line lt and l0 = l. Next we pick a non-zero vector v ∈ l and look at

the perpendicular hyperplane l⊥ + v passing through v. It will intersect

each line lt at one point p(t) = v + q(t). So we have a linear map l ’ l⊥

de¬ned by

d d

v ’ p(t)|t=0 = q(t)|t=0 .

dt dt

Sometimes it is convenient to identify Hom(l, l⊥ ) l— — l⊥ .

The hermitian form (, ) on Cn+1 gives rise to a hermitian form on

any subspace like l or l⊥ , as well as on Hom(l, l⊥ ) if we put for ψ ∈

Hom(l, l⊥ ) the norm

||ψ(v)||

||ψ|| = , v ∈ l, v = 0.

||v||

Thus we got a hermitian form on the tangent space to CPn . Let us

write it down explicitly.

φ

We take U0 ‚ CPn which is identi¬ed with Cn via Cn ’ U0 , φ(z1 , ..., zn ) =

[1 : z1 : · · · : zn ]. Let us take a tangent vector u = (u1 , ..., un ) at a point

102 CHAPTER 3. HODGE THEORY

(w1 , ..., wn ) ∈ Cn and consider the parametric curve w + tu, which cor-

responds to the point [1 : w1 + tu1 : · · · : wn + tun ] ∈ U0 ‚ CPn . The

derivative at t = 0 of this curve in CPn gives us a linear map

± : l =< w >’ l⊥ Cn+1 /l,

where l is spanned by w. The map ± sends w to

( ui wi )(1, w1 , ..., wn )

¯

i

±(w) = (0, u1 , ..., un ) ’ ,

1 + n |wi |2

1

which is a familiar formula for the projection of (0, u1 , ..., un ). So we

obtain a hermitian form

ui wi |2

| ¯

n

|ui |2 ’

2

||±(w)|| 1 1+ |wi |2

H(u, u) = = =

n

||w||2 |wi |2

1+ 1

n

|ui |2 )(1 + |ui |2 ) ’ | i ui wi |2

( ¯

i=1

= .

(1 + n |wi |2 )2

1

It is suitable to have a matrix expression of H. If w is represented as

a column-vector and for a matrix A we denote by A— its transposed

complex conjugate then

n n

2 ’2

|wj |2 )Id ’ w · w— ],

H = (1 + |wi | ) [(1 +

i=1 j=1

where Id is the identity matrix and now

—

H(v1 , v2 ) = v2 Hv1 .

From the matrix notation one can see that H is positive de¬nite, since

H has all positive eigenvalues.

We will often use the following

k

LEMMA 3.2.5 Let (X, J, ω) be a K¨hler manifold and let Y ’ X be

a

a complex submanifold. Then (Y, k — J, k — ω) is also a K¨hler manifold.

a

Proof. In fact, k — J and k — ω simply are the restrictions of the operator J

and the form ω to the tangent space T Y . Using the identity ω(v, w) =

g(Jv, w) we can easily see why k — ω is non-degerate. It follows from the

fact that J preserves T Y and the riemannian metric g on X remains

non-degenerate when restricted to Y .

¨

3.2. KAHLER MANIFOLDS 103

COROLLARY 3.2.6 Every projective manifold is K¨hler .

a

Let us give another interpretation of a K¨hler form.

a

PROPOSITION 3.2.7 Let (M, J) be a complex √ manifold of dimen-

sion n equipped with a hermitian form H = g + ’1ω. Then M is

K¨hler if and only if for any x ∈ M we can ¬nd complex coordinates

a

(z1 , ..., zn ) near x such that

√

’1 n

ω=’ dzi §d¯i + R,

z

2 i=1

where R vanishes to order 2 at x.

√

’1 n

Proof. (Sketch.) First, if ω = ’ 2 i=1 dzi §d¯i +R, then ω is closed,

z

because dω(x) = dR(x) = 0, since R vanishes to order 2 at x and x can

be chosen arbitrarily.

