One attempt to the <span class="mathjax-formula">$K3$</span> modular function - II

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

H IRONORI S HIGA

One attempt to the K3 modular function - II

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4

^e

série, tome 8, n

^o

1 (1981), p. 157-182

<http://www.numdam.org/item?id=ASNSP_1981_4_8_1_157_0>

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HIRONORI

SHIGA

§

^{0. In the}

previous

paper

([9],

^we

^quote

^{it as the}

^part I )

^{the author}

pointed

^out that the Picard’s modular function

(see [1])

coincides with the

one for a certain

family

^of

elliptic

^{K3 surfaces with 2}

complex parameters.

The

elliptic

^{modular function is} characterized as a function which

gives

the moduli of

elliptic

^{curves. And}

already

^{there are} various extensions of the

theory

^{of the}

elliptic

modular function.

Perhaps

^{one fruitfull extension}

is the one for the Abelian varieties. And the Abelian

variety

^{seems to be a}

natural extension of the

concept

^{of the}

elliptic

^{curve. But the author}

thinks that the

K3

surface is another extension. Then he tried to find a new modular function of several variables

by considering

^a

family

^{of K3}

surfaces. The

part

^{I was the first}

experiment

^{for this}

supposition.

^{In this}

paper ^we

study

^several

types

^of

elliptic

^.g3 surfaces and the

period mapping

for them. And we show that the

period

^is

given

^{as a ratio of two solutions}

of

Appell’s hypergeometric equation .F’1.

We

proceed

^our consideration as the

following.

^We

study

^the

elliptic

^K3

surface X with

following properties (see ^[3]

^{for the}

general theory

^{of the}

elliptic

surface and

[4], [5]

^{for the .g3}

surface):

(i)

^{.X has a}

holomorphic section,

(ii)

^the functional invariant

f

is the constant function

0, (iii)

^{X has five}

singular

^fibres.

By

^{a JOK3}

surface

we mean an

elliptic

^K3 surface with the

properties (i)

and

(ii).

In §

^{1 we find a}

representing equation (1-5)

^{for a JOK3 surface with}

maximum number of

singular

fibres. Next we show that there are 9

types

of JO.K3 surfaces with the

property (iii) (Proposition 1-4).

It will be called a

surface of

^class

(j). Any

JOK3 surface has an

automorphism

o of order

Pervenuto alla Redazione il 31 Gennaio 1980.

three which preserves all fibres. ^Let ll be a domain in P2 defined

by($

⁼

=

[$1, 2 , 3]; i =1= j, 0 for i =1= j}, where [$1 , 1 $2, ^$.1]

^{is a}

homogeneous

^coor-

dinate on P2. ^A

surface 8;

^{of class}

(j)

is determined

by

^a

point $

^{on A. We}

denote it

by Sj($).

^There

uniquely

^{exists a}

holomorphic 2-form T

^on

Sj($),

up ^to constant

factors, because 8;

^{is a .K3 surface. We determine this form}

in

Proposition

^1-5.

In §

^{2 we construct a basis}

system

^{r =}

{Ol, ..., 022}

^of

H2(Sj($), Z)

and a basis

system

^{I* =}

{G,,

^...,

G22}

^of

H2(Sj($), Q)

^with

^{OiGj = ðij (Pro-}

position 2-3). Among themC7,

^...,

C22

^and

G7,

^...,

^G22

^are

given

^{as divisors.}

The intersection matrix A. ⁼

(Gi Gj )1 si, sg is

determined in

Diagram

^4.

In §

^{3 we consider a}

period integral

for a surface

8;($).

^Because

C,,

^...,

C,,

^are

algebraic cycles,

^{we have}

q;(8)

^{= 0}

for i

= 7,..., 22.

^{And it occurs a relation}

^{(3 -4)}

among ?IIL 7 - - - 7 716 - Con-

sequently

^the

period mapping 0

⁼

[,q, y ..., q,, ]

^{reduces to a}

mapping

^{from A}

to P2. The

image Ø(A)

^is contained in a

domain Dj (j = 1,

...,

9),

^where

92,

^j

is determined

by Diagram

^{6 and}

(3-6).

^The

domain Qj

^is

biholomorphically equivalent

^{to a}

hyperball (Proposition 3-1).

And we define a

monodromy covering

^domain

A

over A at the

biginning

of this section. Then 0 becomes

injective ^{on A} (Proposition 3-2).

In

§

^{4 we} consider the

monodromy

^of

q,($).

^{And we} determine the

Appell’s hypergeometric

^function

.I’1

^which coincides with

ni(03BE) ^(Proposi-

tion

4-1).

In §

^{5 we}

study

^the behavior of 0 on the

hyperplane H ij

⁼

($; = $; ;

i

=1= j, i = 1, 2,

³

and j

⁼

0, 1, 2,3}

^{in P2. And we}

^point

out the domain

Ao

to which the

mapping 0

^can be extended. As a consequence of this inves-

tigation

^{we know that 0 induces a}

biholomorphic equivalence

^between

A

⁼ P2 and

(S?jlG)*,

where G indicates the discontinuous group induced from the

monodromy

^and

(Q¡/G)*

^{is the}

Baily-Borel compactification

^of

4ll; jG, for i

⁼

1, 2, 3,

⁴

(Proposition 5-2).

