for which πξ is a Riemannian submersion (when µ’1 (ξ) ‚ M is given the induced sub-

manifold metric). Unfortunately, it is not , in general, true that gξ is compatible with „¦ξ .

(See the Exercises for an example.)

If you examine the proof given above in the general case, you™ll see that the main

problem is that the ˜horizontal space™ Hm need not be stable under J . In fact, what one

knows in the general case is that Hm is g-orthogonal to both Tm Gξ ·m and to J Tm G·m .

However, when Gξ is a proper subgroup of G (i.e., when ξ is not a ¬xed point of the co-

adjoint action), we won™t have Tm Gξ ·m = Tm G·m, which is what we needed in the proof

to show that Hm is stable under J .

In fact, the proof does work when Gξ = G, but this can be seen directly from the fact

that, in this case, the ˜shifted™ momentum mapping µξ = µ’ ξ still satis¬es G-equivariance

and we are simply performing reduction at 0 for the shifted momentum mapping µξ .

Reduction at Kahler coadjoint orbits. Generalizing the case where G·ξ = {ξ},

¨

there is a way to de¬ne K¨hler reduction at certain values of ξ ∈ g— , by relying on the

a

˜shifting trick™ described in the Exercises of Lecture 7:

In many cases, a coadjoint orbit G·ξ ‚ g— can be equipped with a G-invariant metric hξ

for which the pair („¦ξ , hξ ) de¬nes a K¨hler structure on the orbit G·ξ. (For example, this

a

is always the case when G is compact.) In such a case, the shifting trick allows us to de¬ne

a K¨hler metric on Mξ = Gξ \µ’1 (0) by doing K¨hler reduction at 0 on M — G·ξ endowed

a a

with the product K¨hler structure. In the cases in which there is only one G-invariant

a

K¨hler metric hξ on G·ξ that is compatible with „¦ξ (and, again, this always holds when G

a

is compact), this de¬nes a canonical K¨hler reduction procedure for ξ ∈ g— .

a

Example: Kahler reduction in Algebraic Geometry. By far the most com-

¨

mon examples of K¨hler manifolds arise in Algebraic Geometry. Here is a sample of what

a

K¨hler reduction yields:

a

Let M = Cn+1 with complex coordinates z 0 , z 1 , . . . , z n . We let z k = xk + ± y k de¬ne

real coordinates on M. Let G = S 1 act on M by the rule

e±θ · z = e±θ z.

Then G clearly preserves the K¨hler structure de¬ned by the natural complex structure

a

on M and the symplectic form

±

„¦ = tdz § d¯ = dx1 § dy 1 + · · · + dxn § dy n .

z

2

The associated metric is easily seen to be just

2 2 2 2

g = tdz —¦ d¯ = dx1 + · · · + dxn

+ dy 1 + dy n .

z

L.8.8 137

‚

Now, setting X = ‚θ , we can compute that

‚ ‚

’ yk k .

»— (X) = xk

‚y k ‚x

Thus, it follows that

dρ(X) = »— (X) „¦ = ’xk dxk ’ y k dy k = d ’ 1 |z|2 .

2

Thus, identifying g— with R, we have that µ: Cn ’ R is merely µ(z) = ’ 1 |z|2 .

2

It follows that every negative number is a non-trivial clean value for µ. For example,

= µ’1 (’ 1 ). Clearly G = S 1 itself is the stabilizer subgroup of all of the values of

2n+1

S 2

µ. Thus, M’ 1 is the quotient of the unit sphere by the action of S 1 . Since each G-orbit

2

is merely the intersection of S 2n+1 with a (unique) complex line through the origin, it is

clear that M’ 1 is di¬eomorphic to CPn .