Conversely, if M is a K¨hler manifold, and if x ∈ M we may consider

a

a neighborhood of x which is biholomorphic to an open set of Cn . On

this open set we have the linear coordinates (z1 , · · · , zn ). Consider then

the Taylor expansion ω = ω0 + ω1 + ω2 + · · ·, where ωj = fijk dzk §d¯l z

and fijk is a homogeneous polynomial in (z1 , ..., zn , z1 , ..., zn ) of degree

¯ ¯

j. (We also know that ω is of type (1, 1).) Let ω0 = aij dzi §d¯j ,

z

where (aij ) is a hermitian matrix. We actually can assume that

√

’1

ω0 = ’ dzj §d¯j ,

z

2

since every positive-de¬nite hermitian matrix can be transformed to the

identity matrix by a linear change of variables. Now we pay attention

to ω1 = gkl dzk §d¯l , where gkl = m (bmkl zm + cmkl zm ) is a linear

z ¯

form such that gkl + glk = 0, or, equivalently, cmkl = ’bmlk . Next

we introduce the group G of formal biholomorphic maps (z1 , ..., zn ) ’

(w1 , ..., wn ) of type

zj = w j + djkl wk wl .

k¤l

(After the substitution we disregard all the terms of homogeneous

degree 2 or more). In fact, G is an abelian complex Lie group of

2

dimension n (n+1) . Let us also introduce the complex a¬ne space

2

104 CHAPTER 3. HODGE THEORY

X = {ω1 , dω1 = 0}. The dimension of this a¬ne space is not hard

to compute. We notice that the complex dimension of the space of

coe¬cients bmkl is n3 and the coe¬cients cmkl are expressed in terms of

bmkl . The condition that dω1 = 0 adds n2 (n ’ 1) real linear conditions.

More precisely, it means that bmkl is symmetric in k and l. Thus,

dimR (X) = 2n3 ’ n2 (n ’ 1) = n2 (n + 1).

The action of the group G on the space X is ¬xed point free. This fol-

lows from the general fact that if an isometry of a Riemannian manifold

M ¬xes a point x ∈ M and acts trivially on Tx M , then it is the iden-

tity map. This fact also holds true for formal jets of di¬eomorphisms,

as can be seen from an inspection of the PDE™s satis¬ed by a Killing

vector ¬eld. Since dim(X) = dim(G) we conclude that G acts on X

transitively. Thus we can arrange that ω1 = 0.

Example. Let us have the following 2-form on C:

√

’1

’ dw§dw + (w ’ w)dw§dw.

¯ ¯ ¯

2

√

By the holomorphic change of variable z = w + ’1w2 one gets rid of

the linear part of the form.

We also notice that it is not possible in general to get rid of the

quadratic terms in ω. If one uses substitutions

zj = w j + djkl wk wl wm ,

k¤l¤m

which form a group denoted by G3 , then the dimension of the complex

3

space of G3 -orbits in the vector space of ω2 components is n (n+1) . This

2

yields 1 when n = 1 and 12 when n = 2.

3.3 The moduli space of polygons is K¨hler

a

Here we shall give an interesting example of a K¨hler manifold, which

a

is the moduli space of n-gons in R3 with ¬xed side lengths. It turns out

¨

3.3. THE MODULI SPACE OF POLYGONS IS KAHLER 105

also that when all sides have integer lengths, this space is projective.

Here we follow the works of Klyachko [38] and Kapovich-Millson [35].