The author obtained the results in the

part.

^I and in this paper

by refering

^{the esthetic}

principle

^« gonkadori » which is found in the medieval

Japanese anthology.

1. -

Representing equation.

[1].

^{In the}

part

^I

(Proposition 2-1)

^we

already

showed the _necessary and sufficient condition that an

elliptic

^{surface of basic}

type, namely

^a

surface with a

holomorphic

^{section and without a}

multiple singular fibre,

should

^{become a} g3 surface.

According

^to

Pjateckii-0160apiro

^{and Safare-}

vic

([5], ^Corollary

^{2 to Theorem}

1)

^a JO-K3 surface

(X,

^a,

L1)

^{has no}

multiple singular

^{fibre. Then} it becomes of basic

type.

^{Hence we have 11 = P and}

z( x)

⁼

^24, where Z

indicates the Euler characteristic. WTe

employ following

notations:

11: a

compact

Riemann surface with genus p,

f:

^a

meromorphic

^{function on}

L1,

$1,

^...,

^{03BEr : points}

^{on L1 such}

that f # 0, 1,

^{oo on L1’ = L1 -}

($1 ,

^...,

03BEr

(if f

^{- constant we choose}

arbitrary points $1 , ... , 03BEr),

’ :

^{a fixed}

point

^on

L1’,

{hlll---2h2.v}:

^{a canonical}

generator system

^of

ni(4, 8’),

gi: ^a closed arc with terminal

point 8’

^which goes around

03BEi

^{in the}

positive

^sense.

Then we obtain a

generator system (hi , ... , h2’P’

gl ,

..., gr}

^of

^{1l1(L1’, 03BE’ )}

with

only

^{one relation}

Let cv be a multivalued

analytic

^{function on 4’ defined}

by

^the

equality joco = f (if f

^-

0,

^{1 we set m -} exp

(fni)

^and

i, respectively), ^{where j}

^in-

dicates the

elliptic

^{modular function. Let}

a)($’)

^{be a certain branch of (0}

at

03BE’.

^{We denote}

by hiw(03BE’)

^and

giw(03BE’) respectively

the values of a) ob- tained

by

the continuation of

to($) along hi

and gi. These are

given by

modular transformations

Sht

^and

8g,, respectively.

^Let

S(hi)

^and

s(gi)

^be

matrices in

SL(2, Z)

which induce

Sh,

^and

S,,,, respectively,

and suppose that these matrices

give

^a

representation

^of

xi(4’, $’)

^into

SL(2, Z).

This

representation

^{is a}

homological invariant ,D belonging

^to

f.

^It

follows a theorem due to Kodaira

([4],

^Theorem

10.2).

THEOREM. Let us consider the situation mentioned above. There exists

uniquely,

^{as a}

fibre surface,

^an

elliptic surface (X,

n,

4) of

^basic

type

^with

the

given functional

^invariant

f

^{and the}

homological

^invariant

§.

^{And its}

singular fibres

^{are situated over}

81, ... , $r .

By a

^critical

point

^{we mean a}

point $;

^{such that}

x-1(8;)

^{is a}

singular

fibre.

Here we show that the

homological

invariant is

uniquely

^determined

by

^the

types

^of

singular

^{fibres in our} situation. In case of

f

^{== 0 the matrix}

S(hi) (8(gi)) is

one of the

following

⁶

types ([4], § 9):

All matrices in

(1-1)

^are commutative each

other,

^{then we obtain an ex-}

plicit

condition that

{S(h,)7 S(Ui)} ^gives

^a

homological

^invariant

belonging tof==0:

Let *

^{be such a}

homological

invariant. And let

(X,

n,

J)

^{be an}

elliptic

surface of basic

type

^with

given f ==

⁰

and §

^as mentioned in the above Theorem. There is a one to one

correspondence

between the matrices

$(,i)

and the

types

^of

singular

^fibres

n-1(8,)

^{as the}

following:

By a

^JO

singular fibre

^{we mean a}

singular

fibre listed in

(1-3).

^Con-

sequently

^we obtain the

following.

PROPOSITION 1-1. When we choose JO

singular fibres a-3L(qi) (i

⁼

1, ..., r)

so that the

corresponding

^matrices

S(e,) ^{satis f y}

the relation

(1-2),

then there exists

uniquely

^an

elliptic surface of

^basic

type

^with

f

^{- 0 which has the}

appointed type of singular fibre

^over

^{each $, .}

REMARK 1-1. We ^can obtain a similar conclusion for the

case f

^-

- constant.

REMARK 1-2. Let

{YI, Y2}

^{be a basis}

^system

^{of G =}

Z)

^with

the

properties

where co indicates the Abelian differential on

n-i($’).

^{And let}

^Mi

^{be a}

monodromy

transformation of G induced from the arc gi. Then the matrix

in

(1-3) corresponding

^to

n-1(03BEi) represents .111i

^with

respect

^to

{YI7’Y21-

Such

a basis

{Yl, Y21

^{will be called}

^canonical.

