symplectic manifolds that are also complex manifolds in such a way that the complex

structure is “maximally compatible” with the symplectic structure. These manifolds arise

with great frequency in Algebraic Geometry, and it is beyond the scope of these Lectures

to do more than make an introduction to their uses here.

Hermitian Linear Algebra. As usual, we begin with some linear algebra. Let

H: Cn — Cn ’ C be the hermitian inner product given by

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

¯ ¯ ¯

Then U(n) ‚ GL(n, C) is the group of complex linear transformations of Cn that pre-

¯

serve H since H(Az, Aw) = H(z, w) for all z, w ∈ Cn if and only if tAA = In .

Now, H can be split into real and imaginary parts as

H(z, w) = z, w + ± „¦(z, w).

It is clear from the relation H(z, w) = H(w, z) that , is symmetric and „¦ is skew-

symmetric. I leave it to the reader to show that , is positive de¬nite and that „¦ is

non-degenerate.

Moreover, since H(z, ± w) = ± H(z, w), it also follows that „¦(z, w) = ± z, w and

z, w = „¦(z, ± w). It easily follows from these equations that, if we let J : Cn ’ Cn

denote multiplication by ±, then knowing any two of the three objects , , „¦, or J on R2n

determines the third.

De¬nition 1: Let V be a vector space over R. A non-degenerate 2-form „¦ on V and

a complex structure J : V ’ V are said to be compatible if „¦(x, J y) = „¦(y, J x) for all

x, y ∈ V . If the pair („¦, J ) is compatible, then we say that the pair forms an Hermitian

structure on V if, in addition, „¦(x, J x) > 0 for all non-zero x ∈ V . The positive de¬nite

quadratic form g(x, x) = „¦(x, J x) is called the associated metric on V .

I leave as an exercise for the reader the task of showing that any two Hermitian

structures on V are isomorphic via some invertible endomorphism of V .

It is easy to show that, if g is the quadratic form associated to a compatible pair „¦, J ,

then „¦(v, w) = g(J v, w). It follows that any two elements of the triple „¦, J, g determine

the third.

L.8.3 132

In an extension of the notion of compatibility, we de¬ne a quadratic form g on V to

be compatible with a non-degenerate 2-form „¦ on V if the linear map J : V ’ V de¬ned

by the relation „¦(v, w) = g(J v, w) satis¬es J 2 = ’1. Similarly, we de¬ne a quadratic form

g on V to be compatible with a complex structure J on V if g(J v, w) = ’g(J w, v), so that

„¦(v, w) = g(J v, w) de¬nes a 2-form on V .

Almost Hermitian Manifolds. Since our main interest is in symplectic and com-

plex structures, I will introduce the notion of an almost Hermitian structure on a manifold

in terms of its almost complex and almost symplectic structures:

De¬nition 2: Let M 2n be a manifold. A 2-form „¦ and an almost complex structure J

de¬ne an almost Hermitian structure on M if, for each m ∈ M, the pair („¦m , Jm ) de¬nes

a Hermitian structure on Tm M.

When („¦, J ) de¬nes an almost Hermitian structure on M, the Riemannian metric g

on M de¬ned by g(v) = „¦(v, J v) is called the associated metric.

Just as one must place conditions on an almost symplectic structure in order to get a

symplectic structure, there are conditions that an almost complex structure must satisfy

in order to be a complex structure.

De¬nition 3: An almost complex structure J on M 2n is integrable if each point of M has a

neighborhood U on which there exists a coordinate chart z: U ’ Cn so that z (J v) = ± z (v)

for all v ∈ T U . Such a coordinate chart is said to be J -holomorphic.

According to the Korn-Lichtenstein theorem, when n = 1 all almost complex struc-

tures are integrable. However, for n ≥ 2, one can easily write down examples of almost

complex structures J that are not integrable. (See the Exercises.)

When J is an integrable almost complex structure on M, the set

UJ = {(U, z) | z: U ’ Cn is J -holomorphic}

forms an atlas of charts that are holomorphic on overlaps. Thus, UJ de¬nes a holomorphic

structure on M.

The reader may be wondering just how one determines whether an almost complex

structure is integrable or not. In the Exercises, you are asked to show that, for an integrable

almost complex structure J , the identity LJX J ’J —¦LX J = 0 must hold for all vector ¬elds

X on M. It is a remarkable result, due to Newlander and Nirenberg, that this condition

is su¬cient for J to be integrable.

The reason that I mention this condition is that it shows that integrability is deter-

mined by J and its ¬rst derivatives in any local coordinate system. This condition can be

rephrased as the condition that the vanishing of a certain tensor NJ , called the Nijnhuis

tensor of J and constructed out of the ¬rst-order jet of J at each point, is necessary and

su¬cient for the integrability of J .

We are now ready to name the various integrability conditions that can be de¬ned for

an almost Hermitian manifold.

