BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 18, Number 2, April 1988
THE DEFORMATION THEORY OF REPRESENTATIONS
OF FUNDAMENTAL GROUPS
OF COMPACT KÀHLER MANIFOLDS
WILLIAM M. GOLDMAN AND JOHN J. MILLSON
We work over a fixed field k, either the real or complex numbers. A
quadratic cone in a k-vector space E is an affine variety Q C E defined
by equations
B(u,u) = 0
where B
:
E x E
—•
E' is a bilinear map and E' is a vector space. (E may be
identified with the Zariski tangent space to Q at 0.) Let V be an algebraic
variety and x € V be a point. We say that V is quadratic at x if the analytic
germ of V at x is equivalent to the germ of a quadratic cone at 0.
Let T be a finitely generated group and G a k-algebraic group. We identify
G with its set of k-points, which has the natural structure of a Lie group over
k. Then the set Hom(r, G) of all homomorphisms r —• G equals the set of
k-points of a k-algebraic variety 9t(r,G) (compare [9, 10]).
THEOREM 1. Let Y be the fundamental
group
of a compact Kàhler man-
ifold.
Let p € Hom(r, G) be a representation such that its image p(T) lies in
a compact
subgroup
of G. Then
£K(r,
G) is quadratic at p.
Suppose that
{H
p
>
q
,
Q} is a polarized Hodge structure of weight n, that G
is the group of real points of the isometry group of Q and X = G/V is the
classifying space for polarized Hodge structures of the above type (see Griffiths
[8,
p. 15]). Suppose further M is a complex manifold with fundamental group
T. A representation p: T
—•
G determines a flat principal G-bundle P
p
over
M and an associated X-bundle P
p
XQ X. A horizontal holomorphic V-
reduction of P
p
is a holomorphic section of P
p
XQ X whose differential carries
the holomorphic tangent bundle of M into the horizontal subbundle Th(X)
defined in [8, p. 20].
THEOREM 2. Let M
be
a compact Kàhler manifold
with
fundamental
group
T and X = G/V a classifying space for polarized
Hodge
structures. Suppose
that p: T —• G is a representation such that the associated principal bun-
dle over M admits a horizontal holomorphic V-reduction. Then 9l(r, G) 'is
quadratic at p.
Received by the editors October 15, 1987.
1980 Mathematics Subject Classification (1985 Revision). Primary 53C55, 57M05; Sec-
ondary 14C30, 32J25.
The first author was supported in part by National Science Foundation grant DMS-
86-13576 and an Alfred P. Sloan Foundation Fellowship; the second by National Science
Foundation grant DMS-85-01742.
©1988 American Mathematical Society
0273-0979/88 $1.00 + $.25 per page
153
154
W. M. GOLDMAN AND J. J. MILLSON
If
7r:
E
—•
M is a holomorphic family of smooth polarized projective vari-
eties parametrized by M, then the period mapping which attaches to
a: G
M
the polarized Hodge structure on H
n
('ir~
1
(x)) is a horizontal holomorphic
V-reduction of the principal bundle associated to the monodromy representa-
tion.
COROLLARY.
Suppose
p: T
—>
G is the monodromy of
a
variation of Hodge
structure over M. Then 9t(r, G) is quadratic at p.
The idea behind the proof of Theorem 2 can be applied to a number of
other closely related situations. The following result can be used to construct
deformations of discrete groups of automorphisms of complex hyperbolic space
(compare [2] and [6]).
THEOREM 3. Let M
be
a compact Kàhler manifold
tuith
fundamental
group
T and X = G/K a Hermitian symmetric space with automorphism group G.
Suppose p:Y—>Gisa representation such that the associated principal G-
bundle over M admits a holomorphic K-reduction. Then
ÜH(I\
G) is quadratic
at p.
Perhaps the most important feature of such "quadratic singularity" the-
orems is that the quadratic functions are computable algebraic topological
invariants. Thus we obtain a criterion for nonsingularity of
£K(r,
G) near a
representation p. The Zariski tangent space to 9t(r, G) near p equals the space
Z
1
(r;0Ad p) of Eilenberg-Mac Lane l-cocycles of T with coefficients in the T-
module 0Ad p (given the action defined by the composition r
—•
G
—•
Aut(jj)).
The quadratic cone is defined by the cup-product where the Lie bracket
[ ,
]
:
0 x g
—•
g is used as a coefficient pairing.
COROLLARY. Suppose that M is a compact Kàhler manifold with funda-
mental
group
T and that p: T
—•
G is a representation satisfying the hypotheses
of Theorems 1,2, or 3. Suppose that the cup-product
^(r,0Ad „)
x
Jï^r.BAd
P)
-
#
2
(r,0Ad
P
)
is identically zero. Then <R(r, G) is nonsingular at p.