Let C be a JO

singular

fibre and let

Si (i

⁼

0, 1, ..., 5)

^{be the corre-}

sponding

matrix. Then we have

Next we consider r JO

singular

^fibres

(7iy

^{... ,}

Cr,

^{and let}

Sil’

^{... ,}

Sir

^{be cor-}

responding

matrices. Assume that we have .

’ r

Because of

(1-4)

it follows

Sil ... ^Sir

⁼

S12

^{= E.}

According

^to

Proposition

^2-1

in the

part

^{I and}

Proposition

^{1-1 we have the}

following.

PROPOSITION 1-2. Let us

appoint finite points 81,

^...,

$,

^{on P and the}

types of

^JO

singular fibres for ^{each $i}

^so that the total sum

of

^{their Euler charac-}

teristics is

equal

^to

24,

then there exists

uniquely

^{a JOK3}

surface

with these

singular fibres.

REMARK 1-3.

According

^{to this}

proposition

^{and the relation}

(1-4)

^we

know that a JOK3 surface has at most 12

singular fibres,

and in the maxi-

mum case any

singular

fibre is of

type

^II.

[2].

^{Let us consider the}

following variety X

^{with 12 different}

parameters

i

(i

.

=

1,

...,

12):

^.

where

[1’}o,

(!1’

1’}2]

^and

[q§ , q§ , q§]

^are

homogeneous

coordinate

systems

^on

P2,

and we

identify

^two

points ([1’}o, 1’}1, 1’}2]’ u)

^and

( [q§ , q§ , q§], u’ ) ^by

^the

condition

1

if we use an affine coordinate

(u, v = 771/?Io, W = n2 /no),

^{we obtain an}

affine

representation

^{of X:}

Let x be a

projection mapping

from X to the

u-sphere

^{P. And let us}

consider a fibre surface

(X,

ny

P).

^{It is}

easily

shown that X is

nonsingular

and

n-’(u)

^{is a}

nonsingular elliptic

^curve with the invariant 0 for every u

except $i (i = 1,

...,

12).

^{And also} we can see that

n-1(8;)

^{is a rational}

curve with one cusp

singularity

^at

[1}0, 1}1, 1}2]

⁼

[0, 1, 0], namely

^{it is a}

singular

^{fibre of}

type

II. Then the total sum of the Euler characteristics of

x-i($;)

^is

equal

^{to 24.}

By Proposition

^{2-1 in the}

part

^{I we obtain that}

(Xy

y,

P)

^{is a K3 surface.}

The curve L ⁼

{1}1

⁼ 1]2 ⁼

0) = 1

⁼

2 == 0} gives

^a

holomorphic

^sec-

tion. Moreover we can show that the

form cp

^{= w-2}

^duadv

ⁱⁿ

^{(1-6) gives}

^a

holomorphic

^{2-form on X. Thus we obtain:}

PROPOSITION 1-3. The

nonsingular variety

^X

defined by (1-5) gives

^a

representation of

^{a JOK3}

surface

^with maximum number

of singular f ibres.

Its

unique holomorphic 2-form

^is

given by T

^{= w2}

^duadv using

^the

au f f ine representation (1-6).

REMARK 1-4. Let us consider the above surface

(1-5).

And suppose that critical

points $i,,,

^...,

$i A;

^{coincide with a}

^{point $,,.}

^{Then the mono-}

dromy

matrix for the arc go with

respect

^{to a} canonical basis

{Y1’ Î’2}

ⁱⁿ

Remark 1-2 is

given by Sk . Let q

^{be an}

integer with q - k (mod 6)

^and

o q

^{6. Because}

81

is of order

6,

^we

have 8§J

⁼

8f. By

^the

correspond-

ence

(1-3)

^{we know that the fibre}

x-i($o)

^{has to}

correspond

^to

SQ..

[3].

^{Next we consider a JOK3 surface with 5}

singular

^fibres

Ci,

...,

C5 .

Such a surface will be called

of type

^F.

According

^to

Proposition

^{2-1 in}

the

part

^{I we have}

In the

part

^{I we}

already

^{studied a} certain class of surfaces of

type F, namely

the surface with 4

singular

^{fibres of}

type

^{IV and one}

singular

^fibre

of

type

^{IV*. We} denote such a combination of

singular

^fibres

by

^4IV’ + ^IV*.

PROPOSITION 1-4. l’liere ^are nine classes

of

^the

surface of type

^If:

PROOF. If we consider the relation

(1-4)

^and

(1-7),

^{we obtain the above 9}

combinations

by

^an

elementary

calculation.

By Proposition

^{1-2 there ex-}

ists a surface with such a combination of

singular

^fibres.

q.e.d.