Let Pn be the space of all n-gons with distinct vertices in Euclidean

space R3 . An n-gon P is completely determined by its vertices which

are joined in cyclic order by edges - the oriented line segments. We

identify two n-gons if and only if there exists an orientation preserving

isometry of R3 such that it maps one polygon to another preserving

the cyclic order of edges. Let E(3) stand for the group of all such

isometries; there is a subgroup T 3 of translations which as a Lie group

is di¬eomorphic to R3 . There is also a subgroup SO(3) of E(3) of

origin preserving transformations. Let us ¬x an n-tuple of real positive

numbers a = (a1 , ..., an ) and consider the subspace Y ‚ Pn of polygons

with sides a1 , ..., an modulo above isometries. This space is non-empty

if the inequalities

ai ¤ a1 + a2 + · · · + ai’1 + ai + ai+1 + · · · + an , 1 ¤ i ¤ n

ˆ

are satis¬ed (hat appears on omitted terms). The multiplicative group

R+ of positive real numbers acts on the space Pn by simultaneous scal-

ing of all edges and for two proportional sets a and »a, » ∈ R+ the

corresponding moduli spaces are isomorphic. It happens that the sin-

gular points of Y are exactly those degenerate polygons which have all

the vertices on a single line.

Let us give another description of the space Y . If we ¬x the ¬rst

vertex of a polygon P then the rest of the data is completely determined

by the directions of the consecutive edges. At each vertex the direction

of the next edge is given by a point of the two-sphere S 2 . Therefore we

can think of the space Y as given by the quotient of the subset X of

(S 2 )n := S 2 — · · · — S 2

n

by the diagonal action of the group SO(3). The subset X of (S 2 )n

is simply de¬ned by the closing condition a1 x1 + · · · an xn = 0, where

xi is the coordinate on the i-th multiple of (S 2 )n (we think here of

S 2 embedded in the standard fashion into R3 ). This equation simply

means that we actually have a polygon - the end of the last edge is

exactly the ¬rst vertex. The results of Deligne-Mostow [21] imply that

106 CHAPTER 3. HODGE THEORY

the space Y is the so-called weighted quotient of (S 2 )n with weights

(a1 , ..., an ) with its product symplectic structure (ω)n , where ω is the

standard symplectic structure on S 2 given by its volume form. Thus,

according to [21] the smooth locus of the space Y is itself a symplectic

manifold and its symplectic form is integral if all the numbers ai are

integers. Indeed, the tangent space to X can be easily identi¬ed. Let

P = (u1 , ..., un ) be a point of X (here each ui is a vector in R3 of length

ai ). We have

TP X = {(v1 , ..., vn ); vi ∈ R3 , vi · ui = 0, ai vi = 0}.

i

The condition vi · ui = 0 simply tells us that vi is tangent to S 2 at ui

and the condition i ai vi = 0 is the mathematical way of saying that

the closing condition should be satis¬ed if we deform the polygon P a

bit. One can simply interpret vi as in¬nitesimal deformations of ui in

the perpendicular direction. (Since side lengths are ¬xed, we do not

deform in parallel direction.)

The Lie algebra of the group SO(3) is so(3), which is the same

as R3 with the cross product serving as the Lie bracket. Therefore,

we can write that T Y = T X/R3 , where w ∈ R3 acts on the n-tuple

(v1 , ..., vn ) ∈ TP X as (w — v1 , ..., w — vn ). We also can describe the

complex structure J on the space Y . Let (v1 , ..., vn ) ∈ TP X as above;

we let J(v1 , ..., vn ) = (r1 — v1 , ..., rn — vn ), where ri = ui /ai . It is easy

to check that J 2 = ’1, because vi ⊥ ui .

The moduli space of polygons has to do with the representations

of SU (2). Let n ≥ 1 and let Vn be the irreducible representation of

SU (2) of dimension n + 1. It turns out that the space of invariants of

the action of the group SU (2) on the space

Va1 — Va2 — · · · — Van

is non-zero if and only if

(1) the sum i ai is even

(2) all the polygon inequalities are satis¬ed:

ai ¤ a1 + a2 + · · · + ai’1 + ai + ai+1 + · · · + an , 1 ¤ i ¤ n.

ˆ

3.4. HODGE DECOMPOSITION IN DIMENSION 1 107

3.4 Hodge decomposition in dimension 1

Let M be a K¨hler manifold and let Aj (M ) be the space of real-valued

a

smooth di¬erential j-forms on M . The de Rham cohomology groups

H j (M, R) are cohomology of the complex