For G compact, the quadraticity of tangent cones to
ÜH(I\
G) was proved in
Goldman-Millson [7]. When p is a reductive representation of the fundamental
group of a closed surface, it was shown in Goldman [5] that the tangent cone
to 9t(r, G) is quadratic. Recent work of Carlos Simpson (Higgs bundles and
local
systems. II, Princeton Univ. preprint) implies quadratic singularities for
a large class of representations, including reductive representations of surface
groups.
The proofs of Theorems 1, 2, and 3 involve showing that two analytic germs
are equivalent. We replace an analytic germ
(V,
x) first by the complete local
k-algebra 0{y,x) °f V at x; it follows from Artin [1] that the isomorphism
class of 0(v,x) determines the equivalence class of the analytic germ of V at
x. Furthermore this isomorphism class is determined by the functor which
associates to an Artin local k-algebra A the set Hom(0(y
)X
), A) of k-algebra
homomorphisms Hom(0(v»> A), or equivalently the set of A-points of V over
DEFORMATION THEORY OF COMPACT KÀHLER MANIFOLDS 155
x € V. Thus we prove two analytic germs are equivalent by showing their
corresponding functors
Ajf
cal
—•
Sets are naturally isomorphic where
Aj^
cal
denotes the category of Artin local k-algebras.
This will be accomplished by introducing several auxiliary "deformation
theories." A transformation
groupoid
is
a category
C
arising from the action of
a group on the set Obj
C
of objects of
C.
If
C
arises from an action of a group H
and Y is another if-set then we denote by
C
M Y the transformation groupoid
arising from the diagonal action of H on Obj CxY. If F: C
—•
C' is a functor
of transformation groupoids preserving actions of the corresponding groups
on a set Y, then there is a corresponding functor F tx\Y: C M F
—•
C' ixi y ;
then F M y is an equivalence of categories if F is. If
C
is a category we denote
by Iso C the collection of isomorphism classes of objects; an equivalence of
groupoids C
—•
C' induces an isomorphism of sets Iso C
—•
Iso C'.
Fix an Artin local k-algebra A with maximal ideal m; let G
A
denote the
group of A-points of G with its natural structure as a Lie group over k—indeed
G
A
equals the semidirect product of G with the nilpotent normal subgroup
exp(g ® m). If po € Hom(r,G) then the set of A-points of 9t(r,G) over p
0
equals the fiber q+
l
{po) of the map
g*: Hom(r,G
A
)—Hom(I\G)
induced by projection q: G A
—*•
G. Let ZA(PO) (resp. Z'
A
(Po)) denote the
transformation groupoid with set of objects q^ipo), and with morphisms
conjugation by elements of exp(g<g>m) (resp. only the identity morphisms).
Let M be a manifold, x G M and let T =
7Ti
(M,
X).
Let (P, uo) be the flat
principal G-bundle over M with holonomy representation po- Let PA = P*G
G
A
be the principal G^-bundle obtained by enlarging the structure group of
P by G
—•
GA and let i: P
—•
PA be the corresponding inclusion; then the
flat connection
u>o
on P induces a flat connection üo on PA- Let
7A(WO)
(resp.
?A(
U
O))
denote the transformation groupoid with objects the flat connections
S)
on PA such that
i*u>
= uo and morphisms the gauge transformations PA —•
PA
which act trivially on P C PA (resp. and fix the fiber over x). Choose a
basepoint p in the fiber of PA over x. If F is a gauge transformation (resp.
infinitesimal gauge transformation) of
PA,
let e
v
(F) denote the element of G
(resp.
g) corresponding to the action of F at p. If a; is a flat connection on
PA,
let hol
p
(u;) € Hom(r,G>i) denote the holonomy homomorphism of u at
P-
LEMMA. The correspondence (holp,£
p
) defines equivalences of categories
£*(wo)
-•
£A(PO)
and
KM - *ît(A>)
which depend naturally on A.
Thus we replace the functor A i-> Iso £^(po) corresponding to the germ
of 9l(r, G) at po by the naturally isomorphic functor A »-• Iso
7A(
U
O)-
We
next reinterpret the functor A i-+ ?A(C*;O) in the general context of differential
graded Lie algebras.
156
W. M. GOLDMAN AND J. J. MILLSON
Let L denote a differential graded Lie algebra; following Deligne [3] we
define a functor A »-• C(L;A) from A£
cal
to the category of transformation
groupoids as follows. The objects of
C(L;
A) are the elements 77 6 L
1
0 m
satisfying the deformation equation
(1)
dry
+ £[77,77] = 0
and the morphisms are elements of the nilpotent group exp(L° ® m) acting
affinely on L
1
<8>
m
by
(2) exp(tA)
:
77
~
exp(*
ad A)(iy) +
7
~
ex
P
(
*
ad
X
\d\).
ao.