By a surface of

^class

(j) ( j = 1,

...,

9)

^{we mean the}

j-th

^{surface in Pro-}

position

^{1-4. Let us consider the}

following

^affine

variety

^V with 5 different

parameters $,

ⁱ

(i

⁼

1, ..., 5) :

where we assume that the

values vi satisfy

^{the condition}

Let S be a minimal

nonsingular

^{model of}

V,

and let a be a

projection mapping

^{from S to the}

u-sphere

^{P. Then} the fibre surface

(8,

71:,

P)

^{is an}

elliptic

^{surface with five}

singular

^{fibres over u ---}

81,

^...,

$s,

and its func- tional invariant

f

^is

equal

^{to 0. Because} of Remark 1-4 and the relation

(1-4)

^{we have}

y(S)

^{= 24.} This surface has a

holomorphic

section ..L ⁼

-

fv

^{= w =}

0}. By Proposition

2-1 in the

part

^{I we obtain that}

(S,

i,

P)

is a .K3

surface,

^{hence it is a} surface of

type

^F.

Let us consider the

2-form 99

^{= w-2}

^duadv.

^{It is}

holomorphic

^{on the}

regular

fibre. Then we

investigate

^{its behavior on the}

singular

fibre. Let p be ^a

projection mapping

^{from V to the}

u-sphere.

^{It is easy to see that}

p-l(Ei)

^is

nonsingular

^on the affine

part, and

^is

holomorphic

^{there. If}

we set v’ ⁼

l /v

^{and w’ =}

ww,

^{then we obtain a}

representation

^{of V:}

From this

representation

we can see that

p-l(03BEi)

has various

types

^of

singularities depending

^on vi .

We have normal forms of the isolated

singularity

^on

p-’(03BEi)

^as

Diagram

^1.

These are rational double

singularities.

Then every ^curve which ^occurs

as a consequence of the resolution of the

singularity

^has the selfintersec- tion number - 2.

Already we

^know

that

^is

holomorphic

^{at any non-}

singular point

^on V. It does not occur that a

meromorphic

^{form has its}

pole only along exceptional

^{curves of second kind.}

Hence q;

^is

holomorphic

on S.

REMARK 1-5. If zve

have vi

⁼

6,

the fibre

p-1(03BEi)

^{has a}

singularity

x2

+ y3 + zll

^{= 0 at}

infinity.

^{This is a}

simply elliptic singularity

^of

type.R,,.

Diagram

¹

Then it occurs a

nonsingular elliptic

curve E from this

singularity.

^The

form 99 has its

pole along E,

^{and we have}

n-i($;)

^{= E.}

From the above consideration ^we have the

following.

PROPOSITION 1-5. We have a

representation of

^a

surface of type

^{F as the}

minimal

nonsingular

^model

of

^the

variety

^V

defined by (1-8)

^and

(1-9).

^And

its

holomorphic 2-form

99 is

given by

^w-2

duadv.

2. -

Homology

^basis.

[1].

^{Let us consider a} JOK3 surface

(S,

n,

P).

^Let

f$,, $1 , ..., I $ool

be the

totality

^of the critical

points.

^And we assume that

$o

^{= 0 and}

=

^oo.

Let Zi (i = 0,1,

... ,

r)

^{be an arc}

connecting $;

^{and oo so}

that

^does

not intersect any other

lj.

we

employ

^the

following

^notations:

,J: the base Riemann surface

P,

03BE’ :

^{a fixed}

point

^on

L10,

fy,, y,l: a

canonical basis of

.gx(C, Z).

Let _gi

(i

⁼

0, 1,

..., r,

oo)

be the closed arc defined

in §

¹

[1]

^{so that}

any gi

(i

⁼

0, 1,

...,

r)

^{does not}

intersect

^{for i}

# j

^{and that}

g-1

^intersects

Zo, ... , Zr

^{in this} order. And let ai

(i = 1, ..., r)

^{be an oriented arc which}

starts from 0 and goes

to $,

^without

intersecting any 1, (j

⁼¹⁼

i).

Let g

^{be an} element of

a, (,d 1, 1)

^{so that it induces a trivial}

monodromy

of

H1(C, Z).

^{If we make a} continuation of a

1-cycle y

^{on C}

along

g, ^we obtain ^a

2-cycle

^on S. We denote this

2-cycle by g X y, and g

will be called

a base acrc of

g X y

^We define the orientation of

g X y

^as the ordered

pair

^of

the orientation of g and the ^one of y.

[2].

^{Let us consider a} JOK3 surface

(S,

7r,

P)

^{with 12}

singular

^fibres

olr-’($i) (i

⁼

0, 1,

...,

oo).

^{We set}

The intersection matrix if ⁼

(GiGj),:,j;5,,

^is

^given

^{as the}

^following:

This is examined

by

^{the same method}

developped

^{in the}

part I §

^3.

Here we note that

EGi

^{= 0}

(i = 1, ... , 20 )

^{is obtained}

by

the direct observation of

Gi

constructed

by

^{use of the}

representation (1-6). By

^an

elementary

calculation we obtain that if is invertible. We have

b2

^{= 22} for a .K3 surface. Hence

{Gl,

^...,

G,,21 ^gives

^{a basis}

^system

^of

H2(S, Q).

Because the fibre space

(n-1(L1o), n, Llo)

^is

^trivial,

^{we can determine a}

canonical basis

{YI(U), Y2(U)} ^of.

^{any fibre}

^n-1(u)

^{over a}

point

^{u of}

40 by making

^a continuation of

{Yl’ Î’2}.

^{Let us} consider the

representation (1-6).