2

Since the coadjoint action is trivial, reduction at ξ = ’ 1 will de¬ne a K¨hler struc-

a

2

ture on CP . It is instructive to compute what this K¨hler structure looks like in local

n

a

coordinates. Let A0 ‚ CP be the subset consisting of those points [z 0 , . . . , z n ] for which

n

z 0 = 0. Then A0 can be parametrized by φ : Cn ’ A0 where φ(w) = [1, w]. Now, over A0 ,

we can choose a section σ: A0 ’ S 2n+1 by the rule

(1, w)

σ —¦ φ(w) =

W

where W 2 = 1 + |w1 |2 + · · · + |wn |2 > 0. It follows that

wk wk

± ¯

— —

φ („¦’ 1 ) = (σ —¦ φ) („¦) = d §d

2 W W

2

dwk §dwk

± ¯ dW

+ (wk dwk ’ wk dwk ) § 3

= ¯ ¯

W2

2 W

W 2 δjk ’ wj wk

± ¯

dwj § dwk .

= ¯

W4

2

I leave it to the reader to check that the quotient metric (i.e., the one for which the

submersion S 2n+1 ’ CPn is Riemannian) is given by the formula

W 2 δjk ’ wj wk

¯

dwj —¦dwk .

g’ 1 = ¯

W4

2

In particular, it follows that the functions wk are holomorphic functions with respect to

the induced almost complex structure, verifying directly that the pair („¦’ 1 , g’ 1 ) is indeed

2 2

a K¨hler structure on CPn . Up to a normalizing constant, this is the usual formula for the

a

Fubini-Study K¨hler structure on CPn in an a¬ne chart.

a

L.8.9 138

Of course, the Fubini-Study metric induces a K¨hler structure on every complex sub-

a

manifold of CP . However, we can just as easily see how this arises from the reduction

n

procedure: If P (z 0 , . . . , z n ) is a non-zero homogeneous polynomial of degree d, then the

set MP = P ’1 (0) ‚ Cn+1 is a complex subvariety of Cn+1 that is invariant under the S 1

˜

action since, by homogeneity, we have

P (e±θ · z) = e±dθ P (z).

˜

It is easy to show that if the variety MP has no singularity other than 0 ∈ Cn+1, then the

K¨hler reduction of the K¨hler structure that it inherits from the standard structure on

a a

C is just the K¨hler structure on the corresponding projectivized variety MP ‚ CPn

n+1

a

that is induced by restriction of the Fubini-Study structure.

“Example”: Flat Bundles over Compact Riemann Surfaces. The following

is not really an example of the theory as we have developed it since it will deal with

“in¬nite dimensional manifolds”, however it is suggestive and the formal calculations yield

an interesting result. (For a review of the terminology used in this and the next example,

see the Appendix.)

Let G be a Lie group with Lie algebra g, and let , be a positive de¬nite, Ad-invariant

inner product on g. (For example, if G = SU(n), we could take x, y = ’tr(xy).)

Let Σ be a connected compact Riemann surface. Then there is a star operation

—: A1 (Σ) ’ A1 (Σ) that satis¬es —2 = ’id, and ±§—± ≥ 0 for all 1-forms ± on Σ.

Let P be a principal right G-bundle over Σ, and let Ad(P ) = P —Ad g denote the

vector bundle over M associated to the adjoint representation Ad: G ’ Aut(g). Let

Aut(P ) denote the group of automorphisms of P , also known as the gauge group of P .

Let A(P ) denote the space of connections on P . Then it is well known that A(P )

is an a¬ne space modeled on the vector space A1 Ad(P ) , which consists of the 1-forms

on M with values in Ad(P ). Thus, in particular, for every A ∈ A(P ), we have a natural

isomorphism

TA A(P ) = A1 Ad(P ) .

I now want to de¬ne a “K¨hler” structure on A(P ). In order to do this, I will de¬ne

a

the metric g and the 2-form „¦.

First, for ± ∈ TA A(P ), I de¬ne

±, —± .

g(±) =

Σ

(I extend the operator — in the obvious way to A1 Ad(P ) .) It is clear that g(±) ≥ 0 with

equality if and only if ± = 0. Thus, g de¬nes a “Riemannian metric” on A(P ). Since g is

“translation invariant”, it “follows” that g is “¬‚at”.

