Complex Analysis and Algebraic Geometry: A Collection of Papers Dedicated to K. Kodaira

Xis etale and hence Xt is a surface of the same type as X (in fact, KXl = 0 and q(Xl)>2, hence by table I, q(Xt)=2). We can deduce quickly that <j) and hence <j>t are all finite morphisms : in fact, if not, let £ c l b e a curve such that <j)(E) is a point ee Alb X. Then considering the Stein factorization X—->7—>Alb of 0, we see that E can be blown down in a birational map X—>7, hence (£ 2 )<0. Now 1) A suitable multiple of an ample curve C on any surface Y is a hyperplane section of Y for some protective embedding and all hyperplane sections of varieties of dimension > 1 are connected.

Enriques' Classification of Surfaces in Char. /?, II

41

for each /, l~l(e) consists of /4 points et 6 Alb X, and 0~1(^) contains a curve Ei that is contracted by t. These curves are disjoint since t (Et) =ei=£ej=l (Ej). Thus Xt has /4 disjoint curves Et with (£?)<() ; thus B2(Xl)>l*. But for all surfaces of the same type as X, B2=6. This is a contradiction i f / > 1, hence is finite. Next,fixZo, an ample sheaf on Alb X. It follows that Lt=^>t (A>) is ample on X,, with Hilbert polynomial independent of/. By the Main Theorem of Matsusaka-Mumford [7], there is also a number N independent of / such that LfN is very ample for all /. Therefore the infinite set of k-varieties Xl can all be embedded in a fixed PM with fixed degree. Since there are only finitely many A>varieties of this degree (as k is finite), it follows that all the pairs (Xh LfN) are isomorphic to finitely many of them ! Now consider the facts— a) for any variety X and ample sheaf Z, the group of automorphisms f of X such that/*Z is numerically equivalent to L is an algebraic group ; esp. it has only finitely many components (Matsusaka [6]), b) The group Al of translations by points of order / acts on Xt since by definition, it is the fibre product Xx A i b (Alb, /) ; moreover each g € At carries Lt into a sheaf algebraically equivalent to Lt. Let (Xt, LfN) be isomorphic to infinitely many other (Xv, LpNYs. Then Av acts on Xi. Let Gi(zAut(Xi) be the group of automorphisms / s u c h that/*/,* is numerically equivalent to Lt. Then AvczGi which implies that the order of GL is infinite, hence G°t (the connected component) is positive dimensional. But if G°t contains a non-trivial linear subgroup, then when this acts on Xh it would follow that Xt was a ruled surface : since KXl = 0, this is absurd. Therefore G°t is an abelian variety. On the other hand, Alf=(Z/lfZy, and subgroups of fixed bounded index in Av are inside G°. Therefore dim G?>2. It follows that XL consists in only one orbit under G?, hence Xl is a coset space G?/7/, hence Xt itself is an abelian variety. Finally X itself is now caught in the middle between 2 abelian varieties : >Alb X. With a suitable origin, Xt —>Alb X is then a homomorphism, hence if K is its kernel, we find : Alx^Ai - {(x,x+k)\xeXhkeK}. But Xtx xXtdXiX AihXXt and XLxxXi is (i) £tale over Xu and (ii) the graph of an equivalence relation on Xt. (i) implies that X%XxXv = {(x,x+k) \xeXt,ke K>) f for some subset K'aK, and (ii) implies that K is a subgroup. It follows that X=XijK'y hence X is also an abelian variety.

42

E. Bombieri and D. Mumford

References

[ 1 ] Bagnera, G. and De Franchis, M. : Le nombre p de Picard pour les surfaces hyperelliptiques, Rend. Circ. Mat. Palermo 30 (1910). [ 2 ] Deligne, P. and Rapoport, M. : Les schemas de modules de courbes elliptiques, in Springer Lecture Notes 349 (1973). [ 3 ] Enriques, F. and Severi, F.: Memoire sur les surfaces hyperelliptiques, Acta Math. 32 (1909). [ 4 ] Kodaira, K. : On the structure of compact complex analytic surfaces I, Amer. J. Math. 86 (1964). [ 5 ] Lang, S. : Elliptic Functions, Addison-Wesley, 1973. [ 6 ] Matsusaka, T. : Polarized varieties and fields of moduli, Amer. J. Math. 80 (1958). [ 7 ] Matsusaka, T. and Mumford, D. : Two fundamental theorems on deformations of polarized varieties, Amer. J. Math. 86 (1964). [ 8 ] Mumford, D. : Geometric Invariant Theory, Springer-Verlag, 1965. : Lectures on curves on surfaces, Princeton Univ. Press, 1966. [ 9] [10] : Enriques' classification of surfaces I, in Global Analysis, Princeton Univ. Press, 1969. [11] : Pathologies III, Amer. J. Math. 89 (1967). [12] : Abelian Varieties, Tata Studies in Math., Oxford Univ. Press, 1970. [13] Raynaud, M. : Specialisation du foncteur de Picard, Publ. Math. IHES 38, p. 27. [14] Safarevic, Let al : Algebraic Surfaces, Proc. Steklov Inst. Math. 75 (1965). [15] Tate, J. : Genus change in purely inseparable extensions of function fields, Proc. AMS 3 (1952), p. 400. [16] : Endomorphisms of abelian varieties over finite fields, Inv. Math. 2 (1966).

Scuola Normale Superiore, Pisa Harvard University (Received January 14, 1976)

Classification of Hilbert Modular Surfaces

F. Hirzebruch and D. Zagier I.

Introduction and Statement of Results

1. 1 In the paper [6] non-singular models for the Hilbert modular surfaces were constructed. In [9] it was investigated how these algebraic surfaces fit into the Enriques-Kodaira rough classification of surfaces ([11], [12]). But this was only done for the surfaces Y(p) belonging to a real quadratic field of prime discriminant. We shall solve the corresponding problem for real quadratic fields of arbitrary discriminant. We shall use the notation of [6] and [9] and refer to these papers very often. 1. 2 Let K be the real quadratic field of discriminant D and o its ring of integers. The Hilbert modular group G=SL2{o)l {1, — 1} acts on $Q x $ where Q is the upper half plane. The complex space Stf-jG can be compactified by finitely many cusps. This gives a compact normal complex space of dimension 2 denoted by Stf-jG which has finitely many singularities (resulting from the cusps and the elliptic fixed points of G). If one resolves these singularities in the canonical minimal way, one gets a non-singular algebraic surface Y(D). Thus for any discriminant D of a real quadratic field (i. e. D= 1 mod 4 or D = 0 mod 4, where Z)^>5 and D or Z)/4 respectively is square free) an algebraic surface Y(D) is defined. (Here we have changed the notation of [6] § 4. 5. There Y(D) was called Y(d) where d is the square free part of D.) 1. 3 The rough classification of algebraic surfaces without exceptional curves was recalled in [9] (Chap. I, Theorem ROC). Since the surface Y(D) is regular (see [1] Part I, [2] or [9] Prop. II. 4), it is either rational or admits a unique minimal model which is a Resurface, an honestly elliptic surface (fibred over the projective line) or a surface of general type. Thus there are four distinct possibilities, and we wish to decide for every D which of these four cases happens. It was proved recently that Y{D) is simply-connected ([17] and A. Kas, unpublished). Therefore, the Enriques surface (which is an honestly elliptic surface) cannot occur as minimal model of any Y(D) and the class (rational, blown-up K3 surface, blown-up honestly elliptic surface, general type) of Y(D) can be characterized by the Kodaira dimension A; (7(2))) (defined as the maximal dimension of the images of Y(D)

44

F. Hirzebruch and D. Zagier

under the pluricanonical mappings). Thus Y(D) is rational if and only if K(Y (D)) = — 1, if and only if K(Y(D)) = 0, a blown-up K3 surface a blown-up honestly elliptic surface if and only if ic(Y (D)) = 1, of general type if and only if K(Y(D)) = 2. In Chap. II we shall recall the formulas for the arithmetic genus of Y(D). Since Y(D) is regular, we have x(Y(D)) = l+pg^l. It is easy to see that x(Y(D)) tends to oo for D—>oo and certain estimates (Chap. IV) and explicit calculations will show that (1)

x{Y(D)) = 1 <* D = 5, 8, 12, 13, 17, 21, 24, 28, 33, 60.

It was proved in [6] § 4, 5 that Y{D) is rational for these values of D. Since the arithmetic genus of any rational surface equals 1, the ten values of D given in (1) are exactly the values for which Y(D) is rational. The following result was proved in [9]. For convenience we express it in terms of the Kodaira dimension. If p is a prime congruent to 1 mod 4, then *(Y(P)) = - 1 ^X(Y(P)) = l<*p = 5, 13, 17, 0 « x(Y(p)) =2^p = 29, 37, 41, K(Y(p)) = W «(Y(p))= l *(Y(p))= 2 To generalize such results to any discriminant we have to calculate c\{Y(D)) which equals K»K where K is a canonical divisor of Y(D). Namely, if Y(D) is not rational and c2l(Y(D))y0, then Y(D) is of general type. This follows from the rough classification theorem : For the unique minimal model Ymln(D) of Y(D) we have «KFmIn(/>)) ^ C\(Y{D)) > 0. Therefore, Fmin(Z>) cannot be a ^3-surface or an honestly elliptic surface, because for such a surface c2l=K2=0. Since c2(Y(D)) tends to oo for Z)—>oo, we can reduce the classification to a finite list. This requires certain estimates. In Chap. IV we will prove :

Theorem 1. IfD>285, then c2(Y(D))>0. (The proof depends on computer calculations.) There are exactly 50 discriminants with c2(Y(D))<^0 ; they are.listed in Chap. IV. We consider these cases by hand and can settle all of them using the methods of [9] (in particular, Proposition I. 8 and I. 9). Many cases are already covered by (1) and (2) above. The result is the following theorem (Chap. V). Theorem 2. The Hilbert modular surface Y(D) is rational for D = 5, 8, 12, 13, 17, 21, 24, 28, 33, 60, blown-up K3 for D = 29, 37, 40, 41, 44, 56, 57, 69, 105,

Classification of Hilbert Modular Surfaces

45

blown-up honestly elliptic for

D = 53, 61, 65, 73, 76, 77, 85, 88, 92, 93, 120, 140, 165, of general type otherwise (i. e. D = 89 or D ^ 97, but Z)^105, 120, 140, 165). 1. 4 The Hilbert modular group G belonging to Q(^/D) acts also on $X!Q~ where £r is the lower half plane. Compactifying ($ X $~)/G and resolving all singularities of the compactification ($X!Q~)/G in the minimal way lead to an algebraic surface Y_(D). Here we get Theorem 3. The Hilbert modular surface Y_ (D) is rational for D = 5, 8, 12, 13, 17, blown-up K3 for D = 21, 24, 28, 29, 33, 37, 40, 41, blown-up honestly elliptic for D = 44, 53, 57, 61, 65, 73, 85, of general type otherwise (i.e. D = 56, 60, 69, 76, 77 or D ^ 88). 1. 5 Let ft be an ideal in the ring o of integers of K. We introduce the group SL2(o, ft) consisting of all matrices

£ eSL2(K) such that a, deo and fieft'1, fsb.

The actions of SL2{o, ft) and SL2(o) on £ 2 are equivalent (i. e. the groups are conjugate in GL2(K)) if ft=Aa2 where X is a totally positive element of K and a an ideal in o (see [6], 3. 7 (40)). The action of SL2(o) on $X$Q~ and the action of SL2{o, ft) on $ 2 are equivalent if ft=(A) where J is an element of o of negative norm. The following four conditions on the field K are equivalent: i) There exists an element A of negative norm and an ideal a in o with (i)=a2. ii) The number — 1 is the norm of an element of K. iii) The discriminant D is a sum of two natural square numbers. iv) The discriminant D has no prime factor = 3 mod 4. If one of these conditions is satisfied, then the actions of SL2(o) on ig2 and $QX$Q~ are equivalent under an isomorphism of £ 2 and $ X $~ given by an element of GL2(K) whose determinant is positive but has negative norm. The converse is also true (compare 2. 2). For this whole section 1.5 we refer the reader to Hammond [3]. For the group SL2(o, 6) we consider $2ISL2(oJ)), its compactification $2/SL2{o,h) and the algebraic surface Y(D, ft) obtained by resolving all singularities (cusps and quotient singularities) of $2ISL2(o, ft) in the minimal way. The surface Y(D,ft) is also simply-connected [17]. The surfaces Y_(D) and Y(D, ft) are isomorphic if &=(J), where A is an element of o of negative norm. The surfaces Y(D) and Y_{D) are isomorphic if one of the above conditions i)-iv) is satisfied. 1. 6 We consider the involution T on $2>2jG induced by (z\, Z2) \-> (z2, Z\) and study the minimal resolution of (fQ2IG)jT. Here we cannot calculate the invariants c\ and x, because we do not have complete information on the fixed points of G\jG»Tin general. However, if D=p is a prime, the fixed points are known [16].

46

F. Hirzebruch and D. Zagier

The question, for which primes the surface {^2jG)jT is rational, was completely answered in [6] ; there are 24 such primes, the largest being 317. For p>\7 we define in Chap. II a certain non-singular model YT{p) of this surface and give its numerical invariants ; this is needed to determine how the surface fits into the rough classification scheme. In particular, we need to estimate c\{YT(p)). In Chap. IV, we prove Theorem 4. c\(YT(p))>Q for />>821. This reduces the classification question to a finite list. In fact, all cases can be settled here, too, but for this we must refer to [8]. The result was announced in [7]. 1. 7 For the surfaces Y(p) (and also for the YT(p)) the arithmetic genus % surprisingly determines the Kodaira dimension : K = min [ 2 , z - 2 ] . For arbitrary D this is no longer true : for Z) = 85, 140 and 165 we have %(7(Z)))=4, but the surface is nevertheless a blown-up honestly elliptic surface (K= 1). In all cases studied up to now, however, X = I <£> ic = —1 (rational) 2& = 0 (blown-up K3) X = K X = 3 => K = 1 (blown-up honestly elliptic) X ^ 5 => K = 2 (general type). It would be interesting to know whether this is an accident or whether there is some general property of simply-connected algebraic surfaces, valid for all Hilbert modular surfaces, which ensures, for example, that the surface is rational if x = 1 and that it is K3 if it is minimal and % = 2. It is known that there exist simply-connected algebraic surfaces with arithmetic genus one which are not rational (I. V. Dolgacev, Dokl. 7(1966)). II.

Numerical Invariants of Hilbert Modular Surfaces

2. 1 The basic term for the calculation of invariants of Hilbert modular surfaces is the volume of §2IG with respect to the normalized Euler volume form ([6] § 1 (5)) (o = (2JT)-yrb>22 dx,/\dy,/\dx2Ady2. (1) If K=Q(i/D) is the underlying field and G the Hilbert modular group, then (2) where £K(s) is the C-function o f K. We have

(3)

U~ * 2 3Z)mod4

Classification of Hilbert Modular Surfaces

47

(compare [6], § 1 (11), (12)). Here a^n) is the sum of the divisors of n. Let ar(G) be the number of points in $2/G for which the corresponding points in ^>2 have isotropy groups of order r. Then the Euler number of £>2/G is given by the formula ([6] § 1 (21)) (4) We restrict ourselves to discriminants ^ 1 3 . Thus we exclude D = 5, 8, 12. Then

the ar(G) vanish for r > 3 . We write (5)

=fls + (G)+fl,-(G).

fls(G)

Herea3+(G) is the number of quotient singularities of $2/G of type (3 ; 1, 1) whereas 03~(G) is the number of quotient singularities of type (3 ; 1,-1). Compare [6] § 3 (13). We have complete information [15] on a2(G), a3+(G), as-(G). We will state the result in terms of the discriminant D. It is convenient to introduce also the square free part d of D : D =d if d = 1 mod 4 D =U if rf = 2 mod 4 orrf= 3 mod 4. By A(-N) we denote the class number of the imaginary quadratic number field of discriminant —N. We have

(6)

a2(G) =

(7)

d= d= d= d=

1 2 3 1

\h{-3D)

if

mod 3

Ah{-Dj3) 3h{-Dj3)

if if

D = 3 mod 9 D = 6 mod 9

\h{-3D)

if

D =£ 0 mod 3

h(-DI3) 0

if if

CO VO

U(-d)

if if if if

III III

(8)

h{-4d) 3h(—4d) lOh(-d)

mod 4 mod 4 mod 8 mod 8

mod 9 mod 9

The Euler number of $2/G is now calculable. It is not difficult to write a computer program for C,K( — l) as given by formula (3), for the class numbers h( — N) and finally for the Euler number of (9)

^ 13,

The second important invariant of the 4-dimensional rational homology manifold ^2/G is the signature. It has no volume contribution. In the formula for sign $2/G only contributions from the quotient singularities of order 3 and from the cusps enter (compare [6] § 3 (43), (44)).

48

F. Hirzebruch and D. Zagier

(10)

For D ^ 13, we have

sign £2/G = 4w—-|-

|

Here w is the total parabolic contribution in the sense of Shimizu. According to [4] Theorem 2. 1, it can be expressed in the following form : (11)

-^h(Dl)h(D2)u(Dl)-lu(D2)-1

w=

with the summation taken over all decompositions D=D{*D2 with D{
(12)

x(T(D))

=^(e(&IG)+vgn(&IG)).

2. 2 We now consider the action of G on $ X $~. The rational homology manifold ($ X $~)/G admits an orientation reversing homeomorphism onto $2/G. Therefore

For the non-singular model Y_(D) mentioned in the introduction we have (14)

^

The formulas (7), (8), (10) imply that sign(x(T(D))' If we exclude Z>= 12, then sign ($ 2 /G)=0 if and only if D is not divisible by a prime = 3 mod 4. For Z)=12 the signature vanishes ([6] § 3. 9). As Hammond showed (see 1. 5), the signature of Q2jG vanishes (D=\2 again excluded) if and only if the actions of G on $ 2 and on £ X $~ are equivalent, one direction of this equivalence being clear by the second formula of (13). 2. 3 A "cusp" is described by a pair (Af, V) where M is a complete Z-module in the real quadratic field K and F a subgroup of finite index in the (infinite cyclic) group of all totally positive units e with eM=M. To such a pair (Af, V) we associate in a topological way ([6] § 3) a rational number <5(Af, V). Then 4w(M, V) =d(M, V), (16) where w(M, V) is the Shimizu number of a cusp given by evaluating the L-function

Classification of Hilbert Modular Surfaces

49

L(M, V, s) for s=l (see [6] 3. 5). The sum of the w(M, V) for all cusps of the Hilbert modular group of the real quadratic field with discriminant D is the number w given in (11). We wish to recall here the expression ([6] 3. 2 Theorem) for d(M> V) using the continued fraction describing the resolution of the "cusp singularity" of type (Af, V) : There exists a totally positive number a in K such that aM = Zwo+Z-l, where w0 is reduced, i.e. 0 < w0 < 1 < w0. Here x \-> x' denotes the non-trivial automorphism of K. The number w0 has a purely periodic continued fraction development

where ((bOy...9 br_x)) is the primitive period which (up to cyclic permutations) depends only on the strict equivalence class of the module and conversely determines this strict equivalence class. (We recall that by definition the modules M and M are strictly equivalent if and only if there exists an element /3 of K of positive norm such that M=fiM. They are called equivalent if there exists an element /3 of K such that M=pM.) We define (17)

()

Z( O i=0

and (18) l(M) = r. Thus l(M) is the length of the period which we shall also call the length of the module. If y is an element of K with negative norm, then d(rM) = 38(M) = In particular, 8(M)=0 if there exists a unit s of K with negative norm such that v

;

sM=M. To prove (19) we observe that w0 (see above) admits an ordinary continued fraction (20)

i

l

1

Wo = co+ — + ±

(c^z, c^\

which is not necessarily purely periodic. We denote the shortest period of even length by

for

t>0)

50

F. Hirzebruch and D. Zagier

(21) («!,-• •,<%.). Thus it is either the primitive period or twice the primitive period, the latter if and only if the primitive period has odd length. The period (au---,a2s) (up to cyclic permutations) depends only on the equivalence class of M and also determines this equivalence class. A period (21) determines two periods in the sense of continued fractions with minus signs, namely

(22)

((2oJ

and

(23)

( ( V ^ 2 5 *3+2, V ^ , * + 2 , - , 2 ^ 2 , ax+2)) a2—\


a2s—\

(compare [6] 2. 5 (19) and 3. 10). These two periods coincide (up to cyclic permutation) if and only if the period (21) is twice the primitive period of (20), i. e. if the primitive period has odd length. The periods (22), (23) determine the strict equivalence classes contained in the equivalence class of M. There is only one such equivalence class if and only if the primitive period of (20) is odd because this happens if and only if (22) and (23) coincide. Therefore the primitive period of (20) is odd if and only if there exists a unit e of negative norm with sM—M. The formulas (19) are an easy consequence of (22), (23). We also observe that 8(M) is up to sign the alternating sum of the at. If we have a cusp of type (M, V), then Fis a subgroup of finite index in the infinite cyclic group £/£ of all totally positive units e with eM=M, and we have V) = [U+M:

V]4(M).

For the cusps of the Hilbert modular group the modules M are always strictly equivalent to ideals in the ring o of all integers of K. The strict equivalence classes mentioned above correspond to narrow ideal classes, the equivalence classes to ordinary ideal classes. Let C+ be the group of narrow and C the group of ordinary ideal classes of o. Then a !-• a"2 (where a is an ideal in o) induces homomorphisms Sq : C-+C+ and Sq : C+->C+ (see [6] 3. 7 (42)). There are h cusps for the Hilbert modular group SL2(o)/ {I, —1} where h equals \C\ and is the class number of K. These cusps are of type (a"2, U2) where U denotes the group of units of o. Let U+ be the group of all totally positive units ; then U+ = U2 if and only if there exists a unit of negative norm, otherwise [U+ : U2] = 2 . In the first case \C\ = \C+\=h, in the latter |C +| = 2-|C| = 2A. Let eoeU be the fundamental unit (s o >l). Then 5(a~2, U2) = 2d(cr2) 8(cr\ U2)= d(a-2)=0 /(a" 2, U2) =-2l(a-2) l(a-\U2) = /(a-2)

if N(e0) = 1, if N(e0) = - 1 , if N(e0) = 1, if

Classification of Hilbert Modular Surfaces

51

The numbers d and / depend only on the strict module class. Therefore, 8 and / can be regarded as functions on C+. For the total parabolic contribution we have in view of (16) and (25)

a> = 4a€C - S W a ) ) , (Sq:C->C+) \

(26)

which because of (11) is a relation between continued fractions and class numbers of imaginary quadratic fields. (Compare [6] 3. 10 (55)). The pair (Af, V) determines a singularity whose minimal resolution is cyclic. The number of curves in this resolution equals l(M, V) (see [6] 2. 5 Theorem). The Hilbert modular surface $2/G for the field K of discriminant D is compactified by h points. They are singularities in the compactification ffi/G which when resolved minimally give rise to h cycles of curves. The number of all these curves will be denoted by lo(D). We have El(Sq(a))

27

<>

if Neo= - 1

£

or equivalently (28)

k{D) = 2 l(Sq(a))

(Sg: C+^C+).

a€C+

The Hilbert modular surface {&X{Q~)/G is also compactified by h points. These cusps are of type (ya~2, U2) where y is an element of A" of negative norm. We denote by IQ (D) the number of curves needed to resolve all these cusp singularities minimally. Then

and by (19) and (26) (29)

lo(D)-lo(D) = I2w.

2. 4 Let Y(D) be the surface obtained from $2/G by minimal resolutions of all the singular points (see Chap. I). If we assume Z)^13, we have only quotient singularities of order 2 or 3. Those of order 2 are resolved by one curve ; those of order 3 by one or two curves depending on whether the type is (3 ; 1, 1) or (3 ; 1, —1). As in [9] (Proposition II. 2 and (7)) we conclude e(Y(D)) =e(&IG)+a2(G)+a3+(G)+2a;(G)+l0(D) for D ^ 13. Noether's formula states that c2l{Y(D))+e(Y(D)) = l2x(Y(D)). (12), (29), (30) we obtain

Using (9), (10),

52

F. Hirzebruch and D. Zagier

If we consider the action of G on § X £>~ instead of $ X £>, then <23+(G),fl3"(G)interchange their role. The same is true for lo(D)9 lo(D). This implies

(32)

^

^ P

We have (33)

ti(Y_(D))

^ c\{Y{D)),

e(Y.(D)) ;> e(Y(D))

and in fact c>(Y_(D))-c*(Y(D))=l;(D)-lo(D)

«(7_(Z)))-«(7(Z)))

if D m 0(3)

=ro{D)-lo{D) -^-)

if

Z> = 0(3)

if

0^0(3)

if 2) = 0(3).

The corresponding inequality for the arithmetic genus was mentioned before (15). 2. 5 As mentioned in Chapter I, the surfaces Y(D)/T will be investigated for prime discriminants in a later paper [8]. However the necessary estimates for c\ will be done in this paper. Let/? be a prime =1 mod 4. The surface Y(p) has some exceptional curves which can be blown down to give a surface Y°(p). We always assume jfr>17 to ensure that Y{p) is not rational and exceptional curves do not meet. (For details see [6] § 5 and [9]). The involution (zu Zi) V-> (£2? Z\) induces an involution T on Y°(p) which has no isolated fixed points. The fixed point set is a non-singular curve F°p. We have (34)

e{Y\p)IT)

=\{e{Y\p))+e{F°p)).

The Euler number e(F°p) is given by a classical formula. Namely, the curve F°p is the compact non-singular model ofiQir$(p) where r$(p) is the normal extension of ro(p) by the element

L \- This element induces an involution on Wp 0 J which has h( —4p) fixed points according to Fricke(loc. cit in [6]). P u t s = l \{p=l mod 3 and £ = 0 if p=2 mod 3. Then FQ(p) has 2e fixed points of order 3 and 2 fixed points of order 2 and two cusps. Therefore

and (35)

e(F%) = * ( -

Classification of Hilbert Modular Surfaces

53

Put <5=1 if/>=l mod 8 and 5 = 0 if/> = 5 mod 8. Then Y°(p) was obtained from Y(p) by blowing down 4 + 2 5 + s curves. By (30) we get (36) e( and by (34) and (35)

e(Y°(p)IT) 24

5 4

e

6

For the arithmetic genera of Y°(p) and Y°(p)/T we have the following formulas (cf. [6] 5. 6 (20), (21)) A ( 4 / ) )

(37)

(38)

x(Y°(p)IT) = | ( z ( 7 ( / , ) ) _ i ^

^ +

-i. + A

By Noether's formula

c\{Y\p)IT) = \2%{Y°{p)IT)-e(Y\p)IT) which yields

(39)

c]{Y\p)IT) = 2 C x ( - l ) - i t f M L - 1 L / * ( - 3 / > ) 5p . 13 , ., , Q . 13 -^4+-6-e+45+8 + ^

Since K=Q(V~p~) has a unit of negative norm, lo(p) and /o~(/0 coincide. The class number h{p) is odd. Thus Sq : C—>C is bijective and /0(j&) equals the number of all reduced quadratic irrationalities of discriminant p which was denoted in [9] by l(p). In [6] it was shown that many curves on Y°(p)/T can be blown down. The ^£-=— blowdowns (for/?> 17). If we use the basic configuration of curves on Y°(p) (see [6] 5. 4 (8)) we get on Y°(p)/Texceptional curves which come from the h( — 3p)/2 "crosses" and the h(—4p)/2 curves of self-intersection number —2 on Y°(p). (The "crosses" were denoted in [9] p. 18 by C,, C\, the (-2)-curves by Dt.) We have T{Ci)=Ci and T(Di)=Di. The images of Ct and Dt are the exceptional curves in Y°(p)/T we are looking for. The surface obtained from Y°{p)jT by these blow-downs will be denoted by YT(p). We have (for p> 17)

(40)

c\(YT(p)) = 2U-\)

+^

^

+

-^h(-3p)-^lo(p)-^-

54

F. Hirzebruch and D. Zagier

III.

The Hurwitz-MaaG Extension of the Hilbert Modular Group, Skew-Hermitian Curves on Y(D)

3. 1 Let K=Q(y/D) be as before a real quadratic field and o its ring of integers. We consider the matrices

, with entries in o such that ad—be is a totally

positive unit of o. These matrices constitute a group which we divide by its center IL

<26 C/> where Uis again the group of all units of o (cf. introduction). We get

the extended Hilbert modular group Ge. We have GJG~U+jU2. It is a group of order 1 or 2. Now we take the matrices

, with entries in o such that w=ad—bc is totally

positive and a/y/w, b/y/w, c/y/w, djy/w are algebraic integers not necessarily in o. The group of all these matrices has to be divided by its center| L

Nz€O>. We get

a group Gm which is a normal extension of G. It was introduced and studied by Hurwitz [10] § 3 and MaaB [13]. Obviously, the square of every element of GJG is the identity element. If we associate to

, the ideal (y/w) of o (consisting of all \_c a\

elements xeo such that x\yfw is an algebraic integer) we get a homomorphism iz : GJG—+C which maps GJG onto the kernel of Sq : C-+C+. The group GJG is the kernel of n. Thus [Gm : G] equals the order of the kernel of Sq : C+—>C+ which is 2l~x where t is the number of primes dividing the discriminant D* We remark that every line and column of , generates the ideal (y/w) in o. \_c a]

3. 2 The group GJG operates on
cycles is an orbit of the GJG action on Y(D).

Classification of Hilbert Modular Surfaces

55

3. 3 We shall discuss the curves on the Hilbert modular surfaces defined by skew-hermitian matrices. By a skew-hermitian matrix we mean a matrix of the form (1)

\

v

|_ —*

m\

wnere

^ € o and al9 a2 e Z.

QtfJ D \

Its determinant is (2) N= a{a2D+XX. The matrix (1) is called primitive if there is no natural number > 1 dividing a{, a2, X. For a given natural number N the curve FN in $2/G is defined to be the set of all points of SQ2JG which have representatives (zi, Z2)^Q2 for which there exists a primitive skew-hermitian matrix of determinant N such that (3) a
(4)

^VDxx'—X'x+Zx' + aWT) = 0.

Since the matrix (1) can be diagonalized over the field K, this holds if and only if N is a norm in K. This is a condition only on N, so either all components of FN pass through a cusp or none of them do. The reduced quadratic irrationalities of discriminant D are of the form

where M and N are natural numbers, 0<w', and M2—D = 0 (mod 4N). There are only finitely many. Their continued fractions are purely periodic. Thus the reduced quadratic irrationalities of discriminant D are arranged in cycles which correspond bijectively to the elements of C+ (see [6] 2. 6 and 4. 1 (5)). For a given cycle we index the reduced quadratic irrationalities as wk = — t r p — where k runs through Z/IZ with / being the length of the cycle. We illustrate such a cycle as follows :

(5)

56

F. Hirzebruch and D. Zagier

(wt = hThe k-ih line of (5) represents the rational curve Sk of self-intersection number — bk in the resolution of the corresponding cyclic singularity. To the kth corner of (5) we associate the quadratic form p2Nk_lJrpqMk-\-q2Nk of discriminant D. In the resolution of the cusps of ffijG we have exactly the cycles associated to the squares in C+ , i.e. to the image of Sq : C+—>C+. In Y{D) there are 2l~l cycles of curves (see 3. 2) belonging to a given element of Sq(C+) ; in each cycle we have a local coordinate system (uk, vk) centered at the kth corner of the cycle. (The curve Sk is given by vk = 0 and Sk_{ by uk = 0.) To relate this coordinate system as in [6] 2. 3 (11) to the coordinates (zi, z2) of $ 2 by equations 2mzx = Ak_x log uk+Ak log vk 2niz2 = Afk_, log uk+A'k log vk we must transform the cusp to oo. This is done by an isomorphism between Y(D) and Y(D, ho), (see 1. 5) where b0 is an ideal in o such that bo"1 represents the element of Sq(C+) corresponding to the cycle (compare [6] 3. 7). Two such isomorphisms differ by an element of Gm/G. For integers p, q^O (not both 0) we consider the local curve uqk = vp. It has (p, q) branches (6)

ul«>'» = ^vpkKP'q)

with

C(/>>9) = 1

of which
(7)

N=

ptN^+pqM.+fN,.

We identify the triples (k\0, 1) and (A;+l|l, 0). For any triple (k\p, q) belonging to an element of Sq(C+) we have 2l~l local curves up = v\ in Y(D). They are transformed to each other under GmjG. It is not difficult to prove the following lemma. L e m m a . For given N the union of all the primitive branches (6) satisfying (7) [restricted to a sufficiently small neighborhood of all the resolved cusps of Y(D)) equals the intersection of FN with this neighborhood. The equation (3) defines a curve in $2 which is the graph of the fractional linear transformation

from £> to $. Thus the curve (3) can be identified in a specific way with §. Then the irreducible component of FN (given by (3)) has '^jF as its non-singular model where F is the subgroup of the Hilbert modular group G consisting of all elements of G which map the curve (3) to itself. The non-singular compact curve JQJF is obtained from
Classification of Hilbert Modular Surfaces

57

+

(S3"1 representing a cycle (5)) the triples (k\p, q), where (k\0, 1) is to be identified with (A;+l|l, 0), are in one-to-one correspondence with the (integral) ideals B6S3 (see [6] 4. 1). We have W={N) with N as in (7) and (p, q)=n(b) where w(B) is the greatest natural number such that b/n(h) is an integral ideal. The set of all ideals bco which belong to an ideal class $8eSq(C+) is the principal genus $£. By the lemma we have (9)

a(FN) = 2- 1 S p

The curve FN has a cusp (ff(-PV)^l) if and only if iV is a norm in A^ (see (4)). Thus (9) is in agreement with the well-known fact that a natural number is a norm in K if and only if it is the norm of an ideal in the principal genus. If N is a norm in K, then the sum in (9) can be taken over all integral ideals h with W = (N). They are automatically in the principal genus. In some cases (9) gives information on the number of components of FN. First we need a definition. iV is called admissible if it is the norm of an ideal ft in the principal genus which is primitive, i.e. w(b) = l. This happens if and only if N is a norm in K and every prime factor of N decomposes or ramifies in o, the ramifying prime factors having exponent 1 in N. Proposition. If N is admissible and not divisible by the square free part dofD, then FN has 2t~1~r components where r is the number of primes dividing (D, N). If N is admissible and divisible by d, then FN has 2t~r {thus 1 or 2) components. The group GmjG operates transitively on the set of components.

We indicate the proof. If (/?, q) = \ then (6) can be represented by the "diagonal" in tQ2ISL2(o, b) where b is the primitive ideal with norm N corresponding to (k\p, q). Compare [6] 4. 1. Therefore, in this case, the non-singular model of the component of/v represented by (6) is QjF where F = FO(N)I {1,-1} or where F is a certain extension of index 2 of F0(N)/{1,-1}. The latter case happens if and only if Nis divisible by d. As is well-known, the cusps ofQ/F can be represented by rational numbers ajc with (a, c) = l, £>0 and c\N. For any divisor c of N we have^((c, Njc)) cusps. If d\N and F is an extension of index 2 of F0(N)/{1,-1}, then a cusp with denominator c is identified with a cusp of denominator cd\(c, d)2. The given equation (6) from which we started is a description of the embedding of $t}jr in Y(D) near the cusp of JQ/F given by c=N. For a given divisor c of Nit can be shown that SQJF near a cusp with denominator c is imbedded in Y{D) by an equation (6) where (k\p, q) corresponds to the ideal 6=(&.c)/(65 c)2 which has norm N and for which (/?, q) =n(b) = (c, Njc). All ideals with norm N are obtained in this way. As can be checked, we get for given c for the various cusps with denominator c all the
o(ro(N)) = !>((,, NIc)).

58

F. Hirzebruch and D. Zagier

Formula (9) now implies the proposition if N is not divisible by d. If N is divisible by d, then all components have -^-a(F0(N)) cusps. Again (9) implies the result. 3. 4 Suppose we have two different skew-hermitian curves in £>2, one given by (3) with determinant N and the second one by with determinant M. They intersect in £ 2 if and only if the matrix

V-bo/D pi

lla^_

^D

X J

has a fixed point in £ (compare (8)) which happens if and only if 4NM— tr(B)2 > 0. 2 It is easy to check that tr(B) —4NM is divisible by D and its quotient by D is a discriminant (i.e. = 0 or 1 mod 4). Therefore, if the two curves intersect in £ 2 , then the following condition holds. (10)

There exists xeZ with (x2-4NM)/D

such that \x\
w wo* satisfied for N^M,

and 4NM—x2

= 0 mod D

then FN and FM do not intersect in Q2/G.

L e m m a , ^f (10) is not satisfied for M=N, then two different components of FN do not intersect in SQ2JG and moreover FN is non-singular in Y(D) outside the resolved cusps.

Proof Assume that (10) is not satisfied for M=N. If a component of FN is given by (8) with Q/F as its non-singular model, then the isotropy group of the Hilbert modular group G at a point x of $ 2 satisfying (8) is contained in F. It also follows that there is only one skew-hermitian curve of determinant iVin £ 2 passing through x. If the isotropy group of G at x is trivial, then FN is non-singular in the point of Y(D) represented by x. If the isotropy group is of order r, then it is of type (r; 1, 1). This follows from (8). (For Z>>12 we have r=2 or 3 ; see [15].) The curve FN passes in Y(D) transversally through the curve of self-intersection number — r which gives the resolution of the quotient singularity. (Condition (10) and the lemma were suggested to us by P. Hahnel and H.-P. Kraft.) The necessary and sufficient condition that FN be non-singular in the neighborhood of a resolved cusp given by a cycle (5) is that for all/>, q satisfying (7) one of the exponents pl(p> q) or q/(p9 q) in (6) be equal to 1. Thus : If (10) is not satisfied for M=N and if in the lemma in 3. 3 all pairs p> q are such that p\q or q\py then FN is non-singular in Y(D).

In particular Fx is non-singular in Y(D) and has 2l~l components. 3. 5 If N is a prime, then the curve FN is non-empty if and only if N is a norm in K, and iVis a norm in K if and only if the t characters/; (t =!,...,*) do not take

Classification of Hilbert Modular Surfaces

59

a value — 1 at TV. Here we define the %i as follows. We write D as product of prime discriminants

= 11 A, for example 60= (—3).(-4).5. Then

li{N)

forAr dd

= 'A {if)

°

and 0 U2)=(Jf)= 1 V ' -1 Using the proposition in 3. 3, we get Proposition. If N is a prime and D^N, equals

(ii)

forA^O (4) forA^l (8) f o r A = 5 (8).

AN, then the number of components of FN

4-n(i+*W). A

i=\

3. 6 Let Nbe a prime. We wish to study the curve FN% in Y(D). If N decomposes in o, i.e. {D/N) = l, then N2 is admissible and we have the proposition in 3. 3 ; the curve FN> has 2'"1 components. By (9), FN, has 2c~l{N—l) cusps if (D/N)J=l and 2f"1(iV+l) cusps if (D/N) = l. If o=Zrf 0 +^ where w0 is reduced, then one of the local coordinate systems for the cusp at oo is given by 2nizx = w0 2mz2 =

There are N— 1 cusps of FNt corresponding to (13) ao = C where C^ = 1, C ^ l (compare (6), (7) ; we have N0=l,p = 0, q=N, and the iVin (7) has to be replaced here by N2) or to skew-hermitian forms (14) Nzx — Nz2 = r{wo—w'o) where (r, N) = 1, (w0—^ = \^D). The curve So (given by z;0=0) intersects the N—l branches (13) of FN» transversally. The component of FN* given by (14) has
I f

( ^ ) = -1>

then

r = r(N)l{1,-1}

and a{F) =

N-l.

Here F'{N) is defined as follows. Consider the multiplicative group of thefieldFN*

60

F. Hirzebruch and D. Zagier

as subgroup of GL2(FN), take the intersection with SL2{FN). Its inverse image in

SL2(Z) is r(N). If#|Z>, i V ^ 2 a n d i ) ^ N,4N, thenF = F^N)/{\,

-1}

and a(F) = N-l. If D even, tf = 2, D * 8, thenT = r i (2)/{l, - 1 } = r o (2)/{l, -1} (16) and(/(r) - 2 . If D = Nor D = 4N(N± 2), thenT = rf(N)/{l9 -1} IfZ) = 8, ^ = The group F{(N) consists of those matrices in SL2(Z) which are of the form + L J modulo N and Ff(N) is an extension of index 2 of F^N). The proof of (15) and (16) is carried out by applying the method of [6] p. 270 to equation (14). To bring F into the above form one must conjugate in GL2(K). Using (9), (15), (16) (and the Proposition in 3. 3 for the case (D/N) = l) we get P r o p o s i t i o n . If N is an odd prime, then the curve FN* has 2l~l components, except in the case D = N or D=4N where it has 2 or 4 components respectively. If N = 2, then FA has 2l~l components if D is odd or if D = 8. If D is even (D=fc8) then F± has 2l~2 components. Remark. For Nj(D the skew-hermitian curves (14) all belong to the same component of FN* and GmjG operates transitively on the set of components. If N\D (iV=£2), then two skew-hermitian curves (14) belong to the same component if and only if the two values of (r/N) are both equal to + 1 or both equal to — 1. In [7] § 3 it was stated that the curve FN((N/p)^ — l) on Y(p) (p prime) is irreducible. This has to be corrected as pointed out by Hammond. It will be shown in a forthcoming dissertation by Hans-Georg Franke (Bonn) that FN is irreducible if N^O (p2). \i N=0 (p2) then FN has exactly two components. 3. 7 An exceptional curve on an algebraic surface is a non-singular rational curve of self-intersection number — 1. If the surface is regular and not rational, then any two exceptional curves are disjoint and can be blown down simultaneously. In this section we assume that Y{D) is not rational. Thus we exclude 10

discriminants (Chap. I (1)). How many exceptional curves can be found on Y(D) using skew-hermitian curves ? For a discrete subgroup F of PL2 (R) with §/F of finite volume the number (17)

cY(F) =

ix() r«>2

was introduced in [6] 4. 3. We recall that e denotes the Euler number, ar[F) the number of /^-equivalence classes of fixed points of order r of F and a(F) the number of cusps. If a component E of a skew-hermitian curve in Y(D) has the

Classification of Hilbert Modular Surfaces

61

non-singular model £ / / \ then where cx*E denotes the value of the first Chern class of Y(D) on E. Since Y(D) is not rational, c^E^l implies that c^E=l and E is an exceptional curve (see [6] 4. 4 Corollary I). The curve F{ has 2L~l components (see the Proposition in 3. 3). Each component passes through a quotient singularity of order 2 and one of order 3 on ffi/G and is on Y(D) an exceptional curve which gives rise to a configuration

-3

-2

(18) -1

of non singular rational curves. These configurations are disjoint to each other. Using FY we have found on Y(D) in this way 3-2'" 1 curves which can be blown down.

The groups F which occur for the components of F2, F3 are Fo(2) and Fo(3) respectively (to be divided by {1, —1}). For the components of F4 we have F = T0(4) or F=F'{2) if D is odd (to be divided by {1, —1}). (These groups were treated in [6] 5. 5 if D is a prime.) If D is even, the group for F± is JT 0 (2)/{1, —1}. If 3\D (Z)^ 12), then the components of F9 have the group Fl(3)=F0(3) (to be divided by {1, - 1 } ) . For these groups F (namely To(2), To(3), T'(2), T0(4), always divided by {1, —1}) the value of c{(F) equals 1. Since Y(D) is supposed to be not rational, all components of F2, F3, FA and (if 3|Z>) F9 give exceptional curves. Each

component of F2 passes through a quotient singularity of order 2 on
(19)

-1

Every component of F2 gives two curves which can be blown down. The curve F4 has 2l~l or 2l~2 components. In the latter case we have a configuration (19) for each component because the group is Fo(2) (see (16)). Therefore F^ gives always 2l~l curves to blow down, F{ and F4 together give 2t+i curves to blow down. For F3 and F9

the corresponding group has no fixed point of order 2. There is no configuration (19). No additional blow-downs occur in this way. For D=\05 a special situation occurs. We have

F. Hirzebruch and D. Zagier

62

4.9.9-3 2 = 3. 105 Thus condition (10) is satisfied. In fact, it can be checked that the 4 components of F9 meet in quotient singularities of order 3 on ffi/G and this leads on 7(105) to a configuration like this

(20)

-3

-3

-1

-1

-1

-1

which gives two extra curves to blow down. In fact, for Z)=105 there are six quotient singularities of order 3 on ffi/G, all of type (3 ; 1, 1). Four of them lie on the 4 components of Fx. The two others give rise to the two curves of self-intersection number —3 in (20). Two intersecting components of F9 never occur for other D (with 3\D) as can be checked by condition (10). By the propositions in 3. 5 and 3. 6 we know the number of components of F2, F3, F4, F9y hence we can collect the information on exceptional curves in the following theorem. T h e o r e m . Suppose Y(D) is not rational. Then /3(Z)) curves on Y(D) can be blown down where (21)

= 2*+i+n(i+z«(2))+4-n(i+Zi(3)) 2

for

0

for D* 105.

^=

We call Y"(D) the surface obtained from Y(D) by blowing down these fl(D) curves. We define Y°(D) only if Y(D) is not rational. Clearly c\{Y\D))=d{Y{D))+p{D)(22) We conjecture that Y°(D) is the minimal model. For D equal to a prime, this was conjectured in [9]. In fact, van der Geer and van de Ven have checked the conjecture for several prime values of D where Y°(D) is of general type. When Y°(D) is not of general type, then the conjecture holds because c2l(Y°(D))=0 as we shall see. 3. 8 For the surface Y_(D) introduced in Chapter I similar considerations hold. We have a curve FN given by all primitive equations (3) with ala2D-\-XXf = — N. This curve passes through a cusp if and only if — N is a norm in K. If — 1 is a norm in X, then Y(D) and Y_(D) are isomorphic. In this case (provided Y(D)

Classification of Hilbert Modular Surfaces

63

is not rational) we can blow down /3(Z)) curves on Y_(D) ; the resulting surface Y°_(D) is then isomorphic to Y°(D). If condition (10) is not satisfied for N, M (Nj=M), then FN and FM do not meet on (# X §")/G. If condition (10) is not satisfied for M=N, then FN is non-singular on Y_(D) outside the resolved cusps. The lemma in 3. 3 holds in the same way except that one has to take the triples (k\p,q) belonging to an element of (VD)Sq(C+) where (VD) denotes here the element of C+ represented by the ideal (\/D). The natural number iV is called admissible for Y_(D) if — N is a norm in K and all primes dividing N decompose or ramify in 0, but where a prime which ramifies occurs in N only with exponent 1. The number of components of FN(N admissible for Y_(D)) is as in the proposition in 3. 3. If —1 is not a norm in K, then F{, F4, F9 are empty on Y_(D), so we can only blow down F2 and F3, and this gives (if Y_ (D) is not rational)

(23)

/}_(/>) = n(l+ z < (_2)) + i - n ( l + Z l ( - 3 ) )

blow-downs. (Note that &( —iV) = (sign Dt) Xi(N)-) Again we conjecture that the surface Y°_(D) obtained by these blow-downs is minimal. IV.

Estimates of the Numerical Invariants

4. 1 The purpose of this chapter is to prove the facts x(Y(D)) = 1 o D = 55 8, 12, 13, 17, 21, 24, 28, 33, 60, c\(Y(D)) < 0 =>Z> c\(Y.(D)) <0=>D< 136, c\{JT(p)) < 0 => p < 821

(p= l(mod 4) prime)

(compare Chapter I), thus reducing the problem of classifying all Hilbert modular surfaces to the consideration of a finite list. Since all of the invariants have been calculated (by computer) up to at least Z)=1500, it will suffice to prove (1) (2)

D > 1500=>z(7(Z>)) > 1, x(Y.(D)) > 1, c]{Y{D)) > 0, c\(Y.(D)) > 0, />>1500=>*?(y r (£))>0.

There are precisely 50 discriminants for which the four inequalities of (1) are not all satisfied ; complete numerical data on these discriminants is given in section 4.5. 4. 2 As explained in 2. 1, the dominant term in the formula for all of these numerical invariants is

<3>

w!)

*2=Z>(mod4)

From ox{n)>n-\-\ we deduce easily that (4) C K (-l)

64

F. Hirzebruch and D. Zagier

this result can also be obtained by writing

and applying the functional equation of £K(s). From 2. 4 (31) and 2. 1 (9), (10), (12) we have c\{Y{D)) = 4 C * ( - l ) - , 7 - / T ^

l

and

moreover, by 2. 3 (29),

and hence

Z)3/2+?(7(i)))

> ^ j f ^ +1

(D > 20°)-

Hence the inequality x(F(Z)))>l in (1) will follow once we have proved that c\(Y{D)) is positive ; since (by 2. 2 (15) and 2. 4 (33)) the values of x and c\ for F_(-D) are at least as large as for Y(D)> the remaining two inequalities in (1) will also follow. Thus to prove (1) we have to show (5)

4C K (-l)-/ 0 -( J D)-a 3 + (G)/3>0

(Z)>1500),

while for (2) the inequality

(6)

2C / f (-l)-|/ 0 (/>)-|£ + ^ > 0

(/>

will certainly suffice (equation 2. 5 (40)). We introduce a new invariant (7)

l(D) = S+/(a)

(notation as in 2. 3). IfD—p is a prime, then Sq : C+—>C+ is an isomorphism and if, however, D has t distinct prime factors, then Sq has a kernel of order 2c~l and so l (9) l~(D) = 2 KrSq(a)) = 22'-' The advantage of working with l(D) rather than l0 (D) is that it can be evaluated by a formula analogous to formula (3) for C,K( — 1). Indeed, l(D) is the sum of the lengths of all cycles occurring as the primitive period of the continued fraction of some quadratic irrationality w of discriminant D (the discriminant of w is de-

Classification of Hilbert Modular Surfaces

65

fined as b2—4ac, where aw2-\-bw+c=0, (a, b, c) = l). This is simply the number of reduced quadratic irrationalities w of discriminant D (i. e. w satisfying zt>> l>w' >0), since, as discussed in 2. 3, such w have purely periodic continued fractions, and a cycle ((#o,---,£r-i)) of length r gives rise to precisely r reduced numbers

If aw2-\-bw-\-c—0, b2—4ac=D, then the condition (a, b, c) = l is automatically satisfied since D is the discriminant of a quadratic field. Therefore

= #{(«, *, c) eZ> | ^ - 4 a c = A The inequalities are equivalent to a>0, \-b-2a\<0, A 2 0, ^2
+

~

2a


'

so precisely half of the elements (a, £) satisfy 2a-\-k>*JD. Therefore (10) k2
a>0 a D~k'

This formula will be the basis for our estimates of c\. 4. 3 In this section we prove the estimate (6) ; this case is easier than estimate (5) for composite D because of (8). We will prove (for all Z), prime or composite) that (11)

2C*(-l)--g-/(Z>) > ^y-y/D-200

since the right-hand side is > - o j

-3.6

^—for Z)>1500, this implies (6).

By (3) and (10), the left-hand side of (11) equals

(•2) with

(D > 730) ;

66

F. Hirzebruch and D. Zagier

We have —0.6 ,, v ^ n—14 3 Q

(13) v

y

for n < 50, ~ -^ for n > 50.

Indeed, for w<56 we can check this by hand, while, for w > 5 6 ^ = ( 7-~-j>

-ar[(-4H'-4)+T S ('+7-1')] Kd
-

30

(each term d+n/d—15 is >0). For Z)>729 there are at most 4 values of £ (two positive and two negative) for which k=D(mod 2) and 0<(Z)—£2 )/4<50 (because the interval (V£>— 200, V^D) has length<4), so the first line of (13) is used at most four times in (12) ; the second estimate in (13) now gives

where we have used the easy estimates (valid for any positive A and integer D) k(mod2)

k
with A=D—200. This proves the inequality (11). 4. 4 We now want to prove the estimate (5). The number IQ(D) in that equation will be estimated using (9) and (10) ; for the number ^ ( G ) , given exactly by 2. 1 (7), we use the estimate (#>4)

(cf. [14]) to obtain 1 The formula to be proved then becomes (14) 4C x (-l)-0.13V^(logZ)+l)>2'- 2

S

tfo(^p-)

(2>>1500).

Because of the factor 2t~2> the method of 4. 3 does not work here and we must have recourse to far cruder estimates. We would like to thank Henri Cohen, who suggested the method for estimating the right-hand side of (14) and carried out

Classification of Hilbert Modular Surfaces

67

the necessary computer calculations. Lemma. Set e=log 2/log 11 =0.289064826-•-. Then for all n (15)

ao(n) < 5.1039782 rf.

Proof. The function <jo(n)/n£ is multiplicative and {a-\-\)jpas<\ for jfr>ll, a>\ by the choice of e. Hence

If we now estimate QK{ — \) by (4), and the right-hand side of (14) by the product of the number of terms in the sum with the estimate of the individual terms given by equation (15), we find as a sufficient condition for (14) the inequality (16)

-^--0.13Z> 1 / 2 (logZ)+l) > 2'- 2 (V£+l)-5.1039782.(Z)/4) £

with £ = 0.289064826-•• as before. A desk calculator computation now shows that (16) holds if 9,000 and D > t<3 or t = 4 and D > 23,000 or t = 5 and and D> 60,000 or t = 6 and D > 157,000 or t = 7 and D > 420,000.

(17)

But the smallest discriminant with t=l is 4. 3. 5. 7. 11. 13. 17= 1,021,020>420,000, so (17) implies that (16) holds for all D with t=l. A similarargument holds for any t>7, since a D with t>l distinct prime factors is greater than 4. J. 5. /. 11. U. 1b

- g 5 5 3 6 (^

J,

so 2^2<^T7TD, more than sufficient to prove (16) for D>420,000. Therefore (17) implies that (16) (and hence (14)) holds for all D> 157,000, and a computer calculation showed that (14) holds for all D up to this point. 4. 5 As already stated, the calculation of the various invariants for Z)<1500 showed that c\(Y{D))<0 for just 50 discriminants, the largest being Z) = 285. We have tabulated all numerical invariants of Y(D) and Y_{D) for these discriminants. The following notation is used : Topological Invariants :

Z = 6C*(—1) 1QJ- = lo{D)J-{D)

(this is an integer for D > 8) (§§2.3,2.4)

a2, af, a3- = a2(G), a}(G), a^{G) (§ 2. 1 ; for D = 5, 8 and 12 there are also fixed points of order 5, 4 and 6 respectively) e = e(
F. Hirzebruch and D. Zagier

68

Invariants of Y(D) : (2.1(12)) = cj(Y(D)) (2.4(31)) (3. 7 (22) ; c° is not listed if Y{D) is rational, ° = ef(P(/))) = c+j8(Z)) since Y°(D) was not defined in this case) Invariants of Y_ (D) (not given if D is a sum of two squares since then Y_ (D) is isomorphic to Y(D)) : X_=±(e+T)=X(Y_(D))

(2.2(14))

e.=ci(Y.(D)) dL = c](Y°_(D)) = c_+p_(D)

(2-4(32)) (§3.8).

D

Z

h

5 8 12 13 17 21 24 28 29 33 37 40 41 44 53 56 57 60 61 65 69 73 76 77 85 88 89 92 93 97 101 104 105 109 113 120 124

1/5 1/2 1 1 2 2 3 4 3 6 5 7 8 7 7 10 14 12 11 16 12 22 19 12 18 23 26 20 18 34 19 25 36 27 36 34 40

1 2 2 3 5 2 4 4 5 8 7 12 11 6 7 4 14 4 11 18 4 21 14 2 18 12 21 4 6 27 11 20 12 17 23 8 16

1 2 4 3 5 6 8 10 5 12 7 12 11 12 7 16 18 24 11 18 16 21 20 14 18 24 21 22 18 27 11 20 44 17 23 40 34

a2

at

2 2 3 2 4 4 6 4 6 4 2 6 8 10 6 12 4 8 6 8 8 4 10 8 4 6 12 12 4 4 14 18 8 6 8 12 12

1 1 2 2 1 4 3 2 3 3 4 2 1 2 5 2 4 6 4 2 9 2 2 6 6 4 1 4 12 2 5 2 6 6 3 8 2

e

1 1 0 2 1 1 0 2 3 0 4 2 1 2 5 2 1 0 4 2 0 2 2 6 6 4 1 4 3 2 5 2 0 6 3 2 2

T

4 0 4 0 4 0 4 0 4 0 6 2 6 2 6 2 8 0 6 2 8 0 8 0 8 0 10 2 12 0 12 4 10 2 12 8 12 0 12 0 14 6 12 0 14 2 16 4 16 0 16 4 16 0 18 6 18 6 16 0 20 0 20 0 20 12 20 0 20 0 24 12 22 6

X 1 1 1 1 1 1 1 1 2 1 2 2 2 2 3 2 2 1 3 3 2 3 3 3 4 3 4 3 3 4 5 5 2 5 5 3 4

X- c_ c°_

£

- 2 3 - 4 - 3 - 4 - 6 - 7 - 8 - 4 - 9 - 5 - 8 - 6 - 8 - 4 -10 -10 -18 - 5 - 8 -11 - 7 - 8 - 8 - 8 -10 - 4 -10 -10 - 5 0 - 4 -22 - 1 0 -20 - 8

• • • • • • • • 0 • 0 0 0 0 0 0 0 • 0 0 0 0 0 0 0 0 2 0 0 2 4 4 0 4 6 0 2

1 -4

•

2 -1 2 -2 2 -2

0 0 0

2 -4

0

9

0

2 4 3 -5 4 5

2 0 4

4

4

4 -2 4 5

2 4

5

2

4

6 6

8 5

8 6

3

5

8

12 12

9 7

14 14 10 12

Classification of Hilbert Modular Surfaces

D

Z

129 133 136 140 141 156 161 165 168 184 204 220 285

50 34 46 38 36 52 64 44 54 74 78 92 96

V.