We can

regard

^a

general

^fibre

a-’(u)

^{as a} three sheeted

covering

^Riemann

surface over

v-sphere,

and its ramified

points

^{are situated over v = 0 and}

v

= :t {3,

^where

We can realize

{YI, Y2}

^{as the}

^cycles

^obtained

^by

^arcs

connecting

^two

ramified

points over fl

^{and -}

#. Consequently y,,(u)

^and

Y2(U)

^{tend to the}

infinite

point

^{as u tends to a critical}

point 03BEi .

^{Hence we obtain a}

2-cycle

on S as the continuation of a

1-cycle y(u) along

a2. We denote it

by

oe,xy.

We set

where we define the orientation of

C, (i = 1, ... , 20 )

as same as for

G j .

^It

is

easily

^{shown from the} construction that we have

Let G be an

arbitrary

element of

H2(S, Z)

^and

^{set ri}

⁼

GGi.

It follows from

(2-3)

^that

This

implicates

^{G =}

’fIOl +

^...

+ r22 C22’

^{Thus we have the}

following.

PROPOSITION 2-1. The

system (Ci, ... , C22} ^{defined by (2-2) gives}

^{a basis}

system of H2(S, Z). And

^the

system {GI,

^...,

G22} defined by (2-1 ) gives

^{a basis}

system of H2(S, Q). And

^{these two}

systems

^{are dual each}

other, namely

^we

have the relation

(2-3).

[3].

^{Here we consider a surface}

(S,

n,

P)

^of

type

^F.

By

^{a fractional}

linear transformation we arrange ^the

singular

^{fibres so that}

they

^{are situated}

as

Diagram

^2.

Diagram

²

For the moment we fix the

parameters I

⁼

[El, E2, E3]

^{so that we have}

And we assume

that tz (i

⁼

0, 1, 2, 3)

^is

given

^{as a line}

segment.

^We

define

2-cycles GI, Gs, 05

^{on 8 as}

Diagram

^3.

Diagram

³

We can

easily

^{show that}

they

^are

certainly 2-cycles,

^{because we}

already

know the

monodromy

transformation of

H1(C, Z)

induced from gi

(Re-

mark

1-4).

^{We set}

where g indicates the base ^arc of

G2i-l.

Next we construct

algebraic cycles

^{on S. There are 14}

components

^of

singular

^{fibres which does not intersect the}

holomorphic

section L. We denote them

by G7,

^...,

G2o,

^{and set}

^G21

⁼

L, G22

^{= C.}

By

^the construction we can show that we have

Let A be the intersection matrix

(au)

⁼

(GiGj),,i,116

induced from

Gi, G6’

^{And let B be the one}

(bij)

⁼

(Gi+6Gj+6)1i,i16

induced from

G7, ... , G22 .

^{We can determine B}

by considering

^the

geometric figure

^of

the JO

singular

^fibres.

REMARK 2-1. The matrix B is ^a direct sum of several minor intersection matrices each of them is the one induced from a J’0

singular

^{fibre or the}

one induced from

G2,

^and

G22.

^{But we must note that we excluded one}

simple component

^{from each}

singular

^{fibre to}

get G7, ..., ^{G2o .}

As a consequence ^we obtain that B is invertible for any class

(j) ( j

⁼

= 1,

^...,

9). According

^{to the same method}

developped

^{in the}

part I §

³

we can calculate the matrices A ⁼

Aj

for the surface of class

(j) (j

⁼

19..o 9)

^as

Diagram

^4.

We can see that

any Aj

is invertible. Let M be the full size intersec- tion matrix

(GiGj), ;5 i, j;5 22

^From

^(2-6)

^{we obtain a direct sum decom-}

position

^{M = A} (B ^B.

Consequently

^{we know that M is} invertible. Then

we obtain a basis

system {Gt, ..., I G12}

^of

^{H2 (SI Q)}

^{for a} surface of

type

^.F’

with fixed

parameters by (2-4).

Now we

regard [03BE1’ 03BE2’ 03BE3]

^{as a}

point

^{on P2. And we use the fol-}

lowing

^notations:

His

⁼

(I + 03BE1, 03BE2’ 1 $31 ; ^03BE1

⁼

03BEj},

^where

I # I f or i =1, 2, 3 and y =

= 0, 1, 2, 3,

A=

P2 -{the

union of all

7?,J,

A

⁼ the universal

covering

^of

A,

p : the natural

projection

^from

^A

^to

A,

S,($):

^a surface of class

(j)

determined

by parameters$ = [$111 $2 03BE3]

on

A,

Diagram

⁴

where

We can determine a trivial fibre space

,F j

^over

^Ã

^{with a fibre}

S;(()

⁼

Sj($) for p(!) ^{= 03BE.} Using

the trivialization

of 9,

^{we obtain a continua-}

tion of the

cycle Gi.

^{Hence we}

get

^{a basis}

system

^of

H2(Sj(!), Q).

REMARK 2-2. The

algebraic cycles GII ...,022

^are invariant when the continuation is

performed.

From the above

argument

^{we obtain the}

following.

PROPOSITION 2-3. Let

G;(I) (i = 1,

^...,

6)

^{be the}

2-cycle

^on

Sj(!)

