C OMPOSITIO M ATHEMATICA
Y UJIRO K AWAMATA
E CKART V IEHWEG
On a characterization of an abelian variety in the classification theory of algebraic varieties
Compositio Mathematica, tome 41, n
o3 (1980), p. 355-359
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355
ON A CHARACTERIZATION OF AN ABELIAN VARIETY IN THE CLASSIFICATION THEORY OF ALGEBRAIC VARIETIES
Yujiro
Kawamata and EckartViehweg
In this paper we shall prove the
following
theorem which wasconjectured by
S. Iitaka(Bn
in p. 131 in[1])
and provenby
K. Ueno for n = 3[2].
In this papereverything
is defined over thecomplex
number field C.
MAIN THEOREM: Let X be an
algebraic variety
and letf :
X --> A bea dominant
generically finite morphism
to an abelianvariety. If
theKodaira dimension
K(X)
=0,
thenf
isbirationally equivalent
to anétale
morphism
and X isbirationally equivalent
to an abelianvariety.
To prove the main theorem we shall reduce it to the
following
theorem 1.
THEOREM 1: Let A be abelian
variety of
dimension n, let X be areduced irreducible divisor on A and
let X
be a resolutionof
X.If
X isan
algebraic variety of general
type, thenqk(X) ==
dimH°(X, fi À)
>_n
k ,for
k =1 ,
..., n - 1.Moreover, if Pg(X) == qn-i(X)
= n, thenqk (X ) = n ,
kfor
k =1,
..., n - 2, and inparticular ]X(Og)]
= 1.1
The
following
lemma isjust
aspecial
case of a theorem of Ueno(3.3
of[2]).
0010-437X/80/060355-05 $00.20/0
© 1980 Sijthoff & Noordhoff International Publishers - Alphen aan den Rijn
Printed in the Netherlands
356
LEMMA 2: Let the notations and
assumptions
be as in the MainTheorem. Then
dim(H’(X, f?n-1» -
n.PROOF: We want to show that
f *(dz,
A Adzi-1
Adz;+,
1 1B ...A
dzn)
= Wi are generators ofH’(X, Q g ’ ) ,
for aglobal
coordinate system(z,,
...,zn)
of A. Take w EHO(X, Qv’)
and ai E C, such that w Af*(dz)
= ai -f*(dz1
A ... Adzn ).
This isalways possible,
sinceH’(X, f2 x
n
is
generated by f*(dzl
1B ... Adzn). Replacing
cvby w - ¿ I)n- 1 aiwi
wei= 1
may assume that ai = 0 for i =
1,
..., n. Choose a small open subset U C X, such thatfi u
is anembedding. (ZI,
..., zn) is a local coordinate system of U. Since W Adzi
= 0 for i =1,
..., n, w must be 0 on U and hence on X.LEMMA 3: Let the notations and
assumptions
be as in the maintheorem. Let
fo:Xo-A
be the normalisationof
A inC(X).
Let DI, ...,Dm
be the irreducible componentsof
the discriminantL1(Xo/ A)
and let
D, , ... , Dm
be theirdesingularisations.
ThenPROOF: Choose
à;
to be one irreducible component off -’(Di),
suchthat
L1i
is ramified over A. We may assume, that X isprojective
andthat
L11
U ...U L1m
is aregular subvariety
of X. Let Wx =il x.
ThenÙJX(&OX Ox di
mÇ:,,2 x and,
sinceHO(X, úJx)
=H’(X, ù)’) x =
C, we;=i
m
know that
@ H°(à;,
i 1Wài)
is asubspace
ofH1(X, wx)
=Hn-I(x, Ox ) =
H’(X, Q v’ ).
However,HO(D, WV;)
cH°(à;, wà,).
Now we recall the
following
Theorem of Ueno(p.
120 in[1]):
THEOREM 4: Let B be a
subvariety of
an abelianvariety
A. Thenthere exist an abelian
subvariety AI of
Aand
analgebraic variety
Wwhich is a
subvariety of
an abelianvariety
such that(1)
B is ananalytic fibre
bundle over W whosefibre
isAI, (2) K( W)
= dim W =K(B).
AI
is characterized as the maximal connectedsubgroup of
A such thatAl + B ç B.
PROOF OF "THEOREM 1 => MAIN THEOREM":
Let q:
A’---> A be anyétale
covering
andXTI = X X A A’.
ThenX, ---> A’
also satisfies the conditions of the main theorem. LetX,,o
be the normalisation of A’ inC(X,).
Thenà(X,,o/A’)
is thepullback
ofà(XOIA) by q. Suppose ,à(XOIA)
is not empty.Any
abelianvariety
has étalecoverings
ofarbitrary high degree (for example "multiplication
with r >0"). Every subvariety
of an abelianvariety
has Pg > 0. Hence,replacing
Aby
some étale
covering
we may assume, that for every étalecovering q : A’-A
the number of irreducible components ofà(XOIA)
and,à (X,,OIA’)
is the same(Lemma 3).
Put B =
lx
EA ;
x+,à (XOIA) Ç:,à(XOIA)JO. Again replacing A by
anétale
covering,
we may assume that A = B’ x B. LetYo
be the Stein factorisation ofXo A B’
and Y anydesingularisation
ofYo.
SinceXo
is a finitecovering
ofYo
x B we haveK(X)
= 0 >K(Y)
+K ( B ) >_
0and
K(Y)=O.
Assume that
à(Yo/B’) = 0.
Sinceà(XOIA) =----,à’x B
for someposi-
tive divisor à’ C
B’,
the ramification divisor ofXo ---> Yo x B
must be a(rational) multiple
of thepullback
of somedivisor â
ofYo.
ThenK(X) > K( Yo, O(à))
> 0, in contradiction to ourassumptions.
