J. Math. Kyoto Univ. (JMKYAZ)29-1 (1989) 27-43

On algebraic surfaces and threefolds whose cotangentbundles are generated by global sections

By

Hiro-o TOKUNAGA

§ 0 . Introduction

For smooth subvarieties of abelian variety, the following fact is well-known:Let X b e a sm ooth subvariety o f general type of an abelian variety with

dim X = 1 or codim X = 1 . Then the tricanonical map 0 13K xi is embedding.In this paper, we consider a generalization of the above fact in the case of

dim X = 2 or 3 and show that 0 13Kxl is b ira tiona l. Main results are as follows.

Theorem ( A ) Let S be a surface of general type whose cotangent sheaf Q isgenerated by global sections. If h ° ( ,(4 ) . 4, the bicanonical m ap 01 2 K 5 1

o f S isbirational except the case w here S = C, X C 2 , g(C2 ) = 2.

Moreover, If th e A lbanese torus o f S is a simple abelian variety. 012Ksi is

embedding.

Theorem (B ) Let V be a projective threefold of general type whose cotangentsheaf 01, is generated by global sections. If h ° ( Q ) 5 , the tricanonical map013K0 o f V is birational.

Note that if S and V are smooth subvarieties of an abelian variety ,S4 andare generated by global sections.

I w o u ld lik e to th a n k P ro f . K . U eno for his valuable suggestions andencouragement.

Notation and Convention.C , S, V denote a compact non-singular curve, a compact non-singular surface,

a compact non-singular threefold, respectively.g,(X ):= h i (X , (9,) i- th irregularityp g (X ): geometric genusg(C): genus of a curve CK (X ): Kodaira dimension of X

m-th canonical map of Xe x (D — kx): subsheaf of C(D ) with k-fold zero at x.

Received, June 16, 1987

28 Hiro-o Tokunaga

e x : sheaf of germs of holomorphic vector fieldsQ i: sheaf of germs of holomorphic cotangent vector fields

Let D,, D2 be divisors.D, — D2 : linear equivalence of divisorsD, D2 : numerical equivalence of divisorsD, '',;i() D2 : numerical equivalence of Q — divisors

§ 1 . Preliminaries

In this section, we shall prove some lemmas which will be used later to proveTheorems (A) and (B).

Lemma 1.1. Let X be a surface (or a threefold) whose cotangent sheaf Qk isgenerated by global sections. If K(X)> 0, then ImK x i (m e N) is free f rom basepoints.

Pro o f . H ere w e prove th e lemma fo r dim X = 2. T he proof of the casewhere dim X = 3 is similar.

It is enough to show that, for any point p E X , there exist two holomorphic2-forms such that the one has zero a t p and the o ther does not have zero atp. Since q 1 (x) 3 , th e re e x is ts a holomorphic 1-form cop w ith zero a t p, and bythe assumption, there exists a holomorphic 1-form co with cop A w # O . On theother hand, by the assumption, there exist two holomorphic 1-forms co„, co2 whichspan the cotangent space at p. For these 1-forms, take exterior product co, A co,.This is non-trivial holomorphic 2-form which does not have zero at p.

The following theorem proved by Z. Ran is im portan t. F o r a proof, see Ran

Theorem 1.2. Let 0: X --+ Y be an immersion where X is a compact complexmanifold and Y is a com plex torus. Assume that X is of general type. T h en , wehave the following.

( i) Fyry is finite, where Fo is the Gauss map associated with 0.(ii) The canonical bundle K x is ample.(iii) If A c 0 (X ) is an abelian subvariety of dimension k, then:

n — mk <

n — m + 1

where n = dim Y, m = dim X.

By the above theorem, manifolds of general type whose cotangent sheaf aregenerated by global sections have an ample canonical bundle K .

Next, we shall prove some inequalities.

Lemma 1.3. Let S be a surface of general type whose cotangent sheaf 5-4 isgenerated by global sections. T h e n the inequality

A lgebraic surfaces and threefolds 29

ci(S) c2 (S)

holds, where c(S ) is the i-th Chern class of S.In particular, we have

6x(es).

P ro o f L e t W b e the projectivized tangent bundle P(Ts ) where Ts i s theholomorphic tangent bundle of S with a natural projection 7E: W - S. P u t L —

0 ( 1 ) . Since we have

n,e2w(1) f r 4 ,

L is free from base points and fixed components. Therefore, we have

0 < L 3 = c — c 2 .

Hence we obtain the desired inequality,

c

2 >.