^obtained

by a

continuation

of ^Gi.

^Then

F* (f)

⁼

{G1(!),

^...,

G6(!), G7,

...,

G22} ^gives

^a

basis

system of H2(Sj(!), Q).

And the intersection matrix

Aj of Gl(!), ..., G6(!)

is

given by Diagram

^{4. And}

algebraic cycles G7 ..., G22

^are

orthogonal

^to

G1(!),

^...,

G6(!).

Next we construct ^a basis

system

^{T =}

{Ciy

^...,

1 C22}

^of

H2(S, Z)

^which

is dual to r*. Here

again

^{we consider a} surface S of

type

F determined

by

^fixed

parameters $

^with

(2-4).

When we make a continuation of a

1--cycle y

^{on a}

general

^fibre

along

«i ? it tends to a

1-cycle

^on

n-i($;)

^{which is}

homologous

^{to zero.}

By

^{this pro-}

cedure we obtain a

2-cycle on S,

^{we denote it}

by

⁰¹⁵³ⁱ

X y.

REMARK 2-3. In

general

^{such a}

2-cycle

intersects some

components

of the

singular

fibre. And the intersection

multiplicity depends

^{on the}

homotopy

^{class of} y.

We determine

2-cycles Cl,

^...,

06

^{on the surface}

Si

^{of class}

(j) (j

^-

-

1,

...,

9) by Diagram

^5.

Diagram

⁵

Let

{C, ,

^...,

C22 f

^{be a basis}

^system

^{of the} Abelian group of rank ¹⁶ gen- erated

by G,,..., G22

^{over Z. As same as for the}

system

^{F* we obtain a}

system 1’(f)

⁼

(Ci(I), ... , Cs(03BE), C,,..., C21}

^of

^2-cycles

^on

^Sj(f).

By

the construction we obtain that

And we can see that

]’(()

^{is a basis}

system

^of

H,(Sj(f), Z) by

^{the same}

argument

^as

Proposition

^{2-1. Hence we obtain the}

following.

PROPOSITION 2-4. The

system T(I) give.-

^{a basis}

system of H2(Sj((), Z)

which

satisfies

^{the relation}

(2-7).

^And

Cl, ’--1 ~22

^are

given

^as

algebraic cycles.

3. - Period

mapping.

0

In the rest of this paper ^we consider

only

surfaces of

type

^{F. And we}

employ

the notations in

§

²

[1]

^and

[3].

[1].

^We

already

^{defined a}

holomorphic 2-form q;

^{on a} surface of

type

^F

in

Proposition

^{1-5. Now we consider the}

periods

for surfaces

Sj(f)

^{of class}

(j) (j

⁼

1,

...,

9).

We have ^’

because

C7,

^...,

022

^are

given

^as

algebraic cycles.

^{Let us}

identify

^two

points I

and

03BE’

^on

à by

the condition

I’(1)

⁼

I’(l’ ).

^{We obtain a}

covering

^domain

A

over 11

by

^this identification. We

call A

the

monodromy covering

^domain

of ll for surfaces of class

(j).

^{We denote}

by 8

^a

point on A

^with

projection ^03BE.

Now we obtain

single

^valued

analytic

^functions

1]1(£), ..., 1]6(03BE)

^on

^ll.

Let V be the

variety

^defined

by (1-8),

^{and let us consider an}

automorphism

of V. This

automorphism

^{induces an}

automorphism t)

^{of the}

nonsingular

model

S, and e

^{preserves each fibre}

a-11(u).

Now let us construct a canonical

homology

^{basis on C =}

n-i($’).

^We

regard

^{this fibre as a three sheeted Riemann surface over}

v-sphere:

""There

Let y be ^a oo-like closed arc on

v-sphere

which goes round ⁰

and @ respectively

^{in the}

negative

and in the

positive

^senses. Let yl be ^a closed

arc on C with a

projection

y. And set

Y2 == e2Yl.

^{Then we obtain a canon-}

ical basis

{Yl, Y2}

^of

H1(0, Z).

By

^the construction of

01, ...,06 (Diagram 5)

^{we have}

C2i

⁼

e202i-l (i

⁼

1,3,5).

^Hence it follows

Set ( = (171, ... 77122)- And

^let

(mi, 7... 7 (022}

^{be a basis}

^system

^of

^{H 2(S’ Z)}

given by

differential 2-forms with

properties

Such ^a basis

system

will be called a dual basis for 1-’ ⁼

{C"

^...,

022}.

^Set

and let I be ^a matrix

(aiJ)1i,j22.

^{Then we have the}

^period

^relations:

because S is a _K3 surface

([5],

^p.

777).

According

^to

(2-7)

(JJj

(j = 1, ..., 6)

coincides with

Gj

^{as a current.}

Consequently

^{we have}

Set 27 (n1

n3,

275)’

^{In view of}

(3-1)

^and

(3-3)

^{we can} reduce the

period.

relations to

where the matrices

A3

^are

given by Diagram

^6.

Diagram

⁶

Let

S2j

^{be a} domain in P2 defined

by (3-6).

^Because

any Aj

^{has two}

negative

^{and one}

positive eigen values, ^Qj

^is

biholomorphically equivalent

to a

hyperball.