There-fore
à(YOIB’) 0 0 and, repeating
this step if necessary, we may assume B = 0.Let
Bi={xEA;x+DiÇD;}O
f or i=I,...,m. We havenBi = 0.
By
Theorem 4 eachDi
is a fibre bundle over a certainE;
with fibreBi,
for i = 1, ..., m. We have
pg(D;) * pg(E;)
for adesingularisation Éi
ofE;
andby
Theorem 1pg(E;) & codimA B;.
Since the
equalities
must be trueby
Lemma3,
we haveIX(OÊ,)l
= 1by
theorem
1,
for i =1, ...,
m.Let r be a natural number, with
r:z-- 2,
and let r: A ---> A be themultiplication
with r.Using
the notation introduced above,L1(X"o/A)
must have components
D,,i,
i =1,...,
m such that thecorresponding
base space
E,,
satisfiesIX(OÉ,",)l
=degree(r) . IX(OÉ,)l -
2. This is acontradiction.
2
PROOF oF THEOREM 1: Let
(xi, ..., 9 Xnl
be aglobal
coordinate system on A such that the setldxl,
...,dXn} gives
a basis of 1-formson A. Let
a:,Z--->A
be the canonical map and let w;=a*(dxi
A ’ ’ ’ Adxi-l
Adx;+i
A ’ ’ ’ Adxn)
for i =1,...,
n. We shall prove first that these arelinearly independent (n - 1)-forms
onX. Suppose
n
the contrary:
aiwi
= 0 for ai E C. Pick a smoothpoint
p on X.a=1 1
358
Suppose
X is defined in A near pby
anequation
x,, =F(xl,.. -,
Xn-1),where F is a certain
holomorphic
function.Then W = (- I)n-i-1 dF A... -n )
aY.for i = 1,..., n - 1.
Therefore,
Q’W
which
means that there is a non-zero
subgroup
B of A such that B + X C X, which is a contradiction.Put w,
=a*(dxi)
A ... AdXik)
for each set I of k-distinctintegers
1
ik:
n.Since wi
arelinearly independent, f cl }I gives
alinearly independent
system of k-f orms onX. Thus, qk(X) (§J ) .
Before we prove the second part of theorem
1,
we shall prove thefollowing theorem,
due to the first author.THEOREM 5 : Let A and X be as in theorem 1. Let
f :
X -> P"-’ be the rational mapdefined by
the system(w, , ..., wn}. If
X is analgebraic variety of general
type, thenf
is dominant.PROOF: Assume the contrary. Let Y be the
image variety
off
andlet q be a smooth
point
of Y such thatf -’(q)
is also smooth near some smoothpoint p E f-1(q)
of X. Ourassumption
means thatdim
f -’(q) -
1. Considereverything
in the universal cover en of A.Let H be the tangent
plane
of X at p, which we assume is definedby
an
equation xn
= 0.Then, X
is defined near pby
anequation
xn =F(xi,
...,Xn-1),
where(x,, ..., xn}
is aglobal
coordinate system cen- tered at p andaF (0) = 0
for i =1,..., n -
1.f-’(q)
is defined near pXi
f
by
theequations aF
7= 0 for i =1,..., n -
1. Y is contained near q in x. 1a smooth divisor D of pn-1
(near q).
After a suitable linear trans- formation of xl, xn-i, theequation
of D can be written asdF
ax, where G is a
holomorphic
function ofdegree
a 2.By
the rule of derivation ofproducts,
we have onThus, f -’(q)
is invariant under translationsin the
direction of x, and hence contains a translation of an abeliansubvariety
of Agenerated by
the line X2 = ... = xn = 0. Let B be the maximal abeliansubvariety
of A such that p + B is contained in X.We have
proved
that B 4 0. Since there areonly countably
manyabelian
subvarieties,
B does notdepend
on p.Thus,
B + XC X,
acontradiction.
Q.E.D.
PROOF OF THEOREM 1 CONTINUED:
Suppose p,(g)
= n. Let p be asmooth
point
of X and let x,, ..., Xn be as in theproof
of theorem 5.Let w be an
arbitrary
k-form onX.
Write near p W =19I(Xl,---,Xn-I)WI-
Put lC ={l,
..., n -1) -
1. Then w A wic = né¡E(I, lC)g¡úJm
where E is thesign
ofpermutations. Therefore,
we havefor some aI G C . Let J be a subset
of {l,...,
n -1}
such that C ard J =n - k - 2. Since
we have
for some
bf; E .
SinceaF, ..., aF
arealgebraically independent,
ax,
Xn-l
we can compare the coefficients and we get
(1)
ajj = 0for j E J, (2) aIi = 0
for1 U {i} = JC,
and(3)
for each K =fil, ..., ik-I}
such thatil
...4-1
and K Ufil U {i}
=J",
E(K
U{i}, J, j)aKUti,i +,E(K
U{j}, J, 1> aKujjj,j
=0,
thatis, E(K, 1)aKui;j,; =,E(K, j)aKuijj,j.
Put aK= E(K, i)aKU{i},i.
Then, W = L q(O)WI + L
uxcax Adxn. Q.E.D.
nfll K
REFERENCES
[1] K. UENO: Classification theory of algebraic varieties and compact complex spaces, Lecture Note in Math. 439, 1975, Springer.
[2] K. UENO: Classification of algebraic varieties, II-Algebraic threefolds of parabolic type-, Intl. Symp. on Algebraic Geometry, Kyoto, 1977, 693-708.
(Oblatum 2-V-1979) Yujiro Kawamata Eckart Viehweg Department of Mathematics Institut für Mathematik University of Tokyo und Informatik Hongo, Bunkyo, Tokyo, Japan A5, Seminargebâude
D-68 Mannheim