30 12 32 4 8 16 14 4 8 16 28 16 4

34 24 32 40 28 48 50 44 48 52 56 64 60

a2

<>t

12 4 12 20 8 16 16 8 12 12 20 16 16

4 8 2 4 15 8 2 16 12 4 12 4 24

1 8 2 4 0 2 2 4 0 4 0 4 0

T

26 24 24 28 26 32 32 32 32 36 44 44 56

2 4 0 12 10 12 12 16 16 12 12 16 24

X 6 5 6 4 4 5 5 4 4 6 8 7 8

c

- 2 8 - 4 4 - 2 8 -16 0 g 2 -16 4 - 8 4 -20 0 -16 4 - 4 8 - 8 12 - 4 12 - 4 16

69

X- c_ c°_ 3 8 7 8 10 7 10 9 11 11 12 12 12 14 15 20

20 16 18 28 24 28 32 24 44 60

20 16 20 28 24 28 32 28 44 60

The Rough Classification of Hilbert Modular Surfaces

5. 1 In this chapter we prove Theorems 2 and 3 of the introduction (Chap. I). Our basic reference for the classification is the joint paper [9] with Van de Ven. In the proposition below we state the main classification principle. A ( — 2)-curve is a non-singular rational curve with self-intersection number —2. An elliptic configuration on an algebraic surface X is a finite set of irreducible curves on X having the same genera and intersection numbers as the configurations occurring as fibres (without exceptional curves) in an elliptic fibration of some surface ([11] Part II). We give a complete list of the elliptic configurations : A non-singular curve E of genus 1 with EE=0 ; a rational curve E having exactly one singular point (a cusp or a double point) with EE=0 ; a configuration of ( — 2)-curves with one of the following intersection diagrams

intersection number 2

A^

(cycle of any length k^

or of the diagrams, better indicated by their dual graphs (a dot indicates a (— 2)-curve and a line a transversal intersection):

70

F. Hirzebruch and D. Zagier

D'k-i (^curves, k—1^5) 9

*

m

•

•

£' •

m

•

•

•

•

•

•-

E7

Proposition. Let X be a simply-connected non-rational algebraic surface. If X contains an elliptic configuration, then X is a blown-up K3-surface or a blown-up honestly elliptic surface. If X contains an elliptic configuration which intersects a ( — 2)-curve on X not belonging to the configuration, then X is a blown-up K3-surface. IfX is simply-connected, not rational, and not a blown-up KZ-surface and E is an irreducible curve on X such that c^E^O (where cleH2(X, Z) is the first Chern class), then either cx*E=\ and E is an exceptional curve or cl*E=0 and E is either a (— 2) -curve or a curve of genus 1 or 0 with EE=0.

The proof is obtained as in [9] (compare Proposition I. 9). For the second part of the proposition we use [9] (Propositions I. 1 and I. 5) and in particular the fact that cl>E^i2 implies the rationality of X. When passing to the minimal model X' of X, a certain configuration L of rational curves on X is blown down. If c1«£'>0, then either E belongs to L and is an exceptional curve or a ( — 2)-curve on X, or E and L are disjoint, cx • E= 0, and E is a component of the unique elliptic fibration of X' or a ( — 2)-curve on the surface X' of general type. 5. 2 The results and the tables of the preceding chapter have shown that c (Y(D))>0 except for 50 discriminants. The arithmetic genus equals 1 for 10 discriminants (5, 8, 12, 13, 17, 21, 24, 28, 33, 60) ; they are among those 50. The corresponding 10 surfaces Y(D) are known to be rational ([6] 4. 5 Theorem). For the remaining 40 discriminants we calculated c2(Y°(D)) (table in Chap. IV) using 3. 7 (21), (22) and obtained c2(Y°(D))>0 (which implies general type !) except for the 22 discriminants 2

(1)

29, 37, 40, 41, 44, 53, 56, 57, 61, 65, 69 73, 76, 77, 85, 88, 92, 93, 105, 120, 140, 165,

for which we get c2(Y°(D))=0. These 22 have to be investigated by hand. The surface Y_(D) has arithmetic genus 1 for 5 discriminants (5, 8, 12, 13, 17). These surfaces are rational. Namely, except for D=\2 they are isomorphic to Y(D), and for D=12 it was shown in [6] 4. 5 that Y_(D) is rational. For D^5, 8, 12, 13, 17 (i.e. D> 17) there are 23 discriminants for which c2(Y_(D))^0. For these we consider Y°_(D) (see 3. 8) and obtain c2(Y°_{D))>0 except for 15 discriminants (see table in Chap. IV)

Classification of Hilbert Modular Surfaces

71

(2)

21, 24, 28, 29, 33, 37, 40, 41 44,53,57,61,65,73,85 for which we get c2(Y°_(D))=0. These 15 surfaces have to be investigated by hand. All other Y_(D) are rational (5 cases) or of general type. 5. 3 The components of the curves FN in Y(D) or Y.(D) all have the same non-singular model if N is admissible (3. 3 and 3. 8). This model is $/ro(N) if N is not divisible by the square free part d of D. The values of cx{rQ{N)j{\, - 1 } ) (see 3. 7 (17)) are denoted by cx(N) and were listed in [6] 4. 3 for the case that the genus of $/ro(N) is 0 or 1. Let cx be the first Chern class of Y(D) or Y_(D) respectively. Then (3)

for any component E of FN (N admissible, N=£0 mod d) where the sum is over all the branches of FN belonging to E near the cusps (see 3. 3 (6)). For (3) compare [6] 4. 5 (34). The sum in (3) equals the intersection number of E with the Chern divisors of the cusps minus a(ro(N)). For JV=5, 6, 7, 8, 9 the curve Q/ro(N) is rational and d(N)=0. In these cases £,«i?_:0 for all components E of FN. If c^E=0 and E is non-singular, then E is a (—2)-curve. 5. 4 In this section we settle the rough classification of the surfaces Y_(D) using the proposition in 5. 1. The principal cusp of Y_(D) has the resolution cycle belonging to the strict ideal class of (V~D). We consider the reduced quadratic irrationalities

(4)

b W(b)=

~l^)D,

beZ, b = D mod 2, -

The module Zw(b)+Z is strictly equivalent to the ideal (
for |£| —2. Furthermore wm is of the form-^(Af+VZ>) with N (b2)

= — (D—b2). Hence we have on Y_(D) a configuration

f(D-9)

-2)

r~ r 2

7

i(D-4)

-2Y-2

V.

(±6=2,4, - ; 161 <jD-2)

± 6 = 1, 3, •••; I b

\
72

F. Hirzebruch and D. Zagier

depending on whether D is even or odd. The (— 2)-curves belong to the resolution of the cusp. The symmetry in the commutation comes from the canonical involution on Y_(D) induced by (zi, z2) \-> (—£2, —Zi)- Two i's differing only up to sign give the same component of FiD_b2)/4. It is carried to itself by the involution ([6] §4. 5). For the rest of this section 5. 4 we suppose that 2)>17 so that Y_(D) is not rational. Then we can use (5) for the rough classification as follows. If in (5) any of the FiD_b*)/4 (with \b\
is F5, F6, F7i F8 or F9, then Y_(D) is

Namely, let E be the component drawn in (5) of such an F(D_bi)/A. Then c^E^iO, but E is not an exceptional curve, because blowing it down would give by (5) two intersecting exceptional curves which is not possible on a non-rational regular surface.. The proposition in 5. 1 and diagram (5) now show: If Y_(D) is not a blown-up X3-surface, then E is a ( — 2)-curve (in fact it has to belong to the largest \b\ in (5)) and E and the ( —2)-curves of the resolved cusp indicated in (5) give a cyclic elliptic configuration proving that the surface is blown-up honestly elliptic. Let us first consider the values D in (2) for which %(F_(Z)))^3. These are certainly not blown-up ^3-surfaces. For 2) = 44, 53, 57, 61, 85 we get on Y_(D) a cycle of non-singular rational curves of self-intersection number —2 using F7, F7, FQ, F9, F9 respectively. For D = 65 we have a configuration -1

(6)

where the F4 belongs to w(_7)=( — 5+\/65)/( — 7+V65) and [w(_7) + l] = 3. Blowing down the component of F4 drawn in (6) we get again a cyclic elliptic configuration. ForZ) = 73 see [9] (for prime discriminants D the surfaces Y(D) and Y_(D) are isomorphic). In all cases we have a cyclic elliptic configuration. The surfaces are blown-up honestly elliptic. Now we study the eight values in (2) for which x{Y_(D))=2. These are 21, 24, 28, 29, 33, 37,40,41. We wish to prove that the corresponding surfaces are blown-up X3-surfaces. We may assume that the components of F59 F6, F7, F6> F9 occurring in (5) are ( — 2)curves, since if they are not, the surfaces are certainly blown-up jO-surfaces (Pro-

Classification of Hilbert Modular Surfaces

73

position 5. 1). For Z)=21, 24, 29 the curve F5 occurs in configuration (5). Every component of F5 passes through a curve A of self-intersection number —2 coming from the resolution of a quotient singularity of order 2, because F0(5) has fixed points of order 2 in $. Thus (5) leads to an elliptic configuration which intersects A, so the surface is a blown-up X3-surface. For Z) = 33, 37, 40 we have in (5) two different values of b for which (D—b2)/4=6, 7, 8, 9. Hence the surfaces are blownup .O-surfaces. For D=28 we have a configuration

which proves by blowing down F3 that 7_(28) is blqwn-up K3. For D — 41 the same argument works : The curve F8 occurs and one has to blow down F4. Theorem 3 in Chapter I is now completely proved. 5. 5 In this section and in the following one we shall do the rough classification of the surfaces Y(D) and prove Theorem 2 of Chap. I. Since Y(D) is equivalent to Y_(D) if D is not divisible by a prime = 3 mod 4, it remains to study the following 13 discriminants from the list (1) ti(Y(D)) = 2) D = 44, 56, 57, 69, 105 (7) D = 76, 77, 88, 92, 93, 120, 140, 165 (x(Y{D)) ^ 3). For these 13 discriminants we indicate the resolution of the cusps with the notation of 3. 3 (5). The reader should consult these diagrams, which are printed at the end of the paper, during the course of the proofs. I -3

i-i

-2 -31

-3!

-2

-1

-1

-1

-1

F. Hirzebruch and D. Zagier

74

For Z) = 44, 56, 57, 69, 105 we consider the curve So in the resolution of the principal cusp (see 3. 6). The exceptional curves Fl9 F4 and F9 (for 3\D) give on Y(D) the configuration in the previous page. The curve of self-intersection —2 intersecting FA exists if and only if D is even. The exceptional curves F9 exist if and only if 3\D. The dotted components of F9 intersect the ( —3)-curves if and only if D= 105 (see 3. 7). (It has to be checked for D=IO5 that the two components of F9 intersecting So do not meet in fQ2jG.) We have S0»S0=— bo= — [w0] — 1 and cl*S0=—b0+2.

O n passing

to Y°(D) the exceptional curves in the above diagram are blown down successively and we get on Y°(D) (whose first Chern class we denote by ?,) an image curve So which has exactly one singular point (a cusp) and for which cr$0 = - A 0 + 7 + l(if2)iseven)+2(if3|Z))+.2(if/)=105). For Z>=44, 56, 57, 69, 105 the values of b0 are 8, 8, 9, 9, 11 and we get crS0=0 and hence SO*SO=O. Thus the single curve So is an elliptic configuration. The curve S{ has for Z)=44, 57 the self-intersection number —2. For Z) = 69, 105 the curve S{ has self-intersection number — 3 and intersects F3, F4 respectively ; therefore in Y°(D) the image curve S{ has self-intersection number —2. For D = 56 we have Sl*Sl = — 4, but the curve S{ meets the exceptional curve F2 which by 3. 7 (19) leads to two blow-downs, so the image curve SY on 7°(56) has the selfintersection number —2. Thus by the proposition in 5. 1 (and because c2l(Y°(D)) = 0), the surfaces Y°(D) are X3-surfaces for Z> = 44, 56, 57, 69, 105. 5. 6 We now study the discriminants in the second line of (7). In all cases we shall find an elliptic configuration on Y(D) which proves that Y(D) is blown-up honestly elliptic and finishes the proof of Theorem 2 in Chapter I. For Z) = 76 consider the curve F6. It has one component which meets four curves of the resolution of the cusp. This gives rise to an elliptic configuration: o

-2

-2

-2

-2

(The proposition in 5. 1 implies that F6 is a ( — 2)-curve.) For D = ll the irreducible curve Fn passes through the two corners of the resolution of the cusp. The genus of Fn is 1. We have ^(11) = —2 and c^F^O by (3). By the proposition in 5. 1 the curve Fn is an elliptic configuration (crFn = *ii-fii=0).

For Z>=88 the two components of F9 together with 6 curves of the resolved cusp give a cyclic elliptic configuration of length 8. For D=92 the two components of F13pass through the four corners of the resolution of the cusp. We have f,(13) = —2, but (3) implies that ^-£"^0 for each com-

Classification of Hilbert Modular Surfaces

75

ponent E of Fl3. Since E is a rational curve and meets two ( — 2)-curves (coming from two quotient singularities of order 2), it follows from the proposition in 5. 1 that E is a (— 2)-curve. The curve F9 was considered in 3. 6. Here 9 is not admissible ; the group T in 3. 6 (15) is r'(3)/{l, - 1 } for which a2{T)=2, a3(F)=0, o{r)=2, e(QIT)=2 and ^ ( r ) = 0 . The curve F9 has two components. It follows as before that each component of F9 is a (— 2)-curve. As can be checked, F9 and Fl3 intersect in ^2jG in quotient singularities of order 2. (Condition 3. 3 (10) is satisfied : (4-9-13—102)/92 = 4.) We have on Y(D) the following elliptic configurations :

where the "vertical" curves come from the resolution of quotient singularities of order 2. For Z) = 93 the two components of F7 together with 4 curves of the resolved cusp give a cyclic elliptic configuration of length 6. For Z)=120 the curve F6 has one component. It passes through both cusps and gives rise to the elliptic configuration:

-2 -2

-2

-2

-2

For Z)—140 the curve FH passes through the four corners of the 2 resolved cusps. It has one component. The genus of FH is 1. We have £,(14) = — 4, and cl*FH=0 by (3). By proposition 5. 1 the curve FH is an elliptic configuration. For Z)=165 the same argument works with Fl5. 8/

1

\8 -4

-4

F. Hirzebruch and D. Zagier

76

15

1 -9

9

-3 3

3 -3

£=69

9

9

-9 1

9

£ = 77
10

£=76

10

10

D=92

11/

-10 1

1

1 Ml

10 -5 2

2 -5 10

2)=105 (two cusps)

1 -10

£=93

17

10

Ml 13 £ = 120 (two cusps)

TT\

1

/\\

12

1 -12

12

6 -2 12

£ = 140 (two cusps)

£ = 165 (two cusps)

-2 6 -12 1

12

13

Classification of Hilbert Modular Surfaces

77

References [ 1 ] Freitag, E. : Uber die Struktur der Funktionenkorper zu hyperabelschen Gruppen I. II. J. Reine Angew. Math. (Crelle) 247 (1971), 97-117, 254 (1972), 1-16. [ 2 ] Hammond, W. F. : The Hilbert modular surface of a real quadratic field. Math. Ann. 200 (1973), 25-45. [ 3 ] Hammond, W. F. : The two actions of Hilbert's modular group (to appear). [ 4 ] Hammond, W. F. and Hirzebruch, F. : L-Series, modular imbeddings and signatures. Math. Ann. 204 (1973), 263-270. [ 5 ] Hecke, E. : Vorlesungen uber die Theorie der algebraischen Zahlen. Akademische Verlagsgesellschaft, Leipzig 1923. [ 6 ] Hirzebruch, F. : Hilbert modular surfaces. Enseignement Math. 29(1973), 183-281. [ 7 ] Hirzebruch, F. : Kurven auf den Hilbertschen Modulflachen und Klassenzahlrelationen, in Classification of Algebraic Varieties and Compact Complex Manifolds. Lecture Notes in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York (1974), 75-93. [ 8 ] Hirzebruch, F. : Rough classification of the algebraic surfaces arising from the symmetric Hilbert modular group (in preparation). [ 9 ] Hirzebruch, F. and Van de Ven A. : Hilbert modular surfaces and the classification of algebraic surfaces. Inventiones math. 23 (1974), 1-29. [10] Hurwitz, A. : Die unimodularen Substitutionen in einem algebraischen Zahlkorper. Nachr. Akad. Wiss. Gottingen, Math. Physik Klasse, 332-356 (1895). (Mathematische Werke Bd. II, Birkhauser Verlag, Basel und Stuttgart 1963, 244-268.) [11] Kodaira, K. : On compact complex analytic surfaces I, II, III. Ann. of Math. 71 (1960) 111-152, 77 (1963) 563-626, 78 (1963), 1-40. [12] Kodaira, K. : On the structure of compact complex analytic surfaces I, II, III, IV. Amer. J. Math. 86 (1964), 751-798, 88 (1966), 682-721, 90 (1968), 55-83, 90 (1968), 1048-1066. [13] Maass, H. : Uber die Erweiterungsfahigkeit der Hilbertschen Modulgruppe, Math. Z. 51 (1948), 255-261. [14] Newman, M. : Bounds for Class Numbers, in Theory of Numbers, Proceedings of Symposia in Pure Mathematics, Vol. VIII, A. M. S., Providence, 1965, 70-77. [15] Prestel, A. : Die elliptischen Fixpunkte der Hilbertschen Modulgruppen. Math. Ann. 177 (1968), 181-209. [16] Prestel, A. : Die Fixpunkte der symmetrischen Hilbertschen Modulgruppe zu einem reellquadratischen Zahlkorper mit Primzahldiskriminante. Math. Ann. 200 (1973), 123-139. [17] Svarcman, O. V. : Simply-connectedness of the factor space of the Hilbert modular group (in Russian), Functional Analysis and its Applications (2) 8 (1974), 99-100.

SFB Theoretische Mathematik Mathematisches Institut der Universitat Bonn (Received October 11,1975)

On Algebraic Surfaces with Pencils of Curves of Genus 2

E. Horikawa 0 Introduction

Inspired by Kodaira's work on elliptic surfaces, several authors have studied pencils of curves of genus 2 on compact complex analytic surfaces. We understand that they have established a "local theory" on such pencils. We refer the reader to [7] for a brief account of results and references. We do not use the results of our predecessors in the following. In this paper we shall study pencils of curves of genus 2 from a little more global point of view. We are more interested in surfaces S which carry these pencils rather than in the pencils themselves. We note that these surfaces are projective algebraic. Our main results are as follows. Let g : S—>J be a surjective holomorphic map onto a non-singular complete curve A whose general fibre C is an irreducible nonsingular curve of genus 2. We let K denote the canonical bundle of S. Then, for a sufficiently ample divisor m on J, the linear system |X+g*m| determines a rational map 0 : S—>W% of degree 2 onto a surface Wx which is a P^bundle over J. Let XS : S—+S be a composition of quadric transformations such that 0 o TS is everywhere defined. We define the branch locus Bl of 0 to be that of 0 o tD', which is independent of the choice of TS. The singularities of Bx are classified into six types (see Lemma 6). We can calculate numerical characters of S in terms of Bx (see Theorems 2 and 3). More accurately these characters are determined by specifying a line bundle F% which satisfies 2F* = [B*]. Our approach is summarized as follows. We first define a rational m a p / : S—>W of degree 2 onto a P l -bundle Plover J, which induces a double covering C-^>Pl. This rational map, in turn, determines its branch locus B on W. By applying "elementary transformations" to W, we normalize the singularities of B. By this process, W is transformed into another P'-bundle Wl over J, which turns out to be the image of the rational map 0 associated with |A^+^*m| for sufficiently ample m. If J = P\ Wi and 0 can be identified respectively with W and / as defined at the beginning. In the final § 5 we shall make a remark on pencils of curves of genus 2 with base points. 1) Supported in part by NSF grant MPS72 05055 A02.

80

E. Horikawa

Corresponding results are valid over any algebraically closed field of characteristic ^ 2 . This paper is an outgrowth of afterthoughts about [3], § 2. In fact, the whole paper depends only on several lemmas there and other more or less standard results. In addition, the surfaces of type II studied in [3] serve as clarifying examples for the present study. 1.

A Fundamental Construction

Let g : S—>J be a surjective holomorphic map of a compact complex analytic surface S onto a non-singular complete algebraic curve A of genus n. We assume that a general fibre of g is an irreducible non-singular curve of genus 2. By results of Kodaira [5], this implies that S is projective algebraic. We also assume that S contains no exceptional curve (of the first kind) in any fibre of g. Let K denote the canonical bundle of S. For any divisor m on J, we set fl(m) = dim H°(S, 0(K+g*m)). Let p be a point on J. Then £(m, p) =a(rt\) —a(m—p) is lower semi-continuous in p. Hence, if/? is a generic point of J, b(m, p) does not depend on the choice of p, and we denote it by b(m). L e m m a 1. For sufficiently ample m, we have fl(m) = pa+% deg HT+TT,

where pa denotes the arithmetic genus of S.

Proof. By the Riemann-Roch theorem and the Serre duality, we have aim) = / > a + l + d e g m + d i m / / 1 ( S , 0(-£*m)). Hence it suffices to prove dim// J (5, 0{-g*m))

=degm-l+;r

for sufficiently ample m. We assume that xn = ^pi is a sum of distinct points and i

that Ci=g'l(pi) are irreducible and non-singular. In view of the exact sequence 0 -» G(-g*m) -» 0 -> 0 0Ci — 0, i

we need to prove dim Ker(W(S, Os)^®H\Ct,

0Ct)) = it.

For this purpose we use the exact sequence 0 -> H> (J, 0A) -> W (S, 0s) -+ H°(J, R>g.Os). This implies that the above kernel is at least ^-dimensional. Furthermore we have H°(Jy (Rlg*Os) (X) 0(—m)) = 0 for sufficiently ample m. It follows that any element of H^S, Os) which vanishes on S C t is in the image of H\A, GA). The following lemma is an immediate consequence of Lemma 1.

Pencils of Genus 2

81

L e m m a 2. For sufficiently ample m, we have i(tn)=2. Let/? be a "variable" point on A and C=g~l(p). We let ^ denote the restriction to C of the canonical bundle K of 5. If C is non-singular, Kc coincides with the canonical bundle of C. Let @j denote the set of those divisors m on A such that £(m)^l, and let nti be a minimal element in &{. Then we take a divisor m2 such that the images of H°(S, OiK+g^m,)) and H°(S9 G{K+g*m2)) generate H°(C, 0(Kc)) for any general fibre C. Such a divisor exists by Lemma 2. We further assume that m2 is minimal among those which have the above property. We set b=m2—TUi. By our choice of nti and m2, we can find C* € #°(£, O(K+.g*mt))9 i=l9 2, such that their restrictions to C form a basis of H°(C, 0(Kc)). The pair (£„ C2) defines a generically surjective rational map of degree 2. Here P(0{—nti)®0( — m2)) denotes the P ! -bundle over J associated with 0 ( - m 1 ) © 0 ( - m 2 ) . This P'-bundle is identified with P(0(b)©0) and we shall denote it by Ib. Let Zf : £—>$ be a composition of a finite number of quadric transformations such t h a t / o XS extends to a holomorphic m a p / : £ —>2'b. We let q : Zb-^>A denote the natural projection and take a section Ao of ^ whose normal bundle is [—b] via an obvious identification. This section will be called the 0-section of Ib. Let R denote the ramification divisor of/, i.e., i f / i s given locally by (zi, £2)—>(^i5 w2), i? is the divisor of the zeroes of the Jacobian determinant d(wu w2)ld(zi, Z^}- We define the branch locus B off to be the direct image/,/? of R as a divisor. We note that B has no multiple component. L e m m a 3. There exists a line bundle F on Ib such that 2F=[B]9 the line bundle associated with the divisor B.

where [B] denotes

Proof. Clearly B is linearly equivalent to 6A0^-q*0 with some divisor 6 on A. Let Z be a very ample divisor on Sb such that q induces a biholomorphic map Z—> A. We assume that Z intersects B transversally at a finite number of points. Then Z=f~l(Z) is an irreducible non-singular curve and the intersection cycle BZ coincides with the branch locus of Z—>Z. From this we infer that the line bundle [6] is divisible by 2. q. e. d. Let F be a line bundle on Ib which satisfies 2F= [B]. We take a covering {£/J of Ib by sufficiently small open subsets and assume that F is defined by a system of transition functions {fj} on the nerve of the covering {[/*}. Then we can take the equations ^ = 0 of B on Ut which satisfy bi=ftjbj on Ut f] Uj. Let wt denote fibre coordinates on F over Ut. We define a subvariety S' ofF by the equations w2i = bi over Ui% We call S' the double covering of Ib in F with branch locus B. Since B has no multiple component, S' is a normal complex space. We note that Sf depends on the choice of F.

82

E. Horikawa

L e m m a 4. We can find a line bundle F on Ib satisfying 2F= [B] such that there exists a birational holomorphic map from S onto the double covering S' of2b in F with branch locus B.

Proof, We set W=2h. Let $-*S'-+W be the Stein decomposition o f / a n d let / ' : S'—>W be the induced holomorphic map. Then S' is normal and f*6§=J\0'S" Since/'is flat ([2], IV, (6. 1. 5)),/;<9 ff, is locally free of rank 2 ([2], IV, (2. 1. 12)). By [2], 0l5 (5. 5. 5), the cokernel of the natural injection Ow—*f*Qs> is invertible. Hence we obtain an exact sequence

with some line bundle F on W. Let {Ut} be a covering of W by sufficiently small open sets Ut and let {1, Wi\ be a basis off'*6s> over each Ut. We may assume that w\ is in F(UU Gw) and we denote it by bt. Then the branch locus B is defined by the equations bi=0. It follows that F satisfies 2F=[B] and that S' is nothing but the double covering of W in F with branch locus B. q. e. d. 2.

Elementary Transformations

Let S' denote the double covering of Ib in F with branch locus B as in Lemma 4. In this section we shall apply elementary transformations to 2b in order to study the singularities of B. We set W=2b. Let x be a singular point of B of multiplicity m. Here the multiplicity means the sum of the multiplicities of irreducible components of B at x. Let p : WX—>W be the quadric transformation with center at x and let L=p~1(x). We set Bx=p*B — 2[m/2]L where [m/2] denotes the greatest integer not exceeding m/2. Let Sx be the double covering of Wx in Fl=p*F— [m/2] [L] with branch locus B{. Then we have a natural birational holomorphic map S{—>S' which is compatible with p (see [3], § 2). Let M be the proper transform of the fibre of W—*A through x. Since M is an exceptional curve on Wl9 we can contract M into a non-singular point. Let q : W^,—•W^ be the contraction of M and set B2=q*Bx. We let n denote the multiplicity of B2 aty=q(M). In view of the equality MBl = 6—2[m/2], n is equal to 7 —2[m/2] or 6 —2[m/2] according as M is a component of Bx or not. In either case we have B{ = q*B2—2[n/2]M. Let F2 be the unique line bundle over W2 which satisfies q*F2=Fl-\-[n/2] [M], Then F2 satisfies 2F2=[B2]. Therefore we can consider the double covering S2 of W2 in F2 with branch locus B2. Again we have a birational holomorphic map SY—>S2 which is compatible with q. Thus, from the birational point of view, we can replace the triple (W, B, F) by (W29 B2, F2) as above. It is well known that W2 is a P ] -bundle over J. We shall call this operation the elementary transformation at x. L e m m a 5. After a finite number of elementary transformations, the triple (Ib, 5 , F) is transformed into another triple ( W , B\ F') such that B' has no singular point of multiplicity

Pencils of Genus 2

83

greater than 3.

Proof. If B has a singular point x of multiplicity ^ 4, we apply the elementary transformation at x. Retaining our previous notation, jy is of multiplicity ^ 3 on B2. Therefore our assertion follows from the theorem on resolution of singularities of curves on surfaces. q. e. d. In view of [3], Lemma 5, we need to study "infinitely near triple points" of B'. Let sx be a triple point of B' and let W[-^W be the quadric transformation with center at sx. If the proper transform B[ of B' has a triple point over sx, let us call it s2. We note that s2 is unique if it exists. Next let W'2-^W[ be the quadric transformation with center at s2, and look for a triple point over s2 on the proper transform of B[. We continue this process until we obtain a maximal sequence sx, % * • -j> *k of triple points over sx. In this situation we call sx a k-fold triple point. If k= 1, we call it a sirhple triple pointy while if A;J>2, we refer to the collection (jl5 s2, •••, J,) as infinitely near triple points. L e m m a 6. After a finite number of elementary transformations, the triple (W\ B\ F') is transformed into another triple (Wl, B*, Ff) such that B* has only the following singu-

larities (0), (I,), (II,), (III,), (IV,) and (V) with jfc^l. (0) A double point or a simple triple point. (I,) A fibre F plus two triple points on it {hence these are quadruple points of Bx). Each of these triple points is (2A:— 1) or 2k-fold. (II,) Two triple points on a fibre, each of which is 2k or (2k +1)-fold. (Ill,) A fibre F plus a (4£—2) or (4k—l)-fold triple point on it which has a contact of order 6 with F. (IV,) A 4A; or (4k-\-1)-fold triple point x which has a contact of order 6 with the fibre through x. (V) Afibre.F plus a quadruple point x on F, which, after a quadric transformation with center at x, results in a double point on the proper transform of F.

Proof Let x be an m-fold triple point of B' with m^>2 and let F be the fibre of W—>d through x. Let/? : W[—*W be the quadric transformation with center at x9 L=p-X(x), Bi=p*B' — 2L and let M be the proper transform of F. By the assumption m^>2, the proper transform B^ of B' has an (m— l)-fold triple point xx on L. Let q : Wx-+W2 be the contraction of M and B2=q^B{. We first assume that F is a component of B'. Then C=B'—F has a double point at x and the proper transform C, of C has a double point at the intersection of L and M. Since we have Bl = Cl-JrL-\-M, B2 contains q(L) as a component. Moreover C2=q^Cl has a quadruple point at q(M). Thus B2 has a singularity of type (V). Next we shall study the case in which F is not a component of B'. This case will be divided into two subcases according as x{ is on M or not. First assume that xx is on M. This implies that B' has a contact of order 6 with F at x. Hence, if m=0 or 1 mod 4, B' itself has a singularity of type (IV,) with A;=[m/4]. While, if m=2 or 3 mod 4, B2 contains q(L) as a component and B°2=

84

E. Horikawa

B2—q(L) has an m-fold triple point x2 on q{L). Furthermore, since B°l^=Bl—L has a contact of order 3 with L at #l5 B°2 has a contact of order 6 with q(L) at x2. Thus Z?2 has a singularity of type (III*) with k=\mj^]. Next we assume that x{ is not on M. This implies that 5 ' has a contact of order 3 with F at x. If B' has another triple point on F, let us call i t j . We may assume thatjy is w-fold with n<^m. If B' has no triple point on F other than x, we set rc=0. We apply an elementary transformation at x. Then the resulting curve B2 contains q(L) as a component and B°2=B2—q(L) has two triple points at q(xl) and q(M). The former one is (m— l)-fold and the latter is (rc-fl)-fold. If we perform another elementary transformation at q(xl), we obtain a curve 5 4 with two triple points xA andyA. One is (m — 2)-fold and the other is (rc + 2)-fold. We note that in this case the fibre through x4 and j 4 is not a component of Z?4. Ifm-\-n is even we obtain singularities of type (I*) or (II*) after a finite number of elementary transformations. We now assume that m^n is odd. In this case the above argument shows that we obtain B6 of the form either B% or B°6 + F where B°6 has (/+l)-fold and /-fold triple points on a fibre F. We distinguish cases according as / is even or odd. Assume that / is even. If, in addition, F is not a component of B6, then B6 has singularities of type (II*) with £ — //2. If F is a component of i?6, we apply an elementary transformation at the (/-f- l)-fold triple point of B°6. Then we obtain the singularities of type (II*) with k = l/2 again. If / is odd, a similar argument works. This time we obtain singularities of type (I*) with k=(l-\-l)/2. This completes the proof of Lemma 6. Definition. We say that a fibre F is a singular fibre of type (I*), (II*), (III*), (IV*) or (V), if B* has the corresponding singularity or singularities on F. 3.

Canonical Resolution

Let Wl, Bl and Fl be as in Lemma 6 and let S* be the double covering of Wl in Fx with branch locus B*. The purpose of this section is to study the canonical resolution of singularities of S*. If B% has a /:-fold triple point with k^2 or a quadruple point x} let p{ : WX—*WX be the quadric transformation with center at x. We set L{=p{l(x), B{=pfBl — 2[m1/2]L1 and Fx=pfFl —[m1/2][L1], where mx denotes the multiplicity oi x. Let Si be the double covering of Wx in F{ with branch locus B{. If Bx has a A>fold triple point with k^2 or a quadruple point x{, we apply the same transformation as above at x{. Then after a finite number of these transformations we obtain a surface W\ a curve B' and a line bundle F' on J4^ such that 2F*=[B>] and such that £k has no infinitely near triple points nor quadruple points. Let S> be the double covering of W^ in F* with branch locus B\ Then S* has only rational double points as its singularities ([3], Lemma 5) and is birationally equivalent to S\ hence to the original S.

Pencils of Genus 2

85

L e m m a 7. The inverse image on W* of each singular fibre of type (I*), (II*), (HI*), (IV*) and (V) has the following configuration. ^3

D4.

Dj.k —9.

/)?*.—1

®2k-2

^2k-\

D?lc

(I*)

D2k

(II*)

Du-iDu -•

D2

X

D3

•, o or w denotes a non-singular rational curve with self-intersection number —2. • is a component of B* which is disjoint from other components ofB* and o is not a component of B\ while a curve denoted by w is a component of B* which intersects with another component of B\ Finally x denotes a non-singular rational curve with self-intersection number — 1 . The left end o or • is the proper transform of the corresponding singular fibre. Two curves intersect transversally at a point if and only if they are connected by a segment in the diagram. The proof of Lemma 7 is a standard calculation of quadric transforms of curves on surfaces and is left to the reader. Let S—>S> be the minimal resolution of singularities of S\ Sis called the canonical resolution of singularities of S*. We let g denote the projection S—>J. Since there is no exceptional curve in any fibre of g : S—-»J, there exists a unique birational holomorphic map S^S which is compatible with g and g (see [8]). The following lemma is an immediate consequence of Lemma 7.

E. Horikawa

86

), (HI*),

L e m m a 8. The inverse image on S of each singular fibre of type (I*), or (V) has the following configuration. E2

£3 £3 —•—

E^

£1

£3

(nt)

Ei

E2

£3

E[

E'2

£3

E5

£6

E2k-2

E'n-i

E2k-2

E

En

2k-l

£0

E,

£1

•

—•

Eu X

z Eo

z E,

£3

£4

Eik-2

(lV k )

(V)

• W

£2

Here •, o and z denote non-singular rational curves with self-intersection numbers — 1 , —4 and —2, respectively. The symbols x and w denote effective divisors, one of the components of w being a non-singular rational curve with self-intersection number —2. Let (f> : Wl-^A be the natural projection and let A% be a section of <[> with normal bundle [—b1]. We denote by a the composition of quadric transformations WS'-^WX as above. L e m m a 9. Let f be the canonical bundle of A and write Fx in the form with some line bundle f on A. Then the canonical bundle K of S is induced by where the summation extends over all singular fibres F( except those of type (0). We have •§i= 2 [«/2] (Da+D'a) for singular fibres of types (Ik) and (II,), and &= 2 [a/2]Z) a /or aS2

a£2

Pencils of Genus 2

87

singular fibres of types (HI*), (IV*) and (V). Proof Let $ denote the canonical bundle of W\ Then K is induced by ®+F> (see [3], § 2). Our assertion follows from this fact (cf. Proofs of [3], Lemmas 5 and 6). q.e.d. Let {/"*„ F2, • • •, FN] be the set of all singular fibres except those of type (0) and let pt=^(rt). We set & = * if rt is of type (I,), (II,), (III,) or (IV,), and ^ = 1 if F, is of type (V). Then we obtain a divisor 2 AA- o n 4 a n d we denote it by c. L e m m a 10. The canonical bundle K of S is induced by where the summation runs over all singular fibres except those of type (0). For each singular fibre Fiy yt is an effective divisor which is a linear combination of Da and Dfa. However Da or D'a with maximum index does not appear in Jt. Furthermore, if F\ is of type (V), D2 does not appear in Jt. Proof For each singular fibre Fh the total transform of Ft on W* is as follows. (*)

AS(ArAr),

(ii,)

D0+°£(Da+iya),

(III,) A+A+22A,, (IV,) ^p (V) Dx+D2+w\. The assertion of Lemma 10 follows from Lemma 9 and the definition of c. q. e. d. 4.

Return to the Original Surface

We return to the original pencil g : S—+J. Theorem 1. Let m be a sufficiently ample divisor on J. Then the linear system 'K-\-g*m\ defines a rational map of degree 2 onto the P{-bundle Wx over A. Its branch locus B% has only those singularities of types (0), (I,), (II,), (HI*), (IV,) and (V) which are listed in Lemma 6. IfJ=P\ W% is biholomorphically equivalent to Ih as defined in § 1 and B% can be identified with B. Proof Let A% be a section of : WX^>A with normal bundle [ —b1]. We write f F = 3[Jj]+^*( —f+f + [bf]) as in Lemma 9. Then, by [3], Lemma 6, the arithmetic genus pa of S or S is given by (1)

/>a = d e g ( 2 f - b ' - 2 c ) - 3 ; r + 2.

By Lemma 1, we have a(nt) = d e g ( 2 f - b t -

88

E. Horikawa

for sufficiently ample m. Hence we have a(m) = dim// o (J^S0( provided that m is sufficiently ample. In view of Lemma 10, this implies that there is a natural isomorphism (2) H»($, 0{&+i*m)) = H°(W*, <j(Ji+0*(f_c+m))). This proves our first assertion and the second assertion follows from Lemma 6. Before proving the last assertion, we prepare with the following L e m m a 11. There is a natural isomorphism (2) for any m.

Proof. Let 2 ^ * denote the pull-back of 2 3 ^ m Lemma 10. From what we have seen above, it follows that 2 3 ^ 1S the fixed part of \K+g*m\ for sufficiently ample m. Therefore 2 3 ^ is contained in the fixed part of |2t+g*m| for any m. Hence it suffices to prove the following L e m m a 12. For any line bundle I on A, we have a natural isomorphism where/: S^>W* denotes the natural projection.

Proof We note that S is a double covering of a surface W in a line bundle F with a branch locus 6. Here Ti^ is obtained from W% by a finite number of quadric transformations. Let n : V—> W be the completion of F as a i^-bundle. n admits two sections Wo and W^ such that [ Wo] = [ W^] -f TT*P. As a divisor of V, S is linearly equivalent to 2 Wo. Let L denote the pull-back on W of the line bundle [JJ]+^*I. We use the following exact sequences on V: 0 -> 0 ( T T * Z - ^ ) -> 0(T:*L) -+ 0&{n*L) -+ 0,

0 -+ 0 ( T T * L - ^ ) -> 0(n*L- Wo) -> Ow{L-F) -> 0, 0-><9(>r*L- rt^) ->0(w*Z) -

The last two yield an isomorphism In view of the first exact sequence, it is sufficient for our purpose to prove H°(Wy Ow{L-F))=0. But this follows from f ( Z , - F ) = - 2 < 0 , where f denotes a fibre of the projection H^—>J. This completes the proof of Lemma 12, hence that of Lemma 11. We now prove the last assertion of Theorem 1. If A=P\ we may assume that W*=2bi. Then we have, by Lemma 11, fl(m) = dim//°(J, 0 ( f - c + m ) ) + d i m / / ° ( J , 0 ( f - c + m - b * ) ) . This implies that b is linearly equivalent to b1 or — b*. Hence W% can be identified with 2b. q. e. d. Lemma 11, combined with (1), has as a consequence the following theorem.

Pencils of Genus 2

89

T h e o r e m 2, With the same notations as above, the geometric genus pg and the irregularity q of S are given by q = ;r + dim Our next purpose is to study the relation between S and S. We note that S contains the exceptional curves Ea and E'a, a = l , 3, 5, •••, over each singular fibre (see Lemma 8). L e m m a 13. S is obtained from S by contracting the exceptional curves Ea and E'a with odd a.

Proof First we note the following fact. Let E be an exceptional curve on S which is contained in a fibre of g : S—>rf. Then E is a fixed component of |2C+g*m| for any

m. Let S be the surface obtained from S by contracting Ea and E'a with odd a and let g : .S5—>J be the natural projection. In order to prove our assertion, it suffices to show that there is no exceptional curve in any fibre of g (see [8]). But, as we have noted above, such an exceptional curve is the image of a fixed component of |i?+g*m|. Hence it is the image of a component of ^71 (see the proof of Lemma 11). This is impossible by Lemmas 8 and 10. q. e. d. This leads us to the following Theorem 3. We have

where v(*) denotes the number of singular fibres of type Proof By [3], Lemma 6, we have

Combining this with (1), we obtain K2=2pa—4-f 671. Now the assertion is an immediate consequence of Lemma 13. q. e. d. We conclude this section with the following theorem. T h e o r e m 4. Assume that K is ample. Then the branch locus Bx has no singularities except those of type (Ij).

Proof From Lemma 8 and the above observations, it follows that a singular fibre of any other type (including that of type (0)) results in a non-singular rational curve with self-intersection —2. q. e. d. 5.

A Remark on Base Points

In this section we consider the case in which S contains an exceptional curve E

90

E. Horikawa

which is mapped onto J(hence we have 7r = 0). In other words, if S—>S0 is the contraction of £, So carries a pencil of curves of genus 2 with a base point. As far as algebraic surfaces of general type are concerned, we have the following T h e o r e m 5. Let S be a minimal algebraic surface of general type which contains a linear pencil \C\ of curves of genus 2 with C 2 >0. Then S has the following numerical characters : pg^2, q=0 and c\=\.

Proof We borrow an argument from Kodaira [6]. Let K denote the canonical bundle of S. Then we have KC+C2=2. Since both KC and C2 are positive, they are equal to 1. Hence we have C(K— C)— 0. Then, by Hodge's index theorem, (K— C)2=K2— 1 is non-positive. Since K2 is positive, we conclude that K2=\. This implies that q=0 andj&^^2 ([1], Theorems 9 and 11 or [6]). q. e. d. The surfaces with pg = 2, q=0 and c\=\ are studied by Kodaira in [6]. They admit holomorphic maps of degree 2 onto a quadric cone in P 3 . The inverse images of lines through the vertex of the cone form a pencil of curves of genus 2 which has one base point (see [4], for details). References [ 1 ] Bombieri, E. : Canonical models of surfaces of general type, Publ. Math. IHES 42 (1973), 171-219. [ 2 ] Grothendieck, A. : Element de Geometrie Algebriques, Publ. Math. IHES 4, 8, •••I960 ff. [ 3 ] Horikawa, E. : On deformations of quintic surfaces, Invent. Math., 31 (1975), 43-85. [4] : Algebraic surfaces of general type with small c\, II, Invent. Math., 37(1976),121-155. [ 5 ] Kodaira, K. : On compact complex analytic surfaces I, Ann. of Math. 71 (1960), 111-152. [6] : Pluricanonical systems on algebraic surfaces of general type, II, unpublished. [ 7 ] Namikawa, Y. : Studies on degeneration, in Classification of Algebraic Varieties and Compact Complex Manifolds. Lecture Notes in Math. 412, Springer, Berlin, 1974. [ 8 ] Shafarevich, I. R. : Lectures on Minimal Models and Birational Transformations of Two Dimensional Schemes, Tata Inst. Fund. Res., Bombay, 1966.

Department of Mathematics University of Tokyo (Received November 5, 1975)

New Surfaces with No Meromorphic Functions, II

M. Inoue 0 0.

Introduction

In our previous note [3], we have constructed a kind of (compact complex) surface S satisfying (a)

b{(S) = 1,

b2(S) ± 0

and (/3)

S is minimal.

As for the significance of these conditions we refer to Kodaira [4], [5] or [2], [3]. In this note, we shall construct other such surfaces and study their properties. They have close connections with the cusps of Hilbert modular surfaces. We shall use some methods devised by Hirzebruch in [1] to resolve the cusp singularities of Hilbert modular surfaces. In particular, we shall refer to [1] for some results on continued fractions and real quadratic number fields. Notations. As usual, we denote by Z, Q, R, R+, Cand C* the ring of rational integers, the field of rational numbers, the field of real numbers, the set of positive real numbers, the field of complex numbers and the multiplicative group of nonzero complex numbers, respectively. We denote by [—oo, oo], [0, oo] and [—oo, 0] the intervals which are mapped homeomorphically onto [—1, 1], [0, 1] and 2 [—1, 0], respectively, under the principal value of —tan"1.

1.

Algebraic Preliminaries

For a real quadratic irrational number x, we denote by x' the conjugate of x, by M(x) the Z-module generated by 1 and x> and by Q(x) the real quadratic number field obtained by adjoining x to Q. Let U(x) = {aeQ(x) | a > 0 , a*M(x) = M(x)}9 U+(x) = {ae U(x) | a-a > 0}. 1) Supported in part by the Sakkokai Foundation.

92

M. Inoue

Then U(x) and U+(x) are infinite cyclic groups and [U(x) : U+(x)] = 1 or 2. # has a unique development as an infinite (modified) continued fraction which is periodic from a certain point on : where ^ e Z , et>2 for f > l and ^ > 3 for at least one j>t. x is purely periodic (namely, /=0) if and only if # > l > # ' > 0 (see [1]). Throughout this note we shall fix a real quadratic irrational number w satisfying co > 1 > a)' > 0. a) is developed uniquely into a purely periodic continued fraction : 0) = [[«o, »i,-», «r_,]] (1) where r is the smallest period and 02, rij > 3 for at least onej. Let Mx and M2 be .Z-modules of rank 2 in Q(co). We say that Mx and Af2 are strictly equivalent if there exists deQ(co) such that <5>0, (T>0 and d»Mx = M2. For the following two lemmas we refer to § 2 of [1]. L e m m a 1. For any Z-module M of rank 2 in Q( 1 > T)' > 0. L e m m a 2. Let ^,, rj2 be irrational numbers in Q((o) satisfying the condition (2), and let be the development of r]x as a purely periodic continued fraction. Then M{r)x) and M(y2) are strictly equivalent if and only if for some teZ,

0
The following proposition plays an essential role in our construction of surfaces. Proposition (1.1). (Hirzebruch). i) For co as above, there exist <w*, PGQ(0, /3'<0, p.M(o>)=M(o>*) and <»*>l>(a>*)'>0. ii) U+((o) = U+(0, f<0 and fM(w)=M(a>*)> then ^y1 belongs to U+((o). iv) Let (3)


New Surfaces with No Meromorphic Functions, II

93

be the development of a>* as a purely periodic continued fraction, where s is the smallest period. If there exist 7], y eQ(w) such that ^ > 0 , / < ( ) , y • M (co) = M (y) and 37 > l > ) / > 0 , then Y]=[[mh mt+l, •••, mt+s.i]]for some te Zy 00, e'<0. We set M=s-M(w). Then by Lemma 1 there exist o>*, 3eQ{oj) such that<5>0, <5'>0, d*M=M{*>l> (a>*)'>0. We set /3 = 3-s. Then /3>0 and P - M ( ( o ) = d-M = M ( o > * ) ,

i8

/

= ( d - e ) ' = d'*ef

< 0,

q. e. d.

We take o>* and /3 as in Proposition (1. 1). We take a generator a of the infinite cyclic group U+(w) such that a > l , and define integral matrices N, N* and B by (a., l).N=a.(M).5 = /S.(fl), 1).

(4) Then we have (5)

detJV= det N* = a-a' = 1,

det B - - 1

and B*N=

(6)

N*-B.

Remark, i) a)*, /3 and .S are obtained more explicitly as follows : Let \jw = [[e0, e{, •••, et_u eh •••_, ^ +s _i]] be the development of \/w as a continued fraction, where s is the smallest period. We define 57,, ^2, ••-, yjteQ((o) inductively by I/a) = *o—l/?i,

^-1 = ^ - i — l / ^ 3

2

Then ^ > 0 and ^ = [ [ ^ 3 ••-, *t+ ,_i]]. We set ^* = ^5

/5 = (^1-^2

fy)A»

1] [0, 1 Then we can easily prove that /3>0,

(fl>*, l ) . 5 = i8.(fli, 1),

det5= - 1 .

(Compare (9) and (10) below.) Since det B= — \, obtain ^•Af^) = Af(a>*),

Q)—W'>0

/ i8

and a;* —(co*)'>0, we

< 0.

+

ii) If [U(w) : C/ (o>)] = 2, there exists aoe U((o) such that a o > 1 and a^<0. In this case, we may take co itself as o>* and a0 as /3. Then we have nt = mt for all z, r = s, ^ = N*,B2 = N. For any integers i, k we define ^ = rij, m k = mh

i =j (modulo r), k = / (modulo J ) ,

0 <j < r—1, 0 < / < s— 1,

94

M. Inoue

where nh ml are the natural numbers in (1), (3). Then we know that for any i, k, (7) ni9 mk > 2 (8) n^ mL> 3 for at least one j , I. We define a)^ w% e Q(w) inductively by

Then wiy o>^>0 and a)i = Q)i+r for a n y i,

w% = CD*+S for a n y k.

W e set

Co*)"1

{wx 1 G)Q*G)_X

for i > 1 for i = 0

c^i+i

for

ft>*)~1

(<w,* 1 &>*#<jt>*i

^ ^ — 1

5

for A: > 1 for A: = 0

^?+i

for

AJ< — 1 .

Then (9) implies

_

°k-2> °k-

For a proof of the following lemma, we refer to (2. 5) of [1]. L e m m a 3.

I/a = ar = bs.

Proposition (1. 2).

L-i,orL-i,oj '" L—i, ojProof. From (10) it follows that \nr_u

L

By Lemma 3 and the definition of #i? we obtain ar_x=o)\a,

ar=l/a9

<_! = (o'\a\

a_l=w,

dr = l/a ; 5

fll,

ao=l,

= o)f,

a0 =

Hence

while (4) implies

^ l i r^.,, l i

r^0,

11 ra,oi =

kJ'Lj

L J

r^ 1

kJ" k

New Surfaces with No Meromorphic Functions, II

95

r«, 0 |

A

Since CD—W =£0, we obtain

^= L

j L

Similarly we obtain the second equality,

q. e. d.

Example. Let a>=(3+\/7)/2. Then we may choose OJ* and /3 as follows : co* = (5+\/7)/3, /3=(v/7—1)/3. In this case we obtain w

= [[376]], r = 2, a)* = [[3, 3, 2, 2, 2]], a =

- T 17^

6

1

n0 = 3, nx = 6, ^ = 5, m0 = m, = 3,

2.

1,

r 2 3 , 91

AT*-

m2 = m3 = m^ = 2, 1

Construction I

We take two series of infinitely many copies of C2:

Vt= {{u^vJzC2}, Wk= {{zk,wk)eC2},

ieZ, keZ.

We identify (uu vt) e Vt with (^_l5 vt_x) 6 Vt_{ if and only if

Mi = iM-i,

«i ^ o,

^_, ^ o,

and form their union,

V = \JV<. Similarly we form the union of Wk, ktZ

identifying (Zk-> wk) € Wk with (Zk-n wk-i) € Wk_x if and only if

Proposition (2. 1). V and W are Hausdorff spaces with countable bases and, hence, are complex manifolds with {Vt] and {Wk] as coordinate neighbourhoods.

(For a proof of this proposition, see (2. 2) of [1].) Let C be the subvariety of V defined by C f l ^ = {(«,, vt) | iit'Vi = 0} for any ieZ. Then C consists of infinitely many irreducible components Cu i € Z, where Ct is a non-singular rational curve and C^n V~ {(0, vt)}. Ct intersects (?,_, transversally at

M.Inoue

96

the origin Pt of Vt and (C,)2 = -ni9

d nCj = t for f, j with i-j

=£±1,0.

Similarly, let D be the subvariety of W defined by D n Wk = [(zk, wk) I Zk'Wk = 0} for any * € Z

Then 5 consists of infinitely many irreducible components &k, keZ, where Dk is a non-singular rational curve and Bkf] W^={(0, wk)}. Dk intersects Dk_l transversally at the origin Qk of Wk and ( 3 k ) 2 = —m A , 5

n A = ^ for K I w i t h A:—/ ^ = ± 1 , 0 .

4

Clearly we have W-t)

= {(Z W0) 6 Wo | Zo*Wo * 0).

Proposition (2.2). on

/; Each equality is derived from (10), (11) and (12). We shall show only the first one. Without loss of generality we may assume i>j. Let

, AJ = L—i, oj - L—i, oj

[-i,oj-

Then (10) implies

Hence By (11) we have vi = vej*uj

and

ut = vf •u)

on Vt fl Vj.

Therefore \Ui\ai*\Vi\ai-x

=

\vf

•uhj\ai*\veJ*ufj\ai-1

q. e. d.

a-j-i

We define

p = l^ bk

b

r = \zk\ '\wk\ >->9

q = s =

*!0'-1' on Vi

^r*-1'

on Wk.

New Surfaces with No Meromorphic Functions, II

97

From Proposition (2. 2) we infer that p, q and r, s are non-negative continuous functions respectively on V and on W. Moreover we have C = { P e V\p(P)

= 0 } = { P e V|

q(P)=0],

D=

r(QJ=O} = Let a, b, c, de Z be the components of the matrix B defined by (4) : B =

c,d\'

Proposition (2. 3). For any (u0, v0) e V — C={(u0, v0) e Vo \ uo-vo

Proof. From the definition of r we have C

m 1Jd\

. \71a.

7/^1°^*

o ' ^ o l * i^o 'uo\

\iJ

\b

— \uo\

while (4) implies aw*Jrc = fio) = Hence

Similarly the second equality follows from (4) and the definition of s, q. e. d. We form the union V[jW of V and W identifying {Zo> w0) eW — D i( and only if (13)

(z/0, vo)eV — C with

w0 = v^ul 0.

Proposition (2. 4). V [jW is a Haus dorjf space and, hence, is a complex manifold with {Vu Wk} as coordinate neighbourhoods. Proof. Let F be the graph of the identification (13) in V xW. To prove this proposition it suffices to show that F is closed in V X W. We set Since /3'<0 and q, s are continuous, A is closed in VxW. On the other hand, Proposition (2. 3) implies that F is a closed subset of A. Thus F is closed in V xW, q. e. d. We set for P e V, \p(,p) for PeW, ( q{P) for PeV, a(P) = \s{Pyir for PeW,

98

M. Inoue

where we define s'(P)l/fi' to be oo for PeD. Since /3'<0, Proposition (2. 3) implies that p and a are continuous mappings of V\}W onto [0, oo) and onto [0, oo], respectively. Moreover we have Cu D= {PeV UW \ p{P) =0}, (14) C= {PeV UW \ a{P) = 0}, D= {PeV U W | a{P) = oo}. We note that C^D= and V U W-C-D = {(«0, z/0) € Fo | tto.^ =£ 0}. 3.

Construction II

We introduce an analytic automorphism g oiV\]W as follows : g sends (uu vt) of Vt to fa, vt) of F,_r, ^ sends (zk, wk) of W7, to (zk> wk) of W7,.,, where r and j are the smallest periods in (1) and (3). (That g is well-defined is easily derived from (6) and Proposition (1. 2).) Clearly we have Ct_r,

g(Pt)=Pt.n

Proposition (3. 1). For any Pe Pig(P)) = P(P)a, a(g(P)) = o(Py, where the second equality means oo = oo whenever PeD. Proof. T h e s e equalities a r e trivial if Pe C{jD. H e n c e w e m a y a s s u m e

-C-D={(u0, write

PeV'\jW

v0) € Vo | Uo-Vo^O}. Then g(P) also belongs to V{jW-C-D.

Then Proposition (1.2) implies g(P) = (»„*.< ve0-u{) for P = («o, Hence by (4) and by the definition of p, we obtain P(g(P))

= p(g(P))

=H-4\.\v^ufr

Similarly we obtain the second equality, We define an open submanifold 50 of V \J W as follows : 50= {PeV U W | p{P) < 1}. By (14) and Proposition (3. 1) we have

q. e. d.

We

New Surfaces with No Meromorphic Functions, II

99

50 D C, A Moreover we can easily show that 30 is connected and simply connected. We denote also by the same g the restriction of g to 50. We set log p{P) -log a(P) for P e < 0 - C - A for PeC, +oo — oo for P e 5 . From (14) and Proposition (3. 1) we infer that F is a continuous mapping of 50 onto [—oo, oo] and F(g(P))=F(P) for any P€
Proposition (3. 2). g generates a properly discontinuous group (g) of analytic automorphisms of 50 without fixed points.

Proof In view of (15) and Proposition (3. 1), it is clear that no automorphism from (g) has fixed points in 50. To prove that (g) is a properly discontinuous group we have to show that for any P l3 P2e 50 there exist neighbourhoods Ux of Px and U2 of P2 such that gn(Ux) fl U2^<j> only for finitely many ne Z. By Proposition (3. 1), this is clear if both P, and P2 belong to 50-C-D. If P{eC and P2e5O—C—D, we set £ / I = {P6F(P2) + l], + l}. U2= {Pe50 | F(P)

for any

n e Z.

Similarly we can show the existence of C/1? U2 when PxeD and P 2 e50—C—B. If P, 6 C and P2 € B, we set Ux= {Pe50 | P ( P ) > 0 } 5 U2= {Pe50 | P ( P ) < 0 } . The Ux and C/2 are the neighbourhoods we need. If both of Px and P2 belong to C or belong to 3, the existence of such neighbourhoods is assured by Hirzebruch in [1, (2. 3)], q. e. d. We define Sw to be the quotient space of 50 by (g) : Sm is a complex manifold of dimension 2. Let ft denote the canonical projection of £) onto Sa. We set for t = 0 , l , . - . , r - l , for

* =

0,1,...,J-1.

100

M. Inoue

By (15), C and D are compact subvarieties of Sw consisting of irreducible components Co, Cl5 •••, Cr_j and Z)o, Z),, •••, Ds_u respectively. When r > 2 , Cis a cycle consisting of non-singular rational curves Co, Cl3 •••, Cr_,, where Ct intersects Cj transversally at exactly one point if and only if j = i + l (modulo r) for r > 3 , and Cx intersects C2 transversally at exactly two points for r = 2 . Moreover, from (11) we deduce that the self-intersection number C\ is equal to —nu where nt satisfies (7) and (8). When r= 1, C is a rational curve with one ordinary double point and the self-intersection number C2 is equal to —rco+2,where no>3. We have a similar situation for D. Moreover we know that Cf]B = $ and C, D are invariant under g. Thus we obtain H

h^r-i —2r),

C Z ) = O,

(18) and

Proposition (3. 3). Let [C^Cj] and [Dk»Dt] denote the matrices of the intersection numbers of Cu C2, •••, Cr_x and of D^ Z)2, •••, L)s-i> respectively. Then [C^Cj] and [Dk»Dt] are negative definite. R e m a r k . From iii) and iv) of Proposition (1.1) we can easily derive that Sw does not depend on the choice of o>* and /3 and, moreover, depends only on the strict equivalence class of M(o)). 4.