Let us consider a

mapping

from Â

to P2. In view of de Rham’s theorem we know that 27 ⁱ

(i =1, 3, 5)

does not vanish at a same time. The

mapping 0

will be called a

period mapping for surfaces of

^class

(j).

From the above consideration we have:

PROPOSITION 3-1. The

period mapping 0 for surfaces of

^class

(j)

^{is a}

holomorphic mapping from A

^to

Qj,

^and

Qj

^is

biholomorpfically equivalent

to a

hyperball.

[2].

^{Here we}

employ

^the

following

^notations:

X : an

algebraic

^K3

surface,

^y

99 ^a

holomorphic

^{2-form on}

X,

L : ^a free Z-module of rank 22 with an even

integer

valued uni- modular

symmetric

^{bilinear form}

( , )

^of

signature (3, 19),

1 : a fixed element of .L with

(l, l )

&#x3E; 0.

REMARK 3-1. Such a Z-module L exists

uniquely,

^{up to}

isomorphisms ( [9]r Chap. 5).

Let (J) and oj’ be two elements of

H2(X, Z).

^{If we define}

then

H2(X, Z)

^is

isomorphic

^{to .L}

([5],

^p.

776).

A marked K3

surface

is defined as a

triple (X,

1p,

F) satisfying

^{the con-}

ditions :

(1) 1p

^{is an}

isomorphism

^{from .L to}

H2(X, Z),

(2)

^{F is a line} bundle on X such that

c(F)

⁼

V(1),

^{c indicates the4}

Chern

class,

^and

.F’D &#x3E;

⁰ for any effective divisor D.

Two marked .K’3 surfaces

(X 1,

7 VIL

FI)

^and

(X2 ,

"P2,

F2)

^are identified if there exists a

biholomorphic mapping f

^from

.Xl

^to

X2

^with "PI

= f*oy,.

Let

{el,

^{... ,}

e22}

^{be a} fixed basis

system

^{of .L. And let}

(X,

V7

F)

^{be a}

marked

g3

surface.

And set a basis

system C1,

^...,

C22

^of

H2(X, Z)

^{with a}

property

Let

M(l)

^{be a}

family

^{of all marked E3 surfaces}

(X, V, F)

^with fixed l. Then

we obtain a

mapping 1JF from M(I)

^{to P21 with}

homogeneous

^coordinate

[f}l, ..., 1]22] by setting

,

According

^to

Pjateckii-Sapiro

^{and Safarevic we have the}

following

^Torelli

type

^theorem

([5],

^{see also}

[7]).

THEOREM. The

period mapping P

^is

injective.

Let us define a

marking

^on

S;($).

^For the convenience we write

S(03BE)

for

S;(#).

^Set

and I ⁼

c(D).

We have

(t, 1)

^{= D2 =}

2,

^because

G 2 =-2 G 2

^{= 0 and}

^{G21 G22}

^{= 1. Let}

(mi(I), ... , mzz(#))

^{be a} dual basis of

L(#)

^for

T(#).

^{Take an element w =}

=

xl m(’ ) + ...

+

X220J22(03BE’)

^{of ..Z. We obtain an}

isomorphism

^{from .L}

to

L(£) by defining y(m)

⁼

simi(#)

+ ^...

+ X22W22(03BE).

^{Hence we obtain a}

marked .K3 surface

(S(#),

^"P,

^F).

^{We note that F =}

^c-lo1p(l)

⁼

^{[G21-E- 2G22]}

is induced from a divisor

independent

^{of the}

parameter 03BE.

REMARK 3-2. Because S is a

K3 surface,

^{the Chern}

mapping

^{c is}

injective.

LEMMA 3-1. Let D be a divisor ^{on an}

algebraic

^K3

surface

^s.

Suppose

that there exists ^a divisor D’ with

only simple components

^{which is}

linearly ,equivalent

^{to D. Then we have}

PROOF. We consider the

following

^{sheaf exact} sequence:

By

^the

assumption

^{we have}

Hence ^we obtain the

long

^exact sequence:

And we have

For a K3 surface we have

Hence we have

H’(S, Os[- D])

^{= 0.} In view of Serre

duality

^{and the}

triviality

^{of the canonical bundle}

of S,

^{we have}

LEMMA 3-2..Let

(S;(#i),

^"PI,

Fl)

^and

( Sj’ (03BE2 ),

"P2,

F2 )

^{be two marked K3}

surfaces of

^class

(j). They

coincide each other

i f

^and

only if $i

⁼

G2.

PROOF. The

sufficiency

^{is trivial. Then we show the}

necessity.

^We

write Sand S’

respectively

^for

S;(#i)

^and

Si(03BE2).

^{We have Cl =}

0,

c2 = 24 for a K3 surface.

Hence, by

Riemann-Roch

theorem,

^{we obtain}

By

Lemma 3-1 the second term of the left hand side vanishes.

By

^Serre

duality

^{the third term also vanishes. As we have D2 = 2 for D =}

G21 +

+

2G22,

then it follows

Take a coordinate

covering U Ui

⁼

S,

^and

let f

^{i be a}

representing

iEI

equation

^{of D on}

Ui.