A
If 0 is a Lie algebra and L is a differential graded Lie algebra, then a g-
augmentation of L is a Lie algebra homomorphism e: L —» g, where 9 is
regarded as a differential graded Lie algebra with no nonzero elements of pos-
itive degree. The augmentation
ideal
is L' = Ker e. We obtain an equivalence
of categories
C(L';
A) -•
C(L;
A)
1x3
exp(g
<g>
m)
depending naturally on A.
Two g-augmented differential graded Lie algebras are quasi-isomorphic if
they can be connected by a sequence of homomorphisms of g-augmented
dif-
ferential graded Lie algebras each inducing an isomorphism on cohomology; a
g-augmented differential graded Lie algebra is formal if it is quasi-isomorphic
to one with zero differential. The following basic result states that quasi-
isomorphic differential graded Lie algebras give rise to equivalent deformation
theories:
THEOREM (SCHLESSINGER-STASHEFF [12], DELIGNE [3]). Let k be a
field of characteristic zero and
<p:
L
—•
L be a homomorphism of differential
graded Lie algebras such that the induced maps
H
%
{<p):
H
X
(L)
—*
H
l
(L) are
isomorphisms for i = 0,1 and infective for i = 2. Let A be an Artin local
k-algebra. Then the induced functor
<p+
:
C(L]A)
—•
C{L;A) is an equivalence
of
groupoids
depending naturally on A.
If (L,e) is a g-augmented differential graded Lie algebra with zero
dif-
ferential, then Obj C(L';A) equals the set of A-points over the origin of the
quadratic cone QL C L
1
defined by
[w,
u]
= 0. By applying the above theorem
with the operation
C »-+
C
M
exp(g ® m)
repeatedly we obtain the following general result, relating quadratic singular-
ities to augmented differential graded Lie algebras:
PROPOSITION. Suppose (L,d,e) is a formal ^-augmented differential
graded
Lie
algebra.
Suppose that e: L°
—•
g is surjective and its restriction to
H°(L) C L° is injective. Let Q denote the quadratic cone
QH(L)X9MH°(L)).
Then the functors
A}f
cal
—•
Sets defined by
DEFORMATION THEORY OF COMPACT KÀHLER MANIFOLDS 157
and
A*->IaoC{L
f
',A)
are naturally isomorphic.
These general results on differential graded Lie algebras are applied as
follows. Let ad P = P x<j g be the Lie algebra bundle associated to P by the
adjoint representation and
f2*
(M;
ad P) the differential graded Lie algebra of
ad P-valued exterior differential forms on M with the covariant differential D
associated to u;o- If
w
€ Obj 7A(^O)> then
r\ =
(JJ
-
(Do
€ n
l
(M;z P
A
) = ^{M-^d P)®A
satisfies (1) and the action of morphisms in
7A{^O)
is given by (2)* Thus we
obtain:
LEMMA. Let L denote the ^-augmented differential graded Lie algebra
(n*(M; ad P),e
p
). The functor
defined by w
*->
UJ
—
û)
0
is an isomorphism of
groupoids
depending naturally
on A.
The proofs of Theorems 1, 2 and 3 are completed by applying the above
propositions to the following.
PROPOSITION. Let M be a compact Kàhler manifold and p € Hom(I\G)
be a representation satisfying the hypotheses of
Theorems
1,2, or 3. Let ad Pc
denote the corresponding
flat
g®C-bundle over M and letL = f2*(M;ad Pc)-
Then the gc-augmented differential graded Lie
algebra
(L,D,e
p
) is formal
The proof follows the ideas indicated in [4] proving that the de Rham
algebra of a compact Kàhler manifold is formal. In each case the covariant
differential on Q*(M; ad Pc) decomposes as D = D'+D", where the standard
Kàhler identities
A
D
= 2 A
D
/ = 2 AD» ,
[A,
D')
= iD"*,
[A,
D")
= -iD'*
hold. (For Theorems 2 and 3, one uses Deligne's decomposition of D by
total bidegree—see Zucker [14], also Corlette [2], Simpson [IS].) Since the
spaces of Z?-harmonic, .D'-harmonic, and /^''-harmonic forms coincide and the
'^principle of two types" is satisfied, it follows that
(n*(M;ad Pc),D,e
p
) «- (Ker D',D",e
p
) -> (^*(M;ad P
c
),0,e
p
)
are homomorphisms of g-augmented differential graded Lie algebras which
induce isomorphisms on cohomology.
REMARK. In their paper Symmetry and bifurcation of momentum map-
pings (Comm. Math. Phys. 78 (1981), 455-478), Arms, Marsden and Mon-
crief prove quadratic singularities for level sets of momentum mappings for
certain Poisson actions on symplectic manifolds. Their results may be under-
stood from the above point of view as follows. To an affine Poisson action