Topological Properties

Let (£, C) be the coordinates on HxC where H is the upper half of the complex plane. Let G be the group of analytic automorphisms of HxC generated by / (19)

1

£,:(?, 0 -

We denote by F the subgroup generated by gx and g2. Then G is a properly discontinuous group of analytic automorphisms of Hx C without fixed points and F is a normal subgroup of G. We set —^ f = (o log y 0 +log w0

and for (

Then (f, C) is well-defined modulo F and

f,-i--c) modulo (compare with the proof of Proposition (3. 1)). Thus (f, Q induces an analytic isomorphism of SM — C—D onto Hx CjG. Let (y, xl9 x2) be the coordinates on R+ xRxR and let ^ be the group of homeo-

New Surfaces with No Meromorphic Functions, II

101

morphisms of R+ X R X R generated by h0: (jy, *„ x2) —• ( a j , a*,, — xX A, : {y, *„ x2) —• (jy, *,+<*>, # 2 +G>'),

(21)

A2: (7, *„ x2)-+(y, *, + l,

x2+l).

~§ is a properly discontinuous.group of homeomorphisms of R+xRxR without 7 is a compact real 3-dimensional manifold, fixed points. Let 7=R+xRxRj-§. which is a real 2-torus bundle over a circle. By (19) and (21), (Im £-Im £, Im f, Re f, Re £) induces a homeomorphism 0 of Hx C/G onto /f X T. Composing <> / with 0, we obtain a homeomorphism 0 of S^—C -D onto j R x y . We write 0={0O, 0,) where 0o(P)eR and 0{(P)z7 for P e ^ — C—Z). By (16) /* induces a continuous mapping i^of S^ onto [—oo, oo] such that F" 1 (oo)=C and JF- 1 ( — OO)=Z). (20) implies that F=0O on S^—C-D. Hence F~1(T) is homeomorphic to T for any r 6 /J. Let ^ denote the space obtained from [-co, oo] x 7 by pinching o o x j t o a point P^ and - o o x J t o a point P_oo. Then $ is extendable to a continuous mapping 0 of ^ onto si such that 0(C)=Poo and 0(D)=:P_TO. We set 51 = ^ ( [ 0 , oo]),

^

^ + and J^~ are CO^J over T with vertices PTO and P.^, respectively. S^ fl -5*^ is homeomorphic to 7. 0 maps S^ — C and S^—D homeomorphically onto si+— P^ and onto si~—P_oo, respectively. For a pair (Jf, i4) of topological spaces, we denote by HV(X, A) the y-th homology group of (X, A) with coefficients in Z. Since j ^ + , j^5" are contractible to points Poo, P-oo, w e have

Hv(s£+, P.) - / / . ( ^ - , P..) = 0 for any

v.

Since C, D are strong deformation retracts of their tubular neighbourhoods, 0 induces the isomorphisms : Hv(S+,C)*Hv{si+,P.), HV(S~, D) ex Hv(s£-, P_ro)

for any

v.

HV(S+, C) = HV(S~9 D) = 0

for any

v.

Hence we obtain From this and the exact sequences of homology groups we derive H»(S+) S HV(C), . HV(S~) s HV{D) Thus we obtain (22)

HV(S+) s

Z Zr 0

for for for

v = 0, v= 2 v > 3,

for any

v.

102

M.Inoue

z H ,(Si) =

(23)

Z 0

for for for

s

v = 0, 1 v= 2 v> 3.

From (21) we easily derive for v == 0, 2, 3 for V = r where e,f are elementary divisors of the matrix A -I.

= [z

(24)

z,

Proposition (4. 1). i) Sa is a compact complex surface. ii) nx[Sn) g* Z. Z > » = 0, 1,3,4 iii) H ( S ) l Proof. Since {££, S^} is an excisive couple, we have the Mayer-Vietoris sequence : (25)

•'•^Hv{S+nSi)^Hv{S+)®Hv{Si)^Hv(Sa)->...

where S^ClS^ is homeomorphic to T. i) By (22), (23), (24) and (25), we obtain

Ht(Sn) s H3{7) = Z, while Hn(X)=0 for any real n-dimensional non-compact manifold X. Hence Sm is compact. ii) is trivial because SO is simply connected, iii) By (22), (23), (24), (25) and ii), we obtain

0 — HS(SJ) -+Z-+Zr®Zs^

H2(SJ)

Since H^S^) has no torsion part, iii) follows from this exact sequence, q. e. d. We denote by bv(S) and c2(S) the v-th Betti number and the Euler number of a surface S.

Proposition (4. 2). W , ) = l for * = 0, 1,3,4, b2{Sj) = c2(Sa) = r+s. + We define b and b~ to be respectively the numbers of positive and of negative eigen-values of the bilinear form of the intersection pairing on H2{S(O). By (18) and Propositions (3. 3) and (4.2), we have Proposition (4. 3). i) {Co, Cl5 •••, Cr_1? Do, Z)1? •••, DS_Y} is a Betti base of 2cycles on S^ with respect to rational coefficients. ii) b+= 0 and b~= r+s.

New Surfaces with No Meromorphic Functions, II

5.

103

Analytical Properties

We denote by K the canonical line bundle of S^, by pg the geometric genus, by q the irregularity and by [E] the complex line bundle determined by a divisor E on Sa.

Proposition (5. 1).

K=

[-C-D].

Proof. By (11), (12) a n d (13), the formula Q = (

defines a meromorphic 2-form on SO with simple poles on C and D. Since Q is invariant under g, Q induces a meromorphic 2-form Q on Sw whose divisor (Q) is equal to —C-D. Hence we obtain K=[(Q)] = [ — C—D], q. e. d. We have the following two well-known formulae : K2-2-c2(Sw)

K2+c2(Sw) =

=3(b+-b-),

l2(pg-q+l)

(the index theorem of Hirzebruch and the formula of Noether, for which we refer to Kodaira [4]). Combining these formulae with Propositions (4. 2), (4. 3) and (5. 1), we obtain Proposition (5. 2). pg = 0, q=\ and K2 = ~{r+s). From iv) of Proposition (1. 1) it follows that J and mo-\ h^s-i m (3) do not depend on the choice of
\~nr-i = 3r.

Proof, i) Combining (17), (18) with Proposition (5. 1), we obtain

K2 =

-(no+-..+nr_l-2r)-(mo+-.-+ms_l-2s),

while by Proposition (5. 2) we have K2= — (r-\-s). Thus we obtain i). ii) By remark ii) in § 1, r=s and we may assume n^nii for any i. Hence ii) follows from i), q. e. d. Let £ b e a divisor on S^. Since b+ = 0 and //2(5W) has no torsion part, we obtain that E2<0 and E2=0 if and only if E is homologous to zero with respect to integral coefficients. Moreover, we know that E is homologous to zero if and only if

104

M. Inoue

[E] is aflat line bundle (see Kodaira [5, § 10]). P r o p o s i t i o n ( 5 . 4 ) . Sw contains no irreducible curves other than Co, •••, C r _ 1 ? 2) 0 , •••, Z) s _!. In particular, S^ is minimal and there are no meromorphic functions on S^ other than constants.

Proof. Take an irreducible curve E on Sw and denote by n(E) the virtual genus ofE:

We assume that E^Ct, Dk for all i9 L Since TT(E)>0, K*E<0 and E2<0 where E2=0 if and only if E is homologous to zero, we obtain three cases : i) E2 = - 2 , K.E = 0 and TT(E) = 0, ii) E2= - 1 , K-E= - 1 , and TT(£) = 0 , iii) E2 = 0, K.E = 0 and n(E) = 1. In case i), E is a non-singular rational curve and is contained in S^—C—D, while S^—C—D^HxC/G. This is a contradiction. In case ii), E is a non-singular rational curve and intersects one and only one, say Cj9 of Co, •••, Cr_b Z)o, •••, Ds_{. Hence E is contained in {*(£)—U^~~^)? w m l e c0—MQ—^ is an open domain in F7 = C2. This is a contradiction. In case iii), [E] is a flat line bundle on S^. Hence there exists a holomorphic function f on 50 such that g*f=X.f

for some ^€ C*, ^^= 1.

Since each irreducible component of C and of 25 is a non-singular rational curve, y is constant along C and 25, while C and 5 are invariant under g and X^\. Thus / vanishes on C and D. This evidently contradicts our assumption. By (7) and (8), Ct and Dk are not exceptional curves of the first kind. Hence Sm is minimal, q. e. d. Propositions (4. 2) and (5. 4) imply that SM satisfies the conditions (a) and Q8) stated in the introduction. Example. Take w = ( 3 + / 7 ) / 2 (see the example in § 1). Then b2(SO)) = 7 and S^ contains exactly seven non-singular rational curves Co, C1? 2)0, 2)l5 2)2, D3, D4 with d = -3,

C? = - 6 ,

D2 = D2= - 3 ,

2)1 = D2 = D2= - 2 .

We note that 3{r+s) - 21, 6.

n b + - + « r - i + w o + - + w.-i - 21.

Special Cases with [t^(a>) : f/+(o;)] = 2

We assume [J7(a>) : U+(w)] = 2. Then, by remark ii) in § 1, r=s and we may

New Surfaces with No Meromorphic Functions, II

105

assume ni = mi for all i. In this case S^ admits an analytic involution c induced by an automorphism 1 on 50 defined as follows : c sends {uu vt) of Vi to (ui9 vt) of Wu c sends (zk, wk) of Wk to (zk, Wk) of Vk_r.

We easily see that c has no fixed points and c{Ci)=Di, Let S^ denote the quotient surface of Sw by (c). Then

r

for

i; = 2.

We set ( ? = ( C ( J #)/<<>, £ = ( C , U A)/<<> for z = 0, I,-.., r - 1 . When r>,2, Q is a non-singular rational curve with [Ci)2=—ni. When r = 1, C=C 0 is a rational curve with one ordinary double point and (C)2=— no-{-2. By Proposition (5. 4), S^ contains no irreducible curves other than Co, Cl5 •••, Cr_l. Thus ^ also satisfies the conditions (a) and (/3). Example. Take (y=(3+v / 5)/2. Then [C/(a>) : f/ + (a>)]-2 and ao = a generator of U(w) = ( l + / 5 ) / 2 , a = a? =

( 3 + A/5)/2, CM

= [[3]], r= 1.

In this case, b2(Sa)) = l and ^ contains exactly one curve C. C is a rational curve with one ordinary double point and (C) 2 = — 1 . 7.

Remarks and Acknowledgements

The author is heartily grateful to the mathematicians who have communicated to him the following : i) The author constructed Sw first only for OJ with [U(w) : U+(w)] = 2. Afterwards F. Hirzebruch has communicated to the author that one can construct S^ for any w satisfying a>>l>(i/>0 with the aid of Proposition (1. 1). ii) The second part of Proposition (5. 3) is obtained by Hirzebruch as a consequence of a theorem on the invariant d (see [1, § 3]). T. Shintani has communicated that he also has obtained both parts of this proposition as an application of his deep results on special values of zeta functions of totally real algebraic number fields (see [7]). iii) D. Mumford, I. Nakamura, Y. Namikawa and T. Oda have communicated that one can construct Sw by the methods of toroidal embeddings. Moreover Oda has pointed out that these methods are applicable to the study of degenerations of SQ, to rational surfaces with singularities (compare Kodaira [6]).

106

M. Inoue

References [ 1 ] Hirzebruch, F. : Hilbert modular surfaces, L'Enseignement Math., t. XIX (1973), 183-282. [ 2 ] Inoue, M. : On surfaces of class VII0, Inventiones Math., 24 (1974), 269-310. [ 3 ] Inoue, M. : New surfaces with no meromorphic functions, Proceedings of the International Congress of Math., Vancouver (1974). [ 4 ] Kodaira, K. : On the structure of compact complex analytic surfaces, I, Amer. J. Math., 86 (1964), 751-798. [ 5 ] Kodaira, K. : On the structure of compact complex analytic surfaces, II, Amer. J. Math., 88 (1966), 682-721. [ 6 ] Kodaira, K. : On the structure of compact complex analytic surfaces, III, Amer. J. Math., 90 (1968), 55-83. [ 7 ] Shintani, T. : On evaluation of zeta functions of totally real algebraic number fields at nonpositive integral places, J. Fac. Sci. Univ. Tokyo, Sec. IA, 23 (1976), 393-417.

Department of Mathematics Aoyamagakuin University (Received December 21, 1976)

On the Deformation Types of Regular Elliptic Surfaces

A. Kas The purpose of this paper is to prove the following theorem : T h e o r e m . Let Sx and S2 be elliptic surfaces over Pl with no multiple fibres, with at least one singular fibre, and with no exceptional curve contained in a fibre. Then Sx and S2 are deformations of one another if and only if Sx and S2 have the same geometric genus.

In particular the theorem implies that Sx and S2 are diffeomorphic. As a corollary we show that any elliptic surface satisfying the conditions of the theorem is simply connected. Let T ^ t h e s e t of pairs of polynomials (g2{u), g3(u)) with deg g2(u)<^n, deg g3(u) <6n and satisfying the conditions : (i) D(u)—4g2(u)3—27g3(u)2 is not identically zero ; (ii) the polynomials g2(u), & ( " ) , " • , g(23)(u), g3(u), g'3(u),~;

gf(u) have no

common zero ; (iii) either deg g2(u)>4:n—4 or deg g3(u)>6n—6. It is clear that Tn may be identified with a Zariski open subset of ClOn+2. Let P2 denote a projective plane on which a system of homogeneous coordinates (x : y : z) is fixed. We take two copies W0=P2xC0 and Wx=P2xCx and form their union W=^ H^U W{ by identifying (x : y : z) X u e Wo with (#, :y{ : Z\) X ux e Wx if and only if: UU{

= 1,

u2nxx = x ,

uZnyx =y,

Zx = Z-

Let j8n be the subvariety of Tn X W defined by the equations : fz = x3—g2(u)xz2—g3(u)z\ in TnxW0 fxZx = tf-^2(l/i0*i^-^&(l/«i)*?, in Tnx WX. l Let ¥ : J3n-+P be defined by : ¥ : (x:y:z)xu^u, ¥ : (xx \yx : Zi) Xux-+ux l where P = Co U Ci identifying ueC0 with ux e d if and only if uux = 1. Let n : j8n—> 7^ be the mapping induced by the projection Tnx W—+Tn. For each point (g25 g3) e Tn, let B(g2
g3). Then Bigttga)

together with the mapping ¥\B(g2ig3) :

B(g2,g3)-^P is a basic elliptic surface (i.e. an elliptic surface with a section) with a

108

A. Kas

section defined by : x = z = 0, in B(g2
n Wo n Wx.

B(g2Pl is an analytic family of abelian Lie groups such that the identity element of the group ¥~l(u)CiB*0 is a(u). Let u—+¥~l(EioEj) is the unique extension of the automorphism z—> Z-\-i]ij{¥(z))- Every elliptic surface with no multiple fibres and with no exceptional curve contained in a fibre is of the form Bv. Let f be the normal bundle of the section o(Pl) in B. f is the line bundle on o(Pl) of degree —(pg-\-l). The stalk of f at ueP1 may be identified with the Lie algebra of BxQC\W~x{u). The exponential map determines a surjective homomorphism of sheaves e : 6{\)-^0(B*0). The kernel G of the mapping e is called the homological invariant of B. There is also a more explicit description of G. Let A' be the punctured sphere obtained by deleting from P 1 those points which correspond to singular fibres of B. Then ¥~l(Jf) is differentiably a fibre bundle of elliptic curves. This bundle is determined by its monodromy, i.e. by a representation R : nx(A')-+SL{2, Z). Let G' be the locally constant sheaf over A' with fibre

On the Deformation Types of Regular Elliptic Surfaces

109

determined by the monodromy R. G' may be extended to a sheaf G on Pl by defining r(U, G)=r{Uf]^9 G') for each open set f / c P 1 (i.e., G=i*G' where i: A'-+Px is the inclusion mapping). We have an exact sequence of sheaves : The collection of the complex analytic surfaces {Be(e)}, 6eHl(P\ 0(f)), forms a complex analytic family. Let c : Hl(P\ O(Bl))-*H2(Pl9 G) be the coboundary map of the above exact sequence. Then the surface B71 is a deformation of the basic elliptic surface B(=B°) if c(y)=O. The above results are in [3], Th. 9. 1, page 603, Th. 9. 2, page 609, Th. 10. 1, page 623, Th. 11. 1, page 1, Th. 11. 2, page 2, Th. 11. 3, 11.4 on page 5. There is a duality theorem for the sheaf G (cf. [5]) which implies that H2(P\ G) ^H0(P\ G)=Z®ZjM where MaZ@Z is the submodule generated by the elements (m, n)Rr—(m, w), jen^A'). Let Ca be a singular fibre and let y€7ix(Af) be represented by a simple closed loop around a in PK Then it follows from what we have said that/Z^P 1 , G)=0ifdet (Rr—l) = ±l. From the list of singular fibres and their monodromies ([3], page 604), we conclude that H2(P\ G)=0 if B contains a singular fibre of type II (a rational curve with a cusp). It follows from [2] that if B=B(g2tgz}, then H2(P\ G)=0 if g2(u) and g3(u) have a common zero which is a simple zero of g3{u). L e m m a 1. If every singularfibreof B is of type Il3 then H2(P\ G)=0. Proof According to [2], every singular fibre of B=-Big%tgz) is of type ly if and only if Z)(tt)=4g2(z/)3—27g3(u)2 has no multiple zeros and is of degree > 12ft—1. Thus for all Mn a dense Zariski open subset Va Tn, the elliptic surface Bt (t=(g2, g$)) has only singular fibres of type I,. Moreover it is clear from the definition of G that H2(P\ Gt) is constant for te V where Gt is the homological invariant of Bt. Let £0€ Tn be such that Bto contains a singular fibre of type II. Then H2(P\ Gto) = Z®ZjMQ=0. It is clear that for t in a sufficiently small neighborhood of to, we have MocMt; thus H2(P\ Gt)=0 (compare [4], Lemma 3, page 781). Since V is dense in Tn, the lemma is proved. Now let B=BtQ, toe Tn9 be a basic elliptic surface and let 7jeHl(P\

O(Bl)).

L e m m a 2. There exists a neighborhood U of t0 in Tn, a covering [Et} of Pl and an analytic family {%j(t)} of l-cocycles with coefficients in O(B%m) with respect to covering {Et} such that the %j(t) depend holomorphically on te U and -q is the cohomology class of (%j{t0)).

Proof Let y=c(rj) eH2(P\ Gto). It is clear that we can find a neighborhood U of t0 in Tn, a finite open covering {Et} of Pl and a family of 2-cocycles ( ^ ( O ) depending continuously on t such that (fty*^)) is a representative of y. Under the inclusion mapping 0—*G—>0(f), the (jijk(t)) may be regarded as a 2-cocycle in 0(f) depending holomorphically on t. Thus we may write fr^(O==0.7*(O~~^ifc(O+^./(O where (0^(0) i s a 1-cocycle with coefficients in O(\) depending holomorphically on

110

tz U. Finally set

A. Kas

yiJ(t)=e(0ij(t)).

Finally to prove the theorem, let S^B* be an elliptic surface satisfying the hypothesis of the theorem. Then B=Bto for some t0e Tn, n=pg(S)-\-\. Choose U and Ofa(0)> te U> a s m Lemma 2, and let yj{t) 6 Hl(Pl, 0(B*m)) be the cohomology class of (rjij(t)). Then it is clear that the analytic surfaces {5f°}i€C7 form an analytic family of surfaces. Moreover, for some te £/, we have H2(P\ Gt)=0, and therefore Bfl) may be deformed to a basic elliptic surface. We have already shown that all basic elliptic surfaces of the same geometric genus are deformations of one another. This completes the proof of the theorem. Corollary. Let S be an elliptic surface over Pl with at least one singular fibre such that S has no multiple fibres. Then S is simply connected.

Proof. S is obtained by blowing up an elliptic surface So which has no exceptional curves in a fibre. It suffices to prove that £0 is simply connected. By our theorem, it suffices to exhibit a simply connected elliptic surface satisfying the hypothesis of the theorem with any given value of pg. We claim in fact that any elliptic surface B with no multiple fibres which has a simply connected fibre is itself simply connected. Let Cai, •••, Car be the singular fibres of B and let C be a non-singular fibre. Let B'=B— {]Ca . It is clear that nAB') maps surjectively onto nx(B). Since 1=1

r

B' is a differentiate fibre bundle over Pl — {Jat, it follows that 7tx(B') is generated by elements y^ y2, a u •••> ar where yl9 y2 generate 7tx(C) and at is the lifting of a simple loop around at in P\ However, the existence of a local section of B at at implies that at may be chosen so that at~0 in B. Thus Ttx{B) is generated by yx and TV The loops yx and y2 may be "moved" to a simply connected fibre where they are contractible. Thus nl(B) = {l}. Now if g2(u) and g3(u) are polynomials with a common zero, then the minimal resolution of B{g2ig3) contains a simply connected fibre [2]. We can give a more explicit construction. Let J be a hyperelliptic curve with a canonical involution a : J—->J where a has 2g-f2 fixed points, g=genus A. Let E be an elliptic curve, E—C\L^ with involution r : E—+E, T(Z)=Z—Z- The surface A x E\a X r has 8g+8 ordinary double points. Let S be a minimal resolution and let W : £—>P' be induced by J x ^ X r - ^ J / a ^ P 1 . Then S is a basic elliptic surface with 2g+2 singular fibres of type If (simply connected). It is not hard to verify that p,(S)=g{J).

References [ 1 ] Brieskorn, E. : Singular elements of semi-simple algebraic groups, Proc. Inter. Congr. Math., Nice 1970, Gauthier-Villars, Paris (1971). [ 2 ] Kas, A. : Weierstrass normal forms and invariants of elliptic surfaces, to appear. [ 3 ] Kodaira, K. : On compact analytic surfaces, II-III, Ann. of Math., 77 (1963), 563-626 and

On the Deformation Types of Regular Elliptic Surfaces

111

78, 1-40. [ 4 ] Kodaira, K. : On the structure of compact complex analytic surfaces, I, Amer. J. Math., 86 (1964), 751-798. [ 5 ] Shioda, T. : On elliptic modular surfaces, J. Math. Soc. Japan, 24(1972), 20-59. Department of Mathematics Oregon State University (Received November 5, 1975)

On Numerical Campedelli Surfaces Y. Miyaoka 1.

Statement of Main Results

In this paper, a surface will mean a compact complex manifold of dimension 2. We denote by \mKs\ (m e N) the pluricanonical system on a surface S, and by 0mKs the associated rational map (the pluricanonical map), assuming that \mKs\ is not empty. A surface S is called of general type if, for a large number m, @mKs{S) is a variety of dimension 2 in the projective space PN (iV—dim \mKs\). We shall study a certain class of surfaces of general type. Definition. (1) A minimal surface of general type is called a numerical Campedelli surface if S satisfies the following numerical conditions : pg(S) = dim H°(S, 08(Ks)) = 0, q(S) = dim H'(S,OS) = 0 , K% = 2. (2) A numerical Campedelli surface is called a Campedelli surface if its fundamental group TT^S) is isomorphic to Z/(2)®Z/(2)®Z/(2) (cf. [2]). Then we have the following results. T h e o r e m A. The tricanonical system \3Ks\for a numerical Campedelli surface S is free from base points and fixed components.

Remark. Except for numerical Campedelli surfaces, we can enumerate all surfaces for which the tricanonical maps are not birational (see Bombieri [1] and Miyaoka [4])]). T h e o r e m B. For a Campedelli surface S, the universal covering S of S is birational to a complete intersection of type (2, 2, 2, 2) in P6.

Remark. It is an interesting but, in general, a very difficult problem to determine the complex structures on a given underlying differentiate manifold. However, in the case of Campedelli surfaces, the problem is rather easy and the complex 1) (added in proof) Recently Bombieri has proved that the tricanonical maps of numerical Campedelli surfaces are birational.

114

Y. Miyaoka

structures are completely determined. 2.

Preliminaries

If S is a surface of general type, the following results are well-known. T h e o r e m 1. (Mumford [5]). Ifm is sufficiently large, @mKs is a birational morphism oo

and 0mKs(S)=X=iPYO)(£)Ho(S,

0s(rKs)).

X is a normal variety with only afinitenumber

r=0

of rational double points as singularities. If S is a minimal surface, then S is the minimal

resolution of X. (X is called the canonical model of S.) Theorem 2. (Mumford [5]). Assume that S is minimal. Then we have Hl(S, Os(mKs)) = 0, for m^O, 1, meZ. Theorem 3. (Riemann-Roch Theorem for pluricanonical systems). Letting c\ be the self intersection number for the canonical divisor of the minimal model of S, we have dim H°(S, 08(mK8))

= x(Os) + (c2J2)m(m-1),

for m > 1,

where x(@s) denotes the Euler characteristic of the structure sheaf 0S of S. T h e o r e m 4. (Iitaka [3]). The m-genera Pm(S) =dim mation-invariants.

H°(S, Os{mKs)) are defor-

As an immediate corollary to Theorems 3 and 4, we obtain the following Theorem 5. (Deformation Invariance of the Minimality). IfS is minimal, then any deformation of S is also a minimal surface of general type.

Especially, we have T h e o r e m 5'. A deformation of a numerical Campedelli [resp. Campedelli] surface is also a numerical Campedelli [resp. Campedelli] surface. 3.

Proof of Theorem A

For the proof we need the following results. Definition. An effective divisor D o n a surface F is called l-connected if

A A > 0, for any non-trivial decomposition 0=0^02,

A>0.

Theorem 6. (Ramanujam vanishing theorem [6]). If an effective divisor D on a

On Numerical Campedelli Surfaces regular surface F (i.e. q(F)=0)

is l-connected, then Hl(F,

115 G(—D))=0.

T h e o r e m 7. (Bombieri [1]). Let F be a minimal surface of general type and P a point on F. Let p : F—*F denote a quadric transformation at P and E the exceptional curve over P. If an effective divisor D is numerically equivalent to 2p*KF—2E, then D is I-connected unless K2F=l.

Now we proceed to the proof of Theorem A. Let S be a numerical Campedelli surface, p : S-+S, the quadric transformation at a point P, and E, the associated exceptional curve. Let us consider the following natural exact sequence of sheaves : 0 -* 0s(3p*Ks-E) -> 0s(3p*K8) -*0E-+0. Then it is obvious that \3KS\ is free from base point at P if and only if H\S9 G(3p*(Ks—E))=0. By the Serre duality, we have dim Hl(S9 0(3p*Ks-E)) = dim Hl(S9 O(2E-2p*Ks)). Hence Theorem 7 yields the vanishing of the cohomology group under the condition that \2p*Ks-2E\^. Now assume that \2p*Ks-2E\ = . Since dim H°(S, 0{2Ks)) = 3, this implies that the rational map 02Ks associated with the bicanonical system \2KS\ is a local isomorphism at P. Therefore there exists an effective divisor De \2p*Ks—E\ such that D is irreducible in a neighbourhood of E and that the unique irreducible component Do which simply intersects E satisfies Dl>0. Now we shall take the following exact sequence of cohomology groups : 0->//*(£ O(2E-2p*Ks))->H°(S, O(E))->H°(D, CD(E)) Note that H°(S9 0{2E-2p*Ks))=0 and that dim Hl{S, 0{E)) = dim Hl(S, O(p*Ks)) = dim Hl(S, 0{Ks)) = q(S)=O. Hence, for the proof of Theorem A, it is sufficient to show the equality dim H°(D, 0(E)) = dim H°(S9 0(E)) - 1. On the other hand we have the following natural commutative diagram ())H\D-E,

0) identity

0{E))^H\DE 0) of which the rows are exact. But it is obvious that the virtual genus of Do is not 0. Since the degree of the divisor E on Do is 1, the restriction map r is the zero-map. This implies that dim H°(D9 0(E)) - dim H°(D9 0). Moreover we have dim H°(D9 O) = l. In fact, there exists the following natural exact sequence

116

Y. Miyaoka

0->H°{S9 0{E-2p*Ks))->H°(S, O)-+H°(D, 0) where dim Hl(S, 0(E-2p*Ks))=dim H*{S, 0(3p*Ks))=dim Hl(S, Thus dim H°(D, 0)=dim H°(S9 6) = \ and the assertion is proved. 4.

0(3KS))=0.

The Structure of Campedelli Surfaces

In this section we shall study Campedelli surfaces. If S is a Campedelli surface, the universal covering S of S has the following numerical characters : Z(S, 6s) = MS, Os) = 8, q(S) = 0, Ks = 8KS = 16.

The fundamental group G of S acts on S as the covering transformation group of the unramified covering e : S—>S, and G operates naturally on the vector space H°(S, 0(Ks))vi2L linear transformations. Hence we obtain a canonical representation k : G—>6rZ/(7, C) and the associated projective representation k' : G-^PGL(6, C). L e m m a 1. k' is a faithful representation.

Proof Let geG be an element of ker kf. Since ^ 2 =id, k{g) = ±\A. Hence pg(Sl(g)) = 7 or 0. But pg(Sl(g)) =3, if g is of order 2. Hence ^=id. Let V denote the image of S by the canonical map OK^ associated with the canonical system \K$\. Then k'{g) (gzG) induces an automorphism of V. Thus we obtain a natural homomorphism a : G—»Aut(F), where Aut(F) denotes the automorphism group of V. L e m m a 2. a is injective. Proof A trivial consequence of Lemma 1. L e m m a 3. The canonical system Kg of S is not composed of a pencil.

Proof Assume that Fis a curve. Since q(S)=O, Fmust be a (possibly singular) rational curve. An automorphism of V induces a unique automorphism of the nonsingular model Pl of V. Hence, by virtue of the above lemma, we infer that there exists a faithful representation a' : G—+PGL(l, C). On the other hand, it is obvious that PGL(l, C) does not contain a subgroup isomorphic to (Z/(2))3. This is a contradiction. Since G is a commutative group, we may assume that k(G) is contained in the diagonal subgroup of GL(7, C). Let wl9 •••, w7 be a basis of //°(<S, 0(Ks)) such that g*(wj) = ±Wj for any g e G.

On Numerical Campedelli Surfaces L e m m a 4. 3-dimensional.

The linear subspace W of H°(S, 0{2Kg))

117 spanned by w2, w\> •••, w2 is

Proof. Lemma 3 implies that the transcendence degree over C of the field C{w2jw^ •••, Wj/Wi) is 2. Hence the transcendence degree of C(wllw2, •••, w2/w2) is also 2. This yields the inequality dim W^ 3. On the other hand, since w) is G-invariant, W can be regarded as a subspace of H°(S,0(2Ks)).But the Riemann-Roch theorem gives the equality dim// 0(S9 0(2Ks )) = 3. This completes the proof. L e m m a 5. Let K be an extension of the rational function field C{xu •••, xn) defined by where Qj is a quadric polynomial in xt. Assume that [Kr : C(xl9 •••, xn)] = 2r. integral closure of C[xl9 •••, xn] in Kr is Rr=C[xl9 •••, xn, VQ^i, ••-, y/Q,r]-

Then

the

Proof Trivial. Corollary. Let Kr be as above. Let Q,r+1 be another quadric polynomial in xt. Assume Then VQr+i is a linear combination of x/s and that Kr+l=Kr-

Let w2, w22, w\ be a basis of W. From Lemma 4, we infer that there are quadric relations w) = cijWl+bjWl+CjWl,

j = 4, 5, 6, 7.

The above corollary asserts that, if the complete intersection defined by the above quadrics is reducible, then any irreducible component of it is contained in a hyperplane in P6. Since the image V of S is contained in the complete intersection V defined by the above 4 equations and Fis not contained in any hyperplane, V'=V is a irreducible surface. Thus we obtain the following Corollary.

V is a complete intersection of type (2, 2, 2, 2) in P6.

As an immediate consequence of this corollary, we have T h e o r e m 8.

The canonical

homomorphism

)->Ho(S9

O(mKs))

is surjective.

Proof Let Qv(m) denote the sheaf associated to the hypersurface section of degree m. Since V is a complete intersection of type (2, 2, 2, 2), we have dim H°(V, 0v{m)) > 8 + 8 m ( w - l ) = dim H°{S9 6s(mKs)) Moreover H°(V9 0v{\)) generates H°(V, 6v(m))> This proves the theorem. Now we obtain the following refined version of Theorem B :

Y. Miyaoka

118

T h e o r e m 9. The canonical model X of S is isomorphic to a complete intersection of type (2, 2, 2, 2) in P 6 . The canonical model X of S is the quotient ofX by the action of following subgroup G of PGL(6, C) :

-1 -1

-1 0

-1

-1

-1

-1 -1

-1

-1

The following theorem is a corollary of Theorem 9 and the form of the defining equations. T h e o r e m 10. The moduli space of Campedelli surfaces is a unirational variety of dimension 6. References [ 1 ] Bombieri, E. : Canonical models of surfaces of general type, Publ. Math. I. H. E. S.} 42 (1973), 447-495. [ 2 ] Campedelli, L. : Sui piani doppi con curva di diramazione del decimo ordine, Atti Accad. Naz. Lincei, 15 (1932), 358-362. [ 3 ] Iitaka, S. : Deformations of compact complex surfaces II, J. Math. Soc. Japan, 22 (1970), 247-261. [ 4 ] Miyaoka, Y. : Tricanonical maps of numerical Godeaux surfaces, to appear. [ 5 ] Mumford, D.: The canonical ring of an algebraic surface, Ann. of Math., 76 (1962), 612-615. [ 6 ] Ramanujam, C. P. : Remarks on the Kodaira vanishing theorem, J. Indian Math. Soc. (N. S.) 36 (1972), 41-51.

Department of Mathematics Tokyo Metropolitan University (Received January 7, 1976)

On Singular K3 Surfaces

T. Shioda and H. Inose By a singular K3 surface we mean an algebraic K3 surface (defined over the field of complex numbers) whose Picard number equals the maximum possible number 20. The main purpose of this paper is to prove the following results: T h e o r e m . There is a natural one-to-one correspondence from the set of singular K3 surfaces to the set of equivalence classes of positive-definite even integral binary quadratic forms with respect to SL2(Z) (Th. 4 in § 4). T h e o r e m . Every singular K3 surface has an infinite group of automorphisms (Th. 5 in §5). T h e o r e m . Every singular K3 surface has a model defined over an algebraic number field K, and its Hasse-Weil zeta function is given, up to finitely many Euler factors, by where £K(s) is the Dedekind zeta function of K and L(s, %2) with a suitable Grbssencharacter x2 or %2 ( T h . 6 in § 6 ) .

or

L(s, %2) is the Heche L-function

The correspondence in the first theorem associates with a singular K3 surface X the binary quadratic form given by the intersection product on the oriented lattice Tx of transcendental cycles on X, and its injectivity was proved by PjateckiiSapiro and Safarevic as a consequence of the Torelli theorem for K3 surfaces (cf. [4] §8). We shall prove the surjectivity of this correspondence by constructing a singular K3 surface X with a prescribed Tx. In fact we shall construct a wider class of K3 surfaces containing all singular ones, starting from Kummer surfaces associated with abelian surfaces of product type (§§ 2, 3). We use the corresponding result for singular abelian surfaces, which was proved in [8]. For the proof of the second theorem, we find on a given singular K3 surface X a suitable elliptic pencil W : X^P1 which has infinitely many sections. The present work is based on the general theory of elliptic surfaces, due to Kodaira [3]. It is a great pleasure for the authors to dedicate this paper to Professor Kodaira.

120

T. Shioda and H. Inose

1.

Elliptic Pencils on a K3 Surface

We recall in this section a few facts on elliptic pencils (i.e. structures of elliptic surface) on a K3 surface, which will be used in the sequel. L e m m a 1 . 1 . Assume that an effective divisor D on a K3 surface X has the same type as a simple singular fibre of an elliptic surface in the sense of Kodaira [3] § 6. Then there is a unique elliptic pencil 0 : X—>P\ of which D is a singular fibre. Moreover any irreducible curve C on X with (CD) = l defines a {holomorphic) section of 0. This follows immediately from Theorem 1 of [4] § 3. L e m m a 1.2. If a divisor D on an elliptic surface has its support contained in a {simple) singular fibre, then the self-intersection number D2 is non-positive, and D2=0 holds if and only if D is a multiple of the singular fibre. For the proof, we refer to [6] Ch. VII § 2 or [7] Lemma 1. 3. L e m m a 1. 3. Let 0 : X-+P1 denote an elliptic K3 surface, andDv=0~l{tv) {l
i > v = 24 ( - t h e Euler number of X).

(ii) Assuming moreover that 0 has a section, let r {0) be the rank of the group of sections of 0. Then the Picard number p{X) of X is given by the formula :

(1.2)

p(X)=r{*)+2+'£<<mv-\).

(iii) When r(0) = 0 , let n(0) denote the order of the group of sections of0. Then we have (1.3)

\detTx\ = \detSx\ = *

Here Sx is the sublattice of H2{X, Z) consisting of algebraic cycles [the Neron-Severi group ofX), and TX=SX is the orthogonal complement ofSx in H2(X, Z) {the lattice of transcendental cycles). In fact these three statements hold for an arbitrary elliptic surface 0 : Jf—>J with at least one singular fibre. The assertion (i) is implicit in [3] Theorem 12. 2. As for (ii) and the second equality of (iii), we refer to [7] Corollaries 1. 5 and 1. 7. The first equality of (1. 3) follows from the elementary remark that, for a primitive sublattice S of a unimodular lattice H, |det S\ is equal to |det S^, S± being the orthogonal complement of S in H.

On Singular K3 Surfaces

121

Finally we note the following table for singular fibres from Kodaira [3] pp. 565-566, p. 14 : II

III

b

1

2

3

6+5

9

8

7

b

1

2

3

4

1

2

3

b

2

3

4

6+6

10

9

8

type Ift(tel)

2.

IV I*(*>0) II* III* IV*

Construction of a Certain Elliptic Pencil on a Kummer Surface

T h e o r e m 1. Let Z = K m ( J ) denote the Kummer surface associated with an arbitrary product abelian surface A = ClxC2 (C,, C2 elliptic curves). Then there exists an elliptic pencil 0 : X—>P\ which has a section and {at least) three singular fibres of types (i) II*, I6* and I6* A 2 <2), or (ii) II*, IV* and Io*.

Recall that the Kummer surface Z=Km(i4) is the minimal non-singular model of the quotient surface AjcA of A by the inversion automorphism cA (cA(z) = —z), which has the 16 singular points corresponding to the points of order 2 of A. Letting ut (or u'i) ( l < / < 4 ) be the 4 points of order 2 on the elliptic curve Cl (or C2), we denote by Etj(\CV (v=l, 2). Each 0V has 4 singular fibres, all of type Io* :

(2.1)

2Gj+±Eij~G

in which F (or G) is a general fibre of 0{ (or 02). The symbol ~ indicates the linear equivalence, which coincides with the algebraic or homological equivalence on a K3 surface. The intersection number of these curves are given as follows : FG = 2, >k * ^ /

z?? &ij

=

FEtJ = G £ y = FtG, = 0,

172 /^2 ^ i — l*i =

O —£->

17 J? rkEtj

=

^ Oki,

FGy = i^ TP ^k&ij

GF<=\, =

^ OA
The configuration formed by the rational curves Eijy Fu G3 is called the double Kummer pencil on X=Km(C 1 xC 2 ) (see Fig. 1). Now we shall construct a new elliptic pencil on X. For that purpose, let us consider the following divisor D :

T. Shioda and H. Inose

122

Fig. 1 (2. 3)

D =

(see the bold lines in Fig. 1). It has type II* in Kodaira's list [3] § 6. Applying Lemma 1.1, we obtain an elliptic pencil 0 : X-^P1 having D as a singular fibre. Note that the curve G, is a section of 0, since G,D— 1. To find other singular fibres of 0, we consider the divisors (see the dotted lines in Fig. 1) : = F3+E3I+E32,

(2. 4)

B2 =

These divisors do not meet the fibre D and their supports are connected. Hence the image 0{BX) (^=1, 2) is a point tx of P\ and Bx is contained in the singular fibre 0~l(tx). We have t^t2, because both Bl and B2 meet the section Gl5 and they cannot belong to one and the same fibre. Therefore the proof of Theorem 1 is reduced to the assertion that the types of singular fibres ®~l(tx) ( ^ = 1 , 2) are either (i) I6* and I£ (*, + A2<2) or (ii) IV* and Io*. L e m m a 2. 1. There exist 4 effective divisors Av (1 < v < 4 ) on X satisfying the following conditions : (1) Av is linearly equivalent to a linear combination of F, G and Etj (l
4^

= ^ 3 = 1 , ^ 4 = ^^4=1.

Proof By an easy computation using (2. 2), we see that the space of divisors satisfying the conditions (1) and (2) is generated (up to linear equivalence) by the 4 divisors A'v A\ = En-\-E2±-\-En—F—G, A2 = E22—E \^' °/

A' -"3

p

p

-^33

-^435

A''

p

\

-^44

xx

p ^^34'

In diagonalizing the symmetric matrix [A^A^ we find the following 4 divisors Av (l
On Singular K3 Surfaces n jy | o/^

A

• y i j ^** ^,-1

^43 -^ •^J-4 '^^"' £F

| A v>^

O 77 A-*-'i i

77

-*^/22

123

77

77

77

"^24

~^^33

-^43

J

77

JL^24

^^33

"^^34

<

^44

r-\-CJ—En—E2it—£33 I ^ v-7

^AJJ 11

J2j
^^4':5#

Now each Av is linearly equivalent to an effective divisor. In fact, by the Riemann-Roch inequality on a K3 surface, we have

Hence either Av or —Av is equivalent to an effective divisor A*. Since AfD = 0 by (2), each component of A* lies in some fibre of 0, and we have A*E>0 for any irreducible curve E not lying in a fibre. On the other hand, we have ^4lJEl33=J2^33 = 2 and A3E43=A^E43=2. Therefore AV~A*, and we can assume that Av itself is effective for all v. Let us then decompose Av into its connected components Avi (i=l, 2, •••) ; each Avi is contained in a fibre. By Lemma 1. 2, we have A2vi<0. Since A2vi is even, we can assume A*Y =—2, A2vi=0 for i>2. Again, by the same lemma, Avi is a multiple of a singular fibre for i>2. Therefore Av~Avl-\-m0~1(t). Computing the intersection numbers of both sides with the section Gx of 0, we see m=0. Thus the conditions (1), •••, (4) are satisfied when Av is replaced by Avl. Finally (5) follows immediately from (2. 2). This completes the proof of Lemma 2. 1. q. e. d. L e m m a 2. 2. The singularfibres0~l(tk) (^=1, 2) are given as follows :

(2. 7) 0~\tx) - 2is+£ 3 1 +£ 3 2 +A+^ 2 ,

Q-%) = 2F4+EAl+E42+A3+A4.

l

In particular, each 0~ (tx) has at least 5 irreducible components and is not of type \b{b>\). Proof It follows from the conditions (4), (5) of Lemma 2. 1 that the supports of Ax and A2 lie in the singular fibre 0~1(tl). Moreover, if we put H=2F3-\-E3l-\E32-\-Ax-\-A2, it is easy to check H2=0. By Lemma 1. 2, we conclude that H is a multiple of 0~l(tl). This multiplicity must be 1, because HGl=E3lGl=l and 0~l(tl)Gl=l for the section G{. This proves the first equality, and the second one can be similarly proven. The last assertion follows from the fact that every irreducible component of a singular fibre of type I6 (b>l) appears with multiplicity 1. q.e. d. Now we continue the proof of Theorem 1. From Kodaira's list of singular fibres, [3] § 6, the possibilities for the singular fibre 0~l(tx) (^=1, 2) are the following types : Ib* (J>0), II*, III* or IV*. Let mx be the number of irreducible components of 0~l[tx). Applying the formula (1. 2) to our elliptic pencil 0, we have (2. 8)

p(X) > 2 + (9-l)

+ (ml-\) + (m2-l).

Since p(X)<20 for a complex K3 surface X, this implies that m 1 +m 2 <12, and hence 5<m,, m2<7. This excludes the possibility for 0~l{tx) to be of type II* or III* (cf. the table in § 1). If 0~l(tl) is of type IV* (m, = 7), then 0-1(t2) has m2=5

124

T. Shioda and H. Inose

components, hence it must be of type Io*. This corresponds to the case (ii) of Theorem 1. Otherwise 0-l(tl) and 0~l(t2) are of types I&* and I&*. Setting mx = bx + 5 (X=l9 2) in (2. 8), we have bl + b2<2 in this case. This completes the proof of Theorem 1. q. e. d. Remark. (1) Actually the converse of Theorem 1 is also true. Namely any K3 surface having an elliptic pencil such as that in Theorem 1 is isomorphic to Km(A) for some A = CxxC2. We omit the proof, since it is rather complicated and also will not be used in the following. (2) We shall prove later that bl9 b2<\ in the case (i) (cf. Lemma 3. 1). (3) The case (ii) of Theorem 1 occurs if and only if the abelian surface A is isomorphic to the product Cw X Ca9 C^ being the elliptic curve with the fundamental periods 1 and a)=e27ti/3. 3.

Certain Double Coverings of Kummer Surfaces

Throughout this section, we assume that X is a Kummer surface given with an elliptic pencil 0 : X—•P1, which has a section and at least three singular fibres 0-%), 0-%) and 0-%) of types (i) II*, I6* and I6* (bl + b2<2)9 or (ii) II*, IV*, Io*. We shall construct a certain K3 surface 7 and a rational map n : 7—>X of degree 2, and examine the relation between Tx and TY, the lattices of transcendental cycles on X and on 7. We consider a ramified double covering f: A—>P\ ramified only at the 2 points tx, t2 of P\ and put f'%) = {SOHSU), f-l(tx) = su f-l{t2) = s2. We denote by ¥ : Y—>A the elliptic pencil induced by 0 : X—>PK Note that 7 is a non-singular surface, birationally equivalent to the fibre product of X and A over P\ and that no fibre of ¥ contains an exceptional curve of the first kind. Obviously ¥ has two singular fibres of type II* over s0l and s02. In case (i), the fibre ¥~l(sx) over sx (^=1, 2) is of type l2bv since it is the pull-back of the singular fibre 0~x{tx) of type I6* at the ramification point tx of the double covering f: A—+PK In fact this is clear from the behavior of the homological invariant (or local monodromy) of ¥ around sx:\ n — 1 I = 0 i ^ (c^- E^] P- 604). In case (ii), ¥-l(s{) is a singular fibre of type IV and W~l(s2) is a regular fibre, as is shown by the same argument as above. Definition. We shall call the surface 7, together with the rational map of degree 2 it: Y—>X, the double covering of X with respect to the elliptic pencil 0 : X—+PK L e m m a 3 . 1 . Y is a K3 surface. Moreover the collection of singular fibres of¥ : 7—»J

falls into one of the following : 1) II*, II*, I 2 ,1 2 , 2) II*, II*, I2, I,, I,, 3) II*, II*, I,, Il5 I,, I,, 4) II*, II*, II, II and 5) II*, II*, IV.

On Singular K3 Surfaces

125

Proof. We denote by 0~l(tv) (v = 0, 1, 2, •••, k) all the singular fibres of 0 : X-* P\ and by ev the Euler number of 0~l(tv). In case (i) (resp. (ii)), since £0=10 and £^ = ^ + 6 (^=1, 2) (resp. £1 = 8, £2=6) (cf. the table in § 1), we deduce from the formula (1.1) that (3. 1) bl + b2+(e3+-..+ek) = 2 (resp. £3+---+£* = 0). l If we put f- (tv) = {sul9sv2](v> 3), the singular fibre ¥-1(svi) (z=l, 2) is of the same type as 0~l(tv). Now the Euler number e(Y) of the surface 7 is equal to the sum of those of singular fibres of ¥ : Y—>A (cf. (1. 1)) : e(Y) =2e o +(2* 1 ) + (2A2)+2(e3 + .»+£,)

(resp. e(Y) = 2£0+4)5

where 2bx (^=1, 2) is the Euler number of the singular fibre ¥~l(sx) of type I26r It follows from (3. 1) that £(7) =24. Next we note that the elliptic pencil ¥ has no multiple fibre because it has a section induced by one of 0. By Kodaira [3] Theorem 12. 1, the canonical bundle KY of 7 is given by KY=¥*(l — f), in which I is the canonical bundle of the base curve A and f is a line bundle on A of degree -p,{Y)+q(Y)-l

= -(K>Y+e(Y))ll2

= -2.

Since A~P\ this implies that KY is trivial and q(Y)=0. Therefore 7 is a 1 3 surface. In case (ii), ¥ has no more singular fibres (k=2) and we have case 5). In order to determine the singular fibres of¥ in case (i), we observe that b{<\, b2<\. In fact, if bx or b2=2, then ¥ would have (at least) three singular fibres of types II*, II* and I4, and then the formula (1.2) would imply p(Y)>2l, a contradiction. Then (3. 1) has only 4 solutions : 1') bl = b2=l, k=2; 2') {£„ b2} = {0, 1}, k=3, £3=1 ; 3;) ^ = 42=0, A;=4, £ 3 =£ 4 -l ; 4') bx = b2=Q, k=3, £ 3 =2. Noting that a singular fibre has Euler number 1 (resp. 2) if and only if it is of type i! (resp. I2 or II) (cf. the table in § 1), we see that the cases T), 2r ), 3;) correspond respectively to the cases 1), 2), 3) stated in Lemma 3. 1, and case 4') corresponds to case 1) or 4). This proves Lemma 3. 1. q. e. d. We remark that the surface X can be recovered from 7 as follows. The nontrivial covering transformation of the double covering/: A-^>PX induces an involutive birational transformation of 7, hence an automorphism c of 7 (by the minimality of a K3 surface). The involution c has the 8 fixed points, 4 on each of the two fibres ¥~1(sx) (^= 1, 2), and Jf is the minimal non-singular model of the quotient Y/c (cf. [3] § 8, pp. 585-586, 591-592, 600-602). In particular, the rational map TZ : 7—>Xhas the 8 fundamental points jfrv(l<><8) corresponding to the 8 curves in X, which appear with odd multiplicities in the two singular fibres In order to formulate the following theorem, we introduce some notation. The rational map n : 7—>X induces a homomorphism : (3. 2)

*m : H2(Y, Z) ~ H2(Y- [p,], Z) -+Ht(X, Z).

126

T. Shioda and H. Inose

We denote by Tx the sublattice of transcendental cycles in H2(X, Z), and by px the period on Tx, i.e. the linear functional on Tx defined (up to constants) by

(3.3)

px(t)=jwx

(t€Tx)9

(Dx being a non-vanishing holomorphic 2-form on X. For xl9 x2eH2(X, Z), (x^x2) denotes the intersection number. T h e o r e m 2. Let n : 7—>X be the double covering defined as above. Then 7r* induces a bijection of TY onto Tx such that

(a) (b)

(^JVTr*^) = 2(ji\jV2), px°x* = const. pY.

y^yi^

TY,

The rest of this section is devoted to the proof of this theorem. First we observe that the pull-back 7r*(a>x) of a holomorphic 2-form wx on X is holomorphic on Y—{pv)> hence holomorphic everywhere on 7. By (3. 3), this shows the assertion (b). Next, keeping the same notation as before, we define some sublattices of Hx =H2(X, Z) and / / F = / / 2 ( 7 , Z). Let F be the sublattice of Hx generated by the 8 irreducible components with multiplicity > 2 of the singular fibre @~l(t0) of type II* ; F is a negative-definite even unimodular lattice of rank 8 (cf. [7] Lemma 1.3). Also let Fn (w= 1, 2) be the sublattice of HY obtained from the singular fibre ¥~l(sOn) of type II* in the same way. Since FX@F2 is a unimodular sublattice of HY, we have the orthogonal decomposition : (3.4)

HY =

Fl@F2@L,

L being unimodular and of rank 6. Obviously we have Furthermore let M denote the sublattice of Hx generated by the 8 curves © y (l<^<8). By the definition of n* (3. 2), TZ* maps HY into the orthogonal complement Mx of M in Hx. On the other hand, there is a natural map n* : M^-+HY such that (9.

CV\

/V*v

yO.

\JJ

yn

(3.7)

- T T * V "\

9 (Y

.v^l

(Y

Xj^TT X2J — Z,yXx*X2)

K*K*jy

==

v

c A/f-L\

V^ij %2 ^ ^^* J?

y-{-t*(ty)

(y£HY),

where c* is the induced map of the involution c of 7 (cf. [2] § 1). Moreover, since c interchanges the singular fibres ¥~l(s0l) and W-l(s02), we have Finally it follows from (b) and (3. 7) that both n^ and x* preserve the algebraic cycles: /Q

Q\

/o

\

r«

* / C

r^

A/f J-\

C

L e m m a 3. 2. The action of c* on L is trivial. The map TC* induces a bijection of L onto its image in Hx, and

On Singular K3 Surfaces

127

jv2) for j , , y 2 e L.

Proof The involution t of Y has the 8 fixed points {pv}. By the Lefschetz fixed point theorem, we have In view of (3. 4) and (3. 8), this implies that tr(c*\L)=6. Since the eigenvalues of c* are 1 or —1, we conclude that t*\L is the identity. It follows from (3. 7) that 7r*n*()>)=2y for yeL. Hence n*\L is injective, and moreover we have from (3. 6) j 2 ). q. e. d. Lemma 3. 3. n* (HY) = n* (L) 0 F. Proof It suffices to show that TT^(L) and F are orthogonal to each other. Take xl=K*(yl) (y}eL) and x2=n:^(y2) (y2eFl). From (3. 7) and (3. 8), we see that 7r*(xl)=2yl eL and x*(x2)=y2Jrc*(y2) eFx@F2. Hence we have 2(*,.*2) = {n*xx.n*x2) = (2j 1 .(j 2 +^(j 2 ))) = 0.

q.e.d.

L e m m a 3. 4. Let M' be the minimal primitive sublattice of Hx containing M. Then

det AT = |det M^| = 26. Proof Obviously (M')± = M±, and |detM'| = |det {M'Y\, because M' is primitive in the unimodular lattice Hx. Since d e t M = d e t (&v&fl)=28, we have only to prove that the index [Mf : M] = 2. Let 0v be the homology class of the curve &v. 8

Take an element x=^av6v(aveQ) of M\ x$M. Since x»6veZ, av must be halfy= l integers. We may assume av — 0 or 1/2. Then the number m of av with av=l/2 is either 4 or 8, because x2=—m/2 is an even integer. Assume m = 4 and

4

x=^0vl2.

If we take a divisor Z) in the homology class x (D2=x2=—2), the Riemann-Roch theorem implies that D or —D is linearly equivalent to an effective divisor D'. But D~D', since 2D~£ev.

As D'6v=x8v= — \ for l < v < 4 , Z>' contains

as components : D'= i^&v+D\

D">0. It follows that 2D"~ — 2J@y? which is a

l

l

8

contradiction. Therefore m = 8 and # = 2 0 y / 2 . This shows [M r : M ] < 2 . Con8

versely, the element2#v/2 belongs to M', because we have l

in which Z)^ (^=1, 2) is the sum of irreducible components with multiplicity > 2 in the singular fibre @~l(tk) of type I6* or IV*. q. e. d. T:*(TY) = Tx. Lemma 3. 5. n*{HY) = ML, Proof For the first equality, we know that the inclusion n*(HY) ciM1- holds. By Lemmas 3. 3 and 3. 2, we have

128

T. Shioda and H. Inose

|det n+(HY)\ = |det x*{L)\ = 26 |det L\ = 26, since F and L are unimodular. Then 7r*(HY)=M± by Lemma 3. 4. Next we shall show that 7i*(TY)(zTX:) i.e. if ye TY, then (TT+(J>) •#)=() for all ^ 6 ^ = 7 ^ . Indeed this is obvious if x e M. If x e ML fl^x? then TT*(X) is defined and belongs to SY. Therefore, by (3. 6), we have 2{x,(j>).x) = {K*K*{y).**{x)) = (2yn*{x)) = 0. This proves 7r^(TY)aTx. Similarly we have 7t*(Tx)(zTY. Now, by Lemma 3. 3, we have (noting M(zSx) Tx = SxczM-= n,(HY) = K^L) 0 T. But Tx is orthogonal to FaSx, and hence Tx
Proof We consider the Kummer surface Z=Km(i4), and the rational map a : A—>X of degree 2. It is well-known (cf. [4] § 5) that a* gives a bijection TA-+TX such that (a') (a*tl»aj2)=2(tl»t2) for tl9 t2e TA ; (br) pxoa* = const. pA. On the other hand, X has the elliptic pencil 0 : X-^P\ constructed in Theorem 1. Let n : Y—±X be the double covering of X with respect to the elliptic pencil 0, as defined at the beginning of this section. In view of Theorem 2, the map (p=(a^)~lo7z^ satisfies the required properties (a) and (b). q. e. d. 4.

Classification of Singular K3 Surfaces

For a singular K3 (or abelian) surface F, the lattice of transcendental cycles TY has a natural orientation. Namely we call a basis {yx,y2} of TY oriented if the imaginary part of pY(yl)lpY(y2) is positive. Using an oriented basis {yi,y2} of TY, we put """

U1J2 yi

Let us denote by Q the set of 2 X 2 positive-definite even integral matrices (4. 2)

Q,= ^

*1

(a, A, ^ € Z, fl, cr > 0,

A 2 - 4 ^ < 0).

We define CLi~d2 if and only if Q^=YQ,2r f° r some y e SL2(Z). We denote by

On Singular K3 Surfaces

129

the equivalence class of Q,3 and by Q/SL2(Z) the set of equivalence classes. Note that {Q,Y} is uniquely determined by Y. T h e o r e m 4. The map F(->{Q^F} establishes a bijective correspondence from the set of singular K3 surfaces onto QjSL2(Z). Proof The injectivity of this correspondence was proved essentially by PjateckiiSapiro and Safarevic. (See [4] § 8. Compare also [8] § 5. We pointed out in [8] p. 278 that there was a certain gap in the proof of the Torelli theorem in [4], but it has since been filled in by several people, notably by M. Rapoport.) The surjectivity follows from the corresponding result on singular abelian surfaces [8] and Theorem 3. Namely we take Q^as in (4. 2), and consider the singular abelian surface A : (4.3)

i4 =

where

(4. 4)

r, = (-b+V4)/2a,

r2 = (A+VZ)/2

(J = b2-4ac).

By [8] § 3, we have Q,A = Q,. Let 7 be the double covering of the Kummer surface X=Km(A) with respect to the elliptic pencil $, constructed in Theorem 1. Then, using Theorem 2, we have

dr = \Qx = Ox = d. This proves the surjectivity of the correspondence.

q. e. d.

