PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S.
CARLOS T. SIMPSON
Moduli of representations of the fundamental group of
a smooth projective variety II
Publications mathématiques de l’I.H.É.S., tome 80 (1994), p. 5-79
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MODULI
OF
REPRESENTATIONS
OF
THE
FUNDAMENTAL
GROUP
OF
A
SMOOTH
PROJECTIVE
VARIETY.
II
by
CARLOS
T.
SIMPSON
Introduction
This
second
part
is
devoted
to
the
subject
of
the
title,
moduli
spaces
of
represen-
tations
of
the
fundamental
group
of
a
smooth
complex
projective
variety
X.
We
study
three
moduli
spaces
for
related
objects.
The
Betti
moduli
space
Mg(X,
n)
is
a
coarse
moduli
space
for
rank
n
representations
of
the
fundamental
group
of
the
usual
topological
space
X^.
A
vector
bundle
with
integrable
connection
is
a
pair
(E,
V)
where
E
is
a
vector
bundle
and
V
:
E
->
E
®
Q^
is
an
operator
satisfying
the
Leibniz
rule
and
V
2
=
0.
The
de
Rham
moduli
space
Mjy^(X,
n}
is
a
coarse
moduli
space
for
rank
n
vector
bundles
with
integrable
connection
on
X.
A
Higgs
bundle
[Hil]
[Si5]
is
a
pair
(E,
9)
where
E
is
a
vector
bundle
and
9
:
E
—>
E
®
Q^
ls
a
morphism
of
6^-modules
such
that
9
A
9
=
0.
There
is
a
condition
of
semistability
analogous
to
that
for
vector
bundles,
but
only
concerning
subsheaves
preserved
by
9.
The
Dolbeault
moduli
space
M^(X,
n)
is
a
coarse
moduli
space
for
rank
n
semistable
Higgs
bundles
with
Chern
classes
vanishing
in
rational
cohomology.
In
all
three
cases,
the
objects
in
question
form
an
abelian
category
in
which
we
can
apply
the
Jordan-Holder
theorem.
Let
gr(E)
denote
the
direct
sum
of
the
subquotients
in
a
Jordan-Holder
series
for
E,
and
say
that
E^
is
Jordan
equivalent
to
Eg
if
gr(E^)
^
gr(Eg).
The
points
of
the
coarse
moduli
spaces
parametrize
Jordan
equivalence
classes
of
objects.
The
constructions
of
these
moduli
spaces
are
reviewed
in
§
5.
The
construction
of
MB
is
a
classical
one
from
the
theory
of
representations
of
discrete
groups.
The
cons-
truction
of
Mp^
follows
from
the
construction
of
Part
I,
§
4,
for
the
case
where
A^
=
Q^
is
the
full
sheaf
of
rings
of
differential
operators
on
X.
We
give
two
constructions
of
M^.
One
is
based
on
an
interpretation
of
Higgs
sheaves
as
coherent
sheaves
on
T*
X,
and
uses
the
construction
of
the
moduli
space
of
coherent
sheaves
constructed
in
Part
I,
§
1.
The
other
consists
of
applying
the
general
construction
of
Part
I,
§
4,
to
the
case
A^
==
Sym-(TX).
The
three
types
of
objects
are
related
to
each
other.
The
Riemann-Hilbert
corres-
pondence
between
systems
of
ordinary
differential
equations
and
their
monodromy
repre-
6
CARLOS
T.
SIMPSON
sentations
provides
an
equivalence
of
categories
between
vector
bundles
with
integrable
connection
and
representations
of
the
fundamental
group.
To
(E,
V)
corresponds
the
monodromy
of
the
system
of
equations
V(<?)
=
0.
The
correspondence
between
Higgs
bundles
and
local
systems
of
[Hil],
[Do3],
[Co],
[Si2],
and
[Si5]
gives
an
equivalence
of
categories
between
semistable
Higgs
bundles
with
vanishing
rational
Chern
classes,
and
representations
of
the
fundamental
group.
Together,
these
correspondences
give
isomorphisms
of
sets
of
points
Ms(X,
n)
^
M^(X,
n)
^
M^(X,
n).
In
§
7
we
use
the
analytic
results
of
Part
I,
§
5,
to
show
that
the
first
map
is
an
isomorphism
of
the
associated
complex
analytic
spaces,
and
that
the
second
is
a
homeomorphism
of
usual
topological
spaces.
There
is
a
natural
algebraic
action
of
the
groupe
C*
on
the
moduli
space
Mp^(X,
n),
given
by
^(E,
9)
=
(E,
^p),
and
our
identifications
thus
give
a
natural
action—no
longer
algebraic—on
the
space
of
representations.
The
fixed
points
of
this
action
are
exactly
those
representations
which
come
from
complex
variations
of
Hodge
structure
[Si5].
Although
Mp^
is
not
compact,
the
properness
of
Hitchin's
map
(Theorem
6.11)
implies
that
M^i(X,
n)
contains
the
limits
of
points
tE
as
t
->
0.