Lemma 1.4. Let S (resp. V) be a surface (resp. threefold) of general type whosecotangent sheaf ,W is generated by global sections. If S (resp. V) has a surjectivemorphism f: S (resp. V) —> C to a non-singular curve. Then following inequalitieshold:

g = g(C) q 1 (S) — 2 (resp. q 1 (V) — 3).

P ro o f Here we shall prove the surface case o n ly . A proof of the threefoldcase is s im ila r . By the assumption, g q ,(S). Hence, it is enough to show thatg 0 q 1 (S) — 1. I f g = q1 (S) — 1, o n a general fibre F there is only one linearlyindependent holomorphic 1-form which is a restriction of a holomorphic 1-formo n S . S o , by the assumption, the general fibre of f is a n elliptic curve. Thiscontradicts the assumption that K(S) = 2.

Lemma 1.5. Let V be a threefold of general type whose cotangent sheaf Q, isgenerated by global sections. Then the inequality

12holds.

P ro o f B y T heorem 1 .2 , K v i s a m p le . S o , b y th e K o d a ira vanishingtheorem,

hi(V, (9v (2K v )) = 0 (i = 1, 2).

By the Serre duality,

IP(V, e v (2K v )) = (9v(— K v )) = O.

Hence, by the Riemann — Roch Theorem for threefold, we have

x(Ov (2K )) = h°(9(2K v )) =11q, — 3x(e)v ) .

30 Hiro-o Tokunaga

Let S be a general member of K . B y L em m a 1.1, S is a non-singular surface ofgeneral type and Q is generated by global sections. Therefore, by Lemma 1.3,

6x(e s ). (1.2)

On the other hand, by a cohomology exact sequence

0 H ° ((9v (K,))—) ((9v (2K v )) —> (S, &5 (K 5 )) —>

—>H1 ((9,(K v )) —> 0

and h i ((.9,(K v )) = h 2 (0 ,) (by the Serre duality), we obtain an equality,

pg (V)— h ° ((9v (2K v )) + p g (S)— q 2 (V )= 0.

By (1.1) and h'((9) = 11 1 WO, we obtain an equality,

pg (V) — (41<,37 — 3x(Cv )) — q2 (V )+ q i (V )+ p g (S)— q,(S)= 0 .

Therefore,

A es) = P g (S) — + I = 1 1<137 — 2 X(Cv) • (1.3)

B y the adjunction formula, K ,= 2K v 15 , hence f q = 4K . Therefore, by (1.2),(1.3), we have

41(13, 6X(es)

= 31q, — 12x((- v ).

Hence, — 12x(G).

By Yau's inequality, if K v is ample, x(Ov ) < 0 . Therefore, K,3, 12.

We will summarize the results of connectedness of the canonical divisors ofsurfaces which will be used in §2 and §3.

Proposition 1.6. Let S be a minimal surface o f general type, and D be anef fective div isor of S such that D m K , ( i n 1 ) . T h e n D is a numerically 2-connected divisor except the case:

K = 1 , m = 2 , D =D 1 +D 2 , D, D 2 K s .

For a proof, see Bombieri [3].

Corollary 1.7. Under the above notation, D is numerically 1-connected.

The following theorems will be used in §2 and §3.

Theorem 1.8. Let S (resp. C) be a surface (resp. curve), and f:S — > C be arelatively minimal .f ibration such that the genus of a general fibre is positive. T h e nwe have

deg (f,,cos i c ) . 0.

A lgebraic surfaces and threefolds 31

Moreover, this degree vanishes ff and only if either f is locally trivial (hence smooth)o r S is an elliptic surface w ith constant moduli and the only singular f ibres aremultiple fibres.

For a proof, see W. Barth, C. Peters, A. Van de Ven [1].

Theorem 1.9. L et S (resp. C) be a Kahler surface (resp. curve), and f : S Cbe a fibration. Then f .co s ic i s the direct sum CV (h = h i (C,f,,cos )) and a locallyfree sheaf which satisfies h i (C , g [K J) = 0, where OP = (9 c C) • • • C) Cc .

h

For a proof, see Fujita [5].

Proposition 1.10. L et S be a minimal surface of general type. Then there areonly finitely many irreducible curves C on S such that the intersection number K s Cis bounded and C 2 < 0.

Theorem 1.11 (Ramanujam's vanishing theorem). L et S be a surface and D aneffective numerically 1-connected divisor on S with D2 > 0. Then h i ((Ps (—D)) = 0.

For a proof of Proposition 1.10 and Theorem 1.11, see Bombieri [3].The following theorem will be used in §3.