For a singular K3 (or abelian) surface X, we put (4. 5)

Cx = H*{X, 0x) /image H\X,

Z),

which is an elliptic curve with complex multiplications (cf. [8] § 1). We mention the following result, suggested by Safarevic [5] p. 416. Corollary. For two singular K3 surfaces X and Y, the following statements are equivalent to each other : (1) X and Y are related by an algebraic correspondence. (2) QW^x)=QWTY), where J x = - d e t Qx and J F = - d e t QY. (3) The elliptic curves Cx and CY are isogenous. Proof In view of the construction of a singular K3 surface X with a prescribed [CLx) (Theorem 4), the assertion is reduced to the case of singular abelian surfaces. In that case, it follows from [8] § 3. q. e. d. Remark. Actually (1) can be replaced by a stronger condition : (V) There is a rational map of finite degree of X to Y. This condition (T) turns out to be symmetric in X, Y, like the notion of isogeny for abelian varieties. The proof is based on another construction of a singular K3 surface, not as a double covering, but as a quotient of a suitable Kummer surface.

130

T. Shioda and H. Inose

5.

Automorphisms

T h e o r e m 5. Every singular K3 surface has an infinite group of automorphisms.

Proof. Let 7 be a singular K3 surface with QY = Q. We distinguish the three cases:

In each case, we shall exhibit an elliptic pencil on Y having an infinite group of sections.

Since any section of an elliptic pencil defines an automorphism of 7, this will prove Theorem 5. Recall that the K3 surface 7 is obtained as the double covering of a Kummer surface X— Km(A) with Q,A = Q,. As we have seen in § 3, there is an elliptic pencil ¥ : 7—»J with a section, whose singular fibres are described in Lemma 3. 1. Using the formula (1. 2), we can determine the rank r(¥) of the group of sections of ¥. The result is (5. 1)

r{¥) =0, 1, 2, 2 and 0,

according to the cases 1), 2), 3), 4) and 5) of Lemma 3. 1. In case 1), the formula (1.3) implies that

Since QY is even, we have n{¥) = \ and det d F = 4 . Hence QY~ L

«L which

corresponds to the case (/-). A similar argument shows that, in case 5), we have r2 ii £ I F ~ I 9 L which corresponds to the case (/3). Thus the assertion for the case (a) is a consequence of (5. 1). For the cases (/3) and (y), we need the following lemmas : L e m m a 5. 1. Let A^C^x Cw, C^ being the elliptic curve with the fundamental periods 1 and a)=e2ni/3. Let a be the automorphism of A defined by <J(Z\> Z2) = (<*>£i> ^2Z2)- Then the minimal non-singular model 7 of the quotient surface Aj(a) is a singular K3 surface such

Proof The automorphism a has the 9 fixed points (vt, v3) (1<J, 7<3), where {vt} are the fixed points of the automorphism a{ of C^, defined by ol(z)=o>z. The quotient Aj (a) has the 9 singular points/?^, each of which is locally isomorphic to the singularity JV_3 in the notation of Kodaira [3] p. 583. The minimal nonsingular model 7 of Aj{a) is obtained by a "canonical reduction", in which each pij is replaced by 2 non-singular rational curves EtJ and E'^ with £^£^-=1. Moreover, 7 contains 6 non-singular rational curves, i. e. the image Ft (or G3) of vt X Cw (or Cw X Vj) in 7. These 24 curves on 7 form the configuration of Fig. 2.

On Singular K3 Surfaces

131

-\—G9.

I

''E\\J]

iZ/01

ILOI

Fig. 2

Now it is easily verified that Y is a X3 surface. Furthermore the elliptic pencil ^ : Y-+P\ induced by the projection A—>CW to the first factor, has the 3 singular fibres of type IV* : y/r

1 o v f 4-3F

(i—\ 2 31

and the 3 sections Gl5 G2 and G3. Applying (1.2), we see that ^ ( 7 ) ^ 2 0 and r(W)=0. Then, by (1. 3), we have det TY = 33/n2 with n = n{W)>3. Hence det TY \2 1" q. e. d. = 3, and £^F~ 1 2• L e m m a 5. 2. Let A = Ctx Ciy Ct being the elliptic curve with the fundamental periods 1 and i=y/—\. Let a be the automorphism of A defined by o(z\, Zi) = (iZ\, —iz2)- Then the minimal non-singular model Y of the quotient surface A/(a) is a singular K3 surface such

Proof Since a2 is the inversion cA of A, it has 16 fixed points. Among these, the 4 points (vu Vj) (l
V

\

\

\

x ;x -' Fig. 3

132

T. Shioda and H. Inose

(vi9 Vj) and the 6 other singular points of type N_2 (cf. [3] p. 583). The minimal non-singular model of Aj(a) is obtained by a "canonical reduction", in which each ptj is replaced by 3 curves and each of the 6 other singular points is replaced by a curve. These curves, together with the 6 image curves of v X Ct and C{ X v (v=points of order 2 of CJ, form the configuration of non-singular rational curves on r, which is described in Fig. 3. Now the projection A-^Ct (to the first factor) induces an elliptic pencil 0 : F—> P\ which has the 3 singular fibres of types III*, III* and Io* (delete the 3 horizontal curves Gl9 G2, G3 from Fig. 3). Moreover Gx and G2 are sections of 0. On the other hand, Y is easily seen to be a K3 surface. Hence, applying (1. 2), we have ,0(7) =20 and r(0)=O. Then (1. 3) implies that det TY=2*2^jn(0)2. Since n(0) >2, it follows that det 7V=4 and Q, F ~ R* ?].

q. e. d.

Proof of Theorem 5 (Continued). Case (j8). We shall find a new elliptic pencil ¥ on the K3 surface Y of Lemma 5. 1 such that r(W)>0. With the notation in the proof of that lemma, we consider the two divisors (cf. the dotted or bold lines in Fig. 2) :

A = D2 =

Gl+2En+3E[l+4Fl+3E[2+2El2+G2+2E[3, Ef2l+Ef22+Ef3l+Ef32+2(F2+Ef23+E23+G3+E33+Ef33+F3),

which are respectively of types III* and I6* and disjoint from each other. By Lemma 1.1, there is an elliptic pencil ¥ : Y—>P\ of which Dx and D2 are singular fibres and E2l is a section. Assume that the group of sections of ¥ is finite, i.e. r(¥)=0. Let us denote by mly •••, mk (ml>'">mk) the number of irreducible components of other singular fibres o(¥. Applying (1. 2), we ha.vc'£(mv—l) = l. Hence mx = 2 and mv=\ for v>2. Then (1. 3) implies the equality 3 = det TY =

2-4-2ln(¥)\

which is a contradiction. Thus we have r(¥)>0. Case (y) Similarly we can find an elliptic pencil ¥ on the K3 surface Y of Lemma 5. 2 such that r(W)>0. Namely there is an elliptic pencil ¥ : Y^>Pl having two singular fibres of types I12 and IV* (cf. the bold or dotted lines in Fig. 3) and having at least one section. Using the same argument as in (/3), we can show that r(¥)>0. This completes the proof of Theorem 5. q. e. d. Remark. As to the automorphisms of a K3 surface, the following result might be worth mentioning : (*)

There is a K3 surface with the Picard number 18 whose automorphism group is

finite. In fact, take two non-isogenous elliptic curves C,, C2 and put A = CX X C2. Then TA is a unimodular lattice of rank 4 with signature (2, 2). By Theorem 3, there exists a K3 surface Y (i.e. a. double covering of Km(i)) such that TY~TA. Therefore SY= T$ is a unimodular lattice of rank 18 with signature (1, 17). Then, by a result

On Singular # 3 Surfaces

133

of Vinberg [10], the automorphism group of the Euclidean lattice SY has a subgroup of finite index generated by reflections. Applying Theorem 1 of [4] § 7, we conclude that the automorphism group of the K3 surface Y is finite. 6.

An Arithmetic Application

The elliptic curve Cx (4. 5) associated with a singular K3 surface X has complex multiplications. Hence it has a model defined over some algebraic number field Ko. We assume that Ko contains End(Cx)(x)Q. By Deuring [1], for any finite extension K of Ko, the Hasse-Weil zeta function £(CX/K, s) of Cx over K is given by (6. 1) C(CX/K, s) = QK(SKK(S-1)L(S-1I2, Z ) - ' L ( J - 1 / 2 , *)-', where £K(s) is the Dedekind zeta function of K and L(s, y) is the Hecke L-function with a suitable Grossencharacter ^. (The symbol = indicates equality up to finitely many Euler factors.) Theorem 6. Every singular K3 surface X has a model defined over some algebraic numberfieldKz)K0, and its Hasse-Weil zeta function £(X/K, s) is given by (6. 2) C(X/K9s) = ZK(s)CK(s-l)»i:K(s-2)L(s-l9j?)L(s-l9f). As a consequence £(X/K, s) is meromorphic on the whole j-plane, and satisfies a functional equation. This answers a question raised by Safarevic [5] p. 416. Also we can verify the conjecture of Tate ([9] § 4, Conj. 2) for singular K3 surfaces : Corollary. number of X,

The order of the pole of £(X/K9 s) at s=2 is equal to 20, the Picard

In fact, this follows from the fact that Z,(l, ^2) Z,(l, %2) ^ 0 , which is a special case of [13] p. 288, Theorem 11. Proof of Theorem 6. (I) First we consider the case of Kummer surfaces. Assume that X=¥Lm(A) is the Kummer surface for the singular abelian surface A = C{xC2 (4. 3). We note that CX~CA=CX (cf. [8] (3. 13)) and C2 is isogenous to C,. Then we can find models for Cx and A defined over some algebraic number field KZDKQ such that (a) all points of order 2 of A are ^-rational, and (b) an isogeny of A to Cx is defined over K. In this situation it is obvious that X=~Km(A) is defined over K. Moreover, by blowing up the 16 points of order 2 of A, we obtain a surface A fitting in the following diagram, also defined over K :

A*—?—

A

(6.3) A/tA *

X.

134

T. Shioda and H. Inose

The inversion cA of A induces an involution cA of A such that Xis the quotient of A by cA. Fix a prime number /. For almost all prime ideals p of K (2lJfNp), we can consider the reduction mod p of the diagram (6. 3). For a moment, using the same notation A, X, ••• for the reduced varieties, we consider the diagram (6. 3) defined over the finite field Fq of q=Np elements. For a surface Z over Fq, we denote by Hz the /-adic cohomology group H2(Z, Qt) (Z=Z(g)Fq9 /\,=algebraic closure of Fq), and by Sz the subspace of Hz generated over Qt by algebraic cycles (defined over Fq). Then we have the induced isomorphisms : (6. 4) (6. 5)

a* : Sx * SA. HJSA * HAjSA * Hx/Sx. P* a* In fact, this follows immediately from the fact that cA acts trivially on HA [A : abelian surface). As usual we put (6. 6)

P2(Z, T) = dct(l-F*T\Hz)

{T=q-)t

where F* is the endomorphism of Hz induced by the Frobenius morphism F of Z over Fq. By (6. 4), (6. 5), we have (6. 7)

P2(X, T) =

det(l-F*T\Sx)det(l-F*T\(HxISx))

= (1-qTmA, T). On the other hand, if we set (6.8)

Q(Cx,T) = (\-nT)(l-nT)l(\-T)(l-qT)

(™ = g, |*| = ?•"),

we have (6. 9)

P2(A, T) =

because A is isogenous to CxxCx. (6. 10)

P2(X, T) =

{\-qT)\\-^T){\-^T)t Hence we have (l-qT)20(l-7i2T)(l-7i2T).

Going back to the global situation, we first note that ^(j))=^/|7r| (^ as in (6. 8)) gives the Grossencharacter ^ in (6. 1). Therefore, using (6. 10), we obtain (6. 11)

C(XIK, s) = U[(l-Np-°)P2(Xmodp, Np-)(1 -

as asserted in (6. 2). (II) In the general case, a singular K3 surface Y is obtained as a double covering of a Kummer surface X=Km(A). The elliptic curve CY is isomorphic to Cx , since 2 d F ~ Q , x - Using the same notation as in (I), we note that all irreducible curves in the double Kummer pencil on X (Fig. 1) are ^-rational; hence the divisor D in (2. 3) is also J^-rational. Therefore the elliptic pencil 0 considered in Theorem 1 is defined over K, since it is the rational map (in fact morphism) associated with the complete linear system \D\. Also the double covering/: J—KP1, ramified at /1=(p(5I) and t2=0(B2) (cf. (2. 4)), is defined over K. Hence the double covering Y of X with respect to 0 (§3), together with the rational map n : Y—+X, is defined over K. By blowing up Y at the 8 fundamental points of re, we obtain

On Singular K3 Surfaces

135

the following diagram, similar to (6. 3) : Y -

£

Y

(6. 12)

Replacing K by a finite extension, we can assume that this diagram is defined over K. For almost all prime ideals p of K, we have a reduced diagram mod p, defined over Fq (q=Np). In this situation over Fq, we have (6. 13) Hf = HY@ (8 blown up curves)

(6.14)

HY =

r,@r2@L,

where Fn ( n = l , 2) is the subspace of HY generated over Qt by the 8 irreducible components with multiplicity > 2 in the singular fibre W~x{s^) of type II* (cf. (3. 4)). Since the involution c of Y acts trivially on L (cf. Lemma 3. 2), we deduce the following isomorphism : (6. 15) Hy/Sy =i Hy/Sf ff HXjSX. As in (I), this implies (6. 16) P2{Y mod p, T) = P2(X mod p, T) for almost all p, and hence we have (6.17) H{YIK,s) = 1i{XIK,s). This completes the proof of Theorem 6.

q. e. d.

Remark. Theorem 6 and its Corollary were known for some special singular K3 surfaces, e.g. the Fermat quartic surface defined over Q{e2*m) (Weil [11], [12], cf. [9] p. 105) and the elliptic modular surface of level 4 defined over Q(\/—l) (Shioda [7] p. 57). The former corresponds to the matrix L

J and the latter to

4 0

References [ 1 ] Deuring, M. : Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins, Nachr. Akad. Wiss. Gottingen (1953), 85-94. [ 2 ] Inose, H. : On certain Kummer surfaces which can be realized as non-singular quartic surfaces in P\ J. Fac. Sci. Univ. Tokyo, Sec. IA, 23 (1976), 545-560. [ 3 ] Kodaira, K. : On compact analytic surfaces I I - I I I , Ann. of Math. 77 (1963), 563-626 ; 78 (1963), 1-40. [ 4 ] Pjateckii-Sapiro, I. I. and Safarevic, I. R. : A Torelli theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR, 35 (1971), 530-572. [ 5 ] Safarevic, I. R. : Le theoreme de Torelli pour les surfaces algebriques de type K3, Actes,

136

T. Shioda and H. Inose

Congres intern, math. (1970) Tome 1, 413-417. [ 6 ] Safarevic, I. R. et al. : Algebraic surfaces, Proc. Steklov Inst. Math., 75 (1965) ; AMS translation (1967). [ 7 ] Shioda, T. : On elliptic modular surfaces, J. Math. Soc. Japan, 24 (1972), 20-59. [ 8 ] Shioda, T. and Mitani, N. : Singular abelian surfaces and binary quadratic forms, in "Classification of algebraic varieties and compact complex manifolds", Springer Lecture Notes, No. 412 (1974), 259-287. [ 9 ] Tate, J . : Algebraic cycles and poles of zeta functions, in "Arithmetical Algebraic Geometry", Harper and Row (1965), 93-110. [10] Vinberg, E. B. : Some arithmetical discrete groups in Lobacevskii spaces, in "Discrete subgroups of Lie groups and applications to Moduli", Tata-Oxford (1975), 323-348. [11] Weil, A.: Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc, 55 (1949), 497-508. [12] Weil, A. : Jacobi sums as "Grossencharaktere", Trans. Amer. Math. Soc, 73 (1952), 487495. [13] Weil, A. : Basic number theory, Springer (1967).

Department of Mathematics University of Tokyo (Received November 7, 1975)

On the Minimality of Certain Hilbert Modular Surfaces

G. van der Geer and A. Van de Ven° 1.

Introduction

For some time now Hirzebruch and others have studied certain fields of Hilbert modular functions from a geometric point of view (see [2], [3], [5]). This leads to the introduction of a non-singular algebraic surface Y°(p) for all square-free positive integers /?. In [5] the question is settled how the surfaces Y°(p) fit into the rough classification of algebraic surfaces, at least for those values of/? which are prime and congruent 1 mod 4.2) It turns out that for/?=5, 13 and 17 the surface Y°(p) is rational, that for/?=29, 37 and 41 this surface is an elliptic X3-surface, that for/? = 53, 61 and 73 it is a minimal honestly elliptic surface, and that for/?> 89 the surface Y°(p) is of general type. As already follows from this description, the surfaces Y°(p) are minimal (i.e. without exceptional curves of the first kind) for 2989. Of course, it would be very interesting if this conjecture could be proved. In fact, if you know that a certain (simply connected) surface is of general type, and even if you know in addition its arithmetical genus and its Euler characteristic, but you don't know whether it is minimal, your knowledge does not amount to very much. For example, you don't know the (higher) plurigenera of your surface, and in the case we consider here, these are of particular importance in number theory, for they give the dimensions of certain spaces of weighted holomorphic cusp forms ([3], p. 255). Given this situation, and given the fact that even Hirzebruch believes that the conjecture in question poses a very difficult problem ([3], loc. cit.), we think it of some use to present here a few principles which lead to a proof of the conjecture for many primes, in particular for all primes/? (congruent 1 mod 4) with 89
138

G. van der Geer and A. Van de Ven

p=89 and />=317 to the reader. Finally, in chapter 4 a few comments are given. We wish to thank Mr. R. Brand from our institute, who wrote the computer programs.

2.

Preliminaries

We shall use the notations of [5]. In particular, we refer to [5] for the definition of the surfaces Y(p), the surfaces Y°(p), and the involution c on Y°(p). On Y°(p) we have the curves St and S[=S_i9 arising from the resolution of the cusps, the curves FN (in as far as they are not blown down), the curves Du arising from quotient singularities of order 2 which lie on FPi and not on Fly the curves Ct and C'iy coming from quotient singularities of order 3 on Fp. Those curves, coming from the resolution of quotient singularities of order 2, which are interchanged pairwise by c will be denoted by 0t, i=l, •••, -~-A(—/>) —1—<5, with c(Oi) = Oi+l, i odd ; and those which come from order 3 singularities and are interchanged pairwise by c will be denoted by Tj9 j=l,

•••, -iyh{ — 3p) — 1— e, with c(Tj) = TJ+l,

j odd. Here d and e are defined as in [5], p. 17. For the next three propositions, see [5], p. 6 and 7. Proposition 2. 1. If on a non-singular algebraic surface X with q(X) =0 there exists an irreducible curve C with KC<0 and C 2 >0, then X is a rational surface. Proposition 2. 2. If on a non-singular algebraic surface X with q(X)=0 there exists a curve C with KC<0 and which has at least one singular point or which is not rational, then X is rational.

Also, if on such a surface X there exists an irreducible curve C with KC<—2, then X is again a rational surface. Proposition 2. 3. If on a non-singular algebraic surface X with q(X)=0 two intersecting exceptional curves {of the first kind), then X is rational.

there exist

An important role in the study of the surfaces Y°(p) is played by ( — 2)-configurations on these surfaces, i.e. the configurations consisting of finitely many nonsingular rational curves with self-intersection —2, the union of which is connected. It is known which (— 2)-configurations can exist on a non-singular surface of general type. In particular one has P r o p o s i t i o n 2. 4. Let {Cj, •••, Ck} be a ( — 2)-configuration on a surface of general type. Then the intersection matrix ( C i C , ) , ^ ^ is negative definite.

Proposition 2. 5. Let X be an algebraic surface of general type, and C an irreducible

On the Minimality of Certain Hilbert Modular Surfaces

139

curve on X. Then KC>0, except if either C is an exceptional curve of the first kind {then KC = —1), or if C is a non-singular rational curve with C2=— 2 (then KC=0).

Proof The proposition is well-known for the case that X is a minimal surface ([1], p. 174). As to the general case, it is sufficient to prove : if the statement holds for the surface X, then it also holds for the surface 7, obtained from X by blowing up a point pe X. This, however, is an easy consequence of Proposition I. 1 in [5]. The next proposition is a special case of the lemma on p. 266 in [3]. Proposition 2. 6. Let the surface X be non-ruled, and c : X—+X a holomorphic involution on X, and a : X—>Y the canonical map from X onto the quotient Y=X/c. Suppose Y to be non-singular. Then if E is any exceptional curve of the first kind on X not contained in the ramification curve, then E does not meet the ramification curve on X. Consequently, Er\c(E) = , and there exists an exceptional curve E' on Y, such that a~x(E')=E\Jt(E). Furthermore, if Xo is obtained from X by blowing down E and c(E), and if Yo is obtained from Y by blowing down E\ then t induces an involution K on XQ, with XJic= Yo.

Let AczY(p) be the union of all the curves, appearing in the basic configuration ([5], p. 18), with the exception of Fpy and let B be the image of A on Y°(p). Then we have: Proposition 2. 7. If E is any exceptional curve of the first kind on Y°(p), then Ep\B consists of at least three different points.

Proof There exists on X = X X j^/SL(2^ oK) — (union of all quotient singularities) a curve, isomorphic to E—E^B (here K=Q(VJ)))> If E f]B would consist of not more than two points, then there would exist on X a curve, isomorphic to either Pi, Cor C*. An unramified covering of this curve would be contained in Xx X. But every map from P,, C or C* into Jt is constant (for C and hence for C* this follows for example from Picard's theorem). Apart from the condition, embodied in Proposition 2. 7, there are other conditions for exceptional curves on Y°(p), which follow easily from Propositions 2. 1 to 2. 6. So for/?>89 one has the following conditions for exceptional curves E on Y\p) :

(a)

En*(E)=t;

(b) if Cis a non-singular rational curve with C2=— 2, —3 or —4, then CE<\ ; (c) E intersects at most two non-singular rational curves with self-intersection — 3, and if E meets two of them, then E does not meet any such curve with selfintersection — 2 ; (d) if a (— 2)-curve C and t(C) belong to the same (— 2)-configuration, then no exceptional curve meets C.

140

G. van der Geer and A. Van de Ven

In some cases, like /> = 89, the minimality of Y°(p) follows already from these simple rules. However, in other cases this is not sufficient. Sometimes (like for/? = 193) a very special argument gives the desired result, but most effective seems to be a reasoning used many times in the cases /? = 229 and/? = 293, which in its simplest form can be described in the following way. Suppose Y°(p) is not minimal. Then, by (a), there exist at least two disjoint exceptional curves on Y°(p). Let Y be the surface obtained from Y°(p) by blowing down these two curves. We havep g (Y) =pg(Y°(/?)). On the (hypothetical) surface Y we construct a particular positive canonical divisor K== J^atAu

a,e Z, a, > 1,

the A^s being irreducible curves on Y (the divisor K is chosen differently in different situations). We then single out one of the curves Ai9 say Ao, to be the "final curve". By intersecting K with the other curves At we obtain inequalities of the following type at > (XiOo + Pi,

aiy & e Q , at> 0 .

Finally, we consider the intersection number and find it to be more than KA0, a contradiction. To illustrate this, we describe a simple example. Let Z b e a configuration formed by three curves : Ao, Ax and A2. The curve Ao is a rational curve with one ordinary double point/*. The curves Ax and A2 are nonsingular rational curves. Both Ax and A2 have only the point/? in common with Ao. Al=—3 (hence KA0=3), A2x=A22=—2, AOAX=AOA2^2, AXA2=\. complex surface X a configuration JC exists, then pg{X)<\.

We claim : if on a

Indeed, suppose pg(X)>2. Then there exists a positive canonical divisor K on X, passing through/?. Such a divisor K contains Ao, Ax and A2, and hence it can be written as K = aoAo+axAx + a2

with tf0? a\-> #2>1) RA0>0, RAx>0,

RA2>0.

Intersection with Ax gives

2ao-2ax+a2+RAx = 0 2ao—2ax + a2 < 0,

and similarly intersecting with A2 gives 2ao+ax-2a2<

0.

Finally, intersection with Ao gives -3flo+2(fl,+fl2) < 3 5a0 < 3,

a contradiction. In order to apply this principle, it is obviously very convenient to know that if a canonical curve passes through a certain point, it automatically passes through

On the Minimality of Certain Hilbert Modular Surfaces

141

one or more other points. In this direction we have Proposition 2. 8. Let X be a compact complex surface, and let c: X—>X be a holomorphic involution on X, such that the quotient Xjc is a non-singular rational surface Y. If Kx is the canonical divisor class on X} then c induces the identity on \KX\.

Proof. Let R be the ramification curve on X. By a well-known result of Hurwitz we have Kx = a*(KY)+R, where a : X-+Y is the canonical projection. From this formula it follows in our case that there cannot be a divisor De\Kx\, containing R, for a*(D—R) would be a divisor in |2A"r|, which is impossible, because Y is rational. Now let s be any section in the canonical bundle of X, different from the zero section. By the preceding remark, the restriction sR of s to R does not vanish identically. The holomorphic function C*(SB)SR1 is equal to a constant c, c^O. The section e*(s)—cs of the canonical bundle vanishes on R and hence vanishes identically on X. So we find c*(s)=cs, and c*(s) and s have the same zero divisor. This proves the proposition. The following remark also will turn out to be quite useful for the cases p=229 By a chain of curves on a complex surface we mean a set of irreducible curves Cu •••, Cn9 with the following intersection behaviour : n n

f *•

**

^ * ~~ I 0

17 ^1 =

*

otherwise.

Proposition 2. 9. Let Bl9 •••, Bn be a chain of'( — 2)-curves on the surface X, and let Al9 •••, Am be irreducible curves on X, all of them different from Bu •••, Bn. Suppose AiBl = AiBn=ai^0for

f = l , ••-, m. If f^atAt + ^bjBj+R i=\

with AtR>0,

BjR>0

is a canon-

j=\

ical divisor on X, then j

for allj=l9

—,n.

Proof. If for this canonical divisor all at vanish, then the statement is trivial. Otherwise, we have bj>l for all j = 1, •••, n. Now suppose bj>k for all j=l9 •••,« m

and lk-\-l for all .7=1, •••, n. Indeed, from

{taiAi+±bjBj+R)Bl = 0 we derive f j a ^ — 2 b l + b 2 < 0, i.e.

bt>

142

G. van der Geer and A. Van de Ven

But this and .7 = 1

implies b2>k+l, etc. Thus the proposition follows by induction with respect to k,

Finally, a word or two about the diagrams. These diagrams don't show always the whole picture, that is, curves may have more points in common than shown in the diagram. For example, in the diagram for p = 229, the curve Fp passes through the intersection point of T3 and T4 (which also lies on So). The intersection properties (and other information) used in the diagrams are generally to be found in [3] and [5]. However, § 3 of [4] is needed to see that on 7°(293) the curves i%3, F3l and F31 are intersected by different curves Tu and that different curves 0t meet Fl7 and F37. From [5], p. 20 it follows that Fp (the ramification curve on Y°(p)) is not a rational curve, and hence not an exceptional curve if/>>89. 3.

The Main Result

Theorem 3. 1. The surface Y°(p) is minimal for 89<317. We shall give the proof for four typical cases, namely p=89, 193, 229 and 293. The remaining cases can be dealt with in a similar way. The case p = 89. In this case, no exceptional curve E can meet St (or S't) for i=l, 2, 3, 5, 7, 8, 9 and 10, because of principle (d). Also, no E can meet S4 because of (b) and—after the blowing down of E and c(E)—principle (d) again. It follows from these remarks and from Proposition 2. 7 that any E must have at least three points in common with So U S6 U S'6 U 0 ,

where 0 = 0, (J 02\j03\j04. Now ESO<1, for ES0=c{E)S0, and if ESo were>2, then after blowing down E and c(E) we would arrive at a singular curve So with KS0 <0, which is impossible by Proposition 2. 5. Similarly, ES6<1 (because of (b)) and EO<\ (also because of(b)). Furthermore, ES6=ES'6=l is impossible, for these relations would imply c(E)S6=c(E)S'6=l, and after blowing down E and c(E) we would obtain a ( — 2)-configuration, that cannot exist on a surface of general type. There remains only one possibility to exclude : ES0=ES6=EO=l. But in this case, if say E0{=\, and hence c(E)O2=l, after blowing down Ey c(E), 0x and 02, we would again obtain a singular curve So with KS0<0, a situation which was already excluded before.

On the Minimality of Certain Hilbert Modular Surfaces

143

The case p=193. In this case a very special argument can be used. In fact, we know that Y= yo(193)/* is a blown-up ^3-surface with c?(7) = —18 ([3]). But on this surface we can blow down eighteen curves, namely /),*, Z)2*, C\*, •••, C4*, 52*3, •••, S?8, Fg, F7*, Ff, 52*, F£ and S? (as is customary, we set a*(A)=A*), thus obtaining a minimal surface. Therefore, the image on Y of an exceptional curve on 7°(193), which is again an exceptional curve on Y by principle (a), has to intersect the union of the eighteen curves mentioned before. Using the fact that Y is not rational, this is easily seen to be impossible.0 The case p=229. It is sufficient to prove the following five statements : I Any exceptional curve E on Y°(p) meets at most one of the curves 0l3 02, and also at most one of the curves 71,, T2. If E meets any of these curves, then the intersection number is 1. II The case EOX=ETY=\ cannot occur. III The case £"0, = l, ET{=ET2=0 cannot occur. IV The case ET{=1, EO{=EO2=0 cannot occur. V The case ETl=ET2=EOl = EO2=0 cannot occur. After some preliminary remarks, we shall prove these five statements one by one. Because of the principles (a)-(d) there is no exceptional curve E intersecting T3, T4 or St for z = l , •••, 1y 11. But also Sl0, Sl2, Sl3 and SH cannot meet a curve E. For if E would meet Sl0, then 2?iS10=l (principle (b)) and after blowing down the curves Sl09 S'l0, Sn, S'n we would obtain a non-rational curve, namely Fl5, with KFl5<0, which is impossible. And if E intersects Sl2 (without meeting Sl0, we may assume), then after blowing down Sl2, Sn, Sl0 we would have £13 > 0 for the nonsingular rational curve «S13, which is impossible since Y°(p) is not rational. Also ESH = 1, hence c(E)S'H=l would give KFn<0 for the non-rational curve Fn. Finally, £^3=1 would after blowing down lead to a non-negative definite (— 2)-configuration, namely {-^95 ^135 ^ 1 3 ) ^14? ^145 ^12} •

The ( — 2)-curves 0 3 and 04, intersected by the ( — 2)-curve F5 cannot meet an exceptional curve since otherwise intersecting exceptional curves would be obtained by blowing down. In this way we find that an exceptional curve E has at least three points in common with *S0 U ^s U •S's U ^9 U ^9 U #iU 02U T{\J T2. For such a curve £ we have in view of principle (b) that CE<\ if C=S9i S'9, 0,, 02, Tl9 T2, and CE<2 if C= S6, S'Q. If E meets 0{ or 02 then ES8< 1, and if E meets both a curve 0t and a curve Tt, then E does not meet £8, S'B, S9, S'9 by principle (b). 1) Hirzebruch has informed us that he is able to find minimal models of Y0(p)/c for all primes p, for which Y°(p)lc is neither rational, nor of general type. Therefore, for all these primes, we can prove the minimality of Y°(p) by the same argument.

144

G. van der Geer and A. Van de Ven

We now come to the proof of the statements I-V. Proof of I. This statement is an immediate consequence of our principles (a)— (d). Proof ofII. Suppose EOl=ETl=l. By the preliminary remarks we have ESa= ES'8=ES9=ES'9=0, hence ESO>1. After blowing down we find KS0<2 and T:{SO) > 3 , hence Sl>2. Now since pg — 9, we can find a canonical divisor of the form aS0 +/?, a>l, SJi>l, hence KS0=(aSQ-\-R)S0>3i a contradiction. Proof of III. We know from the remark above that E intersects 50lj58U^8U^9 \jS9 in at least two points. We claim furthermore : if E meets any of these five curves, then E intersects that curve transversally in one point. For S8, S8, S9, S'9 this follows already from the preliminary remarks ; as for So, this can be seen by the sa*me argument as was used in case II. Now let E be an exceptional curve with EOl=ES0=ES8=l. By blowing down four times we obtain KS0=6, KSB=1, SQS8 >2. Since pg=9, there is a positive canonical divisor aS0+bSB+bS'B+R9

with a, b>l, RS0>0, RS8>0, RS'8>0, which contains at least once the curves £„ £'i? &» £9? ^145 S'u, ^9 a n d B2. Intersection with S8 gives b>4a ; then intersection with So and application of Proposition 2. 9 gives 7
ES9=l ES'8=l ESf9^l

Now the first case being very similar to the one just treated, and the third one being very easy, we shall restrict ourselves to the cases EOl = ES8^ES8=l and EOl—ES8=ES9=l. In the first of these cases we have also c(E)S8=c(E)S'8=l, hence after blowing down E, 01? c(E) and 02 we would obtain intersecting exceptional curves, which is impossible. In the case EOl=ES8=ES9=l, after blowing down four times, we can find as usual a canonical divisor aS8+bS9+R with a, b>l, RS8>1, RS9>1. Intersection with S8 and S'8 yields

— 3a+pb+RS8= 1 fia-2b+RS9 = 0 with 0 > 3 (p-3)a+(p-2)b+RS8+RS9= 1. This gives a contradiction, for RS8>1 and RS9>1. Proof of IV. Because of the preliminary remarks it will be sufficient to exclude the following possibilities : 1) E meets So (at least) twice 2) E meets S8 twice 3) E meets So and S8 (both once) 4) E meets 50 and S9 (both once)

On the Minimality of Certain Hilbert Modular Surfaces

5) 6) 7) 8)

E E E E

145

meets S8 and s meets and 9 meets Ss and S'a meets s and S'9, 9

In case 1), after blowing down E and e(E), we get ^S 0 <6 and Sl>— 2. We consider the system of canonical divisors passing through three singular points and one simple point of So. This system has dimension at least 4, and all its divisors contain So, Sl9 S\, Tu T2, T3 and T4. Since (K-So-Si-S'^ T- T2- T3- T<)S0 < 2, there are canonical divisors containing So at least twice. Thus a contradiction arises from (aS0+bSl+bS[+clTl+c2T2+c3T3+c,T4+R)S0 > 6a = 6. In case 2) we can do a similar thing, but now using S8. The case 3) is slightly more difficult. After blowing down E and c(E), we find ^6*0=8, SI— —8. We can find a canonical divisor containing general points of Tl9 T3, Sl9 B2, SH, F9, Fl5 and F25. This divisor can be written as aS0+bSa+bS'B+cSl3+cS'l3+dF9+R with a, b, c, d> 1, RS0>0, •••, RF9>0, and R containing all other curves St, S't, B2, 7",, ••-, T4, Fl5 and F25. The curves Sl0, Su and Sl2 are contained in R at least twice. From KF9<0 we derive 2a—2rf+2<0, i.e. d>a-\-\. Then, intersection of D with Sn gives us 3^>rf+2, hence c> -^ a+1. Intersecting Z) again with F9 wefind—2d ^-a+2<0,

that is, d>^-a+l

Next, X5 8 =2 yields us b>-ja+-j.

All

this information, together with Proposition 2. 9 and the fact that Fl5 is contained in R, gives a contradiction if we intersect D with So. For case 4) the argument is very similar. In case 5) we take after blowing down, a canonical divisor aSB+bS9+R, <2> 1, b>l, RS8>0, RS9>0, and find by successive intersection with S8 and S9 that a>b and that — 3A + 2J + -o-(fl + i) + -n"A+l
146

G. van der Geer and A. Van de Ven

type. Proof of V. We begin by observing that an exceptional curve E can meet So at most once. Otherwise, after blowing down we would obtain KS0<6, 7r(S0)>3, hence S20>—2. Now there is a positive canonical divisor D=aS0-\-R, containing So at least twice and F15 at least once. D contains S{ and S\ both at least a-\-\ times, and T3, T4 and F9 at least a times each. Thus we find (aS0+R)S0 < 6

-2a+2a+2+2a+a 3a+2 < 6,

<6

contradicting a>2. A similar but simpler argument shows that a curve E intersects also S8 at most once. So we are left with the following possibilities : 1) ES0 = ES8 = ES9= 1 2) ES0 = ES8 = ES'9=l 3) ES0 = ES8 = ES'8=\ 4) ES0 = ES9 = ES'9=l 5) ES8 - ES'8 = ES9=l 6) ESB = ES9 = ES'9= 1. In the cases 1), 5) and 6) we take a canonical divisor a(SB+S'B)+b(S9+S'9)+R9 with a, b>\, RS8, •••, RS'9>0, which contains £I3, F n , F25 and B2. Intersection with 5*9 gives

+

^

+ < 1,

b>a.

Then intersection with S8 yields ^ ^

\ < 2, i.e. «<0,

a contradiction. For case 2) we can use the same type of canonical divisor, but this time we intersect it with S9 and S8 to obtain a contradiction. Finally, in the cases 3) and 4) we use divisors aS0+b(SB+S'B) +R and aS0+b(S9+S'9) +R respectively, to obtain our last contradiction. The case p=293. In 1) 2) 3) 4)

view of the principles (a), •>, (d) it is sufficient to exclude the following cases : ES0>3 ES0 = 2, EO = 1, where 0 is one of the curves 0,, •••, 0 8 ES0 = 2, ET = 1, where T is one of the curves 71,, •••, T8 £5 0 = 1, EO = ET = 1, T and 0 as in 2) or 3)

On the Minimality of Certain Hilbert Modular Surfaces

147

5) ES0 = 1, ET = ET' = 1, where T and 7" are among Tl9 T39 T5 and T7, T^ T'. For the cases 1) and 2) it is sufficient to intersect (after blowing down E and c{E)) the curve So with a canonical divisor of the type aS0+R, a>l, RS0>0, to obtain a contradiction. To exclude case 3), we blow down E and e(E), obtaining KS0<8, S20>— 4, and consider a canonical divisor of the type aS0+bFl7-\-cF37-{-R, a, b, c>l, RS0>0, RFl7>0, RF37>0. This divisor contains T, e(T), T9 and TIO all at least a times, and Sx and S[ at least a + b+c times (Proposition 2. 9). Hence we would find :

-Aa+2a+Aa+2b + 2a+2b + 2c < 8, again a contradiction. In case 4), after blowing down six times, a canonical divisor of the type aS0-{bF37+R leads to the usual contradiction. The last case, case 5), requires a little bit more care. After blowing down 2? and c(E) we have KS0=—Sl=l0. The curve S0 intersects three ( — 2)-crosses, namely the pair T, T' ; the pair c(T), c{T') and the pair T9, Tl0. We claim that it is possible to find a canonical divisor D of the form with #,-•-, A> 1, and RS0, •••, Rc(T')>0. In fact, the divisors in the at least 4-dimensional linear system of canonical curves, which pass through the intersection point of T and T\ through four general points of So and through two general points of Fl7, all contain So and Fl7 at least once and £„ S\ at least twice (Proposition 2. 9). The linear subsystem, passing through two general points of .F37 contains F37, and hence S2 and S2 at least three times. So there is at least one canonical divisor in this subsystem passing through two general points of F3l9 hence containing F3l, and this is a divisor of the type mentioned above. Application of Proposition 2. 9 gives/^4, and subsequent intersection with £,, S2, Fl7, Sx and S2 gives £>4, f>5, b>2, e>5, / > 6 . Since F3l, F37 and i%3 intersect different T/s, we may assume that T and e(T) meet one of the curves F3l, F37 or ^43. If T and c(T) meet F3l or JF37, then g>2, h>2, and intersection with So yields a>2. If T and c(T) meet JF43, then intersection with F43 shows that D contains i%3, and as before we conclude that a>2. Proposition 2. 9 implies/>« + 6 + rf+l ; intersection with Sx and ^2 now gives e>a-\-b-\-d-\-l. Intersection of D with F37 and Fl7 leads to

-6d+d+2(a + b + d+l)+2a < 8, — + —i and

148

G. van der Geer and A. Van de Ven

Finally, intersection with So yields ft ft


Some Comments

Once it is known that Y°(p) is a minimal surface, it is of interest to study the pluricanonical maps of the surface, in particular the canonical and bicanonical maps. As an example we sketch the situation for p=89. By a theorem of Bombieri ([1], Theorem 2) the bicanonical system on Y°(p) has no base points. Therefore, \2K\ maps Y°(p) either birationally onto a surface of degree 8 in P5, or generically 2 to 1 onto a surface of degree 4 in P5 (of course not on a surface of degree 1 or 2, for these surfaces are always contained in a proper linear subspace of P5). Now pg(Y°(89)) —3, so in the first case there would be a linear system of dimension 2 of curves of degree 4 on X. But such curves are always rational or elliptic, and Y°(p) would not be of general type. Hence Y°(p) is mapped by \2K\ generically 2 to 1 onto a surface of degree 4 in P5. This surface contains a linear system of dimension 2 of conies, and it is not contained in a hyperplane of P5. But then, by a very classical theorem, it has to be the Veronese surface, which is biregularly equivalent to P2- Therefore, \K\ gives an everywhere denned map from Y°(p) onto P2, generically 2 to 1, and with "exceptional fibres", which are trees of (— 2)-curves. All this also follows from the results of Horikawa ([7]) on minimal surfaces of general type for which pg = -jK2+2

[K2 even) or pg = ^-K2+-j(K2

odd). These

surfaces are in a sense extreme cases, for one has for minimal surfaces of general type always the ^equalities

pg<\K2+2

(X

Also for higher values ofp it is possible to obtain much information on the canonical and bicanonical maps, and in particular on their relations with the involution i on Y°(p). The first of the present authors intends to treat these questions in a forthcoming publication. As was observed already in chapter 2, what we really do in this paper is to prove theorems of the following type : If on a compact complex surface X there exists a certain The methods used here can also be configuration £ of curves, then Pi(X)
used to obtain many more examples, some of them quite pretty, but the authors have to admit that they don't even have an idea which (if any) theorem of this type would imply the minimality of Y°(p) in the general case.

On the Minimality of Certain Hilbert Modular Surfaces

-2

03

CO

04

CO

o2

1 1

149

-3

£><

-3

The configuration The configuration on Y°(89). K2 = 2 ; pg = 3 ; £ S 0 = 4, ^(S0) = l ; S ? = - 2 , « = 1,2,3,5,7,8,9,10.

S42 =

- 1 ; KF3l<8,

T2

on 7°(293). K2=A0;Pg=l2;

TT{F31) =2 ; i^F 37 < 8 , ^ ( F 3 7 ) = 2 ; X F 4 3

-3,

Si

9

-2 -2

Oi - 2

Tx

-3

-2

T2

-3

S7

/ \ \ / / \ \ / / \

-2 -2

The configuration on 7°(229). K2 =28 ;

pg=9; 7,10,11,

2Di

D2

D5

O O O O Cr2

jr ( Tp \

—. ( zp \

7r(F 15 )=2 ;

KS0

A

• If IT

O

( TT \

KF25<4,TT(F25)=Q.

1

V TT

150

G. van der Geer and A. Van de Ven

4-fold contact

The configuration on Y°(l93)A

References [ 1 ] Bombieri, E. : Canonical models of surfaces of general type, Publ. Math. IHES, 42 (1973), 171-219. [ 2 ] Hirzebruch, F. : The Hilbert modular group and some algebraic surfaces. Intern. Symp. Number Theory, Moscow 1971. [ 3 ] Hirzebruch, F. : Hilbert modular surfaces. Enseignement Math. 19 (1973), 183-281. [ 4 ] Hirzebruch, F. : Kurven auf Hilbertschen Modulflachen und Klassenzahlrelationen. In : Classification of algebraic Varieties and Compact Complex Manifolds. Lecture Notes 412, Springer Berlin-Heidelberg- New York (1974). [ 5 ] Hirzebruch, F. and Van de Ven, A. : Hilbert modular surfaces and the classification of algebraic surfaces. Inventiones Math. 23 (1974), 1-29. [ 6 ] Hirzebruch, F. and Zagier, D. : Classification of Hilbert modular surfaces. In this Volume. [ 7 ] Horikawa, E. : Algebraic surfaces of general type with small c\ I and II. Annals of Math. 104(1976), 357-387, and Inventiones Math. 37(1976), 121-155.

University of Leiden (Received December 15, 1975)

Part II

Complex Structures on S2p+1XS2q+1 with Algebraic Codimension 1

K. Akao1 0.

Introduction

In the present paper we study certain complex structures of a compact complex manifold X of dimension n which is homeomorphic to the product of two odddimensional spheres S2p+l X S2q+l withjfr + ^ > 0 . Since the second Betti number of X vanishes, the transcendence degree over C of the field of all meromorphic functions on X does not exceed n—l. In the following we restrict ourselves to the case where X has exactly (n—l) algebraically independent meromorphic functions. A so-called Hopf manifold is an example of such a manifold with/> = 0. E. Brieskorn and A. Van de Ven [2] have constructed a somewhat different kind of complex structure on SlxS2p+l which also has p algebraically independent meromorphic functions. A complex structure on S2p+l X S2q+l with/>>l and q>l was first constructed by E. Calabi and B. Eckmann [3]. It also satisfies the above condition. (See § 2 below.) Recently Ma. Kato [8] [9] has studied complex structures on Sl X S5 with algebraic dimension 2 which satisfy some additional conditions. Our results are generalizations of his to higher dimensional cases. Now we summarize our main results. First in § 1 we study the structure of a compact complex manifold X of dimension n such that ^(X)^{1} or Z, b2(X)=0 and such that a(Z) = n—1. For such an X, we have the following T h e o r e m 1. There exists a finite unramified covering X of X such that X is subject to a holomorphic semi-free action of a 1 -dimensional complex torus E and such that the meromorphic function field of the quotient space XjE is naturally isomorphic to that of X.

This implies in particular that the Euler number of X vanishes if X admits such a complex structure. Now in § 3 we assume that X is homeomorphic to S2p+l X S2q+l (p-^-q >0), and that Jif has a flat fibration of elliptic curves over a non-singular projective variety W. Then we have T h e o r e m s 2 a n d 3. There exists a finite holomorphic map (pfrom X onto a submanifold 1) Partially supported by the Fujukai Foundation. 2) For notation, see below.

154

K. Akao

of a Calabi-Eckmann manifold V (resp. a Hopf manifold if p=0 or q=0) such that the torus action on X constructed in Theorem 1 above is induced from the standard one on V. X is an abelian branched covering of
As corollaries, we compute the values of some numerical invariants of X and IV. Some of the results of this paper were announced in [1]. The author would like to express his hearty thanks to Prof. K. Kodaira, Prof. S. Iitaka and Dr. K. Ueno for their valuable advice. Notation. Now we fix some notation used throughout this paper. For a topological space M: 7Ti(M) = the fundamental group of M, bt(M) = the i-th Betti number of M, X(M) = 2(—l)^(Af) = the Euler number of M. For a complex manifold X: M[X) = the field of all meromorphic functions on X, a(Z) = trans, deg. CM(X) = the algebraic dimension of X, Qx = the structure sheaf of X, 0x(L) = the sheaf of germs of holomorphic cross-sections of a line bundle L on X, c(L) = the first Chern class of a line bundle L on X, h»-P{X) = dim HP{X, 0x), q{X) = H»(X) = dim H'(X, Ox), Aut(X) = the group of all biholomorphic automorphisms of X, Aut°(JST) = the component of the identity of Aut (X), Qx = the constant sheaf on X with Q as its stalk. For a subvariety Y of X: codim x (F) = the codimension of Y in X.

1.

Existence Theorem for a Holomorphic Torus Action

Let X be a compact complex manifold of dimension n. In this section we assume that X satisfies the following conditions : (1) a(X) = « - l , (2) it\(X) is a finitely generated abelian group whose free rank does not exceed one, (3) i2(J?) = 0 for any finite unramified covering X of X. Note that under the condition (3), the following holds: (4) For any irreducible effective divisor D and any compact analytic curve C in Z, D n C =£ 0 implies that D D C In fact, let C—*C be the normalization of C, and xr: C—+X the composite of n and the inclusion Cd+X. Assume that D"$>C. Then, for the line bundle [D] on X corresponding to D, we have <:1(^*([i)]))(^)=^*(c,([i)]))(C; )>0. But, sinceH2(X, Z)

Complex Structures on S2P+l xS2(i+1 with Algebraic Codimension 1

is a torsion group, 7r*£i([Z)])=O, which is a contradiction,

155

q. e. d.

Under these assumptions, we have the following theorem. T h e o r e m 1. Assume that X satisfies (1), (2) and (3) above. Then there exist a finite unramified covering X of X and a holomorphic action of a I-dimensional complex torus E on X, satisfying the following conditions : (i)° the stabilizer subgroup of E at each point of X is finite, (ii) the quotient map -K : X—>X/E = W induces an isomorphism n* : M(X)~M(W)> and W is a Moiiezon space, and (iii) W has only quotient singularities arising from action by finite abelian groups. Moreover such a torus action is unique. Proof First we quote a lemma due to S. Iitaka. Let Fand S be compact complex manifolds of dimension n and (n— 1) respectively, and 7t: V—->S be a surjective holomorphic map whose general fibre is isomorphic to a fixed elliptic curve E. E can be represented in the form E= C\L where L is a lattice in C generated by 1 and o) with Im ea>0. We denote by C the standard coordinate on C, and by [C] the one on E induced by it. Let D be an irreducible subvariety of S with codim 5 D=l 9 such that for a general point/? of Z), x~l{p) is a (possibly multiple) non-singular elliptic curve. L e m m a 1. Under these assumptions, there exist a positive integer m, an open polydisc neighbourhood U of p in S, an m-fold branched covering

U, and local coordinate systems (z)z=(Zi, •••., Zn-\) on U and (w) = (wl, >-, wn_{) on U respectively such that {(z) e U\Zl = 0}, a) DnU= b) U, where G is the cyclic group generated by g : (wl9 ••-, w n . l 9 [ C ] ) - > ( e ^ i , •••, ^ « - i , [ C + l / m ] ) , and p ( w l 9 •••, w n . l 9 [ C ] ) = (p{wu •••, wn_x) with e a primitive m-th root of unity. For a proof of this lemma, see S. Iitaka [7] or Ma. Kato [9]; there the lemma is proved for n=3, but the generalization to any n is straightforward. Now we return to the proof of Theorem 1. Obviously we can assume without loss of generality that n\{X) is a free abelian group, that is, nx(X)~ {1} or Z. Let V be a non-singular protective model of M(X), and (p : X-+V the corresponding meromorphic map from Xto V. By the assumption (1), dim V=n—\. By Hironaka, there exist a compact complex manifold X* of dimension n and surjective holomorphic maps XJ : X*^>X a n d / : X*—•Fsuch that XX is an isomorphism outside of an analytic subset S of X with codimx S > 2 , and such that the following diagram commutes :

1) We call the action of E semi-free if it satisfies the condition (i).

156

K. Akao

Zf*

f*

Note that, via Zf and/, we get the isomorphisms of fields : M(X) c$M(X*) & By Kawai [10] and Hironaka [4], the general fibres of/ are elliptic curves, namely, X* is an elliptic fibre space over F. By changing the models Fand X* if necessary, we may further assume that for each degenerate divisor 7 of X*, codim x t^(F) is greater than one. Here, by a degenerate divisor 7, we mean a subvariety 7ofX* with codim x *7=l such that c o d i m F / ( 7 ) is greater than one. Let DcV be a subvariety of pure codimension 1 such that, for a general point p of D,f~l(p) is not a regular fibre. Assume that f~l(p) is not a multiple elliptic fibre (which means that f~l(p) is not of type m/0 in the notation of K. Kodaira [11]). Then the singular loci of/"1 (2)) "have an irreducible component 2 such that f\s : 2*—>Z) is a surjective, generically finite map. Note also that a.(2)=n—2 = dim I. But then, since Fis supposed to be projective algebraic, f~l(H) f| -£=£0 for any generic hyperplane section of F. Hence applying the above property (4) to each compact analytic curve contained in 21, we have t^(f~l(H))z)^(I) which again shows that, for l l general/>€ Z>, ^{f' (p))dZf(f- (H)). Therefore coding zf{f~l(D))>2. Let Dt (l
U j

Then n is a surjective holomorphic map, each fibre of which is a (possibly multiple) smooth elliptic curve. Note that coding (X—X') >2, hence also nx(X')^n{{X). Let j : V—+C be the functional invariant of this elliptic fibre space (see [10] and [11]). Since -K oj : X'-+C is extendable to the whole of X, j is a constant map, which implies that every regular fibre of n is isomorphic to a fixed elliptic curve E. Then by Lemma 1, n : X'—»F' is a Seifert fibre space in the sense of Holmann [6]. Since j is a constant map, each local monodromy around Dt is finite. Therefore, by taking a suitable finite unramified covering of X, we may assume that every local monodromy is trivial. But then we can extend the flat sheaf Rln'*Qx> on V to the flat sheaf J on the whole V. Since Fis projective and since T:I(X)—>^i(F)—»{1} is exact, TTI(F) is at most a finite group. Again, by taking a suitable finite unramified covering if necessary, we can assume that J and Rl7i'*Qx' a r e constant sheaves. Then the Seifert fibre space n : X'—>V is reduced to a principal Seifert fibre space with Aut°(£')=£1 as its structure group. Hence X' is subject to a holomorphic action of E such that the stabilizer subgroup at each point of X' is finite, and that X'jEc^V. Since codimx (X—X') is greater than one, this action can be extended to the whole of X. This clearly satisfies (i) and (ii), because W=XIE and F are birationally equivalent. In fact, since codimx (X—X')>2, every meromorphic

Complex Structures on S2P+i X £ 2 ? + 1 with Algebraic Codimension 1

157

function on V is extended to an ^-invariant meromorphic function on the whole of X By [5], the action of E has a holomorphic slice at each point of X, which proves (iii). The uniqueness of this torus action is trivial, q. e. d. Corollary 1. Under the same assumptions as in Theorem 1} we obtain %(X)=0.

Proof Obvious. This shows, for example, that S6 admits no complex structure with algebraic dimension 2. Corollary 2. Under the same assumptions as in Theorem 1, we have the following values of numerical invariants of X: X and W: q(X)=q{X) = l9 bl(W)=q(W)=0, and b2(W) = 2 — r a n k ^ (X). Proof Let ft be the natural projection from X to W=XIE. Then, from the above construction, we have *)w )

if q = 0 and q = 1, otherwise,

and Qw (**)

Qw®Qw 0

if q = 0 and q = 2, if?=l, otherwise.

Since W has only quotient singularities, and since every quotient singularity is rational, the non-singular model W of W has the same fundamental group and irregularity as those of W. But nx{W) is a quotient group of 7Ti{X), Since W is a Moisezon manifold, bx(W) is even, so hence bl(W)=bl(W)<\. bl{W)=bl{W)=Q9 which also proves that q(W)=q{ P ^ ) = 0 . Consider the Leray spectral sequence for Qx : E^q = Hp(W, Rqft^Ox)^>Hp+q{X, Ox). q(W)=0 implies El2-°=0. Since dim c ii!j J = l, q(X)<\. If q(X)=0, then from the exact sequence

we get the inclusion Hl(X, 0*)d > i/ 2 (J?, Z). Hence, by the above assumption (3), every line bundle on X is of finite order, which contradicts the assumption (1). Therefore q(X) = 1. On the other hand, we have dim c // 1 (X, 0x) =dim(i/ 1 (X, 0x)% where G is the covering transformation group of X over X. Hence q(X) < 1. But, by the same argument as above, we get q(X)>0, so q(X)=q(X) = l. For proving the last equality, we use the Leray spectral sequence for Qx : 'E?>q=Hp(W, Rqfc*Qx) ^>Hp+q{X,

Q). F r o m (**), dimQ'E°2>\ = 2, a n d dimQfE}°=b2(W).

L e t d2' :

f

E^-^'E22'°

be the corresponding differential map. Note that d i m Q ^ ' ^ d i r n Q ' ^ ^ r a n k 7Ti{X), and that 'Elt0 = fE2Ji=0. This means that dimQ(ker rf2')=rank TTI(X). Then q. e. d. 1 (AT))=O, thus A 2 (M^)=2-rank ^(JT), Corollary 3. Suppose that dim X=3,

and that X satisfies the conditions (2) and (3)

158

K. Akao

above. If X admits a holomorphic action of a 1-dimensional complex torus E satisfying the

above (i), then a(JQ=2. Proof Let S be the quotient surface X/E, and Zf : S—»S the resolution of singularities. By the same argument as in Corollary 2, we obtain q(S)=q(S)=0, which implies bl(S)=bl(S)=0 by [12]. Hence again by the same spectral sequence argument, we have b2(S)<2. Let j^(l2(S)>0. By a theorem of Y. Miyaoka [13], bl(S)=q(S)=0 implies that S is either a Kahler surface or a K3 surface. In either case we get b+(S)>l+2h°'2(S)>3, which is a contradiction. So h°'2(S)=0, and S is a Kahler surface ([13]). Hence, S is actually a Hodge manifold. Therefore 2L(X)>2L(S)=2. Since Xis not a Moisezon space, a(X)<2, so we have proved the assertion. Corollary 4. Assume further that X is homeomorphic to S2p+1 X S2q+1 where p-\-q >0. Then we get H*(W9Q) =H*(ppxP\Q). Proof Put n=dimcX=p+q+l. Note that blr(X)=0 holds except for r = 0 and r—n. First we shall show that b2r+l(W) =0 for all re Z. Consider the Leray spectral sequence for Q* ; E^q=Hp(W, Rq7t*Qx)=$Hp+q(X, Q) where it is the quotient map from X to W=X\E. We get then Qw if q — 0 and q = 2, 0 otherwise. For an odd s, we get the following semi-exact sequence ft

v ps-2,2

. IPs,I

. fs+2,0

v 0

where a and /3 are differentials of this spectral sequence. By (*), E*s'l=E^1, which is a zero module in view of the vanishing of beyen(3t). Hence this sequence is exact at E°2'\ which means that dimQEs2l2n-l. From this, we get b,(W)< bs-i{W) for s odd, using repeatedly the above inequality. By Corollary 2, b{{W) = 0 , so bs(W)=0 for all odd s. Next, consider the sequence n v J72r-2,2 v J?2r,\ . J72r + 2,0 . C\ \)^>£L2 —> £L2 —> £L2 —> U, r+ltp

where a and /3' are differentials. Since El = 0 for all integers r and/?, we get Pq pq E ' =E ' for all/? and q. Hence a is injective except for r=n and /3' is surjective 1) By [Cix] we denote the line bundle on S corresponding to the divisor Cix.

Complex Structures on S2P+1 X S2Q+1 with Algebraic Codimension 1

159

because of the vanishing of beyen(S). Therefore we obtain the equalities 2b2r{W)=b2r+2(W)+b2r_2{W)+b2r+x(X)

for

r*n,

and bo{W) = b2n_2(W) = 1.