This
yields
the
conclusion
that
any
representation
of
the
fundamental
group
may
be
deformed
to
a
complex
variation
of
Hodge
structure
(Corollary
7.19
below).
This
theorem
was
in
some
sense
the
principal
motivation
for
constructing
the
moduli
spaces.
See
[Si5]
for
more
details
on
some
consequences.
The
reason
for
the
terminologies
Betti,
de
Rham
and
Dolbeault
is
that
these
moduli
spaces
may
be
considered
as
the
analogues
for
the
first
nonabelian
cohomology,
of
the
Betti
cohomology,
the
algebraic
de
Rham
cohomology,
and
the
Dolbeault
cohomology
(B^^H^X,^)
ofX.
The
first
nonabelian
cohomology
set
H^X,
G\(n,
C))
is
the
set
of
isomorphism
classes
of
rank
n
representations
of
T^(X).
This
has
a
structure
of
topological
space,
but
it
is
not
Hausdorff.
The
universal
Hausdorff
space
to
which
it
maps
is
the
Betti
moduli
space
Mg(X,
n).
To
explain
the
analogies
for
the
de
Rham
and
Dolbeault
spaces,
we
have
to
digress
to
discuss
the
Cech
realizations
of
the
cohomology
groups
with
complex
coefficients.
The
algebraic
de
Rham
cohomology
is
the
hypercohomology
of
X
with
coefficients
in
the
algebraic
de
Rham
complex
Q^
4.
O.^
4.
...
If
X
=
U
U^
is
an
affine
open
covering
of
X,
and
if
we
denote
multiple
intersections
by
multiple
indices,
then
the
cocycles
defining
Hp^(X,
C)
consist
of
the
pairs
of
collections
({g^},
{
a^})
where
^ap
are
regular
functions
U^p
->•
C,
and
a^
are
one-forms
on
U^,
such
that
Sa^
==:
S^v
T"
S<x.f9
^03)
=
^a
-
^
and
d(a^)
==
0.
MODULI
OF
REPRESENTATIONS.
II
7
Addition
of
the
coboundary
of
a
collection
{
^
},
where
s^
are
regular
functions
U^
->
C,
changes
the
pair
({
g^
},
{
a,
})
to
({
^p
+
^
-
^
},
{
^
+
^J}).
The
group
of
cocycles
modulo
coboundaries
is
H^(X,
C).
The
nonabelian
case
has
formulas
which
are
more
complicated,
but
which
reduce
to
the
above
if
the
coefficient
group
is
abelien.
A
vector
bundle
with
integrable
connection
is
defined
by
a
pair
({^pM
A^}),
where
g^
:
Ua3
->
G\(n,
C)
are
the
gluing
functions
for
the
vector
bundle,
and
A^
are
n
X
n
matrix-valued
one
forms
defining
the
connection
V
=
d
+
A^.
These
are
subject
to
the
conditions
^3Y
^a3
=
gix-f9
A
^
=
<?a3
1
^C?ap)
+
^a'P
1
A3^p,
and
rf(AJ
+
A^
A
A,
=
0.
A
change
of
local
frames
by
a
collection
of
regular
functions
s^:V^->
Gl{n,
C)
changes
the
pair
{{g^},{\})
to
({^
1
<?ap
^a
M
^
'
A^
+
^a
1
^a)}).
The
set
of
pairs
up
to
equivalence
given
by
such
changes
of
frames,
is
the
first
nonabelian
de
Rham
cohomo-
logy
set
H^(X,
G\(n,
C)),
the
set
of
isomorphism
classes
of
vector
bundles
with
integrable
connection
on
X.
A
similar
if
somewhat
looser
interpretation
gives
an
analogy
between
the
abelian
Dolbeault
cohomology
group
H^X,
0^
C
H°(X,
Q^
and
the
first
nonabelian
Dolbeault
cohomology
set
H^X,
G\{n,
C)),
the
set
of
isomorphism
classes
of
Higgs
bundles
(E,
9)
which
are
semistable
with
vanishing
rational
Ghern
classes.
Here
E
is
a
vector
bundle
and
9
e
H°(X,
End(E)
00^
^)-
Such
a
pair
may
be
given
a
cocycle
description
similar
to
the
above
(just
eliminate
the
terms
involving
d).
The
conditions
of
semistability
and
vanishing
Chern
classes
are
new.
Following
this
interpretation,
we
can
think
of
the
first
nonabelian
cohomology
as
a
nonabelian
motive
in
a
way
analogous
to
[DM],
with
its
Betti,
de
Rham
and
Dolbeault
realizations.
It
would
be
good
to
have
/-adic,
and
crystalline
interpretations
in
charac-
teristic
p.
We
treat
everything
in
the
relative
case
of
a
smooth
projective
morphism
X
->
S
to
a
base
scheme
of
finite
type
over
C.
This
creates
some
difficulties
for
the
Betti
moduli
spaces:
we
have
to
introduce
the
notion
of
local
system
of
schemes
over
a
topological
space.
The
relative
Betti
spaces
M^X/S,
n)
are
local
systems
of
schemes
over
S^.