L.8.4 133

De¬nition 4: We call an almost Hermitian pair („¦, J ) on a manifold M almost K¨hler a

if „¦ is closed, Hermitian if J is integrable, and K¨hler if „¦ is closed and J is integrable.

a

We already saw in Lecture 6 that a manifold has an almost complex structure if and

only if it has an almost symplectic structure. However, this relationship does not, in

general, hold between complex structures and symplectic structures.

Example: Here is a complex manifold that has no symplectic structure. Let Z act on

M = C2 \{0} by n · z = 2n z. This free action preserves the standard complex structure on

˜

M. Let N = Z\M , then, via the quotient mapping, N inherits the structure of a complex

manifold.

However, N is di¬eomorphic to S 1 — S 3 as a smooth manifold. Thus N is a compact

manifold satisfying HdR (N, R) = 0. In particular, by the cohomology ring obstruction

2

discussed in Lecture 6, we see that M cannot be given a symplectic structure.

Example: Here is an example due to Thurston, of a compact 4-manifold that has a

complex structure and has a symplectic structure, but has no K¨hler structure.

a

Let H3 ‚ GL(3, R) be the Heisenberg group, de¬ned in Lecture 2 as the set of matrices

«

of the form

1 x z + 1 xy

2

g = 0 1 .

y

00 1

The left invariant forms and their structure equations on H3 are easily computed in these

coordinates as

ω1 = dx dω1 = 0

ω2 = dy dω2 = 0

ω3 = dz ’ 1 (x dy ’ y dx) dω3 = ’ω1 § ω2

2

Now, let “ = H3 © GL(3, Z) be the subgroup of H3 consisting of those elements of H3 all

of whose entries are integers. Let X = “\H3 be the space of right cosets of “. Since the

forms ωi are left-invariant, it follows that they are well-de¬ned on X and form a basis for

the 1-forms on X.

Now let M = X — S 1 and let ω4 = dθ be the standard 1-form on S 1. Then the forms

ωi for 1 ¤ i ¤ 4 form a basis for the 1-forms on M. Since dω4 = 0, it follows that the

2-form

„¦ = ω 1 § ω3 + ω2 § ω4

is closed and non-degenerate on M. Thus, M has a symplectic structure.

Next, I want to construct a complex structure on M. In order to do this, I will

˜

produce the appropriate local holomorphic coordinates on M. Let M = H3 — R be the

˜

simply connected cover of M with coordinates (x, y, z, θ). We regard M as a Lie group.

˜

De¬ne the functions w1 = x + ± y and w2 = z + ± θ + 1 (x2 + y 2 ) on M . Then I leave to

4

˜

the reader to check that, if g0 is the element of M with coordinates (x0 , y0 , z0 , θ0 ), then

L—0 (w1 ) = w1 + w0 L—0 (w2 ) = w2 + ±/2 w0 w1 + w0 .

1

¯1 2

and

g g

L.8.5 134

˜

Thus, the coordinates w1 and w2 de¬ne a left-invariant complex structure on M. Since M

˜

is obtained from M by dividing by the obvious left action of “ — Z, it follows that there is

a unique complex structure on M for which the covering projection is holomorphic.

Finally, we show that M cannot carry a K¨hler structure. Since “ is a discrete

a

subgroup of H3 , the projection H3 ’ X is a covering map. Since H3 = R3 as manifolds,

it follows that π1 (X) = “. Moreover, X is compact since it is the image under the

projection of the cube in H3 consisting of those elements whose entries lie in the closed

Z, it follows that “/[“, “] Z2 . Thus,

interval [0, 1]. On the other hand, since [“, “]

H 1 (M, Z) = H 1 (X — S 1 , Z) = Z2 • Z. From this, we get that HdR (M, R) = R3 . In

1

particular, the ¬rst Betti number of M is 3. Now, it is a standard result in K¨hler a

geometry that the odd degree Betti numbers of a compact K¨hler manifold must be even

a

(for example, see [Ch]). Hence, M cannot carry any K¨hler metric.

a

Example: Because of the classi¬cation of compact complex surfaces due to Kodaira, we

know exactly which compact 4-manifolds can carry complex structures. Fernandez, Gotay,

and Gray [FGG] have constructed a compact, symplectic 4-manifold M whose underlying

manifold is not on Kodaira™s list, thus, providing an example of a compact symplectic

4-manifold that carries no complex structure.

The fundamental theorem relating the two “integrability conditions” to the idea of

holonomy is the following one. We only give the idea of the proof because a complete proof

would require the development of considerable machinery.

Theorem 2: An almost Hermitian structure („¦, J ) on a manifold M is K¨hler if and a

only if the form „¦ is parallel with respect to the parallel transport of the associated metric

g.