Second, I de¬ne „¦ by the rule:

„¦(±, β) = ±, β .

Σ

L.8.10 139

Since „¦(±, β) = g(±, —β), it follows that „¦ is actually non-degenerate. Moreover, because

„¦ too is “translation invariant”, it “must” be “parallel” with respect to g.

Thus, („¦, g) is a “¬‚at K¨hler” structure on A(P ). Now, I claim that both „¦ and g

a

are invariant under the natural right action of Aut(P ) on A(P ). To see this, note that an

element φ ∈ Aut(P ) determines a map •: P ’ G by the rule p · •(p) = φ(p) and that this

• satis¬es the identity •(p · g) = g ’1 •(p)g. In terms of •, the action of Aut(P ) on A(P )

is given by the classical formula

A · φ = φ— (A) = •— (ωG ) + Ad •’1 (A).

From this, it follows easily that „¦ and g are Aut(P )-invariant.

Now, I want to compute the momentum mappping µ. The Lie algebra of Aut(P ),

namely aut(P ), can be naturally identi¬ed with A0 Ad(P ) , the space of sections of the

bundle Ad(P ). I leave to the reader the task of showing that the induced map from

aut(P ) to vector ¬elds on A(P ) is given by dA : A0 Ad(P ) ’ A1 Ad(P ) . Thus, in order

to construct the momentum mapping, we must ¬nd, for each f ∈ A0 Ad(P ) , a function

ρ(f) on A so that the 1-form dρ(f) is given by

dA f, ± = ’

dρ(f)(±) = dA f „¦(±) = f, dA ± .

Σ Σ

However, this is easy. We just set

ρ(f)(A) = ’ f, FA

Σ

and the reader can easily check that

d

(ρ(f)(A + t±)) = ’ f, dA ±

dt t=0 Σ

as desired. Finally, using the natural isomorphism

— —

= A0 Ad(P ) = A2 Ad(P ) ,

aut(P )

we see that (up to sign) the formula for the momentum mapping simply becomes

µ(A) = FA = dA + 1 [A, A].

2

Now, can we do reduction? What we need is a clean value of µ. As a reasonable ¬rst

guess, let™s try 0. Thus, µ’1 (0) consists exactly of the ¬‚at connections on P and the reduced

space M0 should be the ¬‚at connections modulo gauge equivalence, i.e., µ’1 (0)/Aut(P ).

How can we tell whether 0 is a clean value? One way to know this would be to know

that 0 is a regular value. We have already seen that µ (A)(±) = dA ±, so we are asking

L.8.11 140

whether the map dA : A1 Ad(P ) ’ A2 Ad(P ) is surjective for any ¬‚at connection A.

Now, because A is ¬‚at, the sequence

d d

0 ’’ A0 Ad(P ) ’’ A1 Ad(P ) ’’ A2 Ad(P ) ’’ 0

A A

forms a complex and the usual Hodge theory pairing shows that H 2 (Σ, dA ) is the dual

space of H 0 (Σ, dA ). Thus, µ (A) is surjective if and only if H 0 (Σ, dA ) = 0. Now, an

element f ∈ A0 Ad(P ) that satis¬es dA f = 0 exponentiates to a 1-parameter family

of automorphisms of P that commute with the parallel transport of A. I leave to the

reader to show that H 0 (Σ, dA ) = 0 is equivalent to the condition that the holonomy group

HA (p) ‚ G has a centralizer of positive dimension in G for some (and hence every) point

of P . For example, for G = SU(2), this would be equivalent to saying that the holonomy

groups HA (p) were each contained in an S 1 ‚ G.

˜—

Let us let M ‚ µ’1 (0) denote the (open) subset consisting of those ¬‚at connections

whose holonomy groups have at most discrete centralizers in G. If G is compact, of course,

˜—