Theorem 1.12 (H orikaw a). L e t X an d Y b e com pact com plex manifolds,f: X — > Y a holomorphic map, and let p: —04 be a fam ily o f deformations ofX = p - 1 (o), o e M . L et fi * denote the canonical homomorphism 0 0 — > I l l (X ,f *O y ). A ssume that f i * is surjective f o r i = 1 and is injective for i = 2. Then,there exists an open neighborhood N of o in M, a family q: y N of deformationsof Y = q - 1 (o), and a holomorphic m ap 0 :

X I N — q , ov er N w hich induces f ov er

X = r i (o).

For a proof, see Horikawa [6].

Next lemma is important to prove our main theorems.

Lemma 1.13. (c i) L et S (resp . V ) be a surface (resp. threefold) and co,, co 2 belinearly independent holomorphic 1-forms such that co, A co2 0 . T h e n th e re e x is tsa curve C with g(C) 2 which satisfies the following conditions (1) and (2):

(1) There exists a surjective holomorphic map f: S C ( re s p . V C ) .(2) There exist holomorphic 1-forms (p i (i 1, 2) such that

(0 1 = f* P i C°2= f*(p2.

(fl) L e t V be a threefold and co,, co,, co, be linearly independent holomorphic1-forms such that ci), A w j # 0 (i < j) and co, A co2 A 0 3 0 . T h e n th e re e x is ts asurface of general type which satisfies the following conditions (1), (2):

32 Hiro-o Tokunaga

(1) There exists a dominant rational map f: V S.(2) There exist holomorphic 1-forms ço , (i = 1, 2, 3) on S such that

0)1 = f * ( p i ,

For a proof, see Ran [8].

0 )2 = f* (p 2 , (1 )3 = f*(p3 •

Lemma 1.14. L et S be a surface for which p g 2 q i — 4 . Then there existtwo linearly independent holomorphic 1-forms such that w, A co2 0 .

For a proof, see Beauville [2], Lemma 10.7.

§ 2 . Proof of Theorem (A)

When we consider whether the bicanonical map 0 12, 51 is birational or not, it isimportant to know whether S has a structure of a fibre space with curve of genus2 as a general fibre. In case S is a surface of general type whose cotangent sheaf54 is generated by global sections, the structure of such surfaces is simple.

Lemma 2.1. L et S be a surface of general type whose cotangent sheaf S -4 isgenerated by global sections. If S has a structure of a f ibre space w ith curve ofgenus 2 as a general f ibre, then S = C, X C 2 and g(C2 ) = 2.

P ro o f . Let p: S — > C be a fibration with curve of genus 2 as a general fibre.By Leray's spectral sequence, we obtain an exact sequence:

0 - + 11 1 (C, (9,) — > 111 (S, (9s ) — H ° (C, R 1 p* (9s ) 0.

By the relative duality,

(R1P* (9s) - Z W o w

Hence, IP(C, R 1 p,(9s ) = h 1 1C,(P* cosic) 0 (t)c) •On the other hand, by Theorem 1.9, we have

N o w 2 C P" 0 g

where h = ( C , w a s ) and g is a locally free sheaf with 111 (C, e[K c ]) = 0, Sincerank (w o w ) 2,

Therefore,

But by Lemma 1.4, we have

Therefore we have

h = IP(C, R i p ( 9 s ) 2.

171 ((95 ) = g + h _ g + 2 .

h i (e s ) g + 2 .

h = 2 .

A lgebraic surfaces and threefolds 33

B y th e above argum ent, w e o b ta in p,,,cos / c Z (9c. C c , d eg (p,wslc) = O. B yTheorem 1.8, p is locally trivial fibration. S o , S is a fibre bundle with curve F, ofgenus 2 as its fibre. A ssu m e that this fibre bundle is not trivial. Since Aut (F0 )always acts on H

°(f4 0 ) non trivially, this contradicts the fact that 10(00= g + 2.

For any surface of general type, the inequality x(es ) 1 alw ays holds. But, inour case, the following stronger inequality holds.

Lemma 2.2. L et S be a surface of general type with the irregularity qi (S).. 4such that its cotangent sheaf (4 generated by global sections. Then x((9s ) 2 holdsexcept the case where S is a product of curves of genus 2.

Pro o f . I t is e n o u g h to show th a t if x(Cs ) = 1 under th e assumption inLemma 2.2, then S is a product of curves of genus 2. First, by Yau— Miyaoka'sinequality, if x((9s ) = 1, then we have

< 9 .

By Noether's inequality for irregular surface of general type, we have

cf 3q, — 4 8 (q 1

. 4)

Hence, we haveci• = 8 or 9 and pg = q, = 4 .

