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family of lines lt parameterized by t в€€ [в€’Оµ, Оµ]; so for each t we have a
line lt and l0 = l. Next we pick a non-zero vector v в€€ l and look at
the perpendicular hyperplane lвЉҐ + 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 в†’ lвЉҐ
deп¬Ѓned by
d d
v в†’ p(t)|t=0 = q(t)|t=0 .
dt dt
Sometimes it is convenient to identify Hom(l, lвЉҐ ) lв€— вЉ— lвЉҐ .
The hermitian form (, ) on Cn+1 gives rise to a hermitian form on
any subspace like l or lвЉҐ , as well as on Hom(l, lвЉҐ ) if we put for П€ в€€
Hom(l, lвЉҐ ) the norm
||П€(v)||
||П€|| = , v в€€ l, v = 0.
||v||
Thus we got a hermitian form on the tangent space to CPn . Let us
write it down explicitly.
П†
We take U0 вЉ‚ CPn which is identiп¬Ѓed with Cn via Cn в†’ U0 , П†(z1 , ..., zn ) =
[1 : z1 : В· В· В· : zn ]. Let us take a tangent vector u = (u1 , ..., un ) at a point
102 CHAPTER 3. HODGE THEORY

(w1 , ..., wn ) в€€ Cn and consider the parametric curve w + tu, which cor-
responds to the point [1 : w1 + tu1 : В· В· В· : wn + tun ] в€€ U0 вЉ‚ CPn . The
derivative at t = 0 of this curve in CPn gives us a linear map
О± : l =< w >в†’ lвЉҐ Cn+1 /l,

where l is spanned by w. The map О± sends w to
( ui wi )(1, w1 , ..., wn )
ВЇ
i
О±(w) = (0, u1 , ..., un ) в€’ ,
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
| ВЇ
n
|ui |2 в€’
2
||О±(w)|| 1 1+ |wi |2
H(u, u) = = =
n
||w||2 |wi |2
1+ 1
n
|ui |2 )(1 + |ui |2 ) в€’ | i ui wi |2
( ВЇ
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 Aв€— its transposed
complex conjugate then
n n
2 в€’2
|wj |2 )Id в€’ w В· wв€— ],
H = (1 + |wi | ) [(1 +
i=1 j=1

where Id is the identity matrix and now
в€—
H(v1 , v2 ) = v2 Hv1 .
From the matrix notation one can see that H is positive deп¬Ѓnite, since
H has all positive eigenvalues.
We will often use the following
k
LEMMA 3.2.5 Let (X, J, П‰) be a KВЁhler manifold and let Y в†’ X be
a
a complex submanifold. Then (Y, k в€— J, k в€— П‰) is also a KВЁhler manifold.
a
Proof. In fact, k в€— J and k в€— П‰ simply are the restrictions of the operator J
and the form П‰ to the tangent space T Y . Using the identity П‰(v, w) =
g(Jv, w) we can easily see why k в€— П‰ 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 .
ВЁ
3.2. KAHLER MANIFOLDS 103

COROLLARY 3.2.6 Every projective manifold is KВЁhler .
a
Let us give another interpretation of a KВЁhler form.
a
PROPOSITION 3.2.7 Let (M, J) be a complex в€љ manifold of dimen-
sion n equipped with a hermitian form H = g + в€’1П‰. Then M is
KВЁhler if and only if for any x в€€ M we can п¬Ѓnd complex coordinates
a
(z1 , ..., zn ) near x such that
в€љ
в€’1 n
П‰=в€’ dzi в€§dВЇi + R,
z
2 i=1

where R vanishes to order 2 at x.
в€љ
в€’1 n
Proof. (Sketch.) First, if П‰ = в€’ 2 i=1 dzi в€§dВЇi +R, then П‰ is closed,
z
because dП‰(x) = dR(x) = 0, since R vanishes to order 2 at x and x can
be chosen arbitrarily.
Conversely, if M is a KВЁhler manifold, and if x в€€ 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 , В· В· В· , zn ). Consider then
the Taylor expansion П‰ = П‰0 + П‰1 + П‰2 + В· В· В·, where П‰j = fijk dzk в€§dВЇl z
and fijk is a homogeneous polynomial in (z1 , ..., zn , z1 , ..., zn ) of degree
ВЇ ВЇ
j. (We also know that П‰ is of type (1, 1).) Let П‰0 = aij dzi в€§dВЇj ,
z
where (aij ) is a hermitian matrix. We actually can assume that
в€љ
в€’1
П‰0 = в€’ dzj в€§dВЇj ,
z
2
since every positive-deп¬Ѓnite hermitian matrix can be transformed to the
identity matrix by a linear change of variables. Now we pay attention
to П‰1 = gkl dzk в€§dВЇl , where gkl = m (bmkl zm + cmkl zm ) is a linear
z ВЇ
form such that gkl + glk = 0, or, equivalently, cmkl = в€’bmlk . Next
we introduce the group G of formal biholomorphic maps (z1 , ..., zn ) в†’
(w1 , ..., wn ) of type
zj = w j + djkl wk wl .
kв‰¤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 aп¬ѓne space
2
104 CHAPTER 3. HODGE THEORY

