0

is a K¨hler manifold wherever it is a manifold. (In general, at the connections where the

a

centralizer of the holonomy is trivial, one expects the quotient to be a manifold.)

Since the space of ¬‚at connections on P modulo gauge equivalence is well-known to

be identi¬able as the space R π1 (Σ, s), G = Hom π1 (Σ, s), G /G of equivalence classes of

representation of π1 (Σ) into G, our discussion leads us to believe that this space (which is

¬nite dimensional) should have a natural K¨hler structure on it. This is indeed the case,

a

and the geometry of this K¨hler metric is the subject of current interest.

a

HyperK¨hler Manifolds.

a

In this section, we will generalize the K¨hler reduction procedure to the case of man-

a

ifolds with holonomy Sp(m), the so-called hyperK¨hler case.

a

Quaternion Hermitian Linear Algebra. We begin with some linear algebra over

the ring H of quaternions. For our purposes, H can be identi¬ed with the vector space of

dimension 4 over R of matrices of the form

x0 + ± x1 x2 + ± x3 def

= x0 1 + x1 i + x2 j + x3 k.

x=

’x2 + ± x3 x0 ’ ± x1

(We are identifying the 2-by-2 identity matrix with 1 in this representation.) It is easy to

see that H is closed under matrix multiplication. If we de¬ne x = x0 ’ x1 i ’ x2 j ’ x3 k,

¯

then we easily get xy = y x and

¯¯

def

x¯ = (x0 )2 + (x1 )2 + (x2 )2 + (x3 )2 1 = det(x) 1 = |x|2 1.

x

It follows that every non-zero element of H has a multiplicative inverse. Note that the

space of quaternions of unit norm, S 3 de¬ned by |x| = 1, is simply SU(2).

L.8.12 141

Much of the linear algebra that works for the complex numbers can be generalized

to the quaternions. However, some care must be taken since H is not commutative. In

the following exposition, it turns out to be most convenient to de¬ne vector spaces over H

as right vector spaces instead of left vector spaces. Thus, the standard H-vector space of

H-dimension n is Hn (thought of as columns of quaternions of height n) where the action

of the scalars on the right is given by

«1

« 1

xq

x

. · q = ¬ . ·.

. .

. .

xn xn q

With this convention, a quaternion linear map A: Hn ’ Hm , i.e., an additive map satisfying

A(v q) = A(v) q, can be represented by an m-by-n matrix of quaternions acting via matrix

multiplication on the left.

Let H: Hn — Hn ’ H be the “quaternion Hermitian” inner product given by

H(z, w) = tz w = z 1 w1 + · · · + z n wn .

¯ ¯ ¯

Then by our conventions, we have

H(z q, w) = q H(z, w)

¯ and H(z, w q) = H(z, w) q.

We also have H(z, w) = H(w, z), just as before.

We de¬ne Sp(n) ‚ GL(n, H) to be the group of H-linear transformations of Hn that

preserve H, i.e., H(Az, Aw) = H(z, w) for all z, w ∈ Hn . It is easy to see that

¯

Sp(n) = A ∈ GL(n, H) | tAA = In .

I leave as an exercise for the reader to show that Sp(n) is a compact Lie group of

dimension 2n2 + n. Also, it is not di¬cult to show that Sp(n) is connected and acts

irreducibly on Hn . (see the Exercises)

Now H can be split into one real and three imaginary parts as

H(z, w) = z, w + „¦1 (z, w) i + „¦2 (z, w) j + „¦3 (z, w) k.

It is clear from the relations above that , is symmetric and positive de¬nite and that

each of the „¦a is skew-symmetric. Moreover, we have the following identities:

z, w = „¦1 (z, w i) = „¦2 (z, w j) = „¦3 (z, w k)

and

„¦2 (z, w i) = „¦3 (z, w)

„¦3 (z, w j) = „¦1 (z, w)

„¦1 (z, w k) = „¦2 (z, w).

L.8.13 142

Proposition 1: The subgroup of GL(4n, R) that ¬xes the three 2-forms („¦1 , „¦2 , „¦3 ) is

equal to Sp(n).

Proof: Let G ‚ GL(4n, R) be the subgroup that ¬xes each of the „¦a . Clearly we have

Sp(n) ‚ G.

Now, from the ¬rst of the identities above, it follows that each of the forms „¦a is

non-degenerate. Then, from the second set of these identities, it follows that the subgroup

G must also ¬x the linear transformations of R4n that represent multiplication on the right

by i, j, and k. Of course, this, by de¬nition, implies that G is a subgroup of GL(n, H).