^{Then we have}

Let t be a

parameter

of the base curve d = P such that

a-’(O)

^{is a}

general

fibre. Set

Then

99 (k)

⁼

199t k)l ^(k

⁼

^1, 2, ³⁾ gives

^{a basis}

system

^of

HO(S, O[D]). ^Suppose

that we have

( S,

yi,

PI)

⁼

(S’,

’lp2,

F2 ) .

^{Then we can}

regard

^{the latter as}

the surface S

equipped

^{with another}

elliptic fibring

^{with a}

holomorphic

section

G21

^{and a}

general

^fibre

G22.

We denote this

elliptic

^surface

by (S’,

yri,

J,).

^Let

tl

^{be a}

parameter

^on

d1

^{= P. The}

parameter

^{t was}

given

^{as a}

ratio of two

holomorphic

sections of

0[-D].

^{Then also}

tl

^{must be so. Hence}

t,

is a rational function

R(t)

^{of t with the}

degree

^{at most 2. But if the}

degree

of .R is

exactly two, (SI, a,, A,,)

^{must have a} disconnected fibre. This is absurd. Hence

R(t)

^{is a} fractional linear transformation.

Let

{so, E1’ E2’ E3’ eoo}

^and

{Eo, $’, $’, $’, $’ }

be critical

points

of S and S’

respectively.

^{Because the}

projection

^{nl differs from x}

only by

^{a coordinate}

transformation of

P, ($o, ..., $)

coincides with

($§, ..., $’ }

^{as a set of}

points.

^Hence the transformation B

carries $; (i

⁼

0, 1, 2, 3, oo)

^{to some}

$) .

The

2-cycles Cl, C3, 05

^are constructed over the arcs between

$o

^and

$i, , respectively (§ 2, Diagram 4).

^{If we have a}

point

^{such that}

R($;) # $(,

^{then the}

^marking

^of

(S,

nt,

41)

^{can not} coincide with that of

(s7

n,

L1).

Then .R must be the

identity.

^{Hence we have a same}

projec- tion $

^of

ii

^and

(z. Reviewing

^the

assumption (S;(#i),

yi,

Fi)

⁼

(S;(#i),

’f/J2,

P2)

^{we have}

T(#i) = T(#z).

^{Thus we obtain}

E1

⁼

#z . q. e. d.

As ^a consequence of the above lemma ^we obtain:

PROPOSITION 3-2. The

period mappittg 0 for surfaces of

^class

(j) (j

⁼

= 1,

...,

9) is injective.

4. - Differential

equation.

As

already

^shown

in §

³

Proposition

3-2 the dimension of the vector

space generated by qi(I), ..., q(#)

is three. Then we

investigate

^the

monodromy

transformation of E induced from a closed arc on A.

Let $’

⁼

[$, $’, $’]

^{be a fixed}

^point

^{on A. Let us make a closed arc}

^{f1 if}

as

following:

flij

^starts

^{at $’}

^{and vary}

^only

^one

parameter $;

^{so that the}

moving

point $

goes around

Hij = ([$1 , ^{E2’ E3]: ei = $;}

^{for i #}

^j,

^{i =}

^{1, 2,}

^{3 and}

j

⁼

0, 1, 2, 3, oo}

^{in the}

^positive

^sense.

Set

Let k and 1 be two indices in

to, ^{1, 2,} 3} 2013 {i, jl,

^{and let r1 and r2 be oriented}

arcs on

L10 starting at $;

and goes

to Ek and El, respectively.

^Set

Then Z is

generated by

And also we have the relation

(3-3)

^for

ni.

^Let I and J be the

types

^{of the}

singular

^fibres

n-1(Ei)

^and

n-1(Ej), respectively (I,

^{J =}

II, IV, 1*, IV*, 11*).

Let

Si

^and

Sj

^{be the}

corresponding

^matrices determined

by (1-3) (i, j

⁼

=1, ..., 5).

^{And let}

^M;;

^{be the}

monodromy

transformation induced from

Pij.

We can determine

M ii

^{as the}

following:

(Case 1).

^{If we}

have j #

oo,

(Case 2).

^{If we}

have j

⁼ oo,

By using (4-1), (4-2)

^and

(3-3)

^{we can}

represent

^the

monodromy Mij

with

respect

^{to the basis}

(iji , fj" 1}5}.

^Set

LEMMA 4-1. The

monodromy Mij

^is

of

^order

in f inite z f

^and

only if e(i)(!(j)

^{= 1. And we have the Jordan’s normal}

form of Mij

^{as the}

following:

Let q

^{be a}

point

^{on some}

Hij (i = 2, 3, j

⁼

0, 1, 2, 3, oo)

and suppose q is not an intersection

point

with other

H ii.

^{Let us choose a} local coordinate

(x,y)

^{in the}

neighborhood of q

^{so that we have x =}

(Ei - E1 )/e1.

By

^{Lemma 4-1 we can choose the}

following

forms of basis

system

^{of 27}

in the

neighborhood

of q :

where

Fi (i

⁼

1, 2, 3)

^{indicates a}

single

^valued

holomorphic

function in the

neighborhood

of q.