From this equality we easily get the assertion. 2.

q. e. d.

The Construction of Calabi-Eckmann Manifolds

In this section we state a construction of Calabi-Eckmann manifolds following J. Tits [14]. (See also [2].) Let z={Zi, ••-, ZP) and w=(wx, •••, wq) be standard coordinates on Cp and Cq respectively. For each te C, let g\ be the biholomorphic automorphism of (Cp—(0))x {Cq—{0)) defined by g\{Zx,

•••, Z P , w l 9 •••, wq) = (elzx,

•••, e l z P , e u w l 9

•-, e u w q ) ,

where X is a fixed complex number with Im X^O. Let Gx be the one-parameter complex Lie group consisting of all g\ for teC.Gx operates freely and properly on (Cp— (0)) x (Cq— (0)). Hence, by Holmann [5], we can construct the quotient manifold M=(Cp—(0)) X (Cq—(0))/Gx, which is a compact complex manifold of dimension (p+q— 1). Mis homeomorphic to S2p~l X S2q~\ and its complex structure coincides with the one constructed by E. Calabi and B. Eckmann in [3]. We call M a Calabi-Eckmann manifold. Note that M is a complex analytic principal fibre bundle over Pp~x xPq~l, the fibre of which is a non-singular elliptic curve E with 1 and X as periods. Hence a(Af)=dim M—\. Moreover M admits a unique holomorphic free action of E. Remark. The Calabi-Eckmann manifolds are the simplest example of so-called non-Kahler C-manifolds, namely, simply connected compact complex homogeneous manifolds ([14]). 3.

Complex Structures on S 2 "+ ] X S7q+}

In this section we shall study a complex manifold X of dimension n which is homeomorphic to S2p+l X S2q+l, which satisfies the following condition : There exist an (w—1)-dimensional smooth projective algebraic variety W, and a surjective flat holomorphic map n from X to W with connected fibres. If X satisfies this assumption, the conditions (1), (2) and (3) in § 1 are also satisfied. Hence there exists a finite unramified covering X of X such that X admits a holomorphic semi-free action of a 1-dimensional complex torus E. In particular every fibre of n is a non-singular elliptic curve, and a general fibre is isomorphic to a fixed elliptic curve isogenous to E. Note also that if p>\ and <7>1, 3£ coincides with X. Then we have the following theorems.

160

K. Akao

T h e o r e m 2. Suppose that X satisfies the above conditions, and that p>\ and q>\. Then there exists a finite holomorphic map from X onto a submanifold X' of a Calabi-Eckmann manifold V such that the following diagram commutes : X

>X'

>V

I W\X'

W ^

W

>


PNPN

Here,

W has only simple normal crossings for singularities. Furthermore X is an abelian branched covering of X'', and admits a complex torus action compatible with that of V. T h e o r e m 3. Suppose p = 0. Then, under the same assumptions as in Theorem 2, there exists a finite holomorphic map from X onto a submanifold X' of a Hopf manifold V such that the following diagram commutes: X

>X' d * V

71

i

1 ^x' 1 9

W ^

W d*

PN.

In this case, W has the same rational homology group as that of Pq, and n has the same property as in Theorem 2.

Before proving Theorem 2, we need several lemmata. In the first place, by Corollary 2 in § 1, we have q{X) = l, q(W)=bl(W)=0 and b2{W)=2 because ^(

Lemma 2. Hl(X, 0) ~ Hl(X, G*) ~ C Proof Sincep> 1 and q> 1, Hl(X, Z) =H2{X, Z) - 0 . Therefore, the conclusion follows from the exact sequence : ... -+H\X,

Z) ^H\X,

0) -+H*(X, 0*) - > / / 2 ( Z , Z) -> ••• q. e. d.

L e m m a 3. There exist two very ample line bundles Lx and L2 on W such that c(LY) and c(L2) are linearly independent in H2(X, Z), and such that K*(LX) and TT*{L2) generate a lattice in Hl(X, 6*) = C. Proof Since b2(W)=2, and Wis non-singular projective, h°'2(W)=0. Then the vanishing of q(W) implies t h a t H l ( W , 0*)~H2(W, Z). Therefore, using again the fact that W is projective, we can find two very ample line bundles Lx and L2 such that c(Lx) and c(L2) generate H2(W, Q)^H2{W, Z)®ZQ. It is clear that n*LY and n*L2 are linearly independent over Q. We must show that they are independent over R. Let p be a generic point of W and j : n~l{p)CL¥X be the natural inclusion. 7t~l(p) is a smooth elliptic curve. Then we have the following commutative dia-

Complex Structures on S2P+l X S2«+1 with Algebraic Codimension 1

161

gram:

0 - > # ' ( * - ( * ) , Z)^Ht(*-*(p), 0,-Hpy)^H>(x-<(p), G;-Hp))^H>(n-i(p), Z), where both rows are exact. Note that j? : Hl(X, Ox)^>Hx{n-x{p), 0n-iiP)) is an isomorphism, as can be proved by using the Leray spectral sequence in § 1. Since clearly j*tf*A and j*7z*L2 are trivial line bundles, j?a~ln*Lx and j?a-l7z*L2 lie in the image of i*. Since they are linearly independent over JJ, they generate a sub-lattice of i*Hl(n~l(p), Z). But, since n~l(p) is a non-singular elliptic curve, i*Hl(7t-l(p),Z) is a lattice in Hl(ic'l(p)9 0^(P))^Hl{X, Ox)=Hl{X, Of), hence TT*Z,J and TT*Z,2 generate a lattice in Hl(X,

0jjf).

q- e. d.

Proof of Theorem 2. First we choose coordinate coverings {£/>} of X and {M^} of M^such that ic(U$) = Wt. Let laAj be the transition functions of the line bundle La ( a = l , 2) with respect to {W^\ . Then, by Lemma 2 and Lemma 3, we can find non-vanishing holomorphic functions pi and qi on each U% a holomorphic function Xfj on each U^f] U) (=£0), and a complex number £ with Im c^O such that the following equations are satisfied :

where we put / a ,iy=l for e=/. Since La is very ample, we can find a basis {£$} (A=0, 1, -.., -AT) of i/°( W, O(La)) which gives an imbedding j a of W7 into PN% where iVa=dimci/°(M/r, 0(La)) — 1 ( a = l , 2). Then by the above equations, we can construct a holomorphic map from X into F=(C A l + 1 —(0)) X (CN2+l — (0))/G, which qi**Z$l> ~; qi**W)]. Here G= {£} is the maps *€ £/? to [(^7r*CK, - , Pfr*^\ one parameter group defined in § 2, and for z € CNl+l — (0) X C^2+1 — (0), [>] denotes the image of £ in V. The manifold Fis nothing but a Calabi-Eckmann manifold and we get the following commutative diagram: X

>V

i

On the other hand, every positive-dimensional closed subvariety of V is of the form (p~l{Y) for a closed subvariety 7of PN* X PNi ([3]). Therefore (p : X-*p(X) is a finite map. For a general point/* of IV, the elliptic curve n~x(p)=E is a finite unramified covering of (X)=X/r because f operates on Z b y Theorem 1. The statement H*(W> Q)^H*(PpxPq, Q) was already proved in Corollary 4 in § 1. Now we have only to show that the image J by TT of the set of the singular fibres of it: X^W has only simple normal crossings for sin-

162

K. Akao

gularities. By Holmann [5], the action of E admits a holomorphic slice at every point of X. Let p e W be a singular point of J, and Fp the stabilizer subgroup of E at a point of TC~1 (p). Then there exist an open polydisc neighbourhood U of p in W, and a holomorphic slice U at a point of n~l(p), which is stable under Fp and also isomorphic to an (n—l)-dimensional polydisc. Moreover UjFp~ U via rc, and the image by n of the set of fixed points of Fp in U coincides with AC\U. Since FpczE, Fp is a finite abelian group. Therefore we choose a coordinate system Z={Zi, ", Zn-\) on I? such that every element y^Fp is expressed by a diagonal matrix with respect to these coordinates. Let Uf be another (n—l)-dimensional polydisc and w=(wl, •••, wn_x) be a coordinate system on U\ Put g=#|/ 7 />|, and consider the map xs : #—>£/' defined by tyfo, •••, £n_i) = (£f, •••, *£_,). Then tf factors through 7r|# :

and Tf'(A)CL {(w) e U/\wlw2'"Wn_l = 0}. Hence J has only a normal crossing at/?. The smoothness of irreducible components of J can also be proved easily. q. e. d. Proof of Theorem 3. In this case 7r,(Z)^Z 5 SO q{X) = l, and bl(W)=q(W)=0. Since Wis protective, nx(W) is a finite cyclic group. Let Wbe the universal covering of W. Then by Theorem 1, X—XxwW admits a holomorphic torus action. But Hence H*(W)^H*(P«) by Corollary 4 in § 1, and bv(W)\ for 0, and a non-zero complex number a with |a|^ 1, such that a*li~-p\ami!flv(pv3)-x on £/?n U), where we put /«=1 for each i. Let {#*>} (*=0, 1, ••-, N) be a basis of Z/°(W, 0(L)), where A^-l=dim//°(W/ 0 (L)). This gives an imbedding of Winto P^. Then, as in Theorem 2, we can construct a holomorphic map from X to F=C^ + 1 — (0)/(a n ) neZ which maps A; € U% to [ ( ^ T T * ^ , •••,^7r*^iNr))]. F i s a (homogeneous) Hopf manifold, and the rest of the proof is the same as that of Theorem 2. q. e. d. Remark. If X is homeomorphic to Sl X S5, then we can get rid of the assumption of the flatness of n : X^>W. Proof Since n^W) is finite, we may assume obviously that nl{W)= {1}. Then, by Theorem 1, X admits a holomorphic action of 1-dimensional complex torus E.

Complex Structures on S2P+l X^ 2« +1 with Algebraic Codimension 1

163

Let

-

>W

U where XS is a birational morphism. Note that we have b2( V) = 1. Let (p : V—>Vbe the resolution of singularities of V. Then, since dim c PK=dim c F=2, (poXS is a composite of quadratic transformations. Now consider the Leray spectral sequence for V^W and Qv ; E*>*=H*(W, R*zr*Qv)=*H*+*(V, Q). We shall show that £° >2 =0, which means that XS is a finite map. But then since W is non-singular and XS is birational, XS is an isomorphism, and TZ is flat. Since XS is birational, the dimension of the support of RXZS^QV is zero, so E22A = 0. Since W is non-singular, b3(W) = bl(W)=0 by the Poincare duality, hence E32>°=0. This implies that E°2'2=E°J. But since PK is projective, 4 2 ( ^ ) > 1 5 a n d from t n e inclusion //^M 7 , Q)d^H2 (F, Q), we have E\*=E™~H2(W, Q). Since A2(K) = 1, we get E°2>2 = 0. q. e. d. References [ 1 ] Akao, K. : On certain complex structures on the product of two odd dimensional spheres. Proc. Japan Acad., 50 (1974), 802-805. [ 2 ] Brieskorn, E. and Van de Ven, A. : Some complex structures on products of homotopy spheres. Topology 7 (1967), 389-393. [ 3 ] Calabi, E. and Eckmann, B. : A class of compact complex manifolds which are not algebraic. Ann. of Math., 58 (1953) , 494-500. [ 4 ] Hironaka, H. : Review of S. Kawai's paper, Math. Rev., 32, #466 (1966), 87-88. [ 5 ] Holmann, H. : Quotientenraume komplexer Mannigfaltigkeiten nach komplexen Lieschen Automorphismengruppen. Math. Ann., 139 (1960), 383-402. [ 6 ] Holmann, H. : Seifertsche Faserraume. Math. Ann., 157 (1964), 138-166. [ 7 ] Iitaka, S. : On algebraic varieties whose universal covering manifolds are complex affine 3-spaces, 1, in Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo (1973), 147-167. [ 8 ] Kato, Ma. : Complex structures on Sl X S5. Proc. Japan Acad. 49 (1973), 575-577. [ 9] : Complex structures on Sl X S5, J. Math. Soc. Japan 28 (1976), 550-576. [10] Kawai, S. : On compact complex analytic manifolds of dimension 3. J. Math. Soc. Japan, 17 (1965), 438-442. [11] Kodaira, K. : On compact analytic surfaces II. Ann. of Math., 77 (1963), 563-626. [12] : On the structure of compact complex analytic surfaces I. Amer. J. Math., 86 (1964), 751-798. [13] Miyaoka, Y. : Kahler Metrics on Elliptic Surfaces. Proc. Japan Acad., 50 (1974), 533-536. [14] Tits, J. : Espaces homogenes complexes compacts. Comment. Math. Helv., 37 (1962-63), 111-120.

Department of Mathematics Gakushuin University (Received December 15, 1975)

Defining Equations for Certain Types of Polarized Varieties

T. Fujita Introduction

In this paper we improve a result of Mumford [5]. To be explicit, we fix our notation and terminology0. Every variety is assumed to be defined over an algebraically closed field K. For line bundles L, Af on a variety V we denote by R(L, M) the kernel of the natural multiplication homomorphism r(L)®r(M)-+r(L-\-M). A line bundle I o n V is said to be simply generated if r{tL)®r(L)-+r({t+\)L) is surjective for every t^> 1. L is said to be quadratically presented if'it is simply generated and if the natural homomorphism R(sL, tL)(x)r (L)—>R(sL, (t-{-l)L) is surjective for all s,t^l2). Now we state the following T h e o r e m (Mumford). Let L be a line bundle on a smooth curve C of genus g. Then L is simply generated if deg L^>2g+1, and L is quadratically presented if deg L}>3g-\-l.

We improve the above result in the following three ways. First, we can weaken the assumption that C is smooth. Second, we show that L is quadratically presented if deg L^>2g+2. Third, in the complex case, we give a higher dimensional version of these results, an embedding theorem and a structure theorem for certain types of polarized varieties, which will play an important role in our study of polarized varieties (see [la]). As an example of applications, we give in § 5 a criterion characterizing smooth hypercubics. 1.

One-dimensional Case

Throughout this section, let C be an irreducible reduced curve with hl(C, 0) =g. We remark that C is locally Macaulay, i.e., 0x is a Macaulay local ring for any xeC. Hence there is a canonical dualizing sheaf w on C (see [2]). 1) Basically we employ the same notation as in [1], oo

2) L is simply generated if and only if the graded 7C-algebra G(V, L) — @ P(V, tL) is generated by r(V,L). Then L is quadratically presented if and only if all the relations among the elements of F(V, L) in G(V, L) are derived from quadratic ones.

166

T. Fujita

Definition 1. 1. A coherent sheafs on C is said to be quasi-invertible if it is torsion-free and of rank one. Remark 1. 2. co is quasi-invertible. Any non-trivial subsheaf of a quasi-invertible sheaf is also quasi-invertible. A sheaf is quasi-invertible if and only if it is locally isomorphic to a non-zero ideal of Qc. Definition 1. 3. For a quasi-invertible sheaf J, we define deg J and A(J) as follows :

L e m m a 1. 4. Let J'i ~§ be quasi-invertible sheaves and let

0. Then there is an effective divisor D of degree J + 1 such that h°(o)[-D]) =g-A-\. On the other hand, Horn (0[Z>], 3*)*0 since h\y[-D])^h\J)-dzg D = deg J-2A^\. We apply Lemma 1. 4 to infer that hl{¥)^hl(D)=h°(a)[-D])=g-d-l. This contradicts hl{?)=g-± q. e. d. Proposition 1. 6. Let J be a quasi-invertible sheaf such that deg y ^ Then J is generated by its global sections. Proof Let -§ be the subsheaf of $ generated by global sections. Suppose Then deg ^ < d e g J and J ( ^ ) < J ( ^ ) 5 since h\$)=h\J) implies deg ^ — J ( ^ ) = deg •§—£(•§). Hence deg ^>2A{-§) and consequently g=d($)g-\, b) Bs\L\« = ifl^g+\. Proof We may assume l=g— 1 in a) and l=g-\-1 in b). Let F be a line bundle of degree 2g— 1 and let D be a general effective divisor of degree g. Then h°(F—D) = h°(F)— deg D = 0 and consequently A1 (F—D)=0. This proves a). T a k e g + l simple g+\

points/?!, •••, pg+l of Cin general position and put E= 2 A a n d Ea=E—pa 3) By a general line bundle we mean one belonging to a suitable open dense subset U of the space of line bundles of degree /. 4) By BsA we denote the intersection of all the members of a linear system A on a. variety.

Defining Equations for Certain Types of Polarized Varieties

167

). Then each of these divisors is general as a line bundle since deg Ea=g. Thus hl(Ea)=0 by a). This implies that/> a { Bs \E\. Therefore Bs |2?| = 0. This proves b). q. e. d.

L e m m a 1. 8. Let L, M be line bundles on C and let D be an effective divisor such that Hl(L-D)=0. Then = , a) Bs\L\ = tifBs\L-D\ b) r(M)®r(L)-->r(L+M) is surjectiveifT(M-D) (x) r{L) -+r{M+L-D) is surjective and if Hl(M—D) — 0, c) R(M9 L) (x) r(D) -> R(M, L+D) is surjective if Bs \D\ = and if r(M) (x)

r(L-D)-+r(M+L-D)

is surjective.

l

Proof a). H (L—D)=0 implies that the restriction T(C, L)^>F(D, LD) is surjective. Hence Df)Bs\L\ = and Bs \L\aBs \L—D\. This proves a), b). We have the following commutative diagram :

o _> r(M-D) 0_>

(g) r(L) -> r(M) 0 r(L) -> r(D, MD) ® r(L) -^ 0 ->

r(M+L-D)

r(M+L)

-•

r{D,[M+L]D)

0 0 0 It is easy to see that the rows and the first and the third columns are exact. Hence the second column is also exact, proving b). c). By R(M, LD) we denote Ker (r(D, LD) (g) r(C, M)->r{D, [M+L]D)). Then we have the following commutative diagram : 0

0

1 R(M,L-D)

->

r{M) (g) r(L-D) 0—

r(M+L-D)

0

1 R(M,L)

9 ->

1 R(M,LD)

— r(M) r(Z) -> r(M) ® r(D, LD) -> 0 —

r(M+L)

-+ r{D,[M+L]D).

0 Every sequence above is exact. It follows that ^ is surjective. Next we consider the following diagram :

R(M, L) /?(M, Z^ (g) r(/)) -> 0 72(^,4

>R(M,L+D)

>R(M,[L+D]D).

The dotted arrow is defined by a\-^>a®d, where d e F(D) is the section defining D. Then the above diagram is commutative and the rows are exact. Moreover, (p is surjective since Bs \D\ = . Tracing this we prove c). q. e. d. L e m m a 1. 9. Let L, M be line bundles of degree I, m respectively. Suppose that m^>2g, and L is general. Then p : r ( Z ) ( g ) r ( M ) - > r ( Z + M ) is surjective. Proof In view of Lemma 1. 8. b), it suffices to consider the case l=g+l, by

168

T. Fujita

replacing L by L—D, D being a general effective divisor of degree l—g—1. In this case, we have hl(L)=0, h°(L)=2 and#y \L\ = . LeU,, 12 be a basis for H°(L). Then {^ = 0} f] {^=0} =^. Therefore, ^ 2 = > ^ i € P(L+M) for F(L+M — D) is surjective. So we can apply Lem-

ma 1. 8 b) since hl(L-D)=kl(M-D)=0.

Corollary 1. 11. A line bundle of degree ^2g+\

q. e. d. is simply generated.

L e m m a 1. 12. Let L, M be line bundles of degree I, m respectively and let D be a general effective divisor of degree g+l. Then R(M, L)®r (D)—*R(M, L-\-D) is surjective

Proof It is easy to see that Lemma 1. 9 enables us to apply Lemma 1. 8 £). q. e. d. Proposition 1. 13. Let L, M and N be line bundles of degree I, m and n, respectively. Suppose thatfe2g+2, m^2g andn^2g+2. Then R{M} L)®r(N)-+R{M, L+N) is surjective. Proof. We prove this by induction on n. First, suppose that tt<j3g+2. Let D be a general effective divisor of degree g+l. Lemma 1. 12 says that R(M, L)(x)r(D)—* R(M, L+D) is surjective. So it suffices to show that R(M, L+D)®r(N-D)-> R(M, L-\-N) is surjective. For this purpose we want to apply Lemma 1.8 c). We

should show that hl(L-N+2D)=0,

Bs \N—D\ = t and that

r(M)^r(L-N+2D)

—+r(MJrL—NJr2D) is surjective. This follows easily from Lemma 1. 7 and Lemma 1. 9. Second, suppose that ?2^3g+3. Let D be a general effective divisor of degree g+l. Again Lemma 1. 12 says that R(M, L)®r(D)->R{M, L+D) is surjective. Moreover by the induction hypothesis, R(M} L+D)(^r(N—D)—^ R(M, L+N) is surjective. This completes our proof. q. e. d. Corollary 1. 14. A line bundle of degree §:2g+2 is quadratically presented. 2.

Climbing a Ladder

Definition 2. 1. Let (V, L) be a prepolarized variety5). A member D of \L\ is called a rung of (V} L) if it is irreducible and reduced6). A rung D is said to be regular 5) A pair consisting of a variety V and a line bundle L on it. It is called a polarized variety if L is ample.

Defining Equations for Certain Types of Polarized Varieties

169

if the restriction F(V, L)^>F(D, LD) is surjective. A sequence V=Dn'z>Dn_xZ)--'D Dx of subvarieties of Fwith dim Dj=j is called a ladder of {V, L), if each Dj is a rung of (-Dy+i, L). A ladder is said to be regular (resp. smooth) if each rung of it is so. Proposition 2. 2. Let D be a rung of a prepolarized variety (V, L) defined by 8 e F(V, L). Let f a ( a = l , - - - , k) be homogeneous elements of the graded algebra G(V, L) = ®F(V,

tL) with deg £a = da, and let rja be the restriction of fa to G{D, L) = ©T(D,

tL)7\

Suppose that {ya}a=i,-,k generate the algebra G(D, L). Then G(V, L) is generated by 8 and -»aJ a =

\,-,k'

Proof. Let A be the graded subalgebra of G( V, L) generated by [8, f,, • • -, gk] and let At = AnT(V, tL). Let pt be the restriction r(V, tL)-+r(D, tL). Then we have Ker pt = 8F(V, (t—\)L). Moreover, F(V, tL)(zAt + Ker pt since G(D,L) is generated by [rja] • Combining them, we obtain F(V, tL)=At by induction on t. C o r o l l a r y 2. 3. Let D be a regular rung of a prepolarized simply generated, then so is L.

variety [V, L). If LD is

P r o p o s i t i o n 2. 4. Let {V, L), D, 3, {fa} and {rja} be as in Proposition 2. 2. Let gj (lfgj<^/) be homogeneous polynomials in k variables Yx,--3 Yk with deg Ya = dafor l r g a ' '} Vk) = 0 - Then there exist I homogeneous polynomials f, ••• ,f in (A;+l) variadeg Xa = da for l^a^k such that f(0, Yl7 •••, Yk)=gj bles Xo, ..-, Xk with degX0=l, ( 7 , , •••, Yk)for l^j^l and that all the relations among 8, f^ --, f* in G(V, L) are derived from f (8, £ „ •••, f , ) = • • • = / { 3 , f,, Proofs.

L e t Q=K[X0,

•••,f,)=0.

X,, •••, Xk]

a n d R = K[Yl3

••,

Yk].

L e t q : d-^G(V,

L),

r : R—>G(D, L) and n : Q-^R be the ^-algebra homomorphisms defined by q(X0) =8, ? ( X , ) = ^ rij,)^, x(X0)=0 and x(Xj) = Yj for l^j^k. Letting ft : (£-+d be the multiplication by Xo, we have the following commutative diagram : 0

1

K e r q —>

i /< d i

G(V,L)^<

I

0

0

A

Ker q

aI i

i

0

0

I

—•> K e r r

1 —•

R

->0

i

- ^ G(Z>, L) - » 0

I

0

All the columns and the middle and the bottom rows of this diagram are exact. Hence so is the top row. Therefore, there is/ a € Ker q such that n{fa) =ga for each l:ga<^/. Clearly we can take/ a to be homogeneous. Now it suffices to show that 6) This definition means that a rung has no embedded component. 7) We denote here by the same letter L the line bundle Ln on D induced by L. 8) The author would like to express his thanks to Mr. Mori to whom he owes this proof.

170

T. Fujita

the ideal J of Q, generated by {fa} is the same as Ker q. By assumption, we have ^(y)=Kerr. So Ker qaJ+fjt{Ker q) and hence Ker qlJ=ft(Ker qjj). Recall the following fact: Let I be an ideal of a ring A and let M be a finitely generated A-module. for any a el. Then IM^M. The proof is easy and quite Suppose that (l-\-a)M^0

similar to that of Nakayama's Lemma. Returning to the proof of the proposition, we assume y ^ K e r q. Then (l-\-
Existence of a Ladder

Now we want to establish a sufficient condition for the existence of a ladder of a given prepolarized variety. Unfortunately, we have to assume K=C from now on9), since we need a theorem of the Bertini-type. So every variety is an analytic variety. First we recall the following result of Hironaka [3]. T h e o r e m 3. 1. Let A be a linear system of Cartier divisors on a variety V. Then there exists a triple of a smooth variety V together with a birational morphism TZ : F'—>F, a linear system A on V with Bs A''= and an effective divisor E on V such that 7i*A=E-{-A'. Moreover, W=pA,(V')l0) is determined by (F, A) and is independent of the choice of (V, it, A', E).

Remark 3. 2. A general member of A is irreducible if dim W^2 and if codim Bs A^2 (see, for example, [1], p. 114). Definition 3. 3. A is said to be non-degenerate if dim W=dim V. Proposition 3. 4. Let (V, L) be a prepolarized variety and suppose that V is locally Macaulay, dim Bs \L\ ;g0 and that \L\ is non-degenerate. Then (F, L) has a ladder.

Proof Remark 3. 2 says that a general member D of \L\ is irreducible. Moreover, D is locally Macaulay and generically smooth, hence D is reduced. So D is a rung of (F, L). Repeating this process we obtain a ladder. q. e. d. Proposition 3. 5. Let (M, L) be a prepolarized manifold such that dim Bs \L\ fgO, d(M, L)^2d(M, L) — lU). If\L\ is degenerate, then a general member of\L\ is smooth.

Proof. Since every base point of \L\ is isolated, it suffices to show the existence of a member D of \L\ which is smooth at any given point peBs \L\. Assume that 9) Recently the author proved similar results also for the cases in which char A^>0. (1976, Oct.) 10) pA> denotes the rational mapping associated with A. 11) For a prepolarized variety (V, L) with dim V=n we define (cf. [1]) d(V,L)=Ln and n+d(V9L)-!fi(V,L).

Defining Equations for Certain Types of Polarized Varieties

171

such a member does not exist. Let Ml be the monoidal transform of M with center/^ and Ep the inverse image of p. Then mEp is a fixed part of \L\Mx for some integer m^> 2. Applying Theorem 3. 1 to the pair (Mly AY) where Al = \L\Ml — mEpy we have a manifold M' together with a birational morphism TT : M'—*MU a linear system Af with Bs Af = and an effective divisor E such that ?r* Al = E+Af. Let W=pA,(M') and let / / be the natural hyperplane section on W. Then HM, = [A'] and consequently LHn~l {M'} =L(L—mEp—E)n-l=Ln>0, whererc= dim M. Thus dim W= n—l, since |L| is degenerate. Put w = deg W and let X be a general fiber of pA,. Then £ / ) {Z}>0 since Ln — mn=[Al]n^[Al]Hn-1=LHn-l — mEpHn-l=Ln--mwEpX. Therefore, d=LHnl = (H+E+mEp)Hn-l^mEpHn-l = mwEpX^mw^2w. On the other hand, 0^A{W, H)<,n— 1 +w — h°(M, L)=w — d+A — 1. Putting these together, we obtain d^2w7±2(d—A-\-l). This contradicts the assumption d7>2A— 1. q. e. d. Corollary 3. 6. Let (M, L) be a prepolarized manifold such that dim Bs\L\^0, d(M, L)^2A(M, L)- 1 and d(M, L)>0. Then (M, L) has a ladder. Proof. We may assume that \L\ is degenerate, since otherwise we can apply Proposition 3. 4. So a general member D of \L\ is smooth. Since dim Bs |L|^0, we have Bs \lL\ = for />0. \IL\ is non-degenerate since d(M> L)>0. Hence Hl(M, —L) = 0 (see Mumford [6]). Therefore D is connected, and is a rung of (M, L). Now we have d(D, L)=d(M, L) and A(D, L)<,A(M, L). Repeating this process, we obtain a ladder. q. e. d. 4.

Main Theorem in Higher Dimensional Cases

Theorem4. 1. Let ( M, L) be a prepolarized manifold such that d=d(M, L) >0, A — J(M, L)^g(My L)™=g and that dim Bs |L|^0. Then a) (M, L) has a regular ladder if d^>2A— 1, c) g(My L)=A(M, L) and L is simdly generated if d^> d) L is quadratically presented if d^2A-{-2. Proof a) Corollary 3. 6 implies the existence of a ladder {Dj} with dim D~j. We have d(DlyL)=d and g{D{,L)=g. If this ladder were not regular, then J(Z),, L)
172

T. Fujita

Corollary 4. 2. Let (M, L) be as in Theorem 4. 1, a). Then (M, L) has a smooth regular ladder.

Proof. We may assume Bs \L\^. Take a regular ladder {Dj} of (M, L). Let A5 be the subsheaf of £j=ODj[L] generated by global sections, and let J8J=£JIJ4J. Then Supp £j=Bs\L\ and A°(j8,)=A0(j8n), since the ladder [Dj] is regular. Ax is quasi-invertible on Dx and deg Ax-=d— A°(j8,). So A(J4X)=A — h\Bx)
Applications

L e m m a 5. 1. Let (M, L) be a polarized manifold with g(M, L)
fg j , 1 ^t^j— 1. We prove this by descending induction on j . When /=#, our claim for pH*+l(Dj+l, -(t+l)L). So the claim follows easily from the induction hypothesis. Now we have proved the above claim, which means =g{M,L)^0. This implies that {Dj} is regular. Hence J(M, L)=J(Dl9 L)=09 since A(DX, L) ^hl(D{) =0. q. e. d. T h e o r e m 5. 2. Let (M, L) be a polarized manifold such that dim M=n, Ln = 3 and h°(M, L) =n-\-2. Then M is a hypercubic in Pn+l and L is the line bundle associated to a hyperplane section on it.

Proof We have J(M,L) = l. This implies dim Bs \L\^0 (see[l]). Therefore (M,L) has a ladder (Corollary 3. 6). So g(M, L)^\ (Lemma 5. 1). Hence L is simply generated and very ample (Theorem 4. 1, c)). Our theorem follows from this. q. e. d. Corollary 5. 3. Let (M, L) be a polarized manifold such that ?z=dim M, Ln = 3 and KM-\-(n—l)L=0. Then M is a hypercubic and L is the line bundle associated to a hyperplane section on it.

Outline of proof We can calculate %(Af, tL) using the vanishing theorem of Kodaira. Then it follows that H>(M, L)=x(MyL)=n+2. We note that KM-\- (n—l)L=0 holds for a polarized manifold (Af, L) if and only if A(M, L)=g{M, L) = \. So we have similarly the following

Defining Equations for Certain Types of Polarized Varieties

173

T h e o r e m 5. 4. Let (M, L) be a polarized manifold such that dim M=n and Ln = 4. Then the following conditions are equivalent to each other : a) M is isomorphic to a complete intersection of type (2, 2) and L is the line bundle associated to a hyperplane section on it. b) h°(M,L)=n+3. c) KM+(n-l)L=0.

References [ 1 ] Fujita, T. : On the structure of polarized varieties with J-genera zero, J. of Fac. Sci. Univ. of Tokyo, Sec. IA, 22 (1975), 103-115. [la] Fujita, T. : On the structure of certain types of polarized varieties, Proc. Japan Acad. 49 (1973), 800-802 & 50 (1974), 411-412. [ 2 ] Grothendieck, A. : Local cohomology, Lecture Notes in Math. 41, Springer, 1966. [ 3 ] Hironaka, H. : Resolution of singularities of an algebraic variety over a field of characteristic zero I-II, Ann. of Math., 79 (1964), 109-326. [ 4 ] Mori, S. : On a generalization of complete intersections, J. of Math. Kyoto Univ., 15 (1975), 619-646. [ 5 ] Mumford, D. : Varieties defined by quadratic equations, C. I. M. E. (1969) -III, 29-100. [ 6 ] Mumford, D. : Pathology III, Amer. J. Math., 89 (1967), 94-103. College of General Education University of Tokyo (Received December 12, 1975)

On Logarithmic Kodaira Dimension of Algebraic Varieties

S. Iitaka 1.

Introduction and Notation

Let k be an algebraically closed field of characteristic zero. We shall work in the category of schemes over k. Let V be an ^-dimensional algebraic variety. By Nagata, we have a complete algebraic variety F which contains V as a Zariski open subset. We call F— V an algebraic boundary of V. By Hironaka's Main Theorems I and II, we have after a finite succession of monoidal transformations, a non-singular algebraic variety F* and a proper birational morphism ft: V*—+V such that if F* = /i~1(F), then D= V* — V*=p~l(V— V) is a divisor of simple normal crossing type. We call F* a compactification of F* with smooth boundary D. The sheaf of germs of logarithmic q forms of V along D is the subsheaf of the sheaf

of germs of rational ^-forms Qq(*) of F* which consists of those germs which for any point/? e V*, are written as ^

" r{z}w)~^^A'"A-^lLA

dwja) A---A dwJW,

where (z\, •••, Zm, wY, •-, wn_m) is a local system of regular parameters at p such that £j...£ m =0 defines D around/? and aItJ(z, w) eOv*,P> By Qq(log D) we denote the sheaf of germs of logarithmic ^-forms along D. For any integers m ^ O , •••, m n ^ 0 , put We shall prove that dim Tmi,...,mn{V)

is independent of the choice of compactification of V with smooth boundary and of non-singular model of V. In particular, we define the logarithmic irregularity qV = dim H°( F*, Ql log 5 ) , the logarithmic geometric genus pg V = dim H°( F*, Qn log D) = dim H°(V*,Qn(D)) = dim the logarithmic m-genus Pm{V) = dim / / ° ( F * , (Qn(B))®m) = dim H°(m(K*+D)),

H°(K*+D),

S. Iitaka

176

and finally the logarithmic Kodaira dimension KV = tci^K+D, V*). Here, K* denotes a canonical divisor of P* and in general for a divisor Z), we abbreviate H°(V*9 0{D)) by H°D. We write

2.

Examples

Example 1. Let V be a non-singular curve. Then £F= — oo if and only if V=Pl or an affine line A\ = Gai icV=0 if and only if V is a complete elliptic curve or a 1-dimensional algebraic torus Gm=A\-{p}9 icV=l if and only if V is one of the others. Example 2. Let V be a complete non-singular curve of genus g and p0, -"9pti t-\-\ points on V. Putting V=

V-{po,--;pt},

we have

E x a m p l e 3. Let Do, •••,Bt be 1+t lines in P2 and F=P 2 —Z) 0 U ••• U A - Then t

A otherwise

q{V)

— oo

t

0

2

1

t

2

t

V

GmX (A\- {a2, •••,flj)

Write ICV=KV*=/C(K*, V*), which is the original Kodaira dimension of V. F. Sakai [7] defines another kind of Kodaira dimension of F, which we indicate by KV. Letting pm be the rational map associated with mK-\-{m—1)5, he defines KV = max dim

pm(V*).

m>\

In general, KV^LKV
On Logarithmic Kodaira Dimension of Algebraic Varieties

177

and only if icV=n. Example 4. Let V be a minimal complete non-singular algebraic surface with KV^O. Let D = 2 ) 0 U - U A be a union of 1+* irreducible curves. Put V=V—D. Then KV is computed as follows : (i) if V is an abelian variety or a hyperelliptic surface, then tcV^il and icV=l implies that Z)o, •••, Dt are fibers of an elliptic surface

J, (ii) if Vis a K3 surface or an Enriques surface, then icVT^O and lcV=0 implies that each Dj is Pl with self-intersection number — 2. Moreover, KV=\ implies that D is contained in a finite union of fibers of an elliptic surface

J (iii) if KV=1, then V has the structure of an elliptic surface

V)

(see [6] Lemma 5).

Now, we let V be a K3 surface. icV=0 implies that for any v o ^O, •••, ^ ^ 0 , dim H0{V,O(XVJDJ))

= 1.

Hence by Riemann-Roch Theorem, Z>5 = — 2 and the matrix ((Z)^)) is negative definite. Hence the connected components of Z)=Z)0U ••• U A may be indicated by Dynkin diagrams : An, Dn, E6, E7, E8. icV=l implies that the moving irreducible component C of the general member of \mD\, m>0, satisfies C 2 =0 and so the virtual genus n(C) =C 2 /2+1 = 1. Hence we get an elliptic surface a finite union of whose fibers contains D. Second, we assume V to be an Enriques surface, then there is a 2-sheeted unramified covering map

V where V1 is a K3 surface. Since KV=K(D, V)=/c((p*D, V'), we can infer a similar result in this case. The proofs of the other cases are easy. As in [3], we call V a variety of elliptic type if lcV= — oo, of parabolic type if icV=0 and of hyperbolic type if /cV=n. A variety of hyperbolic type might be called a variety of general type. Our purpose here is to prove the theorem to the effect that the group of strictly birational maps of Fonto itself is a finite group if F is of hyperbolic type. Furthermore, we study the structure of algebraic varieties which have many automorphisms in the case where they are not of elliptic type. In a forthcoming paper : Logarithmic forms of algebraic varieties, we shall study the quasi-Albanese maps with some applications (see [11]). 3 Let F, and F2 be w-dimensional non-singular algebraic varieties a n d / a domi-

178

S. Iitaka

nant morphism VX-^>V2. Choose two compactifications with smooth boundaries Dx and D2 of Vx and V2i respectively. T h e n / i s a representative of a rational map / from Vx into V2. By applying Hironaka's Main Theorem II, we may assume that / is a morphism. Under this assumption, we prove Proposition 1. If a rational q-form w on V2 is logarithmic along D2, then f*co is a logarithmic q-form on Vx along Dx. Moreover, if f is proper and birational, then / * : H°Q« log D2 -> H°Q* log 25, is an isomorphism.

Proof By definition,/*a> is a rational ^-form. For any point px€ Vl9 choose a local system of regular parameters (wl3 •-, wn) and for p2=f (px) choose a local system of regular parameters (Z\,'-,Zn) such that if p2eD2, Zi'-Zr—Q defines D2 aroundp2 and ifpx e Dx, wx--ws=0 defines Dx around px. By definition of/, f~l(Dx) dD2 and so (L) Zi = Tlw¥i£i} nij ^ O5 where et is non-vanishing around/?,. Thanks to /dL\ dzi _ yrn dwj det £i \ L ) ^ ,~ ~^~ ' we see that/*o> is logarithmic. Now, assume t h a t / i s proper. ThtnfV{ is closed in V2, hence fVx =fVx=V2. Letting j : V2-^f-[Vly we getf=(f\f-lVl):J,_vthich l is proper. Hence j is also proper. Thus V2=f~ Vl. In other words, f~lDx = D2. Furthermore assume/to be birational. Take a logarithmic ^-form wx on Vx along D{. Then we have a rational ^-form o>2 on V2 such that a>1=/*a)2- Since / is birational and Vx is complete, there exists a Zariski open subset V°2 with codim (V2—Vl) ^ 2 such that the inverse g : V2—>VX of / exists,, that is, f*g is the open immersion VQ2—>V2. Since wx is a logarithmic 1=a»2|^2 is also logarithmic along g~lDx = D2\V$. Hence co2 is a logarithmic ^-form along D2. q. e. d. This proposition holds for forms in Tmu...tmn(V). Thus we have established the birational invariance of dim Tmit...tmn(V). Precisely speaking, a rational m a p / : VX-^V2 is called a strictly rational map if there is a proper birational morphism ju : F3—>F1? F3 being an algebraic variety, such that/-/^ is a morphism. For example, a dominant rational map from a complete Vx into a non-complete V2 is not strictly rational. A rational map from Vx into a complete F2 is always strictly rational. Actually, consider a compactification Vx of Vx and regard / as a representative of a rational map / : Vl—>V2. Then there is a birational morphism p. from a complete variety F3 onto Vi such that / • / / is a morphism. Put V3=p~lVx. Then {i=p\V3 is proper and / • / i is a morphism. Consider a strictly rational map from an algebraic variety Vx into an algebraic variety V2. As in § 1, taking compactifications V? and F2* of non-singular models Vf—>F15 V}-+V2 of Fi and F2, with smooth boundaries Dx and 5 2 , respectively, we obtain a linear map :

On Logarithmic Kodaira Dimension of Algebraic Varieties

179

/ * : H°Q*logD2^H°Q«(*) U

H°Q« log Dx.

By definition, there is a birational proper morphism n : V3—>VX such that g=f*pt is a morphism. Choose a suitable compactification F3* with smooth boundary A such that p is a morphism and g=f*p is a morphism. By Proposition 1, g * : H°Q H°Qq log 5,. Note that in case/is dominant,/* is injective. Thus, in general, if dim Fj = dim V2 — n, and / i s dominant, then l

0

K,-.,myx

^ pmi,...,mnv2.

In particular, if F, is a non empty Zariski open subset of F2, then PmV^PmV2, KV^KV2 and qV^qV2. We say that a birational m a p / : Vl-^V2 is proper i f / a n d / " 1 are both strictly rational. For example, a proper birational morphism is a proper birational map. Note that V= P 17 P if there is a proper birational map Vl-^V2. Proposition 2. Let V be a non-singular complete algebraic variety and D a divisor of normal crossing type. Assume that K~\-D is ample. Put V= V—D. Then any strictly rational dominant map f: V—+V is a restriction of an automorphism of V.

Proof From the argument above, we infer that

/ * : H°m{R+D) -> H°m{K+D) is an isomorphism. Since m(K-\-D) is a hyperplane section for m>0, we get the conclusion. q. e. d. Example 5. Let D be a divisor of normal crossing type in Pn. Assume that . Then Aut{Pn-D) aPGL{n,k). For a variety V in Pn, we write Lin

V = {a € PGL{n3

k);aV=V],

S Bir V = the group generated by strictly birational maps : V—>V. Then under the conditions of Proposition 2, S Bir V = Lin V, which is a finite group (see Theorem 6).

180

S. Iitaka

Let Vx and V2 be non-singular algebraic varieties and / a dominant morphism P^—>F2. Choose compactifications V{ and V2 with smooth boundaries Dx and D2 of Vx and V2> respectively such that the rational map / : Vl-^V2, f\Vl=f is a morphism. Then there is an effective divisor Rf which satisfies

(R)

Rx+Dx~f*{R2+D2)+Rf

Rf is called the logarithmic ramification divisor for f.

We prove (R) by using the notation in the proof of Proposition 1. A rational logarithmic <7-form co on V2 along D2 is a rational section of the line bundle [K2-\-D2]. Hence for any p e V2,

<» = < p { z ) ^ - / \ ••• A 4 ^ A , and

= (
A-Adwn,

0 and PmVl = PmV2 for sufficiently large m. Then the rational map @mi,Vl associated to Im^ + D^l is factored as follows :

Proof As in the proof of Theorem 3 in [3], choose mt so large that Om.yV. : Vt—* 0muV.(Vi) is birationally equivalent to the ^ + Z)rcanonical fibered manifold of Vt for i = l , 2 (see [4]). Choose m=a multiple of L. C. M. (ml9 m2) so that PmVi = PmV2. Then the natural inclusions :

H°m(K2+D2) c H°mf*(K2+D2) c H°m(f*(K2+D2)+Rf) = ^ ( m ^ + A ) ) are isomorphisms. From this follows the conclusion. Using this we can prove easily T h e o r e m 1. If KVl = n, PmVl = PmV2for map from Vx into V2 is birational.

q. e. d.

m > 0 , then any dominant strictly rational

As a generalization of a theorem of Peters [10], we prove T h e o r e m 2. Let Vbe a non-singular algebraic variety with icV7>0. Then any dominant morphism from V into itself is an etale covering map from V onto V.

Proof Let V be a compactification with smooth boundary D. f is regarded as

On Logarithmic Kodaira Dimension of Algebraic Varieties

181

a representative of a rational map / : V—+V. Performing a finite succession of monoidal transformations with non-singular centers c D on V, we get a birational morphism p : V*—»Fsuch that f*p—g is a morphism F*—+V. Then by (R)

m(R*+D*)

~g*m(K+D)+mRg,

where K* denotes a canonical divisor on V* and D*=p~lD ; and

m(^* + S*) ~ p*m(K+B)+mRft. Hence, With the decomposition f=p*g in mind, we write

Then

m(K+D) ~mf*(K+B)+mRf. By assumption /cF;>0, there is an m>0 such that \m(R+D)\ 3 C(m). Write g*C(m) as a sum of effective divisors D* and 8 where & consists of the components that are mapped to Oby/z*. Similarly write p*p*D*=D*+6'. Note that <S, <S'>0. Then, since
dim \D*\ ^ dim \D*+8' + 8\ = dim |/z*/z+i5*+<S| = dim \p*p*D*\ = dim |/z+Z)*|. Hence dim |m(^+D)| = dim |m|*(^+.D)| ^dim |m/*(^+5)| = dim \mf*(K+D) +mJtf\ = dim \m(K+D)\.

Let V1=g~l(V) and ^,=^1^,, which is proper. V* is regarded as a compactification of F with smooth boundary D? as well as that of V. Hence g{ defines the rational m a p | , : F * ^ F * . Let p! : F2—>F* be a birational morphism such that h = gx*p' is a morphism. We assume that V2 is a compactification of F2=j5/~1(F1) with smooth boundary 5 2 . Hence by A^2 denoting a canonical divisor on V2, and Therefore by \D\ fix denoting the fixed part of \D\ in general, we obtain \m{R* Hence g*R&1^0. On the other hand, since g^^V3.nd gfRgl\Vl=gf(Rgin V)=0 we see that Rf=Rgi\V=0. Similarly, Rgi = Rgif) ^ = 0. This implies t h a t ^ is ^tale. On the other hand, Vx has only to satisfy the condition that V^V and Vx be nonsingular. We have proved that any gx : Fj—• V extending f is etale. Hence V{=V q. e. d. and g=f. Therefore / i s the etale covering map from F onto F. The following proposition is remarkable. Proposition 4 (Y. Kawamata). Let V be an algebraic scheme and F a closed subscheme of dimension d. If there is an isomorphism] : V—+V—F, then F=#.

182

S. Iitaka

Proof. We prove this by making use of the homology groups Hm(V) of Hartshorne [2]. It is easy to see that if m>2 dim V=2n, then Hm(V) = 0 and H2n(V) =£0. Therefore, making use of Mayer-Vietoris sequences, we can prove that dimH2n(V)^>s if there are at least s irreducible components whose dimensions are n. Let rf=dim F. Then H2d+l(F)->H2d+l(V)^H2d+l(V-F)^H2d(F)^H2d(V) (exact). Since j is isomorphic, dim HM+l(V)=dim H2d+l(V—F). Hence 0^H2i+1(V)->H2d+l(V-F) implies 0^Hu(F)^H2d(V). Let F' be the closure of jF in V and Fl=Ff\jF. Since F' =FX—F, we know that dim^—F) =d, hence # {^-dimensional irreducible components of F j ># {rf-dimensional irreducible components of F]. Composing isomorphisms : V^ V-F-^V-F,-> V-F2-+...^> V-Fh we get # {^-dimensional irreducible components of F j ^ / + 1 . Hence dim H2d(Fi)—>oo as l-^oo. This contradicts dim H2d(FL) rgdim H2d(V) V turns out to be an isomorphism. Furthermore, if V is qffine3 then

S Bir V = Aut V. Note that any affine variety is strongly minimal in our birational geometry. Example 6. Let R=k[x,y, satisfies 9

l/(xy— 1)]. Then any endomorphism/of R over k

(/W>/(-?)) = 0 or c(xy-l)n

for some

ceh*,meZ.

Actually, what we do is to check that R Spec R=0 and use the Corollary. Then we get the assertion. Similarly, let R=k[x,y, z, l/(x3+y3+z3— 1)] and l e t / b e an endomorphism of R over k. Then 9(/(*)/(JO/U))

= O

or

Next, we consider unramified covering maps between distinct varieties. T h e o r e m 3. Let Vx and V2 be algebraic varieties andfan etale covering morphism from

On Logarithmic Kodaira Dimension of Algebraic Varieties

183

Vx onto V2. Then KVX=KV2.

Proof. Let fj. : V?—>V2 be a non-singular model of F2, that is, F2* is non-singular and fi is a proper birational morphism. Denote by F,* a fibre product of Vx and F2* over F2. Then we have the dtale covering morphism V*X—>V2* and a proper birational morphism F,*—+ F,. Hence, F,* is a non-singular algebraic variety. Since by definition icVx=ieVx* and KV2 = £F2*, we can assume that Vx and F2 are both nonsingular. As in § 1, we choose compactifications Vx and V2 with smooth boundaries Dx and D2 such t h a t / : Vx—+V2 is a morphism, f\Vl=f Since/is proper, f~{D2= Dx. Delete the condition t h a t / i s unramified in the hypothesis and add the condition to the effect that / is proper and dominant with dim F ^ d i m F2. Under this condition, we shall prove that By C we denote an irreducible component of 5 , such that fC=F is a divisor. Take general points px of C and p2 of F such that fpx =p2. Then we have a local system of regular parameters (z\, -, Zn) at jfr2 such that ^ = 0 defines F around p2. Let Ci be a minimal equation of C around p{. Then (Ci, ^2, •••, Zn) could be regarded as a local system of regular parameters at p{. Hence ^^Cfc, e being a unit in 0Vl,pxHence

^

= ( ^ + — # " ) 4 ^ A ^ 2 A - A dZn.

Thus i?/ = 0 around/?!. We shall proceed with the proof of Theorem 3. Using the notation of the proof of Proposition 1, we get, in view of (R)

K2+D2 =

f*(Kl+Dl)+Rf.

Since/is unramified, f*Rf = 0. Hence using the following lemma, we obtain the result. L e m m a 1. Let f : Vl—+V2 be a surjective morphism between complete algebraic varieties. ^2. Let D2 be a Cartier divisor on V2 and E an effective divisor on F, such that codimf(E) Then K{f*D2+E,Vx)=K{B2,

V2).

Proof. By definition, we can assume that Vx and V2 are normal. First, consider the case when k(V2) is algebraically closed in k (Vx). Then it is easy to see that l(m(f*B2+E))=l(mB2). From this we get the result in this case. Next, let/=/*•£, where g : Vx—>F3, p : f3—>f2, be the Stein factorization of/. Then since p is finite, codim g(E)^>2. Hence, by the former considerations, we have

*{f*B2+E, Vx)=K(p*D2, f3). Furthermore, from the Theorem 4 in [3], we get t(p*B2y f3) = * ( A , V2).

q. e. d.

184

S. Iitaka

Example 7. Let S be a projective algebraic surface which is a covering surface of P2. By I we denote the ramification locus of n : S->P\ and let B=n(2). If S is of hyperbolic type, then 2 = fcS ^ K{S-X-1B)

= K(P2-B).

Hence

t(P2-B)

= 2.

6 As special results of the Z)-dimension theory, we have T h e o r e m 4. Let f: V—>W be a strictly rational dominant map. By Vw we denote a general irreducible component of a general fiber off over a general point weW. Then W. T h e o r e m 5. Let V be an algebraic variety with jc=icV^>0. Then there is a proper birational morphism fifrom a non-singular algebraic variety F* onto V such that there is a surjective morphism f : V*^>W, in which W is an irreducible constructible set of dimension it and a general fiber V* is an algebraic variety whose logarithmic Kodaira dimension vanishes. Such a fibered variety is unique up to proper birational equivalence.

Proof We assume V to be non-singular. Hence, consider a compactification of V with smooth boundary D. Then for m>0, @mV : V—•> Wis the D -\- X-canonical fibered variety ([4]). By performing a finite succession of monoidal transformations, we can assume f=0miV to be a morphism. fV=W is an irreducible constructible set of dimension a and K((D + K)\ VW, VW) = 0 for a general point weW. Since D\ Vw is a divisor of normal crossing type in Vw, we see from the definition that Proof of the uniqueness is easy and so is omitted.

q. e. d.

An irreducible constructible set is not so easy to handle. We will find a variety in W. Put F=fD and W°= W-F which is a Zariski open subset of W. Then D


Note that f\f~l W°

is proper. Let <J be a strictly birational map : V—*V. By definition, there is a proper birational morphism JJ, : V*—>V such that (7* = (7«/i is a morphism and that a compactification with smooth boundary Z)* has the following property : (i) ox = a*p is a morphism and (ii) ft:=$m,v* is a morphism. Then /*—/•/! and oxV* c F , and so ox~xT>cT>"". o induces the isomorphism ox : W—+W. Using fi~lD=D*

and /•ai^al»ft=al'f*jp,

we get

oJ{D) = aJ>(D*) = fa>(D*) = fD.

On Logarithmic Kodaira Dimension of Algebraic Varieties

185

Since al is bijective, it follows that al(W°)=&l( W— F)= W— F=W°. Thus, we obtain a group homomorphism pv : S Bir F-> Lin(W°) c Aut(W°) which is defined by f}v(a)= ox. The following exact sequence of groups is important : l->Ker pv->S Bir F - > I m /3->l (exact). One may expect that Im fiv is finite and that (Ker /3F)° is a quasi-abelian variety (see§ 7). A related problem will be partly solved in a forthcoming paper. T h e o r e m 6. Let V be an n-dimensional algebraic variety of hyperbolic type. Then S Bir V is finite.

Proof. By assumption, Ker /3F= 1. (Lin W°)° is a connected affine algebraic group G. If G is not trivial, then Gz)Ga or Z)Gm. We use L e m m a 2. Let V be an algebraic variety on which a connected algebraic group G acts. Then KV ^ icGp + dim V— dim Gp, where p is a general point of V and Gp is the G-orbit of p.

Proof There exists an admissible dense open subset V° of V whose quotient variety by G exists. Then we get a fibered variety : V°—>F°/G, whose general fiber is Gp. Hence by Theorem 4, KV^KV0

^ KGp+dim (V°/G).

q. e. d.

Therefore if G^>Ga, then icW°=— oo and if Gz)Gm, then /cW°<,n—l. On the other hand, jtWQ=icf-lW0 for f\f~lW° is proper and birational, and we get n = icV^ fcf-W° = icW0^

n-\.

This is a contradiction. Thus we finish the proof of Theorem 6. Remark. For a compact complex variety M and a closed analytic subset I in M, we can define K(M— I) similarly by making use of Hironaka's Main Theorems. Let Mx and M2 be compact complex varieties and Ix(zMx, Z2c:M2 closed ana-

186

S. Iitaka

lytic subsets. If dim Mj = dim M2=ic(M2—Z2)=n, then a meromorphic dominant (i.e., non-degenerate) m a p ^ : Mx — 2{-^>M2—221S the restriction of a meromorphic map / : Mx—>M2(see Sakai [6]). In this case, both Mx and M2 are Moishezon varieties and hence we can assume that these are algebraic. Thus S Rim(M-Z) is finite if it(M—I) = n.

If G is an algebraic group, we have the Chevalley decomposition : 1 —>^->G->W—> 1 (exact), in which $ is the connected maximal affine algebraic subgroup and A is an abelian variety. We assume icV^O. Then by Lemma 1 Aut(F) 0 cannot contain Ga. L e m m a 3. If a connected affine algebraic group ~§ does not contain Ga, then -§ is an algebraic torus Glm.

Proof. Since the radical is an algebraic torus,~§is reductive. Hence~§is a semi product of an algebraic semi-simple group and a torus. However, any non trivial semi-simple group Z)Ga. Hence~§tis a torus. q. e. d. Hence, if £F^>0, we have for any connected algebraic subgroup G of Aut(F) 0 , 1 -> Glm-> G-> A -> 1 (exact). A connected algebraic group A is called a quasi-abelian variety, i{ A is an extension of Glm by an abelian variety A as an algebraic group. L e m m a 4. A quasi-abelian variety A is commutative. Proof Let T e A and consider the group homomorphism : Since Glm is rational, WT: Glm-+ Glm. Hence which is discrete. Thus Wl=^WT, hence Glm is contained in the center of A. Moreover, if a, T € J, we get [r, a] = TOT-'O-'

e G'M9

since A is commutative. We have the right coset decomposition : d = I I pGlm.

Then [r, o]^\^pp~lT^ a] = [p, 0"], because p~lr e Glm for a certain p. Hence the morphism A 9 v h^ [r, o~] € Glm

factors through A—^G^, which is trivial. Hence [r, a] = \. This implies A is com-

On Logarithmic Kodaira Dimension of Algebraic Varieties

mutative.

187

q. e. d.