Proof: (Idea) Once the formulas are developed, it is not di¬cult to see that the covariant

derivatives of „¦ with respect to the Levi-Civita connection of g are expressible in terms of

the exterior derivative of „¦ and the Nijnhuis tensor of J . Conversely, the exterior derivative

of „¦ and the Nijnhuis tensor of J can be expressed in terms of the covariant derivative

of „¦ with respect to the Levi-Civita connection of g. Thus, „¦ is covariant constant (i.e.,

invariant under parallel translation with respect to g) if and only it is closed and J is

integrable.

It is worth remarking that J is invariant under parallel transport with respect to g if

and only if „¦ is.

The reason for this is that J is determined from and determines „¦ once g is ¬xed.

The observation now follows, since g is invariant under parallel transport with respect to

its own Levi-Civita connection.

K¨hler Reduction. We are now ready to state the ¬rst of the reduction theorems

a

we will discuss in this Lecture.

It turns out that it™s a good idea to discuss a special case ¬rst.

L.8.6 135

Theorem 3: Kahler Reduction at 0. Let („¦, g) be a K¨hler structure on M 2n .

a

¨

Let »: G — M ’ M be a left action that is Poisson with respect to „¦ and preserves the

metric g. Let µ: M ’ g— be the associated momentum mapping. Suppose that 0 ∈ g— is a

clean value of µ and that there is a smooth structure on the orbit space M0 = G\µ’1 (0)

for which the natural projection π0 : µ’1 (0) ’ G\µ’1 (0) is a smooth submersion. Then

—

there is a unique K¨hler structure („¦0 , g0 ) on M0 de¬ned by the conditions that π0 („¦0 )

a

be equal to the pullback of „¦ to µ’1 (0) ‚ M and that π0 : µ’1 (0) ’ M0 be a Riemannian

submersion.

Proof: Let g0 and „¦0 be the pullbacks of g and „¦ respectively to µ’1 (0). By hypotheses,

˜

˜

˜

g0 and „¦0 are invariant under the action of G.

˜

From Theorem 2 of Lecture 7, we already know that there exists a unique symplectic

— ˜

structure „¦0 on M0 for which π0 („¦0 ) = „¦0 .

Here is how we construct g0 . For any m ∈ µ’1 (0), there is a well de¬ned g0 -orthogonal

˜

splitting

Tm µ’1 (0) = Tm G · m • Hm

that is clearly G-invariant. Since, by hypothesis, π0 : µ’1 (0) ’ M0 is a submersion, it

easily follows that π0 (m): Hm ’ Tπ0 (m) M0 is an isomorphism of vector spaces. Moreover,

the G-invariance of g shows that there is a well-de¬ned quadratic form g0 (m) on Tπ0 (m) M0

˜

that corresponds to the restriction of g0 to Hm under this isomorphism. By the very

˜

de¬nition of Riemannian submersion, it follows that g0 is a Riemannian metric on M0 for

which π0 is a Riemannian submersion.

It remains to show that („¦0 , g0 ) de¬nes a K¨hler structure on M0 . First, we show

a

that it is an almost K¨hler structure, i.e., that „¦0 and g0 are actually compatible. Since

a

π0 (m): Hm ’ Tπ0 (m) M0 is an isomorphism of vector spaces that identi¬es („¦0 , g0 ) with

the restriction of („¦, g) to Hm , it su¬ces to show that Hm is invariant under the action

of J .

Here is how we do this. Tracing back through the de¬nitions, we see that x ∈ Tm M

lies in the subspace Hm if and only if x satis¬es both of the conditions „¦(x, y) = 0 and

g(x, y) = 0 for all y ∈ Tm G·m . However, since „¦(x, y) = g(J x, y) for all y, it follows that

the necessary and su¬cient conditions that x lie in Hm can also be expressed as the two

conditions g(J x, y) = 0 and „¦(J x, y) = 0 for all y ∈ Tm G·m . Of course, these conditions

are exactly the conditions that J x lie in Hm . Thus, x ∈ Hm implies that J x ∈ Hm , as

desired.

Finally, in order to show that the almost K¨hler structure on M0 is actually K¨hler,

a a

it must be shown that „¦0 is parallel with respect to the Levi-Civita connection of g0 .

This is a straightforward calculation using the structure equations and will not be done

here. (Alternatively, to prove that the structure is actually K¨hler, one could instead show

a

that the induced almost complex structure is integrable. This is somewhat easier and the

interested reader can consult the Exercises, where a proof is outlined.)

L.8.7 136

Now, it seems unreasonable to consider only reduction at 0 ∈ g— . However, some

caution is in order because the na¨ attempt to generalize Theorem 3 to reduction at a

±ve

—

general ξ ∈ g fails: Let » : G—M ’ M be a Poisson action on a K¨hler manifold (M, „¦, g)

a

—

that preserves g and let µ : M ’ g be a Poisson momentum mapping. Then for every

clean value ξ ∈ g— for which the orbit space Mξ = Gξ \µ’1 (ξ) has a smooth structure that

makes πξ : µ’1 (ξ) ’ Mξ a smooth submersion, there is a symplectic structure „¦ξ on Mξ