The
de
Rham
and
Dolbeault
moduli
spaces
are
schemes
over
S,
whose
fibers
are
the
de
Rham
and
Dolbeault
moduli
spaces
for
the
fibers
X^.
The
interpretation
in
terms
of
nonabelian
cohomology
suggests
the
existence
of
a
Gauss-Manin
connection
on
Mp^(X/S,
n),
a
foliation
transverse
to
the
fibers
which
when
integrated
gives
the
transport
corresponding
to
the
local
system
of
complex
analytic
spaces
Mg^X/S,^).
We
construct
this
connection
in
§
8,
using
Grothendieck's
idea
of
the
crystalline
site.
In
§
9,
we
treat
the
case
of
other
coefficient
groups.
If
G
is
a
reductive
algebraic
group,
we
may
define
the
Betti
moduli
space
Mg(X,
G)
to
be
the
coarse
moduli
space
8
CARLOS
T.
SIMPSON
for
representations
of
7^(X)
in
G.
We
construct
the
de
Rham
moduli
space
M^(X,
G)
for
principal
G-bundles
with
integrable
connection,
and
the
Dolbeault
moduli
space
Mp^(X,
G)
for
principal
Higgs
bundles
for
the
group
G,
which
are
semistable
with
vanishing
rational
Ghern
classes,
and
extend
the
results
of
§§
7
and
8
to
these
cases.
One
corollary
is
a
result
valid
for
representations
of
any
finitely
generated
discrete
group
Y:
if
G
->
H
is
a
morphism
of
reductive
algebraic
groups
with
finite
kernel,
then
the
resulting
morphism
of
coarse
moduli
spaces
M(Y,
G)
->
M(Y,
H)
is
finite
(Corol-
lary
9.16).
Parallel
to
the
discussion
of
moduli
spaces,
we
discuss
the
Betti,
de
Rham
and
Dolbeault
representation
spaces
R^X.^/i),
R^(X,
A;,
?t),
and
R^i(X,^,%).
These
are
fine
moduli
spaces
for
objects
provided
with
a
frame
for
the
fiber
over
a
base
point
x
e
X
(here
we
assume
that
X
is
connected).
There
are
relative
versions
for
X/S
where
the
frames
are
taken
along
a
section
^
:
S
->
X,
and
there
are
versions
for
principal
objects
for
any
linear
algebraic
group.
In
§
10,
we
discuss
the
local
structure
of
the
singularities
of
the
representation
spaces,
using
the
deformation
theory
associated
to
a
differential
graded
Lie
algebra
developed
by
Goldman
and
Millson
[GM].
By
Luna's
^tale
slice
theorem,
this
also
gives
information
about
the
local
structure
of
the
moduli
spaces.
The
differential
graded
Lie
algebra
controlling
the
deformation
theory
of
a
principal
vector
bundle
with
integrable
connection
or
a
principal
Higgs
bundle
is
formal
if
the
object
is
reductive
(in
other
words,
corresponds
to
a
closed
orbit
under
the
action
of
G
on
R(X,
x,
G)).
By
the
theory
of
[GM],
this
implies
that
the
representation
space
has
a
singularity
defined
by
a
quadratic
form
on
its
Zariski
tangent
space.
Furthermore,
the
differential
graded
Lie
algebras
controling
the
deformation
theories
of
the
flat
bundle
and
the
corresponding
Higgs
bundle
are
the
same.
This
gives
a
formal
isomorphism
between
the
singularities
of
the
de
Rham
(or
Betti)
representation
space
and
the
singu-
larities
of
the
Dolbeault
representation
space,
at
semisimple
points
which
correspond
to
each
other.
This
isosingularity
principle
holds
also
for
the
singularities
of
the
moduli
spaces.
The
homeomorphism
between
Mp^
and
Mp^
is
not
complex
analytic,
so
these
local
formal
isomorphisms
are
not
directly
related
to
the
global
homeomorphism.
Finally,
in
§
11
we
discuss
the
case
of
representations
of
the
fundamental
group
of
a
Riemann
surface
of
genus
g
^
2.
Hitchin
calculated
the
cohomology
in
the
case
of
rank
two
projective
representations
of
odd
degree,
where
the
moduli
space
is
smooth
[Hi
I],
We
do
not
attempt
to
go
any
further
in
this
direction.
We
simply
treat
the
most
elementary
property,
irreducibility
(which
is
more
or
less
a
calculation
of
H°).
We
treat
the
case
of
representations
of
degree
zero,
so
the
moduli
space
has
singularities
corresponding
to
reducible
representations;
we
prove
that
the
singularities
are
normal.
The
technique
is
to
use
the
fact
that
the
Betti
moduli
space
is
a
complete
intersection,
and
apply
Serre's
criterion
(following
a
suggestion
ofM.
Larsen,
and
prompted
by
a
question
ofE.
Witten).
We
have
to
verify
that
there
are
no
singularities
in
codimension
one.
To
prove
irredu-
cibility
it
suffices
to
prove
that
the
space
is
connected,
which
we
do
by
a
simple
argument
derived
from
Hitchin's
method
for
calculating
the
cohomology.