2 \ {(0, 0)}. Let G = S 1 act on M by the action

e±θ · (z1 , z2 ) = e±d1 θ z1 , e±d2 θ z2

where d1 and d2 are relatively prime integers. Let M have the standard ¬‚at K¨hler a

structure. Compute the momentum mapping µ and the K¨hler structures on the reduced

a

spaces. How do the relative signs of d1 and d2 a¬ect the answer? What interpretation can

you give to these spaces?

E.8.2 150

7. Go back to the the example of the “K¨hler structure” on the space A(P ) of connections

a

on a principal right G-bundle P over a connected compact Riemann surface Σ. Assume

that G = S 1 and identify g with R in the natural way. Thus, FA is a well-de¬ned 2-form on

Σ and the cohomology class [FA ] ∈ HdR (Σ, R) is independent of the choice of A. Assume

2