Returning to the ¬rst of the identities, it also follows that G must preserve the inner

product de¬ned by , . Finally, since we have now seen that G must preserve all of the

components of H, it follows that G must preserve H as well. However, this was the very

de¬nition of Sp(n).

Proposition 1 motivates the way we will want to de¬ne HyperK¨hler structures on

a

manifolds: as triples of 2-forms that satisfy certain conditions. Here is the linear algebra

de¬nition on which the manifold de¬nition will be based.

De¬nition 5: Let V be a vector space over R. A hyperK¨hler structure on V is a choice of

a

a triple of non-degenerate 2-forms („¦1 , „¦2 , „¦3 ) that satisfy the following properties: First,

the linear maps Ri , Rj that are de¬ned by the equations

„¦1 (v, Rj w) = ’„¦3 (v, w)

„¦2 (v, Ri w) = „¦3 (v, w)

2 2

= ’id and skew-commute, i.e., Ri Rj = ’Rj Ri . Second, if we set

satisfy Ri = Rj

Rk = ’Ri Rj , then

„¦1 (v, Ri w) = „¦2 (v, Rj w) = „¦3 (v, Rk w) = v, w

where , (which is de¬ned by these equations) is a positive de¬nite symmetric bilinear

form on V . The inner product , is called the associated metric on V .

This may seem to be a rather cumbersome de¬nition (and I admit that it is), but it is

su¬cient to prove the following Proposition (which I leave as an exercise for the reader).

Proposition 2: If („¦1 , „¦2 , „¦3 ) is a hyperK¨hler structure on a real vector space V , then

a

dim(V ) = 4n for some n and, moreover, there is an R-linear isomorphism of V with Hn

that identi¬es the hyperK¨hler structure on V with the standard one on Hn .

a

We are now ready for the analogs of De¬nitions 3 and 4:

De¬nition 6: If M is a manifold of dimension 4n, an almost hyperK¨hler structure on

a

M is a triple („¦1 , „¦2 , „¦3 ) of 2-forms on M that have the property that they induce a

hyperK¨hler structure on each tangent space Tm M.

a

De¬nition 7: An almost hyperK¨hler structure („¦1 , „¦2 , „¦3 ) on a manifold M 4n is a

a

hyperK¨hler structure on M if each of the forms „¦a is closed.

a

L.8.14 143

At ¬rst glance, De¬nition 7 may seem surprising. After all, it appears to place no

conditions on the almost complex structures Ri , Rj , and Rk that are de¬ned on M by the

almost hyperK¨hler structure on M and one would surely want these to be integrable if

a

the analogy with K¨hler geometry is to be kept up. The nice result is that the integrability

a

of these structures comes for free:

Theorem 4: For an almost hyperK¨hler structure („¦1 , „¦2 , „¦3 ) on a manifold M 4n , the

a

following are equivalent:

(1) d„¦1 = d„¦2 = d„¦3 = 0.

(2) Each of the 2-forms „¦a is parallel with respect to the Levi-Civita connection of the

associated metric.

(3) Each of the almost complex structures Ri , Rj , and Rk are integrable.

Proof: (Idea) The proof of Theorem 4 is much like the proof of Theorem 2. One shows

by local calculations in Gauss normal coordinates at any point on M that the covariant

derivatives of the forms „¦a with respect to the Levi-Civita connection of the associated

metric can be expressed in terms of the coe¬cients of their exterior derivatives and vice-

versa. Similarly, one shows that the formulas for the Nijnhuis tensors of the three almost

complex structures on M can be expressed in terms of the covariant derivatives of the

three 2-forms and vice-versa. This is a rather formidable linear algebra problem, but it is

nothing more. I will not do the computation here.

Note that Theorem 4 implies that the holonomy H of the associated metric of a

hyperK¨hler structure on M 4n must be a subgroup of Sp(n). If H is a proper subgroup

a

of Sp(n), then by Theorem 1, the associated metric must be locally a product metric. Now,

as is easy to verify, the only products from Berger™s List that can appear as subgroups

of Sp(n) are products of the form

{e}n0 — Sp(n1 ) — · · · — Sp(nk )

where {e}n0 ‚ Sp(n0 ) is just the identity subgroup and n = n0 + · · · + nk . Thus, it

follows that a hyperK¨hler structure can be decomposed locally into a product of the ˜¬‚at™

a

example with hyperK¨hler structures whose holonomy is the full Sp(ni ). (If M is simply

a

connected and the associated metric is complete, then the de Rham Splitting Theorem

asserts that M can be globally written as a product of such metrics.) This motivates our

calling a hyperK¨hler structure on M 4n irreducible if its holonomy is equal to Sp(n).