By this lemma, we can verify that H o m ( i , i ) and the set {algebraic subgroups of A} are countable sets. Assume that icV^iO. Then the quasi-abelian variety j^^Aut(F) 0 operates on V. By the following lemma, we see that Aut(F) 0 turns out to be an algebraic group. L e m m a 5. Let G be an algebraic group acting effectively on an algebraic variety V. Assume that the set of isotropy groups Gpforpe V is countable. Then there is a non-empty Zariski open subset V^aV such that Gp=l for p e V^. Proof. LetF= {(g,p) eGx V ;g-p=p] and let n be the projection FcGx F—>F. Then 7r~l(p)=Gpxp. Hence, there exists a non-empty Zariski open subset V° such that dim Gp < dim G for p e V°. Let G*=\^JGP where pe V°. By hypothesis, we can write

xF°=G* and xF° is a constructible subset of G, where it is another projection : F°=Gx V°nFczGx V°^G. Therefore it is easy to see that there is a finite set of subgroups Gl5 • • •, Gm of G such that *F° = U Gj. Hence Gj = Gp(j) for some p(j) € V°, and so dim G ;
* W

and consider V*=f-l(w), Vw=p(V*) for a general w z W. By the result in § 3, KVW SicVl = 0 and so icVw = 0, because icV^O. Hence J(p) = the j^-orbit ofpc Vw for a certain pair (p, w) of general points peV and weW. Thus n—icV^>dim J4 = dim Aut(F) 0 . Consequently, we obtain

188

Theorem

S. Iitaka

7. If /cF^>0, the Aut(F)° is a quasi-abelian variety and dim Aut(F)°

Proposition 5. If V is affine and ycF^O, then Aut(F) 0 is an algebraic torus. The proof is easy. Example 8. Let V=Pn—F, F being a Zariski closed set, and assume icV^>0 and dim Aut(F)°=w. Then Fz)G^. In the next section we shall prove V=Gnm. Remark. It may be true that the dimension of the abelian part of Aut(F) 0 is not greater than q(V)=the irregularity of V. 8 We shall prove Theorem 8. If icV^>0 and dim Aut(F)°=dim V=n, then V is a quasi-abelian variety. Proof By assumption, the quasi-abelian variety A=Aut(V)° operates on Fand A=A(p) is a Zariski open subset for a general p 6 F. Note that Ad V is an equivariant imbedding of A. Let V-^V be a normalization of F. Then since Aut( F) cAut( V), we have the Aequivariant imbedding ; AdV. Hence we assume F to be normal. By Reg F we denote the set of non-singular points of F. Then since Aut Vd Aut Reg F, we have the j^-equivariant imbedding : Ad Reg F. First consider the case in which A is an algebraic torus G^. By Sumihiro's Theorem 5 [see [5] p. 20], we have a finite covering of G^-admissible affine open subsets Uu •••, C/m, such that Reg V— U U5. .7 = 1

If Gm^Uj, then UjC^Gra X G^~r, r being positive, by Theorem 4 in [5] p. 14. Hence /cUj= — oo, which contradicts icUj^ic Reg F ^ ^ F ^ O . Thus Reg V=A and hence codim(F— A)^2. By the following lemma, we conclude that V=A. q. e. d. Lemma 6. Let V be an affine variety and suppose that an algebraic variety V contains Vas a Zariski open subset. Then codimF, [V' —V) = \ or V=V'. Proof. Let p e V —V and consider an affine open subset U of V. Then Uf] V is also affine. Hence we assume V to be affine. Moreover, we can assume that V is normal. Then if codim(V'-V)^2, we get T(F', 0 F , ) = r ( 7 , 0v) by the extension theorem. Hence F' = Spec r(V, 0 F ,)=Spec T(F, 0v) = V, a contradiction. Thus we conclude that V — V is purely 1-codimensional. (The author owes this proof to Nagata.) In general, A has the following decomposition : A being an abelian variety. Let u be a general point of A and let Tu=n~l{u). Consider the closure of Tu in F, which is written Tu. Then ^ i w ^ 0 , because

On Logarithmic Kodaira Dimension of Algebraic Varieties

189

Since T~Tuafu, we have a torus imbedding. In fact, if a € TczA(zAut(V), a operates on V. Hence aTu=Tu. By the previous argument, we conclude that TU=TU. Let v be an arbitrary point of A. Then we have some re A such that rv = u. Let ze7i'l(v). Then f e Aut(F) and ?TV=TU. Hence TV=TV. This implies A=V. q. e. d. T h e o r e m 9. Let V be an algebraic variety of dimension n. If Aim Aut(V)Q=n—aV, then V contains a Zariski open subset V° which is a principal A-bundle, where A = Aut(V)° is a quasi-abelian variety. V°czV is an A-equivariani imbedding and iiV® = iiV.

Remark. It seems interesting to know whether V= V° or not.

References [ [ [ [

1] 2] 3] 4]

[ 5] [ 6] [ 7] [ 8] [ 9] [10] [11]

Deligne, P. : Theorie de Hodge II. Publ. Math. IHES 40 (1973), 5-58. Hartshorne, R. : Algebraic de Rham cohomology, Manuscripta Math., 7 (1972), 125-140. Iitaka, S. : On Z)-dimensions of algebraic varieties. J. Math. Soc. Japan 23 (1971), 356-373. Iitaka, S. : On algebraic varieties whose universal covering manifolds are complex affine 3-spaces I. Number Theory, Algebraic Geometry, and Commutative Algebra, in honor of Y. Akizuki, Kinokuniya, Tokyo (1973), 147-167. Mumford, D. and others : Toroidal embeddings I. Lecture Notes in Mathematics, 339 (1973), Springer. Sakai, F. : Degeneracy of holomorphic maps with ramification. Inv. Math., 26 (1974), 213229. Sakai, F. : Kodaira dimensions of complements of divisors. In this volume. Ueno, K. : Classification theory of algebraic varieties and compact complex spaces. Lecture Note in Mathematics, 439 (1974), Springer. Wakabayashi, I. : On automorphism groups of P2— {curves}, in preparation. Peters, K. : Uber holomorphe und meromorphe Abbildungen gewisser kompacter komplexer Mannigfaltigkeiten, Arch. Math., 15 (1964), 222-231. Iitaka, S. : Logarithmic form of algebraic varieties, J. Fac. Sci. Univ. Tokyo, 23 (1976), 525544.

Department of Mathematics University of Tokyo (Received November 22, 1975)

On a Characterization of Submanifolds of Hopf Manifolds

Ma. Kato 0.

Introduction

The Hopf manifolds afford a quite elementary, though quite typical example of non-Kahler compact complex manifolds. Those were defined by H. Hopf in 1948, and investigated completely by K. Kodaira [3], [4] in the case of dimension 2. In this paper, we intend to study higher dimensional Hopf manifolds and their subvarieties. A Hopf manifold of dimension n>2 is defined to be a compact complex manifold of which the universal covering manifold is biholomorphic to the domain Cn— {0} ([3]). Any Hopf manifold contains nowhere discrete subvarieties. These subvarieties have rather special properties (§ 3, see also [2]), It is our aim to give a complete characterization of submanifolds of Hopf manifolds. In the case of dimension 2, the following result is known : T h e o r e m 1 ([1], [2]). A compact complex manifold S of dimension 2 is biholomorphic to a submanifold of a Hopf manifold if and only if S is of class F70, VII0-elliptic or a Hopf surface.

The proof of the theorem depends on Kodaira's classification theory of surfaces. The purpose of this paper is to give a sufficient condition for a compact complex manifold of dimension > 4 to dominate bimeromorphically a subvariety of a Hopf manifold (Main Theorem § 1). In § 1, we give basic definitions and the statement of our main theorem. In § 2, we recall some recent results due to Y-T. Siu [8], [9] and H-S. Ling [5], which play an essential role in the proof of our main theorem. In § 3, we recall some properties of subvarieties of Hopf manifolds and show that the conditions in our main theorem are satisfied if the compact complex manifold is actually a submanifold of a Hopf manifold. In § 4, we give the proof of our main theorem and a remark.

192

Ma. Kato

1.

Definitions and Statement of Main Theorem

Definition 1. By a Hopf manifold of dimension n>2, we shall mean a compact complex manifold of which the universal covering manifold is biholomorphic to the domain Cn~ {0}, where 0 denotes the origin of Cn ([3]). By a primary Hopf manifold, we shall mean a Hopf manifold whose fundamental group is infinite cyclic. Definition 2. Let X be a complex space0. Let g be a holomorphic map of X into itself with unique fixed point 0 e X, g(0) =0. Then g is called a contraction of X if g satisfies the following two conditions : (i) lim gv(x) =0 for all x e X, (ii) for any sufficiently small neighborhood f/of 0 in X, there exists an integer v0 such that gv(U)aU for all v>v0. Any Hopf manifold is a submanifold of a higher dimensional primary Hopf manifold ([2]). By Kodaira [3] p. 694, any ^-dimensional primary Hopf manifold is biholomorphic to the quotient space Cn— {0}/(G), where Gisa contracting holomorphic automorphism of Cn which fixes the origin. Let 7 be a connected analytic subset of a Hopf manifold. Then we can assume that 7 is isomorphic to an analytic subset of a primary Hopf manifold. Let j : Y—>Cn— {0} j{G) be a (closed holomorphic) embedding. It is easy to see that j induces a non-trivial homomorphism j * : ^1(7)->7r1(Cn— {0} j{G))~Zoi the fundamental groups. Denote by yG the element of 7tx{Cn— {0} I(G)) corresponding to the contracting automorphism G. Definition 3. An element y e TT^Y) is said to be j-contr-active ifj\y=y% for some positive integer m.

If a complex space is irreducible, we sometimes call it a variety. A line bundle L on a compact normal variety X can be regarded as an element of Hl(X, 0*). If L is in the image of Hl(X, C*)->Hl(X, 0*), then L is said to be flat, where C* denotes the constant sheaf of the multiplicative group C*=C—0. Note that a flat line bundle L canonically determines a group representation pL : Now we shall state our main theorem. Main T h e o r e m . Let X be a compact complex manifold of dimension w>4. Assume that there exists an effective divisor D on X which satisfies the following four conditions : (i) The support 7 of D is connected and biholomorphic to an analytic subset of a Hopf manifold, 1) By a complex space, we shall mean a reduced Hausdorff complex space.

On a Characterization of Submanifolds of Hopf Manifolds

193

(ii) The line bundle L=[D] is flat. (iii) Im pL~Z. Let i* be the homomorphism i*: IZI(Y)—^TCI(X) induced by the inclusion i: Y^>X, and pLtY =pL°i*(iv) There exists a {closed holomorphic) embedding j : Y—>Cm— (0}/(G) such that

WAr)\ < i for any j-cont?"active element j e itx (Y). Then there exists a bimeromorphic holomorphic map of X onto a subvariety of a Hopf manifold. 2.

Stein Completions and Extensions of Holomorphic Maps

A real-valued function

c*}. (ii) 7T is called (1, 1) -convex-concave if there exist a C°°-function

R and constants t:+, ct and c1 in 7JU { — oo} with c He £»}. In both cases the function

/J. A subset U of Z is called a neighborhood of infinity if there exists a constant ce R such that f/z) {£€ Z ;

We recall here some results obtained by Siu [8], [9] and Ling [5]. T h e o r e m 2 [9]. Let n : Z—*W be a l-convex holomorphic map. Then, for any point weW and any sufficiently small polyhedral domain U in W with center w, TZ~1(U) is a holomorphically convex space. T h e o r e m 3 [5]. Suppose that Z is an (n-\-\)-normal^ complex space and that W is a 1) See [5] for the definition. Note that if X is an irreducible normal complex space of dimension n, then X is /^-normal for any 0
194

Ma. Kato

Stein space of dimension n. Suppose that n : Z—>W is a (1, I)-convex-concave holomorphic surjection with exhaustion function

R which is s-psh on the whole Z. Then there exists a unique (w+1)-normal Stein space Z with the following properties : (i) Z is an open subset of Z, (ii) Each irreducible component of Z intersects Z, (iii) There exists a holomorphic map n : Z—>W such that 7r|Z=7r, (iv) 7t\{Z— {zeZ ; ^(^) > £}) is proper for any ceR.

Note that, by the construction of Z, the set (Z— {zeZ ; (f{z)F(Z, 0) is bijective. Let Ki : Zi-^U be 1-convex holomorphic surjections with exhaustion functions A:+3, (e) there exist neighborhoods Nt of infinities of Zt and a finite ramified covering h : Nl—±N2 which makes the following diagram commutative :

u. Under these conditions we shall prove the following two lemmas. L e m m a 1. There exists a unique proper holomorphic surjection h : ZX—>Z2 such that h agrees with h on a neighborhood of infinity of Z, and that the diagram

is commutative.

Proof By (/3), we can consider the Remmert quotient rj : Zj—»Q^, where Q is a

On a Characterization of Submanifolds of Hopf Manifolds

195

normal Stein space. The surjection rj is proper and every fibre of rj is connected. Since C/is Stein by (a), there exists a unique holomorphic map a : Q-^Usuch that oor)=nx. Since nx is 1-convex, rj is a biholomorphic map on M c = {££ Zj ; c} for some ce 72. We can assume that McaN{. Put Qc = r](Mc) and ^=(^| M c )- ! . Define r' : Q^C-^Z2 by r'=hoX. We shall prove that r' extends uniquely to a proper holomorphic map r of Q^onto Z2. By (7-), there is a closed embedding z : Z2—>Cl. We identify Z2 with i(Z2)czCl. Then r' : Qc-+Z2 is a (vector-valued) holomorphic function on Qc. It is clear that o\Qc: Q^c—>t/is a(l, 1)-convex-concave holomorphic map with exhaustion function A*^ : Qc—»(£, +00) and that £Hs the Stein completion of Q^c. By (8) and Proposition 1, we can extend zf to a (vector-valued) holomorphic function r : Q-+C1. By the uniqueness of the extension we also have l (J=7:2OT. Now we shall prove that r is proper. Take any compact set K in C . { Since Kf]Z2 is compact, aor~ {K)=n2{K f|Z2) is a compact subset in £/. Put H=7r2(KnZ2).Fix a constant rf such that dye. Then r 1 (X)cr'- 1 (r(Q ( i ) n ^ ) (J (Z2 is an open embedding and r ( Q d ) n ^ is a compact subset of r'(Q^), r'~1(r(Q d) fl^) is a compact subset of £^c. It is clear that <7~l(H) n ((£—Q,d) is compact. Hence r - 1 (^) is compact. This proves that r is proper. Put Z 3 =r(QJ, which is a closed analytic subset in Cl by the proper mapping theorem. Furthermore r(Q^ c )cZ 2 nZ 3 . Hence by (s) every irreducible component of Z3 is contained in Z2 in a neighborhood of infinity. Hence we infer that Z2=Z3. Now define h^zorj. Then, by the above argument, h is a unique proper surjection which agrees with h on a neighborhood of infinity in Zx such that TTI = n2°h. q. e. d. L e m m a 2. If we further assume that Z, w a Sta'w J/?«^ awrf that h is biholomorphic, then h is also biholomorphic.

Proof In the above proof, rj : ZX-*Q, becomes an isomorphism. Clearly r' is an open embedding. Therefore r : Q—+Z2 is biholomorphic outside an analytic subset S such that dim S r\0~l(t)=O for any te U. Since Z2 is normal, we infer that r is biholomorphic on the whole of Q by Zariski's Main Theorem. Therefore h = ro7] is biholomorphic. q. e. d. 3.

Subvarieties of Hopf Manifolds

In this section we recall some results in [2] on subvarieties of Hopf manifolds and discuss further properties of the subvarieties (see also [1]). Propositon 2 [2]. Let V be a complex space and g a contracting holomorphic automorphism of V with a uniquefixedpoint 0. Then the compact complex space V— {0} I (g) is biholomorphic to an analytic subset of a Hopf manifold. Proposition 3 [2]. Any n-dimensional subvariety of a Hopf manifold contains subvarieties of arbitrary dimensions less than n.

196

Ma. Kato

Proposition 4 [2]. Let V be a normal subvariety of a Hopf manifold and D an effective Weil divisor on V. Assume that dim V>2. Then there exists an effective Weil divisor E on V such that D-\-E is a Cartier divisor and that the line bundle [D-\-E\ is flat and has the transition functions which are some powers of a certain constant a € C * ( | a | < l ) .

Proposition 4 is proved as follows. Let j : V—»Cm— {0}/(G) be an embedding, where G is a contracting automorphism of Cm with G(0)=0. We identify V with j(V). Let ft: Cm— {0}—>Cm— {0}/(G) be the canonical projection. Put V=iTx(V). By a theorem of Remmert-Stein, V extends to an analytic subset V in Cm. Moreover V is G-invariant. Let y ^ f ^ S u p p D) U {0}, which is a G-invariant analytic subset in F. Then we can show that there exists a holomorphic function f on V which does not vanish identically on any irreducible component of F, and which satisfies/| Fj = O and

(1)

G*f=af

for some a e C*(|a|< 1). Then some power fk (£>0) of/defines an effective Cartier divisor Dr on Fsuch that E : =D'—D is effective. Moreover the line bundle [Df] is flat and has the transition functions which are some powers of a. Denote by Y the zero-locus of/. Let Z : = V- Y, Y: =Supp D'= Y- {0}/(G>, Z : = F - F , and 7?0=fc\Z. L e m m a 3. f: V—+C is a holomorphic surjection.

Proof Take a point x of Y and a small neighborhood N of x in V. Let C be a 1-dimensional complex subvariety in N such that C f] (Ffl N) = {x}. Since f\C : C—+C is not constant, f\C is an open map and f(C) contains 0 € C Therefore f(C) contains a small disk U with center 0. Hence f (V) 3 U. Hence, by (1), q. e. d. L e m m a 4. f induces a holomorphic surjection f* : Z—»J :

=C*I(a).

Proof Clear by Lemma 3. L e m m a 5. For any point red and any sufficiently small open disk U with center r, the holomorphic surjection/^l/*1 (U) :f*l(U)^>Uis \-convex with respect to some exhaustion function
Proof Letjfr:C*—>J be the canonical projection. Fix a point toep'l(v)c:C*. Let £ be a sufficiently small positive number such that the set U : = {te C; \t—to\ <£} is contained in C* and that p\U : (7—>J is one-to-one. Put U=p(O). Then 7f gives a biholomorphic map between T: = {ze V; \f{z)—to\<£} ( c Z ) and f*l(U). Hence f*l(U) is a Stein space. Now we define
On a Characterization of Submanifolds of Hopf Manifolds

197

Proposition 5. 7=Supp D' is connected. Proof. Assume that there exist non-empty analytic subsets 7l5 72 such that 7 = 7, U 72 and 7, fl Y2=j>. Let Nt be tubular neighborhood of Yi such that NY f\N2=. Let N\:=Ni—Yi. By Lemma 5, we can find a finite open covering {J Ua of J a

such that, for each a,f*\fil(Ua) :f-l(Ua)^>Uais 1-convex with respect to exhaustion function <pa and, moreover, /~ l (U a ) is a Stein space. Let Na= {x e fll{Ua) ; <Pa(x)>ca} be a neighborhood of infinity in f~l(Ua) with respect to <pa. Taking ca to be large enough, we can assume that Na(zN°lDN02. It is easy to see that iVan N?^0for i=\, 2. Hence Na is not connected. Since dim f-l(Ua)> dim i + 3, by Proposition 1, the restriction map F(f*l(Ua), Q)^>r(Na, 0) is bijective. This implies that f-l(Ua) has at least two connected components. Let Zla be the union of the connected components of f~l(Ua) which intersect N°t. Put Zi = [jZia. Then a

Zl (i= 1, 2) are non-empty, open and closed subsets of Z, and Z1 fl Z2=. Hence Z is not connected. This is a contradiction. This implies that 7 is connected. q. e. d. Corollary 1. If X is a submanifold of a Hopf manifold, then there exists an effective divisor D on X satisfying the four conditions (i)-(iv) of Main Theorem.

Proof. Df satisfies the conditions (i)-(iii) by Propositions 3, 4 and 5. In the proof of Proposition 4, we see that the condition (iv) is also satisfied. q. e. d. 4.

Proof of Main Theorem

Corresponding to the kernel of pL : nx(X)—*C% we form the infinite cyclic unramified covering 73 : X—+X. Then there exists a holomorphic automorphism g of X such that Zfog=Tf and X/(g)=X. Put 7='D'- 1 (7). Let Yv(v = 0, 1, •••) be connected components of 7. Let i: Y—>X be the inclusion. Put 7XV=7X\¥V and ^ = 2*11^. Consider the commutative diagram : Im p

where the horizontal arrows are exact. If 7 had infinitely many connected components, then 73Vif would be an isomorphism. Then, by (2), pY={po73^)o[iv^o73~^) = I. This contradicts condition (iv). Hence the number of connected components of 7 is finite. Let ?=\J Yv. Then gh acts on each Yv and ?vl(gh) = Y. Lemma 6. Ker Proof Let jeit\{Y) be an element such that j\f^

1. Then either j of y~l is j -

198

Ma. Kato

contractive. Hence by condition (iv) we have \pY(j)\=fcl. This implies j$ Ker pY. q. e. d. Consider the following commutative diagram : X<

^

Yo

^

Y

-

*> Cm~{0\

*- C m -|0)/,

where ft is the canonical projection. L e m m a 7. There exists a lifting j 0 : Fo—>Cm— {0} of j . Proof. It is sufficient to show that 7 * 0 ^ : 7r](Y0)-^7il{Cm— {0}/(G)) is trivial. By (2), pYo-tf0^=poZf^oi^=l. Hence ^ 0^(^,( f o ))cKer pY. Hence, by Lemma 6, f q. e. d. Note that j0 is an embedding of Yo such that jo°gh = G£°jo> where e is not equal to zero. Replacing g by g~l if necessary, we can assume that (3)

Joogh = G'oJ0

(e>0)

on

f0.

L e m m a 8. There exists a holomorphic function f on X which has the following properties : (a) ? = {xe X;f{x) = 0} as a set, (b) g*f = a f for some constant a e C* with \a\ < 1, (c) / : X—+C is a surjection. Proof (cf. [3] p. 701) Let U= {Uj} be a covering of X by small open sets and represent the line bundle L as a cocycle {a^*} e Zl(U, C*), {mjk} e Z[(U, Z), where a0 is the generator of Imp such that |a o |
on Uj H Uk.

Then

is a meromorphic 1-form on X. Define a multiplicative multi-valued holomorphic function fm on X by

j

(xoeX).

Then it is easy to see that f=Zf*fm is a (single-valued) holomorphic function on X satisfying (a). It is clear that g*f=alof for some non-zero integer /. Let 0 be a

On a Characterization of Submanifolds of Hopf Manifolds

199

path in f0 with the initial point pe Yo and the terminal point gh(p) € Yo. Clearly 0 :=-&-(#) is a closed path in Y. Theng*hf= Uxp (7})f=p([0]) f, where [0] denotes the element of' TTI(Y) represented by 6. Since [6] is clearly 7-contractive by (3), \p([0])\
b y c o n d i t i o n (iv). H e n c e \a%l\ = \p([0])\<\.

T h i s implies \alo\<\.

Put a =

l

a 0. Then we obtain (b). The proof of (c) is the same as that of Lemma 3. q. e. d. Let (£l5 •••, Zm) be a standard system of coordinates on Cm with 0 = ( 0 , •••, 0). Let r

(z) — S \Zi\2> O n Fo, we introduce an s-psh function r0 by ro=j£r. By Richberg [7],

r0 extends to an s-psh function p on a certain neighborhood N'o of Yo. We can

assume that N'ongv(N'o)= for all v, 00. Put Uc={xeNf0; p(x)0. Fix £>0 such that (i) f: Kr n f~l(D£)^>D£ is (1, 1)-convex-concave, (ii) K'p\f-l(D£) is relatively compact in Nf0, and

(iii) ^(9O;n./-1(A))cf7 cn/- | (A).

Put K=K' p\f~l(De). By Theorem 3, there exists uniquely an irreducible normal Stein completion Sof K. L e t / : S—>D£ be the extension of/. It is easy to see that the sets Ex= {xeK;p{x) < c] U(S-K) and E2= [{xeR;p(g-<(x))
Note that Tt is a neigborhood of infinity in Et with respect to ipu j=l, 2. Since gl '• Tx—+T2 is an isomorphism, we obtain, by Lemma 2, a biholomorphic map

200

Ma. Kato

G : El—>E2 such that G agrees with gl on a neighborhood of infinity. Hence the following diagram is commutative :

(4)

Now we regard G as a biholomorphic map of Ex into itself. L e m m a 9. n ^ m ( ^ i ) = {0} for a certain point 0 ef~l(0)

f] Ex. Hence G is a contrac-

tion of Ex.

Proof. Since E{ is biholomorphic to the relatively compact open subset E2 of the Stein space El9 Ex can be embedded in an affine space of a certain dimension as a closed analytic subset. Hence, for each m>0, Gm can be viewed as a (vectorvalued) holomorphic function on Ex. Since, for m > l , Gm(El)ciE2 is relatively compact in EY, the sequence {Gm} m7,x is uniformly bounded on Ex. Hence we can choose a subsequence {G™-7}^ of {Gm} m^{ which converges uniformly to a holomorphic function u on E2. Put A = (~^\Gm(Kn Yo) and denote by dA the boundary of Then clearly u(f-l(0)nE2)=dA. Since dA is compact, this implies A inf-l(0)nEl. that the absolute value of each component of u\f~l{0) C\E2 attains its maximum on the open set E2. Hence u\/-l(0)r\E2 is constant. Since u(E2)af~l{0)C)E2, we infer that uou is a constant function. Let & = u(u(E2)).t ThennG m (£,)enG 2m ^( J E 1 ) = {0}. Since Gm'{El)z)Gm'{El)i^(f> for ml<m2, we have n G ™ ^ ) ^ . Hence it follows t h a t n ^ m ( ^ i ) = {6}, where Oef-^fynE^

q. e. d.

L e m m a 10. Tfer^ existfor some m an open holomorphic embedding I: E{—>Cm(I(0) = 0 = ( 0 , •••, 0))^ <2 contracting holomorphic automorphism G of Cm with G ( 0 ) = 0 , ^ G-mvariant normal subvariety V
(ii) /oG=Go/, and (iii) / = F o / flwrf G*F=alF. Proof We can prove this by method similar to that in [1], Here we explain the outline of the proof. Let U' be a sufficiently small neighborhood of 0 in E{ such that there exists a neat embedding j r : U'—*Cm{j'(0)=0) (m=the dimension of the Zariski tangent space at 0), where j'(U') is an analytic subvariety of an open subset of Cm. Moreover, by Lemma 9, we can find an integer i>0 such that GV(U') c Ur for all v>v0. Put U" = VC)GV(U'). ThenG(U")(zU".

Letj"=j'\U".

T>utG'=j"oGoj"-K

v=0

Then, since j " is a neat embedding, G' extends to a local biholomorphic map at 0 6 Cm. G' induces a linear automorphism of the tangent space at 0 € Cm. We can show that the absolute values of all eigenvalues of the linear automorphism are less

On a Characterization of Submanifolds of Hopf Manifolds

201

than 1. Hence, by Sternberg [10] and Reich [6], there exists a local coordinate transformation r at 0 (r(0)=0) such that G :=TOG'OT-1 automatically defines a contracting automorphism of Cm with G(0)=0. Yutj=Toj". Then, taking a small neighborhood U{(zU") of 0, we find an embedding j : U—>Cm(j (0)=0) such that joG=Goj, G(U)czU, and that j(U) is an analytic subvariety of an open subset of Cm. Now put Iv = G~vojoGv (v>0). The holomorphic map Iv is defined on G~V(U) C\EX and Iv=Iv+l on U. Since Ex = {JG~V(U) n£"i, we obtain by the analytic continuation of {Iv} a holomorphic map I: Ex->Cm which satisfies (ii). Put V={jG~v(j(U)).

Then it is easy to see that V is

m

a closed (^-invariant analytic subvariety of C . Moreover, since E{ is normal, V is also normal. It is clear that / : E—*V is locally biholomorphic. To prove (i), it suffices to show that / is one-to-one. Let pl9 p2 € E{ be points such that I (pi) =I(p2). If*(>0) is sufficiently large, Gv(p{) and Gv(p2) are contained in U. Thenj(Gv(pl)) = Gvol(px) =GvoI(p2) =j(Gv(p2)). Since j is one-to-one on £/, we have Gv(p,) =&(p2)Therefore p{=p2. Thus / is one-to-one. Next we define the holomorphic functions Fv =

a-vlfoj-'oG\

Each Fv is defined on G~v(j(U)), which is an open subset of V. Since V= {jG~v(j(U)) and Fv=Fv+l on j(U), we obtain by the analytic continuation of {Fv} a holomorphic function F on V. By (4), it is easy to see that F satisfies (iii). q. e. d. Put B=(Uc-g*iUc))nf-l(Dla]l£), No = Qg^(N'o). V= f-

£ 0 = the interior of B, N'0=\Jg»l(B), and

Then No is a neighborhood of Yo such that gh(\) =ft0. Put

{0}, W=F~l(0) and W= W- {0}.

L e m m a 11. There exists an open holomorphic embedding J: No—+V such that

(ii) J o
(iii) f=FoJ.^ Proof. Let i : K—>E{ be the inclusion. Put j=Ioi. Then 7 : i?-^F is an open holomorphic embedding. For each integer v, define on g~vl(B) the mapping j v into V by (veZ). Jv = G-vojog" Put Cv = g~vl(B) ng~iv+l)l(B). By the definition of G, G is automatically defined on a small neighborhood of Co and satisfies (5)

iog1 = Gof.

By Lemma 10, we have (6)

GoI=IoG.

Ma. Kato

202

Combining (5) and (6), we have Goj=jog' (7) on a neighborhood of Co. Let x be a point near Cv. Then j\+l(x) = G-<*+»ojog™»(x) = = G-»°j{y)

G-^ojog^y)

(y=g«(x)) (by (7))

Since {Cv} veZ are disjoint from each other, by the analytic continuation of {_/'„}, we can define a holomorphic map J: N0-^V. It is clear that J is locally biholomorphic and that (ii) holds. In order to show that J is one-to-one, it suffices to show that J(gu(Bo))nJ{Bo)=t for a l b > 0 . Clearly G*{i{B0))^6{Ex)=E2. On the other hand i(B0) r\E2=$. Hence (3i'(i(B0))r)i(B0)=$. Since 1 is one-to-one by Lemma 10, we have t=Io&(i{Bt))nI(i(Bt)) = G>oJ(B0)nJ(B0)=J(g«(B0))nJ(Bt). It is easy to see that (i) and (iii) hold. q. e. d. Put Nv=gv(N0) (0 V by Jv=Jog-». Note that Jv+log=Jv for 0
N=Zfl(tf)9

NV=^{NV), M=7t(M), Yv=zf\{t), W=x(W), Z^X-t Z2=V-W=V-W, Zx = Tfx{2x), Z2=7t(Z2), i V ? - ^ ( i V - f ) , and # ? = * ( $ - W). Note that each 7y is the A0-fold cyclic covering manifold of Y. The set Nv is a neighborhood of Yv such that Nvp[Nv,= for i ^ i / . Denote by J the elliptic curve defined by C*l(al) and by p the canonical projection C*—>^. By Lemma 4, we have holomorphic surjections/* : Zj—->J and F*\ Z2—*A, where f* and F* are the maps induced by the holomorphic functions f and F defined in Lemma 8 and Lemma 10, respectively. By Lemma 11, we obtain easily L e m m a 12. {Jv} induces an h-fold unramified covering J: N—*M such that J\YV: Yv—>W(0
Now we shall prove the following L e m m a 13. The covering map J\N0Y : N1—+NI extends uniquely to a proper holomorphic surjection J: Z{-^Z2 which makes the following diagram commutative :

On a Characterization of Submanifolds of Hopf Manifolds

203

Proof. Let re/1 be any point. By Lemma 5, for any small disk U in A with center r, Fv : =F^\F~l(U) : F" 1 (£/)—>£/is a 1-convex holomorphic surjection with some exhaustion function , on the whole offil(U). Put fu=f*\fll(U). Then fu :f-l(U)—>f/is also a 1-convex holomorphic surjection with exhaustion function ^l8 Hence, by Theorem 2, shrinking £/ if necessary, we can assume t h a t / ; ' ( £ / ) is holomorphically convex. Then, by Lemma 1°, there exists a unique proper holomorphic surjection Jn :f~l(U)^>Fil(U) which agrees with y|iV? on a neighborhood of infinity in f~l(U) and makes the following diagram commutative :

Patching together these pieces J: Zl—+Z2 of the lemma.

^ we obtain the proper holomorphic surjection q. e. d.

L e m m a 14. There exists a proper holomorphic surjection 0{ : XXProof. Define Ox to be

It is clear by definition that / = j on N°t = NftZ^

q. e. d.

Take the Stein factorization of
(8)

where 1) Apply Lemma 1 to each connected component of

204

Ma. Kato

(i) V, Vo are compact normal varieties and 0, 0O are proper surjections, (ii) each fibre of 0 is connected, and 0 is biholomorphic outside a proper analytic subset of X{, and (iii) each fibre of 0O is discrete, i.e., 0O is a finite ramified covering. Now we shall prove the final lemma. L e m m a 15. There exists a bimeromorphic holomorphic map of X onto a subvariety of a Hopf manifold.

Proof Lifting 0X to 0X : X—> V, we have the following commutative diagram by (8):

where Vo is the Stein factorization of 0{ : X—>V. Since every fibre of 0 is connected and Vo contains no positive dimensional compact analytic subset, the automorphism g of X induces an automorphism Go of Fo, which satisfies Goo0=0og. Hence 0 induces W : X—+V0/(G0). Note that ¥ is biholomorphic outside a proper analytic subset of X. Hence, to prove the lemma, it suffices to show that Vo/ (Go) is a subvariety of a Hopf manifold. Now 0O: V0-^>V is a finite ramified covering. The images of the ramification loci in V extend to complex subvarieties in V= V\J {0}. Then, by virtue of the continuation theorem of Grauert-Remmert, we can obtain a normal variety V0=V0\J {0} attaching one point 0 to Vo and a finite ramified covering 0O: Vo—*Fsuch that 0O\VO=0O. Moreover the automorphism Go extends to an automorphism Go such that G 0 (0)=0. Since Gl0 induces the contracting automorphism G of Fsuch that G(0)=0, and since the number of points of the fibres of 0O is bounded, Go is a contraction of VQ. Hence F0/(G0) is biholomorphic to a subvariety of a Hopf manifold by Proposition 2. q. e. d. Remark. Let X be a compact complex manifold of dimension n>3. Assume the following : (i) n^X)~Z, (ii) X contains a primary Hopf manifold Y=Cn~l— {0}/(G) of dimension n—l, where G is a contracting holomorphic automorphism of Cn~l with G(0)=0, (iii) [7] is flat. Let fa €7^(7) be the generator corresponding to G. In this case, we can easily see that the contractiveness of the elements of TT,( 7) does not depend on the choice

On a Characterization of Submanifolds of Hopf Manifolds

205

of embedding of 7. Put p=plYy There are three cases to be considered : (b) \p(ro)\ > 1, (c) \p(jo)\ = 1(a) \p(ra)\ < 1, Case (a). If rc>4, then all conditions of Main Theorem are satisfied. Hence X bimeromorphically dominates a subvariety of a Hopf manifold. If n = 3 , we have no answer. Case (b). We shall give an example of such manifold X. Let [Zo' Zi''•-'• Zn-i] be homogeneous coordinates on the projective space Pn~K P u t O' = [l : 0 : ... : 0] a n d Pn~2=

{z=[z0'-

Zx : ••• : zn-x]^Pn~l

I Z0=0] • L e t X be

n l

the open subset of P ~ X C defined by X = ( P ^ - 1 - {0'}) x C-Pn Define a holomorphic automorphism g of X by

2

x {0}.

We shall show that the quotient space X=X/(g) is a compact complex manifold which satisfies (i)-(iii) and (b). It is easy to see that X is simply connected. Consider the following function defined on X, ra-l

Then

(9)

4

Hence the action of g on X is properly discontinuous. It is easy to see that the set (10) | f e ^ ( P » - ' - { 0 ' ) ) x C - ( P " - 2 x (0});l xx(X) induced by the inclusion i: Y^>X is trivial. Let TS : X—+X be the universal covering of X. Let g be the holomorphic automorphism of Xsuch that Xj(g)=X. LetfeF(X, O([Y])) be a non-zero section whose zero locus coincides with Y. Then /==-&*/is a single-valued holomorphic function

206

Ma. Kato

defined on X. Taking g~l instead of g if necessary, we can assume that g*f=af (a € C*, |a|< 1). By the same argument as in Lemma 3, we can show t h a t / : X—>C is a surjection. Put Z=X—Y. Then /induces a surjection/* : Z—+A : = C*/ (ex). Let N be an open tubular neighborhood of 7. Since i+ is trivial, we can find a singlevalued branch^ o f / o n N. Put Dr= {te C; |*|J maps the set Dlal-ie — D£ onto J, j&oyj maps N—N' onto J. Hence / ' * : Z'—>J is a surjection, where f'^=f^\Zf. Let r e d be any point. Then f~l(r) a.ndf'~l(v) are analytic subsets in Z and Z', respectively, such that (11)

/;-'(r)=/;'(r)nZ'.

Let 5 be any connected component o f / ^ ( r ) such that Sr\dN'^t, where dN' denotes the boundary of N'. Since f^(S)=r9 / is constant on S. Hence we have SC-fTl(fi(P))> where/^Sn^N'. Therefore in particular we have SadN'. Hence, by (11), f'*l(z) coincides with the union of all connected components of/^ ! (r) contained in Z'. Since Z' is relatively compact in Z, this implies that/'^^r) is compact. Thus any fibre of/'* is compact. Hence, since J is compact, Z1 is also compact. But this is clearly absurd. This shows that case (c) does not occur.

References [ Y] Kato, Ma. : Complex structures on S1 x S\ J. Math. Soc. Japan, 28 (1976), 550-576. [ 2 ] Kato, Ma. : Some remarks on subvarieties of Hopf manifolds. A symposium of complex manifolds. Kokyu-roku. 240 (Kyoto Univ.) 1975. [ 3 ] Kodaira, K. : On the structure of compact complex analytic surfaces, II. Amer. J. Math., 88 (1966), 682-721. [ 4 ] Kodaira, K. : On the structure of compact complex analytic surfaces, III. Amer. J. Math., 90 (1968), 55-83. [ 5 ] Ling, H-S. : Extending families of pseudoconcave complex spaces. Math. Ann., 204 (1973), 13-48. [ 6 ] Reich, L. : Normalformen biholomorpher Abbildungen mit anziehendem Fixpunkt. Math. Ann., 180 (1969), 233-255. [ 7 ] Richberg, R. : Stetige streng-pseudokonvexe Funktionen. Math. Ann., 175 (1968), 257-286. [ 8 ] Siu, Y-T. : A pseudoconcave generalization of Grauert's direct image theorem, I, II. Ann. Scuola Norm. Sup. Pisa 24 (1970), 278-330. [ 9 ] Siu, Y-T. : The 1-convex generalization of Grauert's direct image theorem. Math. Ann. 190 (1971), 203-214. [10] Sternberg, S. : Local contractions and a theorem of Poincare. Amer. J. Math. 79 (1957), 809-824.

Department of Mathematics Rikkyo University (Received December 16, 1975)

Relative Compactification of the Neron Model and its Application

I. Nakamura Introduction

In this article we shall define an analytic Neron model, i.e., an analytic counterpart of Ne'ron's minimal model [11], and prove the minimality of it among principal homogeneous spaces (§ 2, 3). This portion deals with a partial generalization of the notion of analytic fiber system of groups and related results in [4]. Secondly we shall relatively compactify an analytic N£ron model by applying the theory of torus embeddings [3], [5]. One should recall that the usefulness of a relative compactification of Ndron model has been conjectured in [11]. The original construction in [6] was given in a slightly different, elementary, however rather complicated form. To facilitate better understanding, we employ here the notations of torus embeddings. Kodaira has made a deep investigation of elliptic fibrations of surfaces [4]. Iitaka [2] and Ogg [12] gave a numerical classification of singular fibers in a pencil of curves of genus two. Namikawa and Ueno [8], [9] gave their complete classification and made a systematic study of them, following in principle Kodaira. The final objective of this article is to construct a family of reduced singular fibers of genus two in a systematic and geometric way by making use of a relative compactification of the Neron model (§ 5, 6, 7). In this respect, the present article should be viewed as a continuation of [9], which its authors had intended to write at that time. Namely it involves a part of the construction of a family of curves of genus two of parabolic type (cf. [9] for the terminology). Combining the results in [9] with those here, one could construct all singular fibers (§ 8). The results in § 5, 6 and 7 have given the author a clue as to how to introduce a concept of stable quasi abelian variety (7. 4) (cf. [7] [10]). The latter was first obtained by Ueno in the 2-dimensional case [13]. Although the main part of this article was written in Nagoya in 1973, it has been completed during author's stay at the Mathematical Institute of Bonn University in 1975. The author would like to express his hearty thanks to Professor Hirzeburch and other people in Bonn University for their hospitality and kindness in inviting him there.

208

I. Nakamura

Notation

We denote by Z, R and C the ring of integers, the field of real numbers and the field of complex numbers respectively. Also we write C* = C— {0}, D={seC; \s\<e}, D'=D— {0}, e(#)=exp(2;rt x). The Siegel upper half plane g is, by definition, (&g= {Z ;gxg symmetric matrix, Im Z > 0 } . Then the Siegel space ©J is defined as the quotient space (Bg/Sp(g, Z). 1.

Preliminaries

(1.1) First we give preliminaries from the theory of torus embeddings (cf. [3], [5]; compare also [10]). By an rc-dimensional algebraic torus over C we mean the group scheme T= Spec R, R=C[w{, zflf1, •••, wn9 w~1]. Let M and N be the group of characters of T and the group of one parameter subgroups of T respectively. Both M and N are isomorphic to Zn. An element of M can be expressed as wr=J\wriii r={rt) eZn, whereas an element of Ncan be expressed as Xa(t) = (tai), a=(a^) e Zn. M and iVare canonically dual to each other with respect to the pairing < , > : MxN

>Z

(r, a)

• = S r ^ .

Denote M(x)jfj and N^)R by MR and NR respectively. Definition (1. 2) (1) An (affine) torus embedding is an (affine) scheme X of finite type, endowed with an action of T and containing T as an open orbit. (2) A morphism of two torus embeddings/: X{—>X2 is a morphism of schemes satisfying the conditions (i) and (ii) : (i) /induces a group epimorphism/' : TX-^T2 of tori (ii) the following diagram commutes :

where the horizontal arrows denote the action of tori. By a convex rational polyhedral cone a (abbr f c. r, p. cone) in MR (or NB) we mean a

set of the form a = {x € A/^or NR) ; /<(*) ^ 0, i = 1, ••-, r}, lt being linear functionals over Q. In what follows, we deal with only c. r. p. cones a containg no other linear subspaces than 0. Denote by a the dual cone with respect to the pairing < , > . Definition (1. 3) (1) A rational partial polyhedral decomposition of ^ ( a b b r . r. p. p.

Relative Compactification of the Neron Model and its Application

209

decomposition) is a collection (5= {0a} of c. r. p. cones aa in NR such that (i) then a is in *5 ; and (ii) for all aa, and c^, oa D ^ is in t5. (2) A morphism of an r. p. p. decomposition {aa} of A^ to another {a'fi} of iV^ is a Z-homomorphism a : N^>N' of finite cokernel which maps each oa into some

For a given c. r. p. cone a in iV^, we denote Spec C\o fl Af] by Xa, where the ring C\pP\M] is by definition the ring generated by monomials wri reaf]M9 over C Then Xa is a normal affine torus embedding. If o^Oo, then J ^ can be canonically embedded into Xc%. Through this embedding we identify Xai with an open subscheme of XO2. Hence, if we are given an r. p. p. decomposition t?= {oa}, we have a normal complex space X6 locally of finite type which we call the torus embedding associated with {?. A morphism between two torus embeddings (locally of finite type) is defined in the same way as in (1. 2). Moreover we have T h e o r e m ( 1 . 4 ) . The correspondence {aa} —*X(Oa) defines a faithful functor from the category of r. p. p. decompositions to that of normal torus embeddings locally of finite type.

(1. 5) We consider an r. p. p. decomposition t5= [o] of iV^dim NR=g+l) satisfying the following conditions : (1) there exists a g' (>0) not greater than g such that any a is of the form a = {{u} x,y) e 7J*+1 ; x = 0 € R*"', {u,y) e a'}, a' e & where *5'= {of} is an r. p. p. decomposition of N'R, dim NR—g'-\-l, and max (dim a) = g'. (2) t? is invariant by transformations Tk of NR defined by Tk(u) = u, Tk(x) = x, Tk(y) =y+ku. (keZgf) (3) any one-dimensional cone in (5 is one of rk, Tk

= {[u, x,y) ; x = 0, y = ku, u ^ 0}, k 6 Zg'.

Denote by (50 the subdecomposition of <5 consisting of all one-dimensional cones in t5 and {0}. Let X$ (or X€o) be normal torus embeddings associated with t5 (or t?0) respectively. X6o is an open subscheme of X6. We denote by Xa the normal affine torus embedding associated with a. We express X{0] as Spec C[s, s~\ zt\ wf1]Here s corresponds to a character u, and z% (or Wj) corresponds to the coordinate xt (or yj) of x (or ofj). In view of (3), Xa has a regular projection to C^Spec C[s] if dim (7>0. Hence we can define Pa = XaxD,

dk^

/>'(or/>„) = &'(or

PTk

(dim dm)=XnxD'

h We denote the canonical projection from Pa to Z) (or from Pm to Z)') by n. We call

210

I. Nakamura

n : Pz—*D the relative compactification of it: Q^-^D associated with r5. (1.6) To close this section, we shall give an example 50 satisfying the conditions in (1. 5). Let ak be a c. r. p. cone defined by ak = {(u,y) e R*+l ; u-y\ ^ 0 , / , - / < + 1 ^ 0 and °\ = {(u,j>) e Rgl;

u-y'e(l) ^ 0,y e ( i ) -y e ( i + 1 ) ^ 0

wherey'i=yi—kiU, e a permutation of g letters and k=(kt) eZg. Let <2) be the collection of a\ and their faces. The set-theoretical union of all a\ covers /Jo+ X Rg, where i?0+= {w 6 /J ; w^O}. It is easy to check 3) satisfies (l)-(3) for gf—g. The fiber n~x(Q) of P^ consists of infinitely many rational varieties, each of which is isomorphic to one and the same J. J = Jg is a compact normal torus embedding of finite type, associated with a finite r. p. p. decomposition {jGe and their faces) where jOe=a\...ltQ...Q. For instance A2 is isomorphic to a protective plane blown j

up at three distinct points.

2.

Existence of a Quotient

(2. 1) Let H£ be the universal covering of D'= {s€ C; 0<|j|<e}, namely, if we set s=e(l) (seD'), we have H£= {leH; Im />—log ej2n]. We consider a holomorphic mapping from He to the Siegel upper half plane @ff satisfying the unipotency condition : r ( / + l ) = r ( / ) + 2 ? , for an integral symmetric matrix B. In what follows, we write v(s) instead of r(/), and express the unipotency condition as (2.2)

T{f«s)

=T{S)+B.

We call a period matrix satisfying (2. 2) a stable matrix. L e m m a (2. 3). A stable matrix T(S) can be expressed as T(S) =TO(S)+B

log s/2m

where TO(S) is a holomorphic matrix over D and B is a symmetric positive semi-definite integral matrix. Proof. Take an integral vector m(^0) and define f {s)^=mz{s)tmi g(s)=e(f(s)). Since g(s) is single-valued and bounded on D' by assumption, g(s) extends to a holomorphic function on D in view of Riemann's theorem. Let am be the order of zero of g[s) at s=0. Then we can write g(s)=samg0(s) with a non-vanishing holomorphic function g0 on D. By taking branches of log suitably, we may assume/(J) =flmlog sj'2m+k>g go(s) J2ni. Hence am = mBtm. Since am is non-negative for any m, B is positive semi-definite. Moreover any component of r is a rational linear combination of such f (s), hence a sum of an integral multiple of log sj2ni and a holomorphic function over D. Therefore we have T(S)=TO(S)-\-B1 log s\2izi for a holomorphic matrix r0. From this it follows that B=B{. q. e. d.

Relative Compactification of the Neron Model and its Application

211

From now on, we assume B— L «, L where B' is positive definite and of rank g'. We remark that any symmetric positive semi-definite integral matrix can be transformed into the above form by an element u of GL(g, Z), B—*uBcu. We set

(2. 4)

T(S) = ro(s) + [jj °,] log sl2xi, ro(s) = [J W £WJ

where r,, r2 and r3 are holomorphic matrices of sizes (g—g\ g—g'), {g—g'y g') (#', #') respectively. (2. 5) We consider r. p. p. decompositions t5 and t?0 of JV* (dim NR=g+l) satisfying the conditions (1.5) (l)-(3), and the corresponding toroidal embeddings Pts and d. Then we shall define an action of a discrete group F isomorphic to Zg with reference to a stable matrix r (2. 4). Let r=(rc, m) (ne Zg~g>, me Zg>) be an element

ofr. Define a ring homomorphism 51* from C[crn^] to C[o' C\M], o'= T_mB,(o), by

where e(u)=exp(2mu)

and wc = J\wc\ ct being the z-th coefficient of the vector c. i= l

The holomorphic mapping S7 induced by S* from P6 to P6 is an automorphism over D preserving Q. Then we have T h e o r e m (2. 6). The action of F on P6 {or QJ) is properly discontinuous and fixed point free. Hence the quotient space of F\ [or QJ) by F exists.

Proof If g' = 0, then the assertion is trivial. Assume g'>0. Let p be a point of P<, and choose a cone a such that Pa contains p. Let the generators of crfl M be a~ (0, ej9 0) (\<^j<^g—gf) and ak+g_g,= (ak9 0, ck) (l^k^N), where ^ stands for 7-th unit vector of Rggr. We denote the functions corresponding to a5 and ak+g_g, by ^ and gk. We write f/fc=^a*wc*. We notice that there exists at least one relation (it is not necessarily unique, but we choose and fix one) of the form

(2.7)

£ / ^ = 0, lJk^0,

lkk>0,

ljkeZ.

Let C, = min (1, \Zj{p)\, |f*(^)|^0), C2 = 2 max (1, |£4(^)|=£0) and /=max /^. Define U(p, e), £=(£„ e2, £3), by £/(/>, £) = {(x, ^,, fA) 6 ^ ; |J| < £b k,—^-(j&)| < £2J |?*-f*(/>)l < %) where ^ C l . We want to show S_r(U(p, e)) n f^(A e)=^ except for a finite number of j if we choose £j sufficiently small. Assume the contrary. Then we may assume there exist infinite sequences sv, yv and pv such that (i) limsv = 0, sv = 7r(

I. Nakamura

212

),

(ii) r, (iii) pvz where lim

, 0)

(a,

In view of (iii) we have \Zi(P»)-Zj(p)\<£2, (2.8) Hence 4Cf's 2 ,

(2.9)

if

0

if

where dk={Iljkaj)jlkk^:0. The last relation in (2. 9) is obtained by multiplying the last inequality in (2. 8) by the ljk-t\i power of the inequality for all j except A. If j8^0, there exists ck such the flB"ck<0. Then we have Hence we have /3 = 0. Since Im rj>0, we have lim \e{nvTltej-\-m)!'T2tej)\

= 0

or

oo,

y-.oo

according as a Im r^e^Q or <0. Hence, if a^O, we are led to a contradiction of (2. 9). Consequently a = 0, which contradicts (ii). It is obvious that the action of F is fixed point free. q. e. d. (2. 10) Denote by A^ the quotient P^F, and by X the quotient Q^jT. A6 will be called the relative compactijication of X associated with ~6. A'^A^xD' is a family D

of principally polarized abelian varieties whose general fiber has periods (1, T(S)). XxD'=A't. D

The geometric fiber Xo of X at s=0 consists of det B' irreducible components. Each component is a principal bundle with fiber (C*)g~gr over an abelian variety ATl(0) whose periods are 1 and ^(0). Its bundle structure is given by one cocycle in Ext1 (ATli0), (C*)g~g')=Af^og\ which is determined by the g'-tuple of the column vectors of r2(0). The projection n from A^ to D is proper if and only if t5 covers Ro xRg/. The subset A€—X is of codimension two in A6. 3.

The Neron Model and a Proof of Minimality

(3. 1) In this section we shall define a principal homogeneous space (abbr. p. h. s.) in the restricted sense in nearly the same way as in [11], and prove the minimality of (3T, ft) among p. h. s.'s satisfying the condition (3. 3) below.

X'=

Relative Compactification of the Neron Model and its Application

213

Definition (3. 2). (1) A principal homogeneous space (y, Tf) over D is a complex manifold y and a holomorphic mapping XS from y to D satisfying the following : (1) y = y x D' is a family of principally polarized abelian varieties with a stable matrix r(s) (2.9). (ii) There exist a section e and a holomorphic mapping

)) — £ where 1 indicates the identity mapping of ^ . (iii) yg=Tf~l(s) (seD) is a complex Lie group with respect to the group law induced by e, 1,

( / , / ) =f°
Definition (3. 5). (3f, n) is called (analytic) Neron model of (3T, n') where n' is the restriction of TC to Z'=£xD'. D

Before entering into the proof of Theorem (3. 4), we prove a lemma. L e m m a (3. 6). There exists a basis w^ •••, wgl of holomorphic relative l-forms on where g{ = d i m y—l.

y,

Proof The fiber yo at J = 0 is an abelian complex Lie group equipped with the structure induced by e, ip and v(z, s) is holomorphic in z and s in Uo. Since C = £ = 0 on the section e, Qv is holomorphic at ^ = ^ = 0 , hence in Uo because of homogenity of y%. The translation by (0, a) on

214

I. Nakamura

yo is a specialization of the translation by a section e(s, a) near the origin of D. Hence, if we put C(a)(£, s)=^{z, s)+e(s, a), then C(a)U> 0)=^ ( a ) . Thus C(a) is holomorphic in z(a) and j1 in a neighborhood Ua of 3/0 ? where Ua is a translate of Uo by tf(j, a). Therefore we have a relative 1-form dC,la) on f/ a Uj/', which is equal to dC>v on 2/'. Consequently wv = dC>(va) on Ua\jy defines a relative 1-form on j / since

Proof of Theorem (3. 4). Let o* be a stable matrix associated with j / ' . Let o(e2nis) — a{s)+Bl,Bl=\0

B},

5;>0. Denote by ( # b TT,) the p. h. s. associated with a (2.1 0).

Let f be a given morphism from y to 3T. Our proof of Theorem (3. 4) is divided into two steps. (First step) We shall prove that the canonical isomorphism from y to X\ can be extended to a unique morphism from y to Xx. In view of Lemma (3. 6), we can choose relative 1-forms co,, • ••,
f (ov = <5y/i,

I

wv = aVfX

1^v, /i^^.

The canonical morphism from y to ^T( is given by the Albanese mapping. Let fv(z \ s)=e( fC (ov) for a point £ = C(£(a), 5) in Uar\y. There exists a local section (a

\Je(s)

I

e(Sy a) passing through y% such that
(3. 7)

n fe(S>a)wv Jew

= fei'S)cov+ Je(s)

(wv Jr

where y is a relative 1-cycle and the second integration is performed along a path contained in Uo. Hence the integration (3. 7) is the sum of a holomorphic function and an integral multiple of log sj2ni. On the other hand,/ v is single-valued with respect to s. In particular/ v (l:gyfg gi—g[) is always a non-vanishing holomorphic function on Ua for any a, while fv (z(a\ s)=sbvFv(zia\ s) for a non-vanishing holomorphic function Fv(z(a\ s) on Ua If we define

then F is an extension of/. That F is well-defined and the uniqueness of the extension are obvious. (Second step) Now we shall prove that a morphism / from %\ to %' can be uniquely extended to a morphism from 3£x to 3C. Choose global coordinates rj (or C) of Xi (or X) in such a way that the identity section e is given by ^ = 0 (or C = 0)

Relative Compactification of the Neron Model and its Application

215

respectively. The period matrices of X\ and X are given by ff(«0 =


o[s) + \n D'\ l o S sj2m9 a n d

where rl5 z29 r3 and r4 stand for (g-g')x{g-g')9

(g-g')Xg',

g'x(g-g')

and

g' Xgf matrices. The morphism/from 3C\ to 3Cf can be expressed as £=r)A{s) with a holomorphic gxxg matrix A(s) on 2). From this it follows that A(s) = Nl + N2v(s), a(s)A(s) = N3 + NAT(S), where Nt indicates a. g{xg integral matrix. Then we have (3.8)

Let Nt= | j ^ ^ 2 J , where ^ l 5 ,V,2, iVi3 and Nu stand for ( ^ - ^ ) X {g-g')9 (gi—gfi x 5', g[ X (5—5') and ^J Xg' matrices. Since 5 ' > 0 , N22=N2A=0. By virtue of (3. 8), we have TO NuB"\ _ [0 0 ] ri^n + ^ r , , iV12+iV21r2]

-r

°

"" [B\(NX3+N23TX),

It follows that Nl3=N23=0 1

L

n U

21

'

12

AT

21 2

iY 1 4

•^

e

° 1 B\(Nl4+N23T2)y

since Im r ^ O . Consequently we have A(s) =

define a mapping F from ^ to X as follows :

J

where , = (,', , ^ ) , C = ( C , O , ?', ^/r, C and C are ^ - ^ , ^ , ^ - ^ , g' vectors, ^ =5"y*^(C")5 r]Hll)=7]"—fjtlog sj2ni. Here x* A;' denotes componentwise multiplication of vectors x and *', and j - ^ r r r ^ , •-., s~Vn), v=(vu ••-, pn ), e(C'r) = (e(C}'))- T h e n F is an extension of/. That F is well-defined and the uniqueness of the extension are obvious. q. e. d. 4.

A Family of Curves of Genus Two

Definition (4. 1). A family of curves of genus two (or simply a family) (X, n) is a complex manifold X of dimension 2 free from exceptional curves of the first kind, given with a holomorphic mapping n from X onto D such that (1) 7T is proper, and smooth over D\ (2) for any seD', X8=i:~l(s) is a smooth curve of genus two. Definition (4. 2). The geometric fiber Xo at s=0 is called the singularfiberof the family (X9n).

216

I. Nakamura

Given a family (X, n) we have a holomorphic mapping from D' to the Siegel space <S2* by assigning to each se Dr the normalized period v(s) of the jacobian variety of Xs. We denote also by z(s) the holomorphic mapping from the universal covering He of D' to ©2 (2. 1). If Xo is a reduced curve with at worst ordinary double singularities, then the period matrix v(s) is stable [1]. GL(2, Z) operates on the space of all positive semidefinite integral matrices by B—>uBlu, u e GZ,(2, Z). It is a classical result that any symmetric positive semi-definite integral matrix B is GZ,(2, Z)-equivalent to one of the following :

By replacing r(s) by uz(s)% we consider only stable matrices T(S) with B in the above list. In the rest of this paper, we devote ourselves to construction of a family of reduced curves of genus two with at worst ordinary double singularities. To this end, we prove two more lemmas. A matrix r € ©2 determines a principally polarized abelian variety J. Then a theta function of the abelian variety J is defined by

(4. 4)

<9(r, z+a) = X1e{rtrP+f(z+*))

where y e Z\ z= (Zi, Z2), a= (al9 a2) 6 C2. L e m m a (4. 5). Assume J to be the jacobian variety of a curve C of genus two. Then the equation 0(r, z+a) =0 defines a curve on J isomorphic to C. Proof See Weil [14] Satz 2. L e m m a (4. 6). Let T be a stable matrix (2. 4), and a be the diagonal vector of B, = bu. Define fi(k) =min(rBtr+ (a+2kyr)l2 where y runs over Zg', k= (0, k')9 k' e Z*\ ai g'=rank B. Then (1) s~/iik)6(T(s)i z+a log sj^ni) is absolutely and uniformly convergent in the wide sense on dt (1.5), (2) 0(r, z+a log sl4ni) can extended to a meromorphic function on P^ (1. 5). Proof We set ©^s, Z)=@{T(S), Z+a log sjAni). 6X is single-valued on Pf. The second assertion follows directly from (1). In fact, since P6 is Cohen-Macaulay ([3] p. 52) and P^—(Ms of codimension two, any holomorphic function can be extended to P€. On the other hand, 9X is holomorphic on Pf and in view of (1), for any point p of P^ there exists an integer y. and a neighborhood U of p such that sf'&i is holomorphic on (P<,—QJ) f] U. Hence stl6x is holomorphic on £/, and consequently 6i is meromorphic on P. We shall prove (1). By choosing D smaller if necessary, we take ^0 in such a way that Im vo{s) +B log so/2m > 2/I. 1,

on Z>,

Relative Compactification of the Neron Model and its Application

217

where JJ, is a positive constant. We also choose a positive constant d such that By 2nd*\gl, Then we have log s/lm+fz) Im (rfrl2+f(a+2k) ^ fifr+dntn(-log\sls0\)+f(a+2k) log |j|/4^r+r*(Im z) except for at most finitely many y—{n^ m), if \z—Zo\<£i and [xx and d{ are chosen sufficiently small. By taking D smaller we may assume there exists a positive constant 82 such that <52< — log \s/so\ on D. Then we conclude \s-*»8x\ ^ | finite sum in s-^8x\-{-^ntm exp( — 27:(fjt1ntn+dl82mtm)), which proves (1). q. e, d.

