No.
6]
Proc.
Japan
Acad.,
49
(1973)
377
81.
On
Deformations
of
Quintic
Surfaces
By
Eiji
HORIKAWA
University
of
Tokyo
(Comm.
by
Kunihiko
KODAIRA,
M.
J.
A.,
June
12,
1973)
Let
So
be
a
non-singular
hypersurface
of
degree
5
in
the
projective
3-space
P
defined
over
C.
For
brevity,
we
call
So
a
non-singular
quin-
tic
surface.
By
a
surface,
we
shall
mean
a
compact
complex
manifold
of
com-
plex
dimension
2,
unless
explicit
indications
are
given.
We
say
that
a
surface
S
is
a
deformation
of
So
if
there
exists
a
finite
sequence
of
sur-
faces
So,
S,
.,
S,
.,
S
S
such
that,
for
each
k,
S
and
S_
belong
to
one
and
the
same
complex
analytic
family
of
surfaces.
If
S
is
a
deformation
of
a
non-singular
quintic
surface,
S
has
the
following
numerical
characters:
(.)
pq-4,
q=0,
c--5,
where
p,
q
and
c
denote
the
geometric
genus,
the
irregularity
and
the
Chern
number
of
S,
respectively.
In
particular
S
is
an
algebraic
sur-
face
(see
[5],
Theorem
9).
Moreover,
since
So
is
minimal,
Theorem
23
of
Kodaira
[5]
asserts
that
(**)
S
is
minimal.
In
this
note,
we
shall
give
a
statement
of
the
results
on
the
struc-
tures
and
deformations
of
surfaces
which
satisfy
the
conditions
(.)
and
(**).
Details
will
be
published
elsewhere.
1.
Structures.
Theorem
1.
Let
S
be
a
minimal
algebraic
surface
with
p=4,
q=0,
and
c-5.
Then
the
canonical
system
]KI
on
S
has
at
most
one
base
point.
There
are
two
cases"
Case
I.
[KI
has
no
base
point.
In
this
case,
there
exists
a
bira-
tional
holomorphic
map
f:
S-.S’
of
S
onto
a
(possibly
singular)
quintic
surface
S’
in
P.
S’
has
at
most
rational
double
points
as
its
singular-
ities.
Case
II.
KI
has
one
base
point
b.
Let
z"
S--.S
be
the
quadric
transformation
with
center
at
b.
Then
this
case
is
divided
as
follows"
Case
IIa.
There
exists
a
sur]ective
holomorphic
map
f’S
--P
P
of
degree
2.
Case
IIb.
There
exists
a
sur]ective
holomorphic
map
f"
S-.
of
degree
2,
where
denotes
the
Hirzebruch
surface
of
degree
2,
i.e.,
is
a
rational
ruled
surface
with
a
section
o
with
(z/0)=-2.
The
proof
is
based
on
a
detailed
analysis
of
the
rational
map
378
E.
HORKAWA
[Vol.
49,
S--*P
defined
by
the
canonical
system
IK].
The
holomorphic
maps
f
in
the
above
statement
are
derived
from
.
Corollary.
If
KI
has
a
base
point,
then
there
exists
a
sur]ective
holomorphic
map
g"
S--.P
whose
general
fibre
is
an
irreducible
non-
singular
curve
of
genus
2.
In
particular,
the
rational
map
q
defined
by
the
bicanonical
system
12KI
is
not
birational.
Conversely,
we
can
construct
every
surface
of
type
II
as
follows"
First
we
construct
a
double
covering
S’
of
W-P
P
or
.
with
ap-
propriate
branch
locus
on
W.
S’
is
a
normal
surface.
Let
S
be
the
minimal
resolution
of
singularities
of
S’
(see
[1],
p.
81).
We
construct
S’
so
that
S
contains
one
exceptional
curve
E.
Contracting
E
to
a
point,
we
obtain
a
minimal
algebraic
surface
with
p--4,
q--0,
and
c-5.
2.
Deformations.
First,
we
give
some
results
on
small
defor-
mations
of
a
surface
in
consideration.
Proposition
1.
i)
The
classes
of
surfaces
of
type
I
and
of
type
IIa
are,
respectively,
closed
under
small
deformations.
ii)
A
surface
of
type
I
and
a
surface
of
type
IIa
do
not
belong
to
one
and
the
same
family
(with
non-singular
base
space).