X = {П‰1 , dП‰1 = 0}. The dimension of this aп¬ѓne space is not hard
to compute. We notice that the complex dimension of the space of
coeп¬ѓcients bmkl is n3 and the coeп¬ѓcients cmkl are expressed in terms of
bmkl . The condition that dП‰1 = 0 adds n2 (n в€’ 1) real linear conditions.
More precisely, it means that bmkl is symmetric in k and l. Thus,

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

The action of the group G on the space X is п¬Ѓxed point free. This fol-
lows from the general fact that if an isometry of a Riemannian manifold
M п¬Ѓxes a point x в€€ M and acts trivially on Tx M , then it is the iden-
tity map. This fact also holds true for formal jets of diп¬Ђeomorphisms,
as can be seen from an inspection of the PDEвЂ™s satisп¬Ѓed by a Killing
vector п¬Ѓeld. Since dim(X) = dim(G) we conclude that G acts on X
transitively. Thus we can arrange that П‰1 = 0.

Example. Let us have the following 2-form on C:
в€љ
в€’1
в€’ dwв€§dw + (w в€’ w)dwв€§dw.
ВЇ ВЇ ВЇ
2
в€љ
By the holomorphic change of variable z = w + в€’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 П‰. If one uses substitutions

zj = w j + djkl wk wl wm ,
kв‰¤lв‰¤m

which form a group denoted by G3 , then the dimension of the complex
3
space of G3 -orbits in the vector space of П‰2 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 KВЁhler
a
Here we shall give an interesting example of a KВЁhler manifold, which
a
is the moduli space of n-gons in R3 with п¬Ѓxed side lengths. It turns out
ВЁ
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  and Kapovich-Millson .
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 diп¬Ђeomorphic to R3 . There is also a subgroup SO(3) of E(3) of
origin preserving transformations. Let us п¬Ѓx an n-tuple of real positive
numbers a = (a1 , ..., an ) and consider the subspace Y вЉ‚ Pn of polygons
with sides a1 , ..., an modulo above isometries. This space is non-empty
if the inequalities

ai в‰¤ a1 + a2 + В· В· В· + aiв€’1 + ai + ai+1 + В· В· В· + an , 1 в‰¤ i в‰¤ n
Л†

are satisп¬Ѓed (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 О»a, О» в€€ 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 п¬Ѓx the п¬Ѓrst
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 Г— В· В· В· Г— S 2
n

by the diagonal action of the group SO(3). The subset X of (S 2 )n
is simply deп¬Ѓned by the closing condition a1 x1 + В· В· В· 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 п¬Ѓrst vertex. The results of Deligne-Mostow  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 (П‰)n , where П‰ is the
standard symplectic structure on S 2 given by its volume form. Thus,
according to  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 identiп¬Ѓed. 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 в€€ R3 , vi В· ui = 0, ai vi = 0}.
i

The condition vi В· 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 satisп¬Ѓed if we deform the polygon P a
bit. One can simply interpret vi as inп¬Ѓnitesimal deformations of ui in
the perpendicular direction. (Since side lengths are п¬Ѓxed, 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 в€€ R3 acts on the n-tuple
(v1 , ..., vn ) в€€ TP X as (w Г— v1 , ..., w Г— vn ). We also can describe the
complex structure J on the space Y . Let (v1 , ..., vn ) в€€ TP X as above;
we let J(v1 , ..., vn ) = (r1 Г— v1 , ..., rn Г— vn ), where ri = ui /ai . It is easy
to check that J 2 = в€’1, because vi вЉҐ ui .
The moduli space of polygons has to do with the representations
of SU (2). Let n в‰Ґ 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 вЉ— Va2 вЉ— В· В· В· вЉ— Van

is non-zero if and only if
(1) the sum i ai is even
(2) all the polygon inequalities are satisп¬Ѓed:

ai в‰¤ a1 + a2 + В· В· В· + aiв€’1 + ai + ai+1 + В· В· В· + an , 1 в‰¤ i в‰¤ n.
Л†
3.4. HODGE DECOMPOSITION IN DIMENSION 1 107

3.4 Hodge decomposition in dimension 1
Let M be a KВЁhler manifold and let Aj (M ) be the space of real-valued
a
smooth diп¬Ђerential j-forms on M . The de Rham cohomology groups
H j (M, R) are cohomology of the complex
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