a

The reader may be wondering just how common these hyperK¨hler structures are

a

(aside from the ¬‚at ones of course). The answer is that they are not so easy to come by. The

¬rst known non-¬‚at example was the Eguchi-Hanson metric (often called a “gravitational

instanton”) on T — CP1 . The ¬rst known irreducible example in dimensions greater than 4

was discovered by Eugenio Calabi, who, working independently from Eguchi and Hanson,

constructed an irreducible hyperK¨hler structure on T — CPn for each n that happened to

a

L.8.15 144

agree with the Eguchi-Hanson metric for n = 1. (We will see Calabi™s examples a little

further on.)

The ¬rst known compact example was furnished by Yau™s solution of the Calabi Con-

jecture:

Example: K3 Surfaces. A K3 surface is a compact simply connected 2-dimensional

complex manifold S with trivial canonical bundle. What this latter condition means is that

there is nowhere-vanishing holomorphic 2-form Υ on S. An example of such a surface is a

smooth algebraic surface of degree 4 in CP3 .

A fundamental result of Siu [Si] is that every K3 surface is K¨hler, i.e., that there

a

exists a 2-form „¦ on S so that the hermitian structure („¦, J ) on S is actually K¨hler.

a

Moreover, Yau™s solution of the Calabi Conjecture implies that „¦ can be chosen so that Υ

is parallel with respect to the Levi-Civita connection of the associated metric.

Multiplying Υ by an appropriate constant, we can arrange that 2 „¦2 = Υ§Υ. Since

„¦§Υ = 0 and Υ§Υ = 0, it easily follows (see the Exercises) that if we write „¦ = „¦1 and

Υ = „¦2 ’ ± „¦3 , then the triple („¦1 , „¦2 , „¦3 ) de¬nes a hyperK¨hler structure on S.

a

For a long time, the K3 surfaces were the only known compact manifolds with hy-

perK¨hler structures. In fact, a “proof” was published showing that there were no other

a

compact ones. However, this turned out not to be correct.

Example: Let M 4n be a simply connected, compact complex manifold (of complex

dimension 2n) with a holomorphic symplectic form Υ. Then Υn is a non-vanishing holo-

morphic volume form, and hence the canonical bundle of M is trivial. If M has a K¨hler

a

structure that is compatible with its complex structure, then, by Yau™s solution of the

Calabi Conjecture, there is a K¨hler metric g on M for which the volume form Υn is

a

parallel. This implies that the holonomy of g is a subgroup of SU(2n). However, this in

turn implies that g is Ricci-¬‚at and hence, by a Bochner vanishing argument, that every

holomorphic form on M is parallel with respect to g. Thus, Υ is also parallel with respect

to g and hence the holonomy is a subgroup of Sp(n). If M can be constructed in such a

way that it cannot be written as a non-trivial product of complex submanifolds, then the

holonomy of g must act irreducibly on C2n and hence must equal Sp(n).

Fujita was the ¬rst to construct a simply connected, compact complex 4-manifold that

carried a holomorphic 2-form and that could not be written non-trivially as a product. This

work is written up in detail in a survey article by [Bea].

HyperK¨hler Reduction. I am now ready to describe another method of construct-

a

ing hyperK¨hler structures, known as hyperK¨hler reduction. This method ¬rst appeared

a a

in a famous paper by Hitchin, Karlhede, Lindstr¨m, and Roˇek, [HKLR].

o c

Theorem 5: Suppose that („¦1 , „¦2 , „¦3 ) is a hyperK¨hler structure on M and that there

a

is a left action »: G — M ’ M that is Poisson with respect to each of the three symplectic

forms „¦a . Let

µ = (µ1 , µ2 , µ3 ): M ’ g— • g— • g—

L.8.16 145

be a G-equivariant momentum mapping. Suppose that 0 ∈ g— •g— •g— is a clean value for µ

and that the quotient M0 = G\µ’1 (0) has a smooth structure for which the projection

π0 : µ’1 (0) ’ M0 is a smooth submersion. Then there is a unique hyperK¨hler structure

a

(„¦1 , „¦2 , „¦3 ) on M0 with the property that π0 („¦a ) is the pull back of „¦a to µ’1 (0) ‚ M

—

0 0 0 0