5. The Case 8 = 1"° °l(p>0) Lo p] Let T(S) be a stable matrix (2. 3), in which B= |"jj ° j , TO= P/ ^ 1 , r ^ r , ^ ) . r,(j) is holomorphic in ^ on D. We distinguish two cases according as r 2 (0)^0 or r2(0) = 0. In both cases, we take an r. p. p. decomposition €= {ak> r*> {0}} defined by ak = {(u, 0, y) 6 R3; ( 1 + * ) K - J > ^ 0 , - / : w + j ^ 0 } , ^ = {(«, 0, j ) € /J 3 ; it ^ 0, j = ku], * € Z. We consider the relative compactification P (=P€) and J (=-4tf) with the period matrix r, (1. 5) and (2. 10). (1) The case r 2 (0)^0. For simplicity, we assume r{ and r2 are constant and r 3 =0. We set [

'

}

= ^sp'm n,m

where z=(Zi, Z2)> Z=Z\ and w=e(z2), and a=(09 — p). Let 7 ; be the divisor of P' defined by 0 - 0 , Y the closure of 7' in P, Y0=Y f](P)0 Our problem is to find explicit defining equations of Y and Yo. First we shall examine 7 0 on Q. discovered by Qk (ke Z), where Q^—Spec C[x, e(±z), wfl]xD, wh=s~kw. On Q,* we have an expression for s~^k)6 as (5. 2) s-«k>8(s, z, w) = Xsp(m2~m)/2+km~"(k)e{nhJ2+mnT2){wk)m. The convergence of the expansion (5. 2) is absolute and uniform in s and s~kw (in the wide sense) in view of Lemma (4. 6). As s approaches the origin, the limit of (5. 2) is given by the sum of term wise limits. Hence we have (5.3)

(0(r,, Z + Tt)+e(Tu Z+2r2)w.l)w.l

(p=l, k= — l)

0{TIZ)+0{T1,Z+T2)W»

(k=O)

6{zx, z)

(0

[k=p)

218

I. Nakamura

where 0(rl9 z) = Tle(n2vll2+nz) • Since w_x does not vanish on (£_„ 0(vl9 Z+T2) + n

0(rl9 z+2r2)w.l or 0(rl9 z+r2) gives a defining equation of Yo on (Q,_,)o. ( ^ J o = Spec C[e(±£), a>A, ^+i]/(^^*"+i) X-D is covered by Uk and F* defined by c

Uk = Spec C[e(±z), wk]9 Vk = Spec C[e{±z), w^]. (Q,*)o (o r (Q,*+i)o) is an open dense subset in Uk (or F*) respectively. Hence Yo is given by the same equations on (Pak)0 (Ofg £). On the other hand, Yo on (Pa_x)o is given by the equations : ( % „ z)wo~l+0{Tli Z+T2) = 0 on V.X (5. 4) 0(rb Z+T2)+0{T19 z+2r2)w.l = 0 on £/_, (^=1) I 0(T19Z+T2)=0 on [/_, (p>l). Since -4 is covered by Pah{— 1 ^k^p— 1), the description of Fo is now complete. Next we shall examine Y. Let 0* be the holomorphic function on Pak defined by &k(z, wk9 wiU) ©.,(£, w.l9 wo1) where A(m)=p{m2-m)/2 + (k+l)m, B(m)=p(m2-m)/2+km. We notice ©*=<9(0^ k^p— 1), 6.i = (wo)~l0 on P'. Now 0A and 6_x give defining equations of 7 on Pah and P a . In fact, lim 6k coincides with the equations in (5. 3) or (5. 4) according s-»0

as 0<^k^p— 1 or k= — 1. Outside 7o, ®* gives also a defining equation of Y in view of Lemma (4. 5). The remaining problem is to check that Y is smooth. First we examine the problem on Q,. On Q,*, Y is defined by &(s,z,w)=0 (O^k^p-l). We have {d&/dz)(0, z, w) = O\TX, Z)+6\TX, Z+T2)W (k=0) (d9/dw)(0, z, w) = 0(tl9 Z+T2), W = W0 (dd/dz)(0, z, w) = 0'{rl9 z) (0
(dekidz)(z, o, o) = O'(T19 Z) (ae-iiaz)(z, o, o) = O>(T19 Z+T2)

(o^k^p-i) (k=-i),

which proves the smoothness of Y at pk. Let Ck be the closure of Fo n Q,*. Ck is a non-singular elliptic curve if k=0 mod/>, or a non-singular rational curve if k^0 mod p. Ck and Ck+{ meet at p* transversally. r(=Z2) transforms Ck (or pk) into C^ (or J)A,) if k=k' mod j&. Let X be the image of Y by the canonical projection from P^ to ^ Xo consists of p irreducible components, one of which is a smooth elliptic curve and the others

Relative Compactification of the Neron Model and its Application

219

are smooth rational curves if £ > 1 . Ifp= 1, XQ is an elliptic curve with one ordinary double point. In fact, the points pp_x and p.{ are given by p,-i = ( ( 1 + 0 / 2 , 0,0) inP^ J>-i = ( ( l + r , ) / 2 - r 2 , 0 , 0 ) in/>,_,, and they are identified by the action of F. In particular, if p= 1, Xo is obtained by identifying two points (1+^0/2 and (l+r^/2—T 2 of ATl. The self-intersection number C\ is —2 if/?> 1. Finally we shall give the configuration for Xo in the relative compactification A6 of the corresponding N£ron model.

elliptic

(2) The case r2(0)— 0. For simplicity, we assume T1(S)=T1 (/>0), T3(S)=0. (5. 5)

(constant),

T2(S)=S1

We set 8(s, z, w) =

^isp'm2-m)/2e(n2rll2+nmsl+nz)wm.

With the same notation as in (1), Yo is given by

)=0 on

on Qk

dO9 (0
Y0C\ Q,o consists of two smooth curves Co and C'o defined respectively by 0(r b z)=0 and l-\-w=0. If we denote their closures by the same letters, C0 is a smooth rational curve, and Co is a smooth elliptic curve with periods 1, r{. They meet at only one point p: (s, z, wo) = (O, (l+r,)/2, —1). Ck and Ck+l meet at pk : (j, ^ , , a;^i) = (0,0, 0) transversally. CQ does not meet Ck (k^0). The defining equations of Y are given in the same way as in (1) and the smoothness of it except at p can be similarly proven. Now we shall examine the singularity of Y at p. We have

, z, w) = (de/dw) (o, z, w) = o at p.

I. Nakamura

220

Near p, 6 has an expansion of the following form : 6 = 0(T19 JZ)9{S, w)+slwd'(rl, zW(s, a;)+**(•••) where 9(s, w) = J^s^m2-m)wm, &'(s9 w)=d-9(s, w)/dw. We put U = 6(T1, Z), v=d(s, w). m

Then s, u and v form a system of.local coordinates of Ptf at p because 0'(ri> (1 + r i)/2) ^=0, ^'(0, —1)^0. By modifying s slightly, Y becomes analytically isomorphic to uv—sl=0 at p. I f / > 1 we obtain a nonsingular model Jtof X=YjF by replacing p by a chain of (/—I) rational curves. It is easy to check X is minimal. In conclusion Xo has a configuration of type (5. 7), and Yo has a configuration of type (5. 8) in A6. C'o '. elliptic

(5.7)

(/-l)

I

»

(5.8) C Co Co

I

£ elliptic

6.

The Case B =

( P/ q > 0 )

Let r(j)= f r i j J | r25JJl + fn °1 log J/2TTI, rt(j) being holomorphic on D, distinguish two cases according as r2(0)=£0 or r 2 (0)=0. As an r. p. p. decomposition t? we take the following: <5 = [Ok (k € Z2) and their faces} ok = {(^Ji ? J 2 ) € /J 3 ; (l+*0tt-J>* ^ 0, - M + ^ i ^ 0

( i = l , 2)}.

. We

Relative Compactification of the Neron Model and its Application

221

(5 satisfies the conditions (1. 5) (l)-(3). (P6)o consists of Ak (keZ2), each of which is by definition the closure of (Q,k)0 and isomorphic to PxxPx. (1) The case r 2 (0)^0. For simplicity we assume r1 = r3—0, T2(S)=T2 (constant). We set /£ i x

®(s, u>i, w2) = &{s, z+a log si4m)

where Wi—e^Zi)-, Then we have

an

d a={—p> ""?)• ;1^2, kx = k2 = 0

(6. 2) lim 8(s, wl9 w2) =

l+wl9 1,

0 < A,


r

We define F by the equation ®(.y, w,, zt>2) = 0 on P', and let Y be its closure in P6. The equations (6. 2) show Yof]Ak is empty for 00 and Cf+If0 (or C0)y and C0J+i) meet transversally at pt (or qy) where ^ is the point (j, «i5 w^, z;0) = (0, 0, 0, - 1 ) , (0£i
Vj=s-Jw2)

On the other hand, if p = q = 1 (1 -\-u_lJre{r2

lim s&(s, s-0

w2) =

if

on on

p>l,q=l

on on

if p> 1, q> 1

on on

Co, 9 -i

I

(6.3) Co.i

p1 P-!

/

^0, Cl ,o

Cp- i,o

222

I. Nakamura

Hence C_uo (or C0,-i) a n d Co,o meet transversally a t p ^ (or q_l) where p_{ : (s, M_,, ui\ vo) = {O, 0, 0, - e ( - r 2 ) ) , qM : (s9 u0, v.l9 v;l) = (0, - e ( - r 2 ) , 0, 0). Now 7 is smooth and minimal. The configuration of 70 in A6 is shown in (6. 3). (2) The case r 2 (0)=0. For simplicity we assume T1=T3=0, r2(s)=sl(l>0). Then we set 9(s9 wl9 w2) =

Hence we have on Qk r

r\! -

v

\ '

b

* +^25

2J

0 < k{ < p, k2 = 0

M + wl5 1,

*! = 0, 0 < k2 < g 0 < kx
Let C A = 7 0 n dk. C0)0 consists of two smooth rational curves Co,o a n < i ^o,o defined by 1 + ^ = 0, and by 1 + r ^ 2 = 0 respectively. Co,o and Co'o meet at j) : (j, w{ w2) = (0, — 1 , —1). Cit0 and C i+lt0 meet transversally at pt(O0 at p_Y : (j1, M_!, MQ"1, »O) = (O, — 1 , 0, 0) and Co>1 at p0. Similarly Co'o meets Q - i at q_! : (s, uOi z;_l5 &^1) = (0, — 1 , 0, 0) and Co>1 at q0. The singularity of 7 at p is analytically isomorphic to that of the analytic set: uv—sl = 0 at the origin. Hence we can resolve this singularity by replacing p by (/—1) rational curves, so we get a configuration Ip-q-l for the singular fiber of X, the nonsingular model of X = 7 / J T , [8J.

7. The Case B = p ^ ' ^ J (p, q, r>0) Assume T(S)= y

r

lbg sj2m for simplicity.

We put (7. 1)

6(s9 wl9 w2) = @(r, z+a

log ^/4TTZ)

where a=(-(p+r), -(q+r))9 Wi=e(Zt) and i4(n, m) = {p+r){n2-n) 2 (m —m)l2-{-rnm. We take an r. p. p. decomposition 3b (1. 6) and consider P=P^ and A=AZ). Ao consists of pq+pr-\-qr ( = det B) irreducible components, each of which is isomorphic to a projective plane blown up at three distinct points. Let N be a finite Z-module Z2/Z(pJrri r)-\-Z(r, q+r). Then N has representatives k=(ki, k2) satisfying the following inequalities : ki ^ 0, k2 ^ 0, (7. 2)

p-k{+k2

^ 0,

p+r—kY ^ 0,

q+k{-k2

^ 0,

q + r—k2 ^ 0.

For £ satisfying (7. 2), ^(*)=0 (4. 9). Let u^s-'w^ On Qk we have

vi=s~iw2.

Relative Compactification of the Neron Model and its Application

223

= (0, 0) l+VQ+UplV09 1+UpliVi,

k= k= -. 0 k = (p+r, i+r),

(7. 3)

0

lim 6{s, wx, w2) = :

'-i9q+r)9 k = (r, q+r)

0<

k = (r—i, j f + r — t ) ,

0 <

k = (0, q) k=(09q—i)9

1,

0
otherwise.

Let Y' be the divisor on P' defined by 8(s, wu w2)=0, and Y its closure. Let Jk be the closure of (Q,»)o, and Ck=dkf)Y0. Ck is a smooth rational curve. By the action of I\ Cii0, Cp+iti and Cp+rfr+i are transformed onto Cr+iiQ+r, Ciig+< and C0>i respectively. Two of them meet at most at one point and there transversally. The smoothness of 7 can be proven in the same way as before. Y/f has a singular fiber of type Ip-q-r in the terminology of [8]. The configuration of Yo in (P)o is described as follows. (r-1)

U-i)

(7. 4) Finally, the mapping : (ukl9 vk2)—^(wli w29 spWilw2, sqwxw2\ sp+rWi\ sq+rw2l) on Ak,for k satisfying (7. 2), gives rise to a new family of principally polarized abelian varieties, which coincides with Proj R{r)jF (cf. [7] § 2 Remark). Its special fiber is a union of two protective planes, independent, surprisingly, of the choice of p,q and r. Therefore it seems to be natural to expect that this variety have some special property. Similar

I. Nakamura

224

considerations and computations provide us with similar kinds of varieties in the three-dimensional case. The author has obtained thus the concept of stable quasi-abelian varieties. As consequence of it we. can compactify the moduli space of principally polarized abelian varieties with a fixed level structure in the two and three-dimensional cases [7]. 8.

Final Remarks

(8. 1) Now we shall construct a family of parabolic type, not included among the types in the preceding sections. From [9], we have the case Ip-q-r, in which the monodromy is given by [•-1 0 -p-r -r 0 - 1 -r —q-r M= 0 0 - 1 0 0 0 0 - 1 log sj2i:i. If we put s=t2, the

Then we take a period matrix T(S) — y

"I 0 2p+2r 2r pull backri(£)=r(£ ) is a stable matrix, and M = 0 1 2r 2q+2r 00 1 0 ..0 0 0 1 2

2

We set 6(t, z)^Ze(nTl{t)tn/2+nt(z+a)),

where z={zl9z2), and a=({p+r)/49

Then we define a divisor Y' on A'% by 8(t, z)=0. The automorphism a of A'^ a : (£, Z\> Zi)—*( — t9 —Zi, — £2)5 can be extended to a bimeromorphic mapping of A^. It satisfies o*6=6, and induces an automorphism of the closure Y of Y''. Denote the closure of 7 0 n Q,* by C*. Any irreducible component Ck is mapped by a onto C_k, in particular C^,-,, Cp+r>r and Cr,g+r are mapped onto themselves mod P. Moreover a has two fixed points of type N2 ([4] p. 583) on CPt_q mod P9 Cp+Tir mod P and Cr,,+r mod P respectively. At the image of a fixed point of a, the quotient space Yja has an ordinary double singularity, which we can resolve by replacing the point by one rational curve. Notice that Y is smooth. Let X be the nonsingular model of Yja. Then Xo has a configuration of type Ip-q-r [8]. Combining the results in [8] with those here one could construct a family of curves of genus two in the same way as above. (8. 2) A smooth curve of genus three can be also realized by theta functions. In fact, a complete intersection of a theta divisor of level 1 and its translate in the jacobian variety of C, if they are in good position, consists of two smooth curves, isomorphic to C, meeting transversally at two distinct points [15]. Thanks to this fact, we can construct a family of reduced curves of genus three with at worst ordinary double points. The details will appear elsewhere. (8. 3) Given a family X of curves we assume Xo to be reduced and to acquire only ordinary double points as singularities. Then the local monodromy is of the

p.. 0

l

B=B^0

[1]. Let Z* be the union of X' and X0 with its singular

Relative Compactification of the Neron Model and its Application

225

points deleted. Then a canonical morphism from X' to A' can be extended to a morphism from X* to 3C.

References [ 1 ] Clemens, C. H. et al : Seminar on degeneration of algebraic varieties, Institute for Advanced Study, Princeton, (1969-70). [ 2 ] Iitaka, S. : On the degenerates of a normally polarized abelian variety of dimension 2 and algebraic curve of genus two (in Japanese), Master degree thesis, University of Tokyo, (1967). [ 3 ] Kempf, G. et al : Toroidal embeddings I, Lect. Notes in Math., 339, Springer, Berlin, (1973). [ 4 ] Kodaira, K. : On compact analytic surfaces II-III, Ann. of Math., 77 (1963), 563-626, 78 (1963), 1-40. [ 5 ] Miyake, K. and Oda, T. : Almost homogeneous algebraic varieties under algebraic torus action, Manifolds-Tokyo 1973, Proceedings of the International Conference on Manifolds and Related Topics in Topology, 1973, Univ. Tokyo Press, (1975). [ 6 ] Nakamura, I. : On degeneration of abelian varieties, (unpublished) [ 7] : On moduli of stable quasi abelian varieties, Nagoya Math. J., 58 (1975), 149-214. [ 8 ] Namikawa, Y. and Ueno, K. : The complete classification of fibers in pencils of curves of genus two, Manuscripta Math., 9 (1973), 143-186. [ 9] : On fibers in families of curves of genus two 1. Singular fibers of elliptic type, Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki, Kinokuniya, Tokyo, (1973), 297-371. [10] Namikawa, Y. : A new compactification of the Siegel space and degeneration of abelian varieties I-II, Math. Ann. 221 (1976), 97-141, 201-241. [11] Neron, A. : Modeles Minimaux des varieties abeliennes sur les corps locaux et globaux, Publ. Math. IHES 21, Paris, (1964). [12] Ogg, A. P. : On pencils of curves of genus two, Topology 5, (1966), 352-362. [13] Ueno, K. : Degenerating fibers of families of abelian varieties of dimension 2. (unpublished) [14] Weil, A. : Zur Beweis des Torellischen Satzes, Nach. Akad. Wiss. Gottingen, 2 (1957), 3353. [15] Gunning, R. C. : Lectures on Riemann Surfaces, Jacobi Varieties, Math. Notes, Princeton Univ. Press (1972).

Department of Mathematics Nagoya University (Received November 29, 1975)

Toroidal Degeneration of Abelian Varieties

Y. Namikawa In this article we establish a method of constructing somewhat general degenerating abelian varieties as a partial answer to the problem raised in [8]. This subject was once treated by Mumford [4] from a different point of view. Here we make use of the theory of torus embeddings developed by Mumford and other mathematicians ([2], [3]). Our method of construction itself is a direct generalization of Nakamura's in [6]. However by virtue of this generalization we can obtain a synthetic view of late related results including the author's [9], and those of Oda-Seshadri [10] (cf. § 4). It is a very interesting problem to investigate how general our method is. It would be reasonable to expect that our method works not only in the complex analytic category but also in the algebraic or etale category with suitable modifications. We take a torus embedding 3CE of 7/={Gm)n determined by a rational partial polyhedral decomposition (abbreviated to r. p. p. decomposition) 2= {o} of NR = N(^)Z R where iV=Homgroups(Gm) T) (see § 1 for the precise definition). Consider a T-invariant closed set S' of 3CS and an open neighbourhood U of Sf in Xs. We fix the triple (S\ % 3CS) and set U=Uf]T. We shall then consider a family of polarized abelian varieties of dimension g, XS° : J4°—+U. For simplicity we assume moreover the polarization to be principal. The family XS° determines a multiplevalued holomorphic map T: £/—•©# called the period map, where (&g denotes the Siegel upper-half plane of degree g (cf. § 2). We add some mild conditions so that the family can be extended. The problem is to construct degeneratingfibresover S' to extend the family XS° to a family of analytic spaces XS : A—^IL which has some "nice" properties.

Indeed there are many choices in constructing degenerating fibres. Among these possibilities we consider a special kind of fibres having much importance. Namely, letting L be a lattice of rank g, we take an r. p. p. decomposition K of NR x LR such that the projection NR x LR—*NR maps every cone K in K onto a cone a in I. (The surjectivity is necessary for the extended family to be equidimensional.) For the extendability of the family, K should satisfy a condition of compatibility with T. If the condition holds, we call K toroidal degeneration data and we can construct a family T5 : A-+U extending XS° (§ 3). Since A and all fibres of XS are toroidal (cf. [2] p. 54), we say this degeneration is toroidal. Our method seems to be quite general for constructing toroidal degenerations, although we do not know whether

228

Y. Namikawa

all toroidal degeneration can be constructed in this way. Several geometric properties of XS and the fibres can be stated in terms of those of K (§ 3). In the last section, § 4, we give a number of examples to show that most of preceding results ([6], [9], [10]) fall into our category. Because the number of pages is limited, we only indicate the ideas of proofs, which will appear in complete form somewhere else. General assumption : All algebraic varieties in this article are defined over C and are identified with the associated analytic spaces. 1.

Torus Embeddings

In this section we recall briefly the theory of torus embeddings and fix our notation. For details, see [2] (or [9])°. (1.1) LetT=Spec (C[Tl9 •••, Tn, 7Y\ •••, T~1]) be an algebraic torus of dimension n over C, which we often identify with the analytic group (C*)n where C* = C — {0}. An algebraic torus of dimension one is denoted by Gm. Let Af=Homalg.groups(!T, Gm) be the group of characters : for r e M we denote by r T the corresponding monomial in r(ZT, 0). Let iV=Homalgtgroups(6rm, ZT) be the group of 1-parameter subgroups of T. M and N are free abelian groups of rank n and dual to each other by the pairing MxN-+Z 01 UJ (r, a) -> (r, a) where (r, a) is defined by the condition :

r'(*(0) = *,

teGm.

Denote M(x)zJ? and iV(x)zi? by MR and NR respectively. Definition (1. 2). i) An qffine torus embedding (resp. a torus embedding) of T is an affine scheme (resp. a scheme) over C containing ZT as an open subset, equipped with an action of T which extends the action of ZT on itself induced from the translation in ZT. ii) A morphism of two torus embeddings 3C{ and 3C2 is a m o r p h i s m / : 3CX—>5£2 of

schemes satisfying the conditions : a) / induces a group epimorphism g : ZTl-^>ZT2 of the tori contained in them ; b) the following diagram commutes :

(g,f) I if 2 X

JL

j / 2

*«X 25

where the horizontal arrows denote the action of tori. 1) Torus embeddings considered here are only locally of finite type, while in [2] they consider only those of finite type. For the propositions used here, however, the same proofs as in [2] hold.

Toroidal Degeneration of Abelian Varieties

229

Definition (1. 3). i) A convex rational polyhedral cone a in MR (or NR) is a set of the form a = {xeMR(or

NB) ; lt(x) > 0,

/ = 1, ...,ro}

where //s are linear functional defined over Q . We abbreviate it to c. r. p. cone. ii) A rational partial polyhedral decomposition (abbreviated to an r. p. p. decomposition) of NR is a collection 2 = {aa} of c. r. p. cones in NR, each of which contains no linear subspace, such that a) if for oa e 2, a is a face of aa, then a e 2; b) for aae2 and 0^el7, is again in 2. Given an r. p. p. decomposition of NR9 we can construct a torus embedding of T in a canonical manner. First we fix some notation. For a c. r. p. cone a in NR, the dual cone a in MR is defined by a = {r € Afjj; (r, fl) > 0 for all a € a} and is again a c. r. p. cone in MR. For a semigroup 5 in M we denote by C[£] the C-vector subspace of/^(T, 0) generated by Tr, reS. For a c. r. p. cone a in JV^, (jfl A^T is clearly a semigroup. Proposition (1. 4). i) For a c. r. p. cone a in NR which does not contain any linear subspace, 3 ^ = S p e c (C[df]M]) is a normal affine torus embedding ofTf of finite type. ii) For an r. p. p. decomposition I of NR the affine torus embeddings %a, ael} can be patched together to form a normal torus embedding 3£E locally of finite type. Proposition (1. 5). Let 3CX be the torus embedding of T associated with an r. p. p. decomposition I. Then 3CS has the following properties : a) For a c. r. p. cone a in 2 any element ae N in the relative interior of a (i. e.? in a but not in any proper face of a) gives the same limit <2(0)=lim a(t), hence determines a J-orbit t-*0

b) the correspondence o-^>0a gives a bijection between 2 and the set of ZT-orbits in 3CS ; c) ax is a face a2 if and only if Oa^Oat \ d) dim
Many geometric properties of 3£z can be described in terms of I7. For later use we make the following Definition (1. 6). An r. p. p. decomposition 2 of NR is said to be of projective type if there exists a real convex continuous function on the convex hull of \Jo(zNR satisfying the following conditions : a) / has integral values on N f] ((J o) ; b) for XeR+9f(la)=Xf (a); c) f is piecewise linear ;

230

Y. Namikawa

d) for each eel such t h a t

t h e r e a r e a finite n u m b e r of linear functionals lu i=l,

* = {a*NR ;f(a)=li{a)

•••, m,

for all*}.

If an r. p. p. decomposition I is of projective type, one can construct an invertible sheaf £f on 3£s with the above function/. £f is ample when I7 is a finite set. Next we consider morphisms of torus embeddings. Definition (1. 7). i) A morphism of an r. p. p. decomposition I of NR to another 2" of N'R is a Z-homomorphism p : N—*N' of finite cokernel which maps each aae I into some ^ € 2 " . ii) A morphism of r. p. p. decompositions as above is said to be of equidimensional type if p maps each aael onto a\ e 2", Proposition (1. 8). i) ^4 morphism of r. p. p. decompositions p as above determines a morphism of torus embeddings p* : %z—>3£r. This correspondence p-^>p* is functorial. ii) If the morphism p above is of equidimensional type, the associated morphism p* is equidimensional. 2.

Period Map and Degeneration Data

(2. 1) We take a torus embedding 3CS of ZT defined by an r. p. p. decomposition I of NR. Let 2" be a subset of I satisfying the condition : a e 2", o^ax implies ax € I'. Then the union of T-orbits S'={JO((j) is closed in jT^. Let % be an open neighbourhood of S' in 3£s. We henceforth fix the triple (%l9 S\ U) and set U=

(2.2) Let be a family of polarized abelian varieties of dimension g. For simplicity we assume moreover that the polarization is principal and that XS° has a section s : U—+A° which we consider as the set of the identity elements of abelian varieties. Then, as Ueno has shown in [12] § 1, one can define a multiple-valued holomorphic map (called the period map of Zf°) T:U^
where <3g= {re M(g, C) ; r=r, Im r>0} denotes the Siegel upper-half plane of degree g, and a homomorphism (called the monodromy of 73°) of the fundamental group of U into the symplectic group such that for a loop y with base point t0 the analytic continuation T(yt0) of T(t0) along y is subject to T(rto)=*(\r])>T(to),

Toroidal Degeneration of Abelian Varieties

231

where [y] denotes the homotopy class of y and, for Me Sp(g, Z), T e ©*-, M*r denotes the usual action of Sp(g, Z) on g. (2. 3) We here put one more assumption on the period map of Zf° which is not as restrictive as it looks (cf. Remark (2. 4)). In order to state the condition, we need some notation. Let tyg be the vector space of real symmetric matrices of degree g and Yg the lattice of integral matrices in tyg. We denote by g)£ the cone of positive matrices in g)^. Note that (£g=tyg-\-j—\ g)+. Denoting by 7% the set of non-negative integral matrices in Yg, we define the rational closure f)J of g)J as the convex hull of f J in $)g which in fact contains g)J. For 0 f)£ sendingy" to L „ , by which we consider g)J,, as a closed subset of $)g. We consider a map

where e{ ) = e x p ( 2 W - l ( )), and put U=e-l{U). In what follows we assume that the period map T satisfies the following Assumption (U). There exists a Z-linear map B:N->Yg such that BR=B(^)ZR : NR—^g has the following properties : i) every a el is mapped into $)i, ii) for some g" with §
T(z)=Bc(z)+S(e(z)) on {?, where BC=B(^)Z C, and S(t) is a bounded single-valued holomorphic function on U with values in the symmetric complex matrices of degree g.

Remark (2. 4). i) The above condition i) on BR is in fact superfluous (cf. [6] Lemma (2. 3)), and the condition ii) is added only for technical reason. ii) In spite of the seeming complexity of the above condition, it holds in a quite general situation. Namely if 3£s is non-singular and if for every element y of nx (U), &(y)— Id is nilpotent, then the period map satisfies the assumption (U) at least locally. The idea of the proof of this is as follows. Since S is a divisor with at worst normal crossings, we may assume nx(U) is abelian by shrinking % if necessary. Hence the image of the monodromy is contained in a parabolic subgroup of Sp(g, Z). By transforming the period map and monodromy by a suitable element of Sp(g, Z) we may assume that for every y e n^U), @(y) can be written in the form 0 1'

B =

232

Y. Namikawa

Also g" can be taken minimal. Then by the same argument as in [7] Th. 2 or [6] Lemma (2. 3) one can show that T satisfies the assumption (U). The above condition is quite general because one can take an equivariant resolution %s ([2] p. 32) and then take a suitable covering of T sending tt to tf (which extends to a finite morphism of 3CS into itself) so that the pullback family satisfies the assumption as claimed above by virtue of the theorem of quasi-unipotentness of monodromy (cf. [11]). This fact reflects a very special property of the period map. (2. 5) Let -6g be an algebraic torus of dimension g and L the group of 1-parameter subgroups of €g (cf. § 1). Put LR=L(g)z R. Given a period map T satisfying the assumption (U), we can define an action of a lattice X of rank g on NR X LR as follows : Tx:NRxLR->NRxLR ID

W

(a,x)-^(a,x+xBR(a)) for ^ € X where BR=B(x)zR, and x and ^ are considered as row ^-vectors. Definition (2. 6). i) An r. p. p. decomposition K= {ie} of NRxLR admissible (with respect to 7") if it satisfies the following conditions : 1) the first projection

is called

p:NRxLR-+NR induces a morphism of r. p. p. decompositions from K to I of equidimensional type; 2) K is invariant under the action of X; 3) for every tee K and ae NR, tcf)p~l(a) is contained in ImBR(a) (dLR) ; 4) for every o e I, there are only finitely many classes of c. r. p. cones in K mapped by p onto a modulo X. An r. p. p. decomposition K of NR X LR which is admissible with respect to T is called toroidal degeneration data of the family ZS°.

Definition (2. 7). i) An admissible decomposition K is called of complete type if for every a el the union of c. r. p. cones in K contained in p~l{o) is convex in

NRxLR. ii) An admissible r. p. p. decomposition K is called of projective type if there is a real continuous function / with the properties : 1) for every a el the r. p. p. subdecomposition Ka= [ic: p(fc)cza] becomes of projective type by means o f / (cf. (1. 6)) ; 2) for-^eX^ dx=f—f* Tx is a linear function on NRxLR. 3.

Construction of Degenerating Fibres

(3. 1) Let XS° : A°—*U be a family of principally polarized abelian varieties of dimension g over U whose period map T: U—*<&g satisfies Assumption (U), and

Toroidal Degeneration of Abelian Varieties

233

suppose that degeneration data K of Tf° are given. We shall construct an extended family of degenerating abelian varieties over % by means of the given degeneration data. In what follows we employ freely the notation of the previous section. (3. 2) First we note that the original family XJ° is reconstructed by means of T as follows (cf. [12] § 1). On UxiSg, the lattice X acts via

Tx:

Ux-6g-+Ux€g

w

w

(t,w)^>(t9w.e(xT(t))) for i e X. Here e(% T{t)) € $g is uniquely determined in spite of the multiple-valuedness of T. The action is properly discontinuous and fixed point free. The quotient space Ux f)g\X is a fibre space over U and canonically isotnorphic to A° over U, hence we identify it with A° from now on. (3. 3) The construction is done in two steps. In thefirststep, we construct by means of K the torus embedding Jg^, which admits a fibre structure p :fiK—>%zby virtue of the condition (2. 6) 1). Put p~l(U) =J8 and consider the restriction which we called a family of semiuniversal coverings in [9]. Note that the restriction of p to U is isomorphic to the family of algebraic tori pv : Ux€>g-+U considered above. In the second step, we see that the action o f Z o n Ux€g extends to that on j8. In fact, for 16 X, if we define a linear endomorphism of Nx L by sending (a, x) to (a, x-\~iB(a)), it induces an automorphism T'x ofjBK by virtue of the condition (2. 6) 2) (cf. Prop. (1.8) i)). Being bounded on U, S extends to a matrix-valued holomorphic function on U which we denote by the same letter, hence determines a holomorphic map U-^^g sending t to e(xS(t)). We can see easily that the translation (t, w)—* (t, W9e(x$(t))) extends to an automorphism T'xf of J8. The composite TX»T'X is the desired extension of Tv which we henceforce denote by the same letter Tx% The most difficult is to show Proposition (3. 4). The action of X on jB is properly discontinuous and fixed point free.

The idea of the proof is essentially contained in [9] Prop. (13. 1) and [6] Th. (2-6). As a direct corollary of (3. 4), we obtain the following theorem which is one of the main results of this article. T h e o r e m (3. 5). i) The quotient space A = JB/X admits a canonical structure of normal analytic space which is toroidal. The canonicalfibrestructure

234

Y. Namikawa

induced from p is equidimensional (by (2. 6 ) , 1)) and of finite type (by (2. 6) 4 ) ) , and its restriction over U is XS°. We call XS the toroidal family associated with the degeneration data K. ii) Let Kx and K2 be two admissible decompositions. If the identity map of NxL induces a morphism of r. p. p. decompositions from Kl to K2i then there is a natural morphism

over U which is identity over U (essentially by Prop. ( 1 . 8 ) i ) ) . If moreover K{czK2, then (p is an open immersion.

One can describe the degenerate fibres over S' quite explicitely. Let X' be the subgroup of X consisting of the elements ^' = (^lj • ••, ^ , , 0, •••, 0) where g'=g—g". As we have seen in (3. 3), X' acts on 2lx€g> via

Tx, :Ux6g,->UxG8. W

LU

(t,w)-*(t,w.e(x'S'(t))), where S'(t) is the g'-th principal matrix of S(t). (Noteihj4ol0(o) over O(o) whose kernel is a family of algebraic toril). This K is in fact determined by ^

W

= =

(^ \t)i,g' + j)l£i£g',i£j£g"9

iv) relative dim 6z+relative dim ic=g. Concerning geometric properties of XS we can say the following. Proposition (3. 7). If the r.p. p. decomposition K is non-singular ([2] p. 14), then A is non-singular. Proposition (3. 8). If K is of complete type, then XS is proper. 1) Here we need the condition (2. 6) 3).

Toroidal Degeneration of Abelian Varieties

235

T h e o r e m (3. 9). If K is of projective type, then XS is quasi-projective. Only the last claim is not trivial. We need to develop a theory of degenerate theta functions. The idea of the proof can be found already in [9] Chapter V. P r o b l e m (3. 10). Find a suitable condition for each component of a fibre ofxs reduced and smooth in terms of K. 4.

to be

Applications

The last section is devoted to exhibiting a number of examples which cover most of known results in this direction. A) Analytic Neron model. For simplicity we consider an affine torus embedding 3Cv determined by 1= {a c. r. p. cone a and its faces} and let S= [a]. Let K be an admissible r. p. p. decomposition such that 1) for ae
every /ceK over a, p(/cP\(NxL))=o(~)N. Then the toroidal family ZS : A—>U associated with K has the structure of a family of commutative group varieties. If we denote by K' the subdecomposition of K generated by aX {0} (i. e., the smallest admissible r. p. p. decomposition containing ax {0}), then the family Zf' : A'—^U associated with K' is an open subfamily whose fibre A't over t e %L is the connected component of the fibre At of A over t containing the unit. In case 1= {R+, {0}} and K= {icn= {(a, a(0 «))}, neZg", *°= {(0, 0)}}, the corresponding toroidal family A is the analytic Neron model constructed in [6]. Hence it is naturally expected that the above general A also has the minimality property described in ibid. Th. (3. 4). It is a very interesting problem to ask whether any toroidal compactification of the Neron model can be constructed by our method. B) Delony-Voronoi decomposition and stable quasi-abelian varieties ([9]). Let / b e a convex function on f) J X LR defined by f{y,x) =min{?/?+2^}. Then/defines an r. p. p. decomposition K of f)J X LR whose projection to g)J also is an r. p. p. decomposition I of $)g. The latter is called the Delony-Voronoi decomposition, and the former the mixed decomposition. A c. r. p. cone in I (resp. in K) is called a Delony-Voronoi cone (abbreviated to D-V cone) (resp. a mixed cone). If we denote by ZTg the set of symmetric complex matrices of degree g none of the coefficients of which vanishes, (T^ is an algebraic torus by coefficientwise multiplication), 3CS is a torus embedding of ZTg. Consider the map e : ©^—>T«- sending = (exp(2W—1 r iy )), and put T ° = Im e and 3Tv=the interior of the closure of J ° in 3£s. If we let I' denote the set of D-V cones meeting g)J, then it 7°=3CsV\7g. turns out that %°s contains all 0{a) with aeZ', and Clearly the inverse map 7" of e on J ° satisfies Assumption (U), and the mixed decomposition K is admissible with respect to T. Moreover K is projective accord-

236

Y. Namikawa

ing to definition (by elementary calculation we have f(y, X./x~l~2;^#), and it turns out to be of complete type. Therefore the toroidal family (*J g • J¥ g

x)—f{y, x-\-%y) =

^ Jo g

associated with K is a projective family whose fibres we named stable quasiabelian varieties (abbreviated to SQAV). The study of these SQAV led to our notion of toroidal degeneration. The D-V decomposition and the mixed decomposition for g<3 are already explicitly known ([13]). Using this result Nakamura studied closely the structure of SQAV of dimension < 3 ([5]). For g<2 see also [9] (14. 5). C) Compactification of the generalized Jacobian variety of a stable curve (due to O d a

and Seshadri [10]). Let n : €—>D be a family of stable curves of genus g > 0 (for the definition of stable curve, see [1]) over the disc D— {te C\ |f|<e} such that 1) (5 is non-singular and 2) n is smooth over D'=D— {0}. (Hence the fibre Co over 0 may contain a smooth rational component F with f2=—2.) Then we can define the period map T: D'—^Bg satisfying the assumption (U) (cf. [7] § 4). The analytic Neron model XS : A—+D constructed in A) is nothing but the family of generalized Jacobian varieties of (5. Now we use the notations in [10] freely. We take a natural embedding i : H 1 (r(C 0 ), R)^>LR by *->(0 x). For 0€ dCx(R) let Del^ be the associated polyhedral decomposition, which is called the Namikawa decomposition in [10]. We define an r. p. p. decomposition K of R+ X LR consisting of {(a, i(aJ)) ; aeR+} JeI)el0 and {0}. Then the associated toroidal family XS*: J4*—*D is (the analytic counterpart of) the family constructed in [10]. If 0 is non-de-

generate, then A* contains A as an open set by virtue of Theorem (3. 5), ii) and [10] Prop. 7. 6. We remark finally that we can construct such toroidal families even over the local universal deformation space of Co by using our method. It would take too much space to go into the subject, so it will be discussed elsewhere.

References [ 1 ] Deligne, P. and Mumford, D. : The irreducibility of the space of curves of given genus, Publ. Math. IHES, 36 (1969), 75-110. [ 2 ] Kempf, G. et al. : Toroidal embeddings, I, Springer Lecture Notes, No. 339, Springer, Berlin, 1973. [ 3 ] Miyake, K. and Oda, T. : Almost homogeneous algebraic varieties under algebraic torus action, Manifolds-Tokyo 1973, University of Tokyo Press, Tokyo, (1975), 373-381. [ 4 ] Mumford, D. : An analytic construction of degenerating abelian varieties over complete rings, Compositio Math., 24 (1972), 239-272. [ 5 ] Nakamura, I. : On moduli of stable quasi-abelian varieties, Nagoya Math. J., 58 (1975), 149-214. [ 6 ] Nakamura, I. : Relative compactification of the Neron model and its application, in this Volume. [ 7 ] Namikawa, Y. : On the canonical holomorphic map from the moduli space of stable curves

Toroidal Degeneration of Abelian Varieties

237

to the Igusa monoidal transform, Nagoya Math. J., 52 (1973), 197-259. [ 8 ] Namikawa, Y. : Studies on degeneration, Classification of algebraic varieties and compact complex manifolds, Proceedings 1974, Lecture Notes in Math., No. 412 (1974), 165-210. [ 9 ] Namikawa, Y. : A new compactification of the Siegel space and the degeneration of abelian varieties, I-II, Math. Ann., 221 (1976), 97-141 and 201-241. [10] Oda, T. and Seshadri, C. S. : Compactifications of the generalized Jacobian variety, to appear. [11] Schmid, W. : Variation of Hodge structure : the singularities of the period mapping, Inventiones Math., 22 (1973), 211-319. [12] Ueno, K. : On fibre spaces of normally polarized abelian varieties of dimension 2, I, J. Fac. Sci. Univ. Tokyo, 18 (1971), 37-95. [13] Voronoi, G. : Nouvelles applications des parametres continues a la theorie des formes quadratiques, II, J. reine u. angew. Math., 134 (1908), 198-287 and 136 (1909), 67-181.

Department of Mathematics Nagoya University (Received February 14, 1976)

Kodaira Dimensions of Complements of Divisors

F. Sakai The Kodaira dimension of a compact complex manifold plays an important role in classification theory (Iitaka [5], Ueno [18]). The purpose of this paper is to introduce the notion of Kodaira dimension for a non-compact complex manifold. We shall mainly consider a complex manifold which has a compactification. Let Z b e a complex manifold of dimension n which is a Zariski open set of a compact complex manifold Xsuch that D = X—Xis a divisor with at most normal crossings. In this case, the Kodaira dimension K(X) of JSf is defined as follows. Let K% denote the canonical bundle of X. For a positive integer m, let
In case / is a meromorphic map, rank(/) is also defined, but the maximum is taken where/is holomorphic. The image/(X) is a topological space and dimRf{X) = 2 r a n k ( / ) (cf. Remmert-Stein [16]). 1.

Kodaira Dimensions of Non-Compact Complex Manifolds

Let X be a complex manifold of dimension n. Let {Ua} be an open covering of X with holomorphic coordinates (wlai •••, wl) in Ua. The canonical bundle Kx of X is defined by transition functions kafi=det {dw\jdwja). For any positive integer m, a holomorphic section
(1.1)

in

Ua,

is a global m-ple w-form on X Thus we can view an element of 7/°(X, G{mKx)) as a holomorphic m-plerc-formon X For any holomorphic m-ple /z-form co on X, written locally as (1. 1), denote by (a)/\®y/m the continuous (n, w)-form given by (a, A <»)1M = |p«(to«)rwll ( V - 1 P^dwlAd&a,

in

Moreover we set

Clearly || || satisfies the following properties. (i) ||OJ|| = 0 if and only if co = 0, for (ii) IHIH'IIHI ceC. (iii) IK+oiill^C^HI + IWI)

(CI = 1, Cm=2^~l

for m^

If ||o>||
Kodaira Dimensions of Complements of Divisors

241

We shall see later that if X has a compactification, then dim Fm(X)<doo. Fm(X)^ {0}}. For a finite set of elements w0, • ••, wN ofFm(X)9 Set N(X)={m>0\ me N(X), we can define a meromorphic map 0{Q)O>...tQ)N) associated to them by

ID

UJ

w

> ( f l > 0 ( w ) • •••

Define the m-*A rank rk m (Z) of Z by rk m (Z) = max {rank (0,a,o,...,«*,)}, where the maximum is taken over all choices of finite sets o)0, -,(oN of Fm(X),

tf=0,

for

1,2,-

Definition. The Kodaira dimension K(X) of Z is defined by max {rkm(Z)}

if

-OO

if

(Note that ic(X) takes o n e of the values — oo, 0, 1, •••, n.) T h e o r e m 1. 1. The Kodaira dimension ic(X) is a bimeromorphic invariant of a complex manifold X.

Proof Let X' be a complex manifold which is bimeromorphic to Z a n d / : X1—* Z a bimeromorphic map. T h e n / * induces an isomorphism of Fm(X) andF m (Z / ), which implies that K(X)=K(X'). In fact, for an element (oeFm(X),f*a) is a holoHere S(f) is the set of points where/is not morphic m-ple w-form on X—S(f). holomorphic. Since S(f) is an analytic subset of X' of codimension^2 (See [15], p. 333), f*
f (f*w A/*^) ^ I / m = ff

A w) 1/m
Hence f*w e Fm{Xf). Obviously/* is an injection. Considering/" 1 , we get the surjectivity of/*. q. e. d. As a consequence, we may formulate the following Definition. For a singular complex space Z, the Kodaira dimension ic(X) is defined to be /c(Z*), where Z* is a complex manifold which is bimeromorphic to Z. A holomorphic m-ple /z-form w on J* can be written as a)=f(z)(dz)m, a holomorphic function on J*. L e m m a 1. 1 (cf. Kobayashi-Ochiai [9], Addendum). Leta)=f(z)(dz)m

where/is

be a holo-

242

F. Sakai oo

morphic m-ple n-form on J*. Letf(z) = Zl ^nZn be the Laurent expansion off(z). n=-oo

Fm(d*), then an=0for n<>—m. Proof We have, by definition, (c£>A5J) 1/m =|/U)| 2/m (v /I= T/2^)^A^. Let p be the least integer ^ m such that a.p^0. Putf{z) = 2 anzn, and/ 2 (^) = 2 anzn. n ( m l ) n=-(m-l)

n o o

Assume that^(^)^O. Letting rQ be small enough, we can make I/1//2K b < 1 and |/2 | ^ |a_,| /1*|* for * € J*. It follows that

f On the other hand, using polar coordinates z=r^~le, e-»0 Je

we have

Jo

i/M)^"-"*]? 2[logr]r» =

if /»>m, if /» = m,

OO.

Combining these inequalities, we get ||o>|| = 00. This is a contradiction,

q. e. d.

Corollary. ic(C)= —00, /c(C*)= —00. Proof Here we prove /c(C*) = — 00. Let w be an element of Fm(C*)- We can 00

represent o> as 01= ( 2 anzn){dz)m. Clearly o>M* €.FTO(J*). So by the above lemma, n=-<»

we have an=0 for n-^—m. Let w=l/z. Then the domain {£€ C[ |*|>1} is transformed onto the punctured unit disc d*= {we C \ 0<|ze;|
co=

±

an{\lwY(-dwlw*YzFm{dt).

n=-(m-l)

It follows from the above lemma that an=0 for n}> — (m— 1). Consequently, we get (o=0. q. e. d. Corollary. K(C* X J n " l )= — 00, =

«(Cx Jn~l)=

— 00flwrf/c(C n )= — 00, /c(C*n)

—00.

The Kodaira dimension has the following properties. Proposition 1 . 1 . Let X, Y be complex manifolds of dimension n such that XaY. Then K(X)*>IC(Y). In particular, if K(X) = — 00, then K(Y) = — 00. Corollary. If a complex manifold X contains Cn as its open subset, then ic(X) — — 00.

Kodaira Dimensions of Complements of Divisors

243

In case X is compact, this fact is obtained by Kodaira in [11], using Nevanlinna theory. Proposition 1. 2. Let X be a complex manifold of dimension n and Z an analytic subset of X. Ifcodim Z^2, then K(X-Z)=K(X). Proposition 1. 3. Let X, Y be complex manifolds of dimension n. Suppose that there is a proper surjective meromorphic map f: X—+Y. Then we have K(X)

>

K(Y).

Proof First, assume t h a t / i s holomorphic. The set Ef= {xe X\ rank x (/)<w} is an analytic subset of X. Set E'=f(Ef), E=f~1(E'). By Lemma 2. 2 in [1], there exist analytic subsets S (resp. S') ofX—E (resp. Y—E') and a positive integer s such t h a t / : X—E—S-+Y—E'—S' is locally biholomorphic a n d / - J ( j ) consists of s distinct points for everyy e Y—E'—S'. The number s is called the sheet number of/ Take an element (oeFm(Y). Thcnf*weFm(X), because

£(/•• A / ^ ) - = £_,_,(/•« A / ^ ) - = ,/ r __> A «)»• = S f (w A cD)l/m < o o . Clearly / * is injective. Therefore we get ic(X)}>/c(Y). In case / is only meromorphic, the assertion follows from the first case by using the desingularization of the graph of/(cf. Theorem 1.1). q. e. d. Proposition 1. 4. Let X be a complex manifold of dimension n and g a bimeromorphic transformation ofX. Then g* induces an automorphism of Fm(X) for every positive integer m. 2.

Complements of Divisors

In what follows, we shall consider a complex manifold X of dimension n which has a compactification. In other words, X is a complement of an analytic subset of a compact complex manifold. According to Hironaka, there is a smooth compactification Xof X. Namely, Xis a compact complex manifold of dimension n and D=X—X is a divisor which has at most normal crossings. With respect to local coordinates (wl9 •••, wn) of X, D is given locally by wl"'Wj = O9j^n. Denote by [D] the line bundle on X determined by the divisor D. Theorem 2. 1. Let X be a complex manifold of dimension n with a smooth compactification X. Put D = X—X. Then we have an isomorphism Fm{X) s H\X, Proof First we prove

O(mKx+(m-l)[D])).

244

F. Sakai

L e m m a 2. 1. We have an isomorphism : unrv-^f

v . ,

i\rmw

f meromorphic m-ple n-forms on X {holomorphic \

H (X, O(mKT+(m-l)[D})) = { m

X) with at most

^^.pupoles

dong

D)

Proof. The identification is described as follows. Cover X by coordinate neighborhoods {Ua} with holomorphic coordinates (wla, •••, vu%) in Ua. Let {kafi}, {dafi\ be transition functions of the line bundles K^, [D], respectively. Take a holomorphic section
J

^

™

in Ua.

This gives the above isomorphism. q. e. d. We proceed to the proof of Theorem 2. 1. Let

where ^(M;) is a holomorphic function on C/ and st^(m—l) venience sake, we put ^ = 0 for i=j-\-\, •••, n. Then

for z= 1, •••,7. For con-

Using polar coordinates wi=rie>/-l0iy we get n

Let C=max |g|2/m. We obtain the estimate : u ^o

Jo

U=l*/o

2

C t / l m - J i D r ' - " " " < oo, because 1 — (2^/m) > — 1. Since X is compact, we can cover X by a finite number of polycylindrical coordinate neighborhoods. It follows that ||y>||
Kodaira Dimensions of Complements of Divisors

245

D. Since codim £^>2, co extends meromorphically to X. Moreover w has at most (m— 1)-pie poles along D. q. e. d. Remark. In particular, we have FX(X)^F{(X)^H\X,

0{K2)).

Definition. Let X, X and D be as above. Set = dim Fm{X) = dim//°(X, O(mK^+(m-l)[D])) < oo. Corollary. Let X, X and D be the same as in Theorem 2. 1. Let
Qm:Xzw-+Qn{w)

-

(
of X into PN. Then we have

where N{X)=

max {dim0 m (Z)}

if

-OO

if

N{X)=j>,

{m>0\Tm>0}.

Proof. The assertion follows easily from Theorem 2. 1. Remark. In view of the above corollary, in case X is compact, our Kodaira dimension ic(X) agrees with the original one in Iitaka [5] (cf. Ueno [18]). Example. We present a classification of algebraic curves: structure

K

— OO

0

0

P.,Pi-{«i},Pi-K}-K}

0

1

1

elliptic curves

0

m(k—2)— &+l(except k=3, m=2)

A-UW. *^3 1=1

1

<1

k

elliptic curves— {J {at}, k^l

mk—k

z=l k

m(k+2g-2)-k-\-l-g

curves of genus ^ 2 — Q {at}, k^O i=i

Example. Let D be a hypersurface of degree d in the projective space Pn. If D has at most normal crossings, then K Jr„—U)

=

if

rf>w+l,

For a compact complex manifold X, the L-dimension of X, written tc(L, X), of a line bundle L on X is defined as follows ([5], [18]). For a positive integer m, let 0mL be a meromorphic map defined by a basis of//°(X, G{mL)). Then

246

F. Sakai

max {0mL{X)}

K(L, X) =

if

N(LA-

1

if

N(L,X)=t,

where N(L, X)= {m>0| dim H°(X, 0(mL))>O}. For a divisor Z) on Jf, we put *(D,X)=K([D],X).

Our Kodaira dimension has the following relation with the Kx-\- Z)-dimension of

x. Prpposition 2. 1. Let X, X and D be the same as in Theorem 2. 1. Then we have fc(X)^fc(Kx+D, X). Further ific{X)^0, then K{X)=IC(KZ + D, X). Proof. Since H°(X, O(mKx-\-(m— 1) [D])) can be regarded as a subspace of H°(X, O(mKz + m[D]))9 we have by definition, tc{X)^fc{Kx + D, X). If/c(X)^0, there is a non-zero element (peH°(X, O(m0Kx+(m0—\) [D])), for some positive integer m0. Let 2. T h e o r e m 2. 2. Let X be a complex manifold of dimension n which has a compactijication. Then there exist positive numbers a, /3 such that the estimate amK(X) £ j

m

^

1) A compact complex manifold is called a Moisezon manifold if the transcendence degree of itsfieldof meromorphic functions is equal to its dimension.

Kodaira Dimensions of Complements of Divisors

247

holds for every sufficiently large integer m. In particular, ic(X)=n if and only if limsup {rm/mn} > 0. Proof This follows easily from Proposition 2. 1 and Theorem 1 in [5]. Proposition 2. 3. Let X, Y de complex manifolds having smooth compactifications X, F5 respectively. Then /c(Xx Y)=IC(X)+K(Y). Proof Put D = X— X, C=Y—Y. Let TT,, n2 be the projections from I x f to X, F, respectively. It follows that Xx F is a smooth compactification of X X Y and

Xx f-Xx

Y=Dx 7+XxC. Obviously Dx Y=nT(D), XxC^^(C).

to prove that Fm(Xx Y) =Fm(X)®Fm(Y)

Fm(XxY)

It suffices

for every m. Using Theorem 2. 1, we get

^H°(Xx = H°(X, 0(mKx+(m-\)[D]))(g)H°(?,

0(mKy+(m-l)[C]))

(by Kiinneth formula) q.e.d.

= Fm{X)®Fm(Y).

Proposition 2. 4. Let X be a complex manifold with a smooth compactification X. Let f: X-+Y be a surjective holomorphic map, where Y is a compact complex manifold. Then for a general point y inf(X), we have *(X)^K(Xv)+dim

F,

l

where Xv = Xnf- (j>). Proof. This can be proven in a manner similar to that in Ueno [18], Theorem 5. 11. We have the following structure theorem (cf. Iitaka [5]). Theorem 2. 3. Let X be a complex manifold of dimension n with a smooth compactification X. Assume that /c(X)^>\. Then there exist a complex manifold X* and its smooth compactification X*, a projective algebraic manifold F* and a surjective holomorphic map f: X*—>F* which satisfy the following conditions. (i) X* (resp. X*) is bimeromorphic to X (resp. X),

(ii) dimF*-/c(Z) 5 (iii) for a general point yef(X*),

3.

X*=f~l(y)

is a smooth compactification of X* =

Bergman Kernel Forms

Let Z b e a complex manifold of dimension n. Let {Ua} be an open covering of X with holomorphic coordinates (wla, --, wl) in Ua. An m-ple quasi-volume form Q on X is an m-ple {n,rc)-formdefined by a continuous section {fa} of the line bundle \mKx\2 which is C°° and positive outside an analytic subset of X. We write

248

F. Sakai

O = £« {ft W=Tl2ic)dw*aAdw^m,

in Ua.

Further, if {fa} is an everywhere C°° and positive section of \mKx\2, i. e., if {?«} is a metric of mKXi we call Q an m-ple volume form. Denote by Qx/m the quasi-volume form defined by

f

in Ua.

The Ricci form Ric Q of Q is the real (1,1) -form, given locally by Ric Q = ddc log £a, in C/a, where dc=(V— 1 /4^)(3—3). Let a> be a holomorphic m-ple rc-form on X Then Hmo)/\w is an m-ple quasi-volume form, where Using the above notation, we see that (a>A6>)1/m= {fima)A®}I/m. Assume now that X has a smooth compactiflcation X. Put D=X—X. Definition. For me N(X), we define an m-ple quasi-volume form Vm by Vm = VmtX = s u p {fim(o A&}, Ila>|| = l

where the supremum is taken over all elements coeFm(X) such that ||a>||=l. Further we put vm= {Vm}1/m. L e m m a 3. 1.

i vm < oo. JX

Proof. Cover X by coordinate neighborhoods {Ua} with holomorphic coordinates (wla, •••, Wa) in Ua. Let {^a} be a holomorphic section of [D] defining D and {fla} a metric in [Z>]. A length ||d|| of d is given by \\d\\2=\da\2laa in f/a. By Lemma 2. 1> ||^||2(m"1)/m(ft>Aft>)1/m is a continuous m-ple («, n)-form on X, for weFm(X). Hence um = is a continuous (n, w)-form on X (for a proof, see Narasimhan-Simha [14], Appendix). Clearly z;m=||<5||-2(m-1)/mz/m. Thus the assertion follows from the proof of Theorem 2. 1. q. e. d. Definition. We define a hermitian form on Fm(X) by for couo>2e

Since |/im<WiA6>2|^(Om> w e § e t (^u W2)<°°3 because of Lemma 3. 1. This form is positive-definite. Remark. In general, the product (a)l9 co2) can be defined for o)xe a>26H°(X, O(mKx+m[D])). Let wOi -•', o)N be an orthonormal basis of Fm(X) with respect to the above her-

Kodaira Dimensions of Complements of Divisors

249

mitian form, where N-\-l=fm. That is, (wu wJ)=dij. Denote by X* the complex manifold which is conjugate to X. Then i^

Bm,x{w, C) =Til*mi(w) A ^ i ( C ) ,

for

w9 C € X,

i=0

J

is a holomorphic m-ple 2^-form on Z x Z (cf. [7]). Note that Bm>x{w^) is independent of the choice of an orthonormal basis of Fm(X). Letting w=£, we obtain an m-ple quasi-volume form Bm on X: Bm = Bm,x = Bm,x{w,

w). {Bm}l/m.