Theorem
2.
Let
S
be
a
surface
o
type
IIb
of
which
the
canonical
bundle
is
ample.
Let
p"
-M
be
the
Kuranishi
family
of
deformations
of
S=p-(0)
with
0
M
(see
[7]).
Then
i)
M=MoU
M
where
each
M
(i--0,
1)
is
a
40-dimensional
mani-
fold,
ii)
N=
Mo
M
is
a
39-dimensional
manifold,
iii)
St-p-(t)
is
a
non-singular
quintic
surface,
a
surface
of
type
IIa,
or
a
surface
of
type
IIb
according
as
t
Mo--N,
t
M--N,
or
t
N.
We
now
indicate
an
outline
o
the
proof
of
Theorem
2.
Let
S
denote
a
surface
as
in
Theorem
2
and
let
s
denote
the
sheaf
of
germs
of
holomorphic
vector
fields
on
S.
We
have
dim
H(S,
Os)--41
and
dim
H(S,
)-1.
Let
D--(t
e
C’lt]}
with
0
sufficiently
small.
Then
there
exists
a
(0,
1)-form
(t)
with
coefficients
in
Os
depending
holomorphically
on
t
e
D
such
that
M=
{t
e
D"
H[(t),
(t)]--0},
where
H
denotes
the
projection
onto
the
space
o]
harmonic
forms
with
respect
to
a
Hermitian
metric
on
S
and
[,
denotes
the
Poisson
bracket.
Since
dim
H(S,
0s)=1,
we
may
regard
H[(t),,(t)]
as
a
holomorphic
function
on
D.
We
can
prove
that
H[((t),
(t)]
tt
+
(higher
terms),
or
an
appropriate
choice
of
coordinates
(t,
t,.,.,
t)
on
D.
On
the
other
hand,
applying
an
improved
orm
of
Theorem
2’
[3]
to
the
holomorphic
map
g"
SP
in
Corollary
to
Theorem
1,
we
can
No.
6]
Quintic
Surfaces
379
construct
a
40-dimensional
effectively
parametrized
family
p"
3-M1
of
deformations
of
S=p(0)
with
0
e
M
(see
[6],
Definition
6.4).
It
follows
that
H[(t),
(t)]
decomposes
into
a
product
q(t)r(t)
with
q(t)=t2+(higher
terms),
r(t)=t+(higher
terms).
This
proves
the
as-
sertion
i).
Other
assertions
can
be
proved
by
applying
the
general
theory
on
deformations
of
holomorphic
maps
[4].
It
seems
difficult
to
study
the
deformations
of
a
surface
of
which
the
canonical
bundle
is
not
mple.
However,
applying
a
result
of
Brieskorn
([1],
[2]),
we
can
prove
Theorem
:.
Every
minimal
algebraic
surface
with
p-4,
q--O,
and
c-5,
is
a
deformation
of
a
non-singular
quintic
surface.
References
[1]
Brieskorn,
E."
ber
die
AuflSsung
gewisser
Singularitten
yon
holomor-
phen
Abbildungen.
Math.
Ann.,
1{1,
76-102
(1966).
[2
__A:
Die
AuflSsung
der
rationalen
Singularititen
holomorpher
Abbildun-
gen.
Math.
Ann.,
178,
255-270
(1968).
3
Horikawa,
E.:
On
deformations
of
holomorphic
maps.
Proc.
Japan
Acad.,
aS,
52-55
(1972).
4
:
On
deformations
of
holomorphic
maps.
I
(to
appear
in
J.
Math.
Soc.
Japan,
2.5
(1973)),
II
(in
preparation).
5
Kodaira,
K.:
On
the
structure
of
compact
complex
analytic
surfaces.
I.
Amer.
J.
Math.,
86,
751-798
(1964).
6
Kodaira,
K.,
and
Spencer,
D.
C.:
On
deformations
of
complex
analytic
structures.
I,
II.
Ann.
of
Math.,
67,
328-466
(1958).
7
Kuranishi,
M."
On
the
locally
complete
families
of
complex
analytic
struc-
tures.
Ann.
of
Math.,
7.5,
536-577
(1962).