We call it the m-th Bergman kernel form of X. Put bm= L e m m a 3. 2. Bm = sup{fimo)A&},

for

weFm{X).

Proof See [7], [12]. "'" T h e o r e m 3. 1. Let g be a bimeromorphic transformation of X. Then we have o*V =V g

* m—

e*B

*

raj

6

= B

±J

m

^mj

(g*
wl9

a)2eFm(X).

Proof From the fact that ||^*fi>|| = |H| for (oeFm(X), it follows that g*Vm=Vm, which implies the rest of the above equalities. q. e. d. Let S= {(o € Fm(X)\ (o>, w) = 1} be the unit sphere in Fm(X). Since || || is continuous onFm(X), we can define j} = m a x ||o>||,

r = m i n ||o>||.

We have, by definition, ^ ^ r > 0 . L e m m a 3. 3. TV{(*>, CO) ^\\co\\ ^ yjV{(o, w) , for

weFm(X).

Proof Take an element weFm(X). Then a>/V(o>,ft>)€ S. By definition, we get >, (o) \\^7], which implies the above inequality. q. e. d. Proposition 3. 1. r2Vm £ Bm ^ rfVm. Proof. By definition, if (o>, o>) = l, t h e n ||cw||^^. H e n c e Bm = sup {^mo> A ^ l ^ s u p {[im(D A o>} (Q),(O) = \

\\a>\\£ri

= 7]2 SUp {/imft>' A ©'}

=

We can similarly prove the other side of the above inequality.

q. e. d.

Remark. We say that Fm(X) is base point free if for every point xeX, there exists an element o)eFm(X) such that o>(#)=£(). If Fm(X) is base point free, then Vm and Bm become m-ple volume forms on X, and define metrics in the line bundle mKx. For weH0(X, 0{mKx)), denote by \\(o\\v, \\o)\\B the lengths of co with respect

250

F. Sakai

to the metrics defined by Vm9 Bm, respectively. Namely, we have \\co\\vv —

TT TT v m

?

\\CO\ \\CO\

B

and then

(3.1)

IHI

We conclude that coeFm(X) if and only if \\co\\v (or \\co\\B) is uniformly bounded on X, In fact, if ||||^C f vm JX

||F<;||a>||. In view of Proposition 3. 1, the assertion for \\co\\B follows from this. In this case, the map 0m (cf. (2. 1)) is holomorphic and Ric(/?m) is induced from a standard Kahler form of PN by @m. 4.

Extension of Holomorphic Maps

Let Z b e a complex manifold of dimension n. We say that X is quasi-projective if Xis given as a complement of an analytic subset of a projective algebraic manifold. We shall consider the case in which tc(X)=n. First we recall the following Proposition 4. 1. Let X be a quasi-projective manifold of dimension n such that K(X) =n. Then there exists a quasi-volume form W on X such that (Ric W)n^¥.

Proof See [7], Proposition 1. For later application, we define the form W in the following way, which is a slight modification of that in [17]. Let J? be a smooth compactification of X, which is projective algebraic. We may assume that D=X— X \-Dk, and that D has normal is a union of non-singular divisors Dt : D=Dx-\ crossings. Cover X by coordinate neighborhoods {Ua} with holomorphic coordinates (wla, •••, wl) in Ua. Let dt= {dii(X} be a holomorphic section of [/)*] defining Dt and {ait(X} a metric in [AL f° r each i, and let {aa} = lflai>a\ be a metric of [D]. \i~ 1

)

Then ||^||2=|^>a|2/^,a in Ua defines a length of 8t. Let / / b e a very ample line bundle on X. Further we assume that the line bundle H-\- [D] is ample on X. Since K(X) =n, we have, by Proposition 2. 2, K(K^+D, X)=ny and then there exists an effective divisor E in the complete linear system \m(K^+D)— H— D\ for a large integer m. Let {aa} be a holomorphic section of [E] defining E. Let {ha} be a metric of H such that ddc log (haaa) is a positive (1,1) form on X. Define ¥ by in We obtain the desired form, by letting the constant c be small enough and multiq. e. d. plying the metrics {aia} by suitable constants (See [17], for details). Definition. The Poincar£ volume forms on An and J*xJ w ^ ! are defined by

Kodaira Dimensions of Complements of Divisors

F

251

—

where 0= f[ (V— 1 l2n)dZi/\dZi. Note that (Ric V)n=V, in both cases. L e m m a 4. 1. Let X have the same meaning as in Proposition 4. 1. Let f: Jn—•X (resp.f: J* x Jn~1-^X) be a non-degenerate holomorphic map, i. e., a holomorphic map such that r a n k ( / ) = d i m X=n. Then ^ v, f* ¥ where V is the Poincare volume form of An (resp. d* X Jn~l). Proof See the proof of Proposition 2 in [17]. As a consequence of the above lemma (Schwarz' lemma), we obtain the following Proposition 4. 2 (cf. Proposition 3 in [17]). Let X be the same as in Proposition 4. 1. Letf: J* x An~x^>X be a non-degenerate holomorphic map. Then f extends to a meromorphic map from Jn to any compactification of X.

Proof. Let X be a smooth compactification of X as in the proof of Proposition 4. 1. Let {Ua}, D, dt, //, ha, E and aa be the same as in Proposition 4. 1. Since //is very ample, there exists a basis s09 •••, sNofH°(X, 0(H))y with i V + l = d i m H°(X, 0{H)) such that the map 0H : w—>(so(w) : ••• : sN(w)) is an imbedding of X into PN. Therefore, it suffices to prove that the map 0Hof extends to a meromorphic map from An. Set ^ = 1 1 ^ . Let Vi=-^T-{dwlaA-'-AdwZ)m,

in Ua

be an m-ple w-form on X for each i. By Lemma 2. 1, each co^

A ^)1/TO = tfJZ

U (V^T^dw^Adm,

in Ua.

Letting ||^|| be small enough by multiplying the metrics [ai
n Hence >t A o>ty/m ^ -

t

(o»4 A ® t ) 1 M ^

by multiplyng the metric {ha} by a constant as ||jf||<^l. Combining this with Lemma 3. 1, we get

252

F. Sakai

which implies that

L L


for

By Lemma 1. l,f*(ot extends meromorphically to J?, and then to An by letting r—>1. Noting that (f*o>0: ••• :f*(oN) = (f*s0' ••• : / * ^ ) > we infer that (Z^o/extends to a meromorphic map from J n into / V q. e. d. Proposition 4. 3. Z,££ X be as in Proposition 4. 1. Let X be a compactification of X. Then every biholomorphic transformation of X extends to a bimeromorphic transformation of

X. Let X be a complex manifold. Given a Borel subset B in X, choose holomorphic maps/*: An—>X and Borel subsets Bt in J n , such that 5 c U / * ^ ) . Define

where the infimum is taken over all possible choices of/i5 Bi9 and Fis the Poincar£ volume form on J n . We say that X is measure-hyperbolic if fix(B)>0 for all nonempty open subsets i? in X (cf. [8]). T h e o r e m 4. 1. Let X be a quasi-projective manifold of dimension n such that ic(X) =n. Then X is measure-hyperbolic.

Proof Let B be a nonempty open set in X and choose holomorphic maps^ : J —>X and Borel subsets Bi such that 5 c \jfi(Bt). Then, we have by Lemma 4. 1, n

which implies that fiz(B)^fw>0.

q. e. d.

Corollary. Let X be as above. Then there exists no non-degenerate holomorphic

map f: 5.

Representation of the Group of Bimeromorphic Transformations

Let X be a complex manifold. Denote by Bim (X) (resp. Aut(JQ) the group of bimeromorphic transformations (resp. biholomorphic transformations) of X. Moreover in case X has a compactification X, we denote by Bimalg(X) the group of bimeromorphic transformations of X which can be extended to bimeromorphic transformations of X. So Bim alg (Z)=Bim(X) nBim(Z). Now assume that X is a complex manifold of dimension n having a smooth compactification X. In view of Proposition 1. 4 and Theorem 3. 1, we have a unitary representation

Kodaira Dimensions of Complements of Divisors

253

where U (Fm(X)) aGL>(Fm(X)) is the group of unitary matrices with respect to the hermitian form of Fm(X) defined in Section 3. We shall prove the following T h e o r e m 5. 1. Let X be a complex manifold of dimension n having a smooth compactification X. Assume that X is a Moisezon manifold. Then i o m (Bim alg (X)) is a finite group.

As a corollary, we obtain T h e o r e m 5. 2. Let X be a quasi-projective manifold of dimension n such that ie{X) =n. Then Aut(-Af) is a finite group.

Proof. By Proposition 4. 3, every element of Aut(X) extends to an element of Bim(X). So the above theorem implies that pm(Aut(X)) is a finite group. Since K(X) —n, the map Om (defined by (2. 1)) becomes a bimeromorphic map for a large integer m (cf. the proof of Proposition 4. 3.). Hence the representation pm is faithful for this m. It follows that Aut(Z) is a finite group. q. e. d. Corollary. Let X be a complex manifold of dimension n with a smooth compactification X. Put D=X-X. IfKz+[D] is ample, then Bim(X)=Aut(X) is afinitegroup. Example. Let D be a hypersurface of degree d in Pn which has at most normal crossings. If d>n-\-\, then Bim(P n —D)=Aut(P n —D) is a finite group. Remark. Some partial results are obtained in [3], [4]. See also [6]. Proof of Theorem 5. 1. We follow the argument of Ueno [18], Nakamura-Ueno [13]. L e m m a 5. 1 (cf. [18], Proposition 14. 4.). For an element g€ Bimalg(X), if g*a) =Xco holds for some non-zero element co e Fm(X), then X is an algebraic integer. Proof. By hypothesis, g extends to an element g € Bim(Jt). If m= 1, then we have seen that FX(X)^HQ(X, 0{K2)). The assertion is the same as in [18]. In fact, g induces an endomorphism of Hn(X, Z)o and then X is an algebraic integer and the degree of the minimal equation of X is bounded by the n-th Betti number bn(X) of

X. Consider the case m^>2. Let {Ua} be coordinate neighborhoods of Xwithholomorphic coordinates (wla, •••, w%) in Ua. ¥\itD=X—X. Let {da} be a holomorphic section of [D] defining D and let {ka^ be transition functions of Kg- By Lemma 2. 1, we can write
m

,

in Utt9

where {<pa} is a holomorphic section of H°(X, Q(mKz+(m—l)[D])). Let n : M—• X be the P r bundle over X corresponding to the canonical bundle Kg of X. We may assume that M^^UaXPx. We choose coordinates (wa, £«) on UaxPl9 with

254

F. Sakai

inhomogeneous coordinates C« of P l3 such that ^a=kafi/m induces a bimeromorphic transformation g of M by

MlUaC]U^. We see that g

$ : (wa9 C«) - ( I K ) , det(3|/3a;a)-'C«). Let mv be the automorphism of M defined by mv: (wa, C«) - • O«, *. Let F be the subvariety of M defined by Then F is an m-fold covering of X. By a sequence of monoidal transformations of M, we can desingularize F. Let ^ : M*—+M be such a desingularization and let F* be the strict transform of f by nx. V\xtf=no7tx and Y*=f^(X). Then 7* is a generically m-fold covering of X. Moreover, mvog induces a bimeromorphic transformation h of F*. Define a meromorphic rz-form 0 on M by 0 = £adwla A • • • A 2 in Af,^tt. Then 6 can be lifted to a meromorphicrc-formon M* and induces a meromorphic 6 on F*. By definition, we get Further, since mvg*6=v6, we have h*(d)=vd. By the proof of Proposition 1. 3, we see that /*(<*>)ir* €F m (7*), from which follows that ^ 6 ^ ( 7 * ) . Hence 0 is a holomorphic /z-form on F*. Therefore v is an algebraic integer, by the case in which m=\. Moreover [Q(X) : Q]^bn{Y*). q. e. d. L e m m a 5. 2. Let X, g, w and X be the same as in Lemma 5. 1. Then X is a root of unity.

Proof. For the proof of this fact, we need the hypothesis that X is a Moisezon manifold and then we can assume that X is protective algebraic. Because pm is a unitary representation, we see that |^| = 1. The assertion follows from the proof of Proposition 14. 5 in [18]. q. e. d. We proceed to the proof of Theorem 5. 1. For ge Bimalg (Z), pm(g) is of finite order by Lemma 5. 2. To prove the theorem, it suffices to show that the order of pm(g) is uniformly bounded for all g e Bimalg(X) (Burnside). This can be established in a manner similar to that in [18], Theorem 14. 10. q. e. d. Appendix

Let W be a compact complex manifold of dimension n and D an effective (reduced) divisor on W. As we have seen in Section 2, to calculate the Kodaira dimension K(W—D), we need to desingularize D. Here we examine the process of desingularization more precisely. According to Hironaka, we can find a sequence of monoidal transformations m : Wi-^Wi_x with non-singular centers Ct_x for i= 1, •••, / such that

Kodaira Dimensions of Complements of Divisors

255

(i) Wo = W9Wt = W*9 (ii) Dt = the support of ;r*(A-i)> (iii) Dt = D* has at most normal crossings, (iv) n : W*—D* —> W—D is biholomorphic, where n = ^o ••• o/^. Thus W7"* is a smooth compactification of W—D. We use the following notations : Dt — \ht strict transform of Di_l by m ; 2?* = the exceptional locus of n^ i. e., ^il{Ci_x) ; ^ = the codimension of Ci_l in W ^ ; i^ = the multiplicity of the singular locus of Dt_x along Ci_l. Then we have Dt = Dt+Ei9 nf(Di.l)=Di+viEi, KWi = It follows that (A. 1)

Proposition A. 1. We have Tm(W-D) = dim H°(W*,O(mKw*+(m-l)[D*])) ^ dimH°(W9O(mKw+(m-\)[D])). Proof. Set Li = mKWi+(m—l) [A] and Ai = m(^—^) + (^— 1). By (A. 1), we see that Li=7ti(Li_l)+bi [Ei]. It suffices to prove that (A. 2) dim H°(WU 0(Lt)) ^ dim H°(Wt.l9 0(A-O), for each i. If i ^ O , (A. 2) is clear. In case 6i>0, for any effective divisor Z belonging to the complete linear system 1^1, the direct image T=m^{Z) belongs to |A-i|> which implies that Z—7r*(T) is linearly equivalent to btEi and then Z=n%(T)-\biEi, because Et is exceptional. Therefore the map K^ : \Lt\—>|A-i| is an isomorphism. Thus we obatin the equality dim H°(Wi9 0{Lt)) =dim H0(W^l9 0(A-i))q. e. d. L e m m a A. 1. If dim H°(Wi} 6(mKw.+ (m—l) [A])) > 0 for m, then we have K(KWi+Di9 Wt) = K(KWM+Dt,l9

some

positive integer

W^).

Proof Let ri=Kw. + [Di] for each i. By (A. 1), we obtain the equality rt = ^ ( A - O + ^i—vJlEt]. In case (dt—vt)^0, we can show that dimH°(Wi9 0(rt)) = dimH°(Wi_u O(rt-i)) in a similar manner as in the proof of the above proposition. Consider the case in which (^—^)<0. Using (A. 1), we get rfK-,) = mKWi+(m-l)[Di] + [Di]+ {m^-dj + l}^]. By hypothesis, there is an effective divisor Ze \mKw. + (m— 1)A|? a n d t n e n w e g e t

^i, W^i) = *(ri9 Wt).

(cf. Lemma 5, in [17]) q.e.d.

256

F. Sakai

As a consequence of this lemma, we obtain Proposition A. 2 (Iitaka). If it{W— D)^0, then we have K(KW*+D\ W*) = >c(Kw+D, W). Proof Since *{W-D)^09 we have dim H\W\ 0(m^+(m-l)Z)*))>O, for some positive integer m. By Lemma A. 1, we get K(Kwt+D*, W*) = f ^ + A - , , W,-,) and by (A. 2), we have ])) ^ dim H°(W*, O(mKw*+(m-l)[D*])) Therefore by induction, we can prove Proposition A. 2.

q. e. d.

Combining this with Proposition 2. 1, we have the following Theorem A. 1. Let W be a compact complex manifold and D an effective divisor on W. Iffc(W-D)^0, then

Example. Let Hl9 H2, H3, i/ 4 be four lines in P2 which meet as in the diagram below. Set D=Hl-{-H2-\-Hi-\-HA. In this case, we have (KP2+D, P2) = 2, K(P2-D) = 1, K{P2-D) = - co.

References

[ 1 ] Andreotti, A. and Stoll, W. : Analytic and algebraic dependence of meromorphic functions. Lecture Notes in Mathematics 234, Springer, Berlin-Heidelberg-New York, 1971. [ 2 ] Griffiths, P. : Holomorphic mappings into canonical algebraic varieties. Ann. of Math. 93

Kodaira Dimensions of Complements of Divisors

257

(1971), 439-458. [ 3 ] Carlson, J . : Some degeneracy theorems for entire functions in an algebraic varieties. Trans. Amer. Math. Soc. 168 (1972), 273-301. [ 4 ] Fujimoto, H. : Families of holomorphic maps into the projective space omitting hyperplanes. J. Math. Soc. Japan 25 (1973), 235-249. [ 5 ] Iitaka, S. : On D-dimensions of algebraic varieties. J. Math. Soc. Japan 23 (1971), 356-373. [ 6] : On logarithmic Kodaira dimension of algebraic varieties, in this Volume. [ 7 ] Kobayashi, S. : Geometry of bounded domains. Trans. Amer. Math. Soc. 92 (1959), 267290. [8] : Hyperbolic manifolds and holomorphic mappings. Marcel Dekker, New York, 1970. [ 9] and Ochiai, T. : Mappings into compact complex manifolds with negative first Chern class. J. Math. Soc. Japan 23 (1971), 137-148. [10] Kodaira, K. : On Kahler varieties of restricted type. Ann. of Math. 60 (1954), 28-48. [11] : On holomorphic mappings of polydiscs into compact complex manifolds. J. Diff. Geometry 6 (1971), 31-46. [12] Lichnerowicz, A. : Varietes complexes et tenseur de Bergman. Ann. Ins. Fourier 15 (1965), 345-408. [13] Nakamura, I. and Ueno, K. : An addition formula for Kodaira dimensions of analytic fibre bundles whose fibre are Moisezon manifolds. J. Math. Soc. Japan 25 (1973), 363-371. [14] Narasimhan, M. S. and Simha, R. R. : Manifolds with ample canonical class. Invent. Math. 5 (1968), 120-128. [15] Remmert, R. : Holomorphe und meromorphe Abbildungen komplexer Raume. Math. Ann. 133 (1957), 328-370. [16] and Stein, K. : Eigentliche holomorphe Abbildungen. Math. Zeitschr. 73 (1960), 159189. [17] Sakai, F. : Degeneracy of holomorphic maps with ramification. Invent. Math. 26 (1974), 213-229. [18] Ueno, K. : Classification theory of algebraic varieties and compact complex spaces. Lecture Notes in Mathematics 439, Springer, Berlin-Heidelberg-New York, 1975.

Department of Mathematics Kochi University (Received February 9, 1976)

Compact Quotients of C3 by Affine Transformation Groups, II

T. Suwa15 Let G denote a group of affine transformations of the three-dimensional complex vector space C3. Assume that the action of G is free and properly discontinuous and that the quotient C3jG is compact. This is the second part of a study of compact complex threefolds of the form C3/G. The main results were announced in [4]. Each element g in G is represented by a 4 x 4 marix g

~ [o

where A(g) (the holonomy part) is in GL(3, C) and b(g) (the translation part) is a column vector in C3. The action of'g on C 3 = {z\z = '(Zi, Z2, Zi)} is given by

LiJ

L o

In the first part [5], it has been shown that, if G contains no elements whose holonomy parts have three different eigenvalues, then G contains a nilpotent subgroup N of finite index. Thus if G satisfies the condition given above, the quotient C3\G is finitely covered by C3/N, where N is nilpotent. In the present paper, the quotients C3/N are classified. In section 1, the number of linearly independent closed holomorphic 1-forms is computed in each case. We determine, in section 2, the structure of C3jN by analyzing the Albanese map, which is defined by Blanchard [1]. The quotients C3jN are classified in Tables I and II. They include some new non-Kahler manifolds. In section 3, we study holomorphic forms on C3jN. 1.

Closed Holomorphic 1 -Forms on C3/N

We may assume ([4]) that, for every element g in N, (1)

"1 an{g) al3(g)~ A(g) = 0 1 a23(g) _0 0 1 (61(g), b2{g), b3(g)). For a pair (gl, g2) of elements of N, we let [gl, g2] 21' F o r f u n c t i o n s fl< : N ^ > C , l < i < r , w e d e f i n e [al9 •••, ar] : N r = N x ••• X N

1) Research supported in part by NSF grant GP 43614 X.

260

T. Suwa

—>C by [al9 •••, ar] (gl9 •••, £ r )=det (at(gj)). For each fixed r, the functions : Nr—*C form a C-algebra in a natural manner. We have 0 0

|>12,

4B]

!

°

0

j

(&, &)

* [4B,

*s] (&,& 0

0 0 0 1 where * = ([>12, *2] + I>i3, *s])(£i, & ) ~ f e 4B] (&, &)-(*s(^i)+*3 (&)). Let £/,= {£,}, 1 <*"<3, denote the i-th factor of C3=Cx Cx C. For each fixed £3, the commutator group Nil) = [N, N] of N acts on UxxU2x {z$} effectively and properly discontinuously as a group of translations. Hence we have Na)aZ4=ZQ)Z®Z(§)Z, where Z denotes the ring of integers (Wolf [6], 2. 5. 4. Lemma). Note that N, being the fundamental group of a compact space, is finitely generated. Let 0 denote the structure sheaf of a complex manifold. In this section, we compute the number
/< € H\C\ 0),

\
Since the pullback g*w of w by ge N is given by g*o>=fl{gz)dzl f2(gz))dz2+(fl(gz)al3(g)+f2(gz)a23(g)+f3(gz))dz3, the N-invariance conditions are (3) /,(*) =/,(**), Uz) =fi(gz)al2(g)+f2(gz), and (4) (5) Uz) =Mgz)aa(g)+fl(gz)aB(g)+Mgz), for all^6 N. Also, since co is closed, we have

By (3), we may think of/I as a holomorphic function on C3/N, which is compact. Hence/j(^) =cx is a constant. From (4), we have-J^(z) 0Z2

= -%^-(gz). QZi

Hence~^-=c2 0Z2

is a constant and we have f2(z) =^2+^1(^3), where cpx is a holomorphic function of £3. From (4), we infer that
Compact Quotients of C 3 by Affine Transformation Groups, II (11)

^ 2 3 + ^9*3 = 0,

(12)

^l«13 + ^ 2 3 + ^ ( ^ 3 + ^ ) + <7*3 + *8*3 = 0>

(13)

2c
261

= 0.

Definition 1. A permissible coordinate transformation (p. c. t., hereafter) of C3 is a coordinate transformation (analytic automorphism)

(pg
tne

(2, 3)-entryof q. e. d.

Case I) Suppose # (1) = 0. L e m m a 2. In this case, by a suitable coordinate transformation of C3, every element of N is reduced to a translation (cf. [2] Theorem 2. 1). Proof By Lemma 1, we may assume that a23=0. Since C3/N is compact, there is an element g0 with b2(g0)^0, and b3(g0)^0. Letting a = al2(g0)lb2(g0) and £ = 0i3(£o)/*s(&)> consider the coordinate transformation Then

Z2, Z3),

q. e. d.

By the above lemma, the quotient C3/Nis a complex 3-torus. Therefore d=3 and dzi, dz2, dz3 form a basis for H\C3jN, dO)=H°(C3, dO)N. Case II) Suppose N(l)^0 and [a23, £3] = 0. By Lemma 1, we may assume a23=0. Thus the functions al2, al3, b2, and b3 are homomorphisms of iV^ into the additive group C. Let N{ and A be, respectively, the kernel and the image of the homomorphism g—>(b2(g)9 b3(g)) of JVinto C2=U2X U3. We have N^czN^ Moreover, since iVhas no fixed points on C3, we have Nx= {ge N\al2(g)=al3(g)=b2(g)=b3(g)=0]. Therefore, NY acts on UX = C effectively and properly discontinuously as a group of translations. Thus Nx(zZ2. Let V denote the real vector subspace of U2X U3 spanned by J. Note that J is a finitely generated free abelian group.

262

T. Suwa

L e m m a 3. dim RV=A and hence rank J > 4 . Proof. We have C*IN={C3INl)l(NINl) = (UlINlxU2x UJKN/NJ. If dim* 7 < 4, we would have U2xU3=Vx W for some positive dimensional real vector subspace WofU2x U3. Then, since N/N&J acts trivially on W, C*IN=({UlINl X V)/ X W, which is not compact, q. e. d. L e m m a 4. [b2, b3] ^ 0. Proof If [£2, b3] = 0, then b2=ab3 for some ae C, contradicting Lemma 3, q. e. d. Since a23=0 and b3^0, we have, from (10) and (11), cb—c9=0. Then (13) implies £8z=0. The iV-invariance conditions reduce to (9)'

Wn + Czbi + Ctbs = 0,

(12)'

c{al3+c4b2+cA = 0.

From these we get £i([tfi2, ^2] + [<^i3? # 3 ])=0. Since JV(1)^:O, we have ^ = 0. Lemma 4, (9)', and (12)' imply c2=c^=c7=0. The constants £3 and £6 are arbitrary. Therefore d=2 and ^ 2 , rf^ form a basis for H°(C3/N, dO). Case III) Suppose [tf23, £ 3 ]^0, and 0 12=O. From (10), (11), and (13), we get c2 The invariance condition is

—C4=C5=CQ=C9=0.

(12)" Since al2=0,

^1^13 + ^23 + ^ 3 = 0. t h e functions al3, a23, a n d b3 a r e h o m o m o r p h i s m s . T a k e a s e t g l 5 g2, •••,

gr of generators of N and set DiJk=[al39 a23, b3] (gi9 gj9 gk), \
k
L e m m a 5. If Dijk = 0 for all i,j, k, then the situation is reduced to that of case II. Proof. From the condition, we infer that the equation (12)" has a non-trivial solution for (cl9 c3, c7). Since [a23, b3]^0, we have ^ ^ 0 . Thus we have al3=aa23+pb3 for some a, fie C. Applying the p. c. t.
q. e. d.

In view of Lemma 5, we assume that Dijk^0 for some i,j, k. Then from (12)", we have cl=c3=c7=0. Therefore d=l, and dzz is a basis for / CaselV) Suppose [a23, b3]^0 and al2^0. From (10), (11), and (13), we get c2 =C4=c5=cQ=c9=0. Also, from (9), we have ^ = 0. Then (12) implies £ 3 =£ 7 =0. Therefore, d=\ and dz3 is a basis for H0(C3/N, dO). Summarizing the above we have the following T h e o r e m 1. i) By a suitable coordinate transformation of C3, every case is reduced to one of the followings :

Compact Quotients of C 3 by Affine Transformation Groups, II

(I) N^ = 0, (II) # 0 ) ^ o 5 a23 = 0 and [b29 b3] * 0, (III) a12 = 0 and [al3, a23, b3] ^ 0 , (IV) al2 * 0, and [a23, b3] * 0. ii) In each case, we can choose the following set as a basis for H0(C3jN, H°(C\dO)N: (I) dz» dz2, dz3; (II) dz29 dz3; (III) and (IV) dz3. 2.

263

dO) =

Structures

In this scetion, we determine the structure of the quotient C3/N by analyzing the Albanese map of C3jN. Recall that the Albanese variety Alb(Af) of a compact complex manifold M is defined as follows (see Blanchard [1]). Consider the space D dual to H°(M, dO), and let J denote the image of the homomorphism F: HX(M, Z) —>D given by the integration of 1-forms on the 1-cycle p, i. e. F(y)= I . Also let A denote the smallest closed subgroup of D containing A whose component of 0 is a complex vector subspace of D. Then Alb(M) is the complex torus DjA. Fix a point z0 on M. The Albanese map M—>Alb(M) is defined by £—• I Jzo

^

(mod. A). Let 7} : N=n{(C3jN)—>//,(C3/iV5 Z) be the canonical surjection (Hurewicz homomorphism), and let iV^ker Forj. We have the exact sequences :

(*)

n

|

II For)

1 _ > NX ^ N > A - > 0. Note that A is a finitely generated free abelian group. We will see below that iVjCZ4 and that rank iV^-frank A = 6. Thus the group N has at most 6 generators. We denote by Tk a ^-dimensional complex torus. Case I) NO) = 0. As was proved in section 1, the quotient C3/N is T3, and Alb(C3/N) = C3/N. Case II) JV(1)^O, a23=0, and [b2, b3]*0. We identify D with C/2X f/3 by taking the basis dual to (dz2> dz3) for D. Then we have For}(g) = ( f dz2, \ dz3) = (b2(g)i b3(g)). Hence Nx coincides with the kernel of the natural action of Non U2 X U3. Moreover, if he Nly then al2(h) =al3(h) =0, since otherwise h would have a fixed point on C3. Lemma 6. N{czZ2. Proof. See section 1, Case II). L e m m a 7. We may assume that there are elements gx and g2 in N which are represented

I . Suwa

264

as follows :

Si =

-1 0 0 0

0 1 0 0

a

i9I-

0 1 1 0 0 1

> & =

'I 0 0 0

0 1 0 0

0 0 1 0

& 0 1 1

Proof We have [b2, b3] (gl9 g2)^0 for some gl9 g2 in N. By a suitable linear transformation on U2xU3, we may assume (#2(gi)5 ^3(£i)) = (l> 0) and (b2(g2)9 = (0, 1). Consider the p. c. t. of C3. I n terms of the new coordinate system, g{ and g2 are represented by •1

0 «i p 0 1 0 1 0 0 1 0 0 0 0 1

and

"1 a2 0 j82 0 1 0 0 g2 = 0 0 1 1 0 0 0 1

Finally if we apply the p . c. t.
Z2, ZS), then g{ andg 2 are q. e. d.

Let V denote the real vector subspace of U2 X U3 spanned by A as in § 1. By Lemma 3, d i m i J F = 4 and rank J > 4 . Let hu •••, hs(s<2) be a Z-basis for NY and choose g3, •••,^ r (r>4) so that {b^g,), A3(gi)), ••-, (* 2 (^) ? 43(5r)) form a Z-basis for J = I m F o ^ I m F . Then N is generated by Ab •••, hs, g{, •••, ^ r . Finally let k{, •••, A:« (^<j) be a Z-basis for iV(1). Lettig mu 1 4. Profl/. Suppose rank J = 4 . Then J is a lattice in U2 X C/3, and N/N^A acts properly discontinuously on U2X U3. Since NX=Z, the quotient C3IN=(C3INl)l(NINl) = {{C/Z) XU2X U^KN/Nt) is a C*-bundle over T2, which is a contradiction, q. e. d. We have [gt, gj]=h%i', for some nt^Z^ l
Hence we have

( K , b2] + [al3, bs])(gi9 gj) = nih 1 < z, j < r.

In particular, we have (15)

a = nl29

(16)

al2(gt)— abs(gt) = nil9

(17) L e m m a 9. For

C,

(18)

al2+X(al3—nb2)

= 0 , n = n12.

Compact Quotients of C 3 by Affine Transformation Groups, II

Proof. Let nx and n2 be defined by nl(gi)=nil (16), and (17) in (14), we have (19)

265

and n2(gi)=ni2. Substituting (15),

(K, b2] + [n2, b3]-n[b2, b3])(gi, gj) = ntj.

From (19), we have (20)

[»„ A2, A3](ft, gjy gk) = nijbs(gk) +nJkb3(gt)

(21)

[n2, b29 A3](ft, gj, gk) = — (ntjb2(gk) +njkb2(gi)

+nkib3(gj)9 +nkjb2{g3)),

(22)

([»„ «2, A3] —n[n l9 b29 AJ) (gi9 gj, gk) = n$k9

(23)

([nl9rc2,A2] + « K A2, A,]) {gi9 gj, gk) = - « , %

where n^k=nijnkl+nJkna+nkinJl9 l
k
Recall (A2(ft), A3(ft)) = (l, 0) and (A2(&), As(ft)) = (0, 1) (Lemma 7). From Lemma 8, we must have (25)

ni% = ni% = m^ = 0,

1 < ij, k < r.

Since nx=a{2—nb3 and n2=al3, we have from (19) 0 = mij=[nl9 n2] (ft, g^—nn^ = ([al2-nb3, al3]-n([al2-nb3, = ([«i2, ais]—n[al29 b2])(gi9gj)

A2] + [fl13, A3]—n[A2, A 3 ]))(ft,^) = [al2, aa—nb2](gi9gj).

Hence we infer that the equations ^i2(ft)+^(«is(ft)— nb2(gt)) =0,

(26)

1 < i < r,

have a non-trivial solution for (cx, c2). If ^, = 0, then nb2(gi)=al3(gi)=ni2. Hence d i m ^ F ^ S , which is a contradiction. Thusc,^0. Hence for some Xe C, fli2(ft) + Kan(gi)—nh(gi))=Q, l Zs) = (^i9 Z2, Zz) = (Z\> Z2, Z2—*~xZz). Then, in terms of the n e w coordinate system, every element g 6 N is represented by

(27)

1 0 g = 0 0

1 0 0

0 1 0

TO] %(g)

K(g) I

"i ai2{g)+tou(g)

0

1

0

0

0

0

b{(g) 0 1 0

Wg) b*(g)-*-%(g)

1

From (18), we have a\2(g) =nXb'2(g). Applying the p. c. t.
nX g"

q. e. d.

T. Suwa

266

Let N2 and Ax denote, respectively, the kernel and the image of the homomorphismg—>b3(g) of N into the additive group U3—C. Also let Vx denote the real vector subspace of U3spanned by Ax. Since C3/Nis compact, dim RVX = 2 and rank A{>2. L e m m a 11. If g e N29 then al3(g) =0. Proof Take ge N2. Then, for any h e N9 we have bx([g9 h])=al3(g)b3(h). If al3(g) =^0, then, since rank Ax>29 we have rank iV (1) >2, which is a contradiction, q. e. d. By Lemma 11, the group N2 acts on Ul X U2 effectively and properly discontinuously as a group of translations. Hence N2aZ\ Let rank Ax=p and choose/ l 5 ..., fp€ N so that bs(f)9 •••, bs{fp) form a Z-basis for J,. Also let /1? •••, lq be a Z-basis for JV2. Clearly the group N is generated b y / l 5 ll9 •••, /, L e m m a 12. N/N2c^Al acts properly discontinuously on U3. Proof We have [fi9fj] =hlmij, for some mtj e Z. Since 3. Then we have /n.;. = 0, l
0 0 1 0 0 1 0 0

b \H)~

0 0 1 \h) 0 > Ji = 0 0 1 0 0 •1

r

a

Yi\

0 1 0

1

where*) (*,(/,), *2(/,)) = (1, 0), (*,(/,), 4,(4)) = (0, 1), * , ( / ) = !, b) (*,(/«), < 4 , «r^ linearly independent over R, and b3(f), j=l, 2, are linearly independent over R, c) (b2(lt)9 0), 2 < z < 4 , and {b2(f), b3(f)), j = l , 2, ar* /m^r/j independent over Z,

d)[al3,b3](flJ2)=meZ-0, ?>)HX{C*IN,Z) =Z5®Zm. Proof 1) We have C3IN=(C3IN2)l(NIN2) = (((UlxU2)IN2) xU5)l{N/N2).

By

Lemma 12, the canonical projection CzIN—^U^Al is a bundle map onto the maniSince C3/N is compact, and A{ and iV2 are fold U3\AX with fiber {UxxU2)jN2. groups of translations, U3/Al= Tl and (C/2X U3)/N2= T2 are complex tori. Clearly A = U2xAl9 Alb (C3IN) = U2x U3\A = U3\A{ and the Albanese map is the bundle map. 2) From 1), we h a v e p = 2 , and q=4. The assertion b) also follows from 1). By suitable linear transformations of U2 X U3 and £/3, we may assume the situation is as in a). Lemma 8 implies c). Since al2=a23=0, N(l) is generated by [f,f2]> Moreover, since N(l) c N29 we may let [f,f2] =/r> f ° r some m e Z— 0, which yields d). 3) In view of the

Compact Quotients of C3 by Affine Transformation Groups, II

exact sequence (*), it suffices to show that rj(l2)9 ?]{k), y(h)> v{f)i ^ ( J Q 4

267 are

linearly

2

independent over Z. Suppose we have 2 m^(/^)+ Zi ^ ( / / ) = 0 - Then we have t=2 5= II we I?*.have II fytN«\ Hence 0 = *3te) = £ "A(/,), which yields n, = q.0, e.j =d.l , 2. Then rn^O, i=2, 3, 4, t2 l l

Remark. From 2a, c), we have rank J = 5. II ii) Suppose NX=Z2. In this case we have the following Theorem 3. In case II, if NX=Z2, 1) the Albanese map of C31Ngives a Tl-bundle structure on C3/N over Alb(C3/N) = T\ 2) the group N is generated by "I 0 0 0

0 0 1 0 0 1 0 0

(k)

0 0 1

gi

13

0

~ 0

0

1 £ 3 (V

_0

0

0

_

0

b2(g

£=1,2,

1 "

a) A, (hi), i== 1,2, ' linearly independent over R, (b2(gj)9 h{gj))9 1 <j<4, independent over i2, b) [^, ^ ] , l < z , j < 4 , generate N(l\

Proof 1) Since A ^ Z 2 , Ux/Nx=Tl is a complex torus. We have C3IN=(C3INX)I U3)I(N/NX), and the projection C3jNx-^U2X U3 is proper. Hence (N/Nx) = (TlxU2x NjNx~A acts properly discontinuously on U2 X U3, and C3/Nis a T^-bundle over T2 = U2xU3/d. Clearly Alb(C31N) = (U2x U3)/A, and the bundle map is the Albanese map. 2) From 1), we have s=2, and r=4. The assertion a) also follows from 1). Since a23=0, aX2, aX3, b2, and b3 are homomorphisms. Hence we have b). 3) if N(l) = Z , we may assume hfx^kXi mx 6 Z—0. In view of the exact sequence (*), it suffices to prove that r](h2), ^(gi), •••, ^(^4) are linearly independent over Z. Suppose 4

\»Mgj)=0-

ij{(gj),

Then

g=%Ugl'e N^cN,.

Hence (0, 0) = (b2(g), bs(g)) =

b3(gj)). Therefore fy = 0, 1 < 7 < 4 . Then we have also n=0. If N^ =

2

Z , we may assume hfi=kii m^Z— 0, z= 1, 2. We can similarly prove that y(gj)9 1 < 7 < 4 , are linearly independent over Z, q. e. d. Remarks. 1. By Lemma 6, we may let aX2{gx) =al2(g2) =al3(g2) —0. 2. By suitable linear transformations of Ux and U2X U3, we may let bl(hx) = l, (*»(«.)» *s(5.)) = ( l , 0 ) , and (A2(&), As(&)) = (0, 1). Also let b1(ha)=w. Then, in terms of the entries, the condition 2b) is given as follows. If Na)=Z, ciX2(gx), aiiigj—anigjbsigi), aX3{gt), i=3, 4, and ([al29 b2] + [al39 b3]) (g3, g4) are integers. The integer mx in 3) is the g. c. d. of those integers. If iV(1) = Z 2 , those complex numbers generate a subgroup of rank 2 in the free abelian group generated by 1 and a). The integers mx and m2 in 3) are the invariant factors of this subgroup.

268

T. Suwa

Case III) al2=0 and [a13, am b3]^0. We identify U3 with D by taking the basis dual to dzzforD. Then we have Foiq(g) = f dz3=b3(g). Hence N{ coincides with the kernel of the natural action of iVon U3. For each fixed £3, N{ acts on UxxU2X {£3} effectively and properly discontinuously as a group of translations. Hence we have the following lemma (cf. [5], 2. 5. 4. Lemma). Lemma 13.

NxaZ\

Let V denote the real vector subspace of U3 spanned by A. Since C3/N is compact, we have dim / J F=2 (cf. the proof of Lemma 3). Hence rank J > 2 . Let hl9 •••, hs and kl9 •••, kt be, respectively, Z-bases of NY and Ar(1)(cAri) such that ki=hfi, mt eZ—0, l2, so that b^gy), •••, b3(gr) form a Z-basis for J. From the equation (2), we have A([g9 h])=I, lb([g9 h]) = ( K , b3] (g, A), K , As] ( f t A), 0), for all g, he N. I l l i) N(l)=Z. By a suitable coordinate transformation, we may assume A(kl) = t /? b(kl) = (bl(kl), 0, 0). Then we have [ 0, 0) and tb(k2) = (O, 1,0). Lemma 14. 7/" A € iVl5 then al3(h) =a23(h) = 0 . Take h$Nx. Then [A, gJ^A;^™* f° r some nu m^zZ. Hence we have =«i J 3(gi)=rni. Since rank J > 2 , we have al3(h)=a23(h)=0, q. e. d. Lemma 15. rank J = 2 and rank JV,=4. Proof, If rank N{ = 4, N/N^ acts properly discontinuously on U3. Hence rank A = 2. Suppose rank N:<3. Then, since C3/Nk compact, we must have r=rank A> 3. We have [£<,&]=*?"*?" for some nij9 m^eZ. This implies [>13, bs](gt9 gj) ^ritj and [fla, As](ft, gj) =miJ and we have ntjb3{gk) +njkb3(gi) +nkjb3(gi) = 0 and m^b^g,) +mjkb3{gi)+mkib3(gj)=0. Hence ^ = 0 , l
il

an t=l

d fl2s(^) = i ; «ifl2s(ft). Thus we have [fl13, A3] = [«235 *s] = 0, which is

a contradiction,

q. e. d.

Thus N i s generated by ft, g29 hl9 •••, A4- Since [hi9 hj] = [gi9 hj]==l9 JV0) cannot be Z2. I l l iii) N™=Z*, NX=Z\ I n this case, 013(A)=023(A)=O for all A 6 ^ 1 . We may

Compact Quotients of C 3 by Affine Transformation Groups, II

269

2 assume that A3C?i) = l- BY t h e P- c- *• V>(Zi, Z* Z3) = {Zi 2 2 — ^ ^ t £3), g, is reduced to a translation. Also we may assume that b(hl) = (l, 0, 0), lb = (0, 1, 0), and tb(h3) = (wl, co2, 0). For g, he N9 we have [g9 gl] = h^ig)h2n^f^g)9 and [g, h]=hfg>h)h%(g'h)h%g>h\ for some «(^), m(g)9 l(g)9 n(g9 h), m(g9 h) and l(g9 h) eZ. Hence we have

(28)

G)

+"<**> El+'<**>

(29)

Let ^ = ^ + J > « wlx=pl,+iqP, ju=l, 2, 3, v = l , 2, «=V—1 • We may assume ?,#0, then the vectors v, = (l, 0), » 2 =(0, 1), v3=(a)1, Q>2), and f 4 =(0, t) are linearly independent over R. Choose these vectors as a real basis for £/, X U2. Let rt, 1
«te)

where a=——sn-\-s23,

rte) *.te)

/3=——r^+r^, y=——U\-\-u2. From (28) and (29), we

<7i

^i

1\

h a v e a = 0 , ft=cu3 for some c e i J . Consider t h e c o o r d i n a t e transformation 3) = sented by 1 0 0 g = 0 0 0 L0 r

(30)

0 1 0 0 0 0 0

0 0 1 0 0 0 0

0 aa(g) 0 a25(g) 0 av(g) 1 0 0 1 0 0 0 0

aK(g) a26(g) ax(g) 0 0 1 0

Ate)

Ate) l

where j 8 « = - ^ « 1 + ^ - y 4 18,=^ A = a , , A = < l -

Lemma 16. iV'1 = iV1. Thus, ifg€.N\, atj{g)=Q, l < i < 3 , j = 5 , 6. /Voo/. We have i 8 4 (A1 )= i 8 5 (A i )=A(^)= 0 - Hence iV.ciV',. Conversely, if

<»te)=«»te)=0- H e n c e

tf'iCA7,,

q. e. d.

270

T. Suwa

Theorem 4. In Case III, rank Nw>3.

If NX=Z\

then Nm=Z3, and

2) differentiably, C3/N is a real 3-torus bundle over the real 3-torus, 3) the group N is generated by "10 0 1 0 0 0 0

0 0 1 0

1 < ij < 3, gj = 0 1 b>(g,) 1 1 0 where a) (^ b2(hi)), l < t < 3 , are linearly independent over R, b) b3(gj), 1 < J < 3 , linearly independent over Z, c) 't,gj]> \
4)

Hl(CiIN9Z)=Z^Zni

Proof. First we prove 2). Introduce, on C 3 =/? 6 , the f-coordinate system described above, and let M^= {fj denote the z-th factor of R6. From Lemma 16, ^5 = Nl = Z3 acts on W^ X W2X W3 properly discontinuously as a group of translations. Hence we have C3IN={C3INl)l(NINl) = {R6IN[)/(NIN[) = (T]lxW,xlV,xlV6)/ (NIN[), where T%=Wl xW2x W3/N[ is the real 3-torus. Since the projection C3-> W^xW5X W%l(NjN\) is proper, NjN\ acts on W± x W5 X W6 properly discontinuously as a group of translations (see (30)). Since C3/N is compact, (W4x W5X W6)j{NjN\) must also be the real 3-torus. Therefore, C3/N is (the total space of) a ^ - b u n d l e over 7V 1) We have JcxN/N^NIN'^Z5. Therefore, Alb(C3IN) = U3/J = 0. 3) From 2), we see that N is generated by hl9 h2, h3, gl9 g2, g3. The assertions a) and b) also follow from 2). Finally, since al2=0, al3, a23, and b3 are homomorphisms.

Hence N(l) is generated by [gi9 gj], li3> b3]{gi, gj), [a23, b3](gi, gj)), 1 Fis a complex analytic family of compact complex manifolds, and is differentiably a fiber bundle. Ill iv) Suppose Nl=Z*. In this case, we have the following Theorems. In Case III, if NX = Z\ then Na) = Z\ and 1) the Albanese map gives a regularfiberspace structure on C3/Nover Alb(C3/N) = Tl. Eachfiberis a complex 2-torus, but the complex structure may vary,

Compact Quotients of C3 by Affine Transformation Groups, II

271

2) the group N is generated by 'I 0 0 0

1 0 al3(hi) b^hi)' 0 1 a23(hi) b2(hi) 0 0 1 0 0 0 0 1

0 al3(gj) 1 a23(gj) 0 1 0 0

b.igj) b2(gj) bs(gj) 1


1,2,

where*) fli3(Ai)=fl23(Ai)=0, l < z < 3 , b ) (*,(*<), b2{ht)), \
-3)

2,

Hl(C*IN,Z)=Z*®Zni®ZmJ®Zmi.

Proof. For each fixed £3, iV, acts on U{x U2X {£3} properly discontinuously as a group of translations. Since N{ = Z\ the canonical projection C3/Nl-^U3 gives a regular fiber space structure on C3/Nl whose fibers are complex 2-tori 7^3, £3 € C/3. The group N/N{ acts on C/Nl-^U3 as a fiber preserving automorphism group, and the projection is proper. Hence NjN^A acts on U3 properly discontinuously as a group of translations. Therefore, C3/'N= (C3/'iV,)/(N/N{) is a regular fiber space of complex 2-tori over Tl = U3/J. Clearly Alb {C3jN) = C/3/J, and the map C3\N^U3\A is the Albanese map. 2) From 1), we see that N is generated by hl9 h2, h3, A4, g{, g2. We claim N(l) = Z3. If NW = Z\ we may assume hT*=ki9 mteZ—0, l < i < 4 . Hence flI3(Ai)=a23(^i)=0j l < z < 4 . Since «12 = 0, a13, <223, i 3 are homomorphisms. Hence N(l) is generated by [hi9 Ay], l < z , i < 4 , [Ai3 ^ ] , l < e < 4 , 7 = l , 2, and [g{, g2]. But since [A<, A,] = [A<,&] = 1, A^(1) cannot be Z 4 . Therefore N(l) = Z3, and we may assume h%ti=ki, l < / < 3 , which yields a). The assertions b) and c) follow from 1). Finally the condition tf12=0 implies d). 3) We have h™i=ki, l < z < 3 . Hence it suffices to show that 37(A4), ^(gi), and r](g2) are linearly independent over Z. Suppose nr 2

^lni7](gi)=0,

n, nteZ, then the element g^h^g^g?2 is in Na\

Hence O = bs(g) =

2 n

ibs(gt)> which yields n{ = n2=0. Since any non-zero power of A4 is not in

we have n =

Na\

q. e. d.

Remarks. 1. Since h4 has no fixed points on C3, if al3(h4) ^ 0 or a23(hA) ^ 0 , then {h)b2(h<)-a23(h<)bx(h<)*0. 2. We may assume (*,(*,), A2(A,)) = (1, 0), (i^AJ, *2(A2)) = (0, 1). The period matrix of the torus T\% is ( n

,

'

\U

1

o>2

l 3

3

) , where o)l = bl(h3), a)2 = b2(h3),

«2^3H-<w4/

oj3=bl(h4), o)4=b2(h4), cxi = ai3(h4), i=l, 2, dition 2d) is given as follows. The pairs ([fli3> b3](gl9 g2), [a23, b3](gl, g2)) generate a group generated by (b^hi), b2{ht)), l < z < 3 . invariant factors of this subgroup.

3. In terms of the entries, the con(al3(h4)b3(gi), «23(A4)Z>3(^)), 2 = 1 , 2, and subgroup of rank 3 in the free abelian The integers miy l < z < 3 , i n 3) are the

Case IV) fl12^0, and [^3, b3]^0. We identify U3 with Z) by taking the basis dual to dz3 for Z). Then we have Fo7](g)= I

dz$—b3(g). Hence N{ coincides with the

272

T. Suwa

kernel of the natural action of N on U3. Let V denote the real vector subspace spanned by J . We have d i m i 2 F = 2 and rank J > 2 . L e m m a 17. If he Nly then al2(h) = 0 . Proof. Suppose al2(h)^t0. Then we have a23(h) = 0, since otherwise h would have a fixed point on C 3. Thus for any g 6 N,

[K g] =

1 0 al2(h)a23(g) 0 1 0 0 0 1 0 0 0

*" 0 0 1

Since [h, g] has no fixed points on C3, we get al2(h)a23(g)=0. 0, which is a contradiction,

Hence we have a23= q. e. d.

L e m m a 18. x Proof By Lemma 17, for each fixed z3e U3, Nx acts on UxxU2X discontinuously as a group of translations. Hence JVjCZ4,

{z*} properly q. e. d.

Let /fl5 •••, hs and kl9 •••, kt be, respectively, Z-bases of Nx and N(l)((zNx) such that ki=hf\ rriieZ—0, l 2 , so that b^g^ ••-, b3(gr) form a Z-basis for J . Finally let A/r(2) = [iV, iV(1)] be the commutator group of N and N(l\ It is easy to see that for any g e N(2\ al2(g) =al3(g) =a23(g) =b2(g) =b3(g) = 0. Hence N(2) acts on Ux properly discontinuously as a group of translations. Thus N(2)aZ2. Lemma 19. N(2^0. Proof Since [a23, b3]^0, we have b2{k)i=.Q for some A([g, * ] ) = / , 'b{[g, k]) = (a12(g)b2(k)-al3(k)b3(g), 0, 0). would have al2=ab3, for some a e C*. This implies A([g, = {2ab3{g)[a23, b3]{g, g'), 0, 0), for all g, g' in N. Since 0, which is a contradiction, L e m m a 20. rank N«>>2. Proo/. Since N^czN™, if N™=Z, This yields [^3? ^ = 0,

k e N(l). Then for any g € N, Suppose N<*>=0. Then we [g, g']]) = / , ^ ( [ 5 , [g, g']]) ^ » = 0, we have [a23, As] = q. e. d.

we would have A2(A;)=0, for all A; in iV(1). q. e. d.

L e m m a 21. rank N(l><3. Proo/. Suppose N(l)=Z\ Then NX=Z4, and we may assume ki=hf-i, mte Z—0, l < f < 4 . Hence 023(A)=O, for all Ae iVj. Moreover, if NX=Z\ noting Lemma 17, we can show that C3/N is a regular fiber space of complex 2-tori over Tl = U3/J by the same argument as in the proof of Theorem 5, 1). Thus r = r a n k d=2, and the group N is generated by A15 ••', A4, ^ and ^2- Since a23 and ^3 are homomor-

Compact Quotients of C 3 by Affine Transformation Groups, II

phisms, we have b2{ki)=ni\a23, b3] (gl9 g2), nteZ,

273

l < / < 4 . Then b2{hi) = -—-b2{kl) nii

= —^-[^23? b3] (gi,g2)- This means that (bi(ht)9 b2(ht))> l < z < 4 , cannot be linearly TYli

independent over R, and hence N{ fails to act properly discontinupusly on £/, X U2X {0}, a contradiction, q. e. d. Lemma 22. 7frank i\T,<3, then rank J > 3 . Proof. If rank iVj<3, the projection C3/Nl—^U3 gives a fiber space structure on 3 C jNx with noncompact fibers ((£/, X U^jN^) x {£3}, £3<= £/3. Suppose rank J < 2 . Then the fiber preserving automorphism group NjN^A would also act properly discontinuously on U3, and C3IN=(C3INl)l(N/Nl)^>U3l/l would be a fiber space q. e. d. over the manifold U3\A with noncompact fibers, a contradiction,

Lemma 23. If N™=Z\ then NX=Z\ Proof. Since 0^N(2)czN(l), we may assume b2(kl)=0. Then we have [<223, b3]» (&>£./)^^A(£2)> ntjeZ, l 3 , and hence % = 0, l h)b2(k2), for some «(^, A) 6 Z. Since rank J > 3 , this implies a23(h)=0. The group N is generated by Al5 ••-, As, ^ l5 •••5^rr, and a23 and i 3 are homomorphisms. Thus we conclude [a23, ^ 3 ]=0, which is a contradiction, q. e. d. Lemma 24. If N™ = Z3, then N{=Z*. Proof Suppose N{ = Z3. Then, from kt=ht\ l < z < 3 , we have a23(h)=0, for all heN{. Moreover, by Lemma 22, rank J > 3 . We claim Nm=Z. Suppose Ni2)=Z2. Then, since N(2)czNa\ we may assume that A(ki)=I, b2(ki)=0, i = l, 2. Thus we get [023, b3](giigj)=niJb2(k3)i n^eZ, \
ai2(gi)b2(kp) —al3(kp)b3(gt)

=

nipbx(l).

This yields [A2, flls](A:^ kq)*b3(gi) = (nipb2(kq)—niqb2(kp))bl(l). Since rank J > 3 , we have [i2,flls](A:^A:g)=0, l<j^ ? q<3. Hence al3(kp)=ab2(kp), l^p<3, for some ae C, and we have, from (31), b^k^^a^g^—ab^g^^n.pb^l). We claim 012(&) = ab3{gt), l w e would have b2(kp)=npp, l
[«23, *3](ft 5^) = p=\ S«
p=\

for

some n

ijptZ'

Thus

we

g e t [^23, b3](gu gj)=miJp,

nijpnp. Therefore we conclude [a23, b3]=0 as before. Hence al2(gi) =

ocb3(gi), \
Moreover, since al2^0, a^O. We have [gt, [gt, gj]] = leiJ, for some

T. Suwa

274

ttjeZ, l 3 , we have ^ = 0. Then from (32), [fla, A3] (&,&)=0, l
flB(Ai)=0, \
M^ks+W,

fla&ks+W),

linearly independent over R,for any £ 3 e U3=C, d) b3(gj), j=l, over R,

\
are

2, «r^ linearly independent

?/

1\ —

From Lemmas 20-24, we have NX = Z\ and N(l) = Z2 or Z 3 . The same argument as in the proof of Theorem 5, 1), shows 1) and 2c, d). Since 0^iV (2) c JV(1)cJVb we have 2b). The assertion 3) is also proved as in Theorem 5, q. e. d. Remark. It does not seem easy to find the integers mt from gly g2, A1? •••, A4. The group N(2) is isomorphic to Z or Z2 and is generated by [gi9 [g{, g2]], i=l, 2. 3.

Holomorphic Forms

p

Let Q denote the sheaf of germs of holomorphic jMbrms on the quotient C3/N. In this section we compute hpt0=dim H°(C3/N, Qp), the number of linearly independent holomorphic /?-forms on C3/N. We may identify H0(C3/N, Qp) with the space H°(C3, QP)N of iV-invariant holomorphic /?-forms on C3. Note that the pullbacks g**/^, i = l , 2, 3, are given by (33)

g*dz2

L e m m a 25. A 30 =l. Proof. Take co 6 H°(C\ Q3)N and let «=/(<;)afe, A=f (gz)dz\A
Take a>€H°(CiIN,Q')=H''(C\QT,

and let o>=±fi{z)dZi,

f^H\C\0),

i= 1, 2, 3. The iV-invariance conditions for co are given by (3), (4), (5) in § 1. Also take 6eH\C*IN, &)=H*(C*, QZ)N and let 0=k{z)dz2/\dzi+fh(z)dZt/\dZi + h3(z)dZiAdz2, ^6// 0 (C 3 , 0). The iV-invariance conditions for 0 are

Compact Quotients of C 3 by Affine Transformation Groups, II

(34) (35) (36)

275

h,(z) = K{Z)=h2{gz)-an{g)h,{gz), h3(z)=h3(gz).

Case I) Clearly, A''°=A 20=3. Case II) In this case, we have Lemma 26. Letfe H°(C\ 6). If we have (37) f(z) = a(g)+f(gz),

forallgeN,

:

for some function a : N—+C, then f (z) =az2-\-fiZs+Y)for some a, /3, je C. Proof We have -^— (z) = -J- {gz). Hence we have/(^) =clzl+(p(z2, £3), where uZ