Now, if S does not have any morphism to a curve of general type, by Lemma 1.13,and Lemma 1.14, we have

pg 2 q 1 — 3 > q, = 4 .

Therefore, a surface which satisfies the assumption in Lemma 2.2, and x(es ) = 1has a structure of a fibre space p: S C with g(C) = 2. O n the other hand, wehave

c, 4(g(C)— 1)(g(F) — 1)

= 4(g(F) — 1)

where F is a general fibre and c, is second Chern class of S. Hence, by Noether'sequality, we have

c? = 8 , c2 = 4 , g(F)= 2 .

Therefore by Lemma 2.1 we conclude that S = C x F. 0

R em ark. By using Beauvelle's result (see Beauville [2], Theorem), we simplifythe proof of Lemma 2.2.

Corollary 2.3. L et S be a surface which satisfies the assumptions in Theorem(A). Then we have .1q. 12 except the case where S is a product of curves of genus2.

34 Hiro-o Tokunaga

Pro o f . It follows immediately from Lemma 1.3 and 2.2.

From now on, we assume that S C , X C 2 (g(C ,)= 2, i = 1, 2).

Proof o f Theorem (A). It is enough to show that if 0 12, s1 is not birational, Shas a structure of a fibre space with curve of genus 2 as a general fibre. A ssum ethat 0 12, s1 is n o t a birational m orphism . Then there exists a Zariski open set Uo n S such that for any point x e U, there exists a po in t y e U with A

1 2 KChoosing a smaller U if necessary, by Proposition 1.10, we may =12K(.0. assume0 tha t U S \ UC, where C runs over all irreducible curves C on S with K s Cand C 2 < 0 . I f (6 is the set of such curves, contains all irreducible curves withK C = 0 o r K s C = 1. I n o u r case, since K s i s ample, there is n o curve withK s C = 0.

C laim 2.4. L e t x , y b e the points as above and n: — > S be obtained bycomposition of blowing-ups of S at x an d y . Then In*K s — 2Ex — 2Ey 1 0, w hereEx and E y are the exceptional curves of the first kind appearing in the blowing-ups atx and y, respectively.

Proof of Claim 2.4. Since Qk is generated by global sections and 11°W D 4,there exist at least two linearly independent holomorphic 1-forms coa , a),, such that

w (x) = cob (x) = 0.

Moreover, these forms have also zero a t y. In fact, if coa (y) 0, then sinceis generated by global sections, there exists a holom orphic 1-form such that(wa A w ) ( y ) 0. The holomorphic 2-form coa A co is not identically zero, and haszero a t x . This contradicts the assum ptions. So, coa (y) = co,,(y) = 0. I f co,,,satisfy Wa A W b # 0, then this proves Claim 2.4. Now, we can find 1-forms (Da , co,such that co a A W b * 0, as follows.

C ase 1 : For any linearly independent, oh, co 2 e H°(S2), co, A (D , 0.

In this case, we have C O , A Wb # 0.

C ase 2 : There exist linearly independent oh, co2 e H°(,(2,) with ah A co, 0.In th is case, by Lemma 1.13, there exists a surjective morphism p: S —> C,

g(C) 2 and w 1 , co2 are the pullback of 1-forms on C . Moreover, by Lemma 1.4,there is a relation g, g + 2 between the irregularity g , of S , and the genus g ofC . Therefore, we take coa in such a way that it is contained in p*H °(W ), and wetake co b in such a way that it is not contained in p* H°( Q ) . F or these 1-formswa, wb, we have COa A Wb 0 .

By Corollary 2.3, a n d S C , x C 2 ( g ( C i ) = 2, i = 1 ,2), w e have lq > 12.Hence,

(n* K s — 2E„ — 2Ey )2 4 > 0 .

Therefore, there does not exist an effective divisor D e In*K s — 2E„ — 2Ey 1 whichis numerically 1-connected. In fact, if D is a numerically 1-connected divisor, byTheorem 1.11, we have

(S, (Os ( — (7t* K s — 2Ex — 2E,,))) = 111(S, D)) = 0 .

Algebraic surfaces and threefolds 35

So, in the cohomology exact sequence,

•• • —› H 0((9,(27c* Ks )) —> 1-10(CE ) CD II° W O

—> H° (Cs (2n* Ks — E — E y ) —> • • • .

we can show,

•• H IV s ( 27E* Ks )) H ° (( E ) '0 11° 1(90 (exact) .

T his contradicts th e assum ption that 012,icsi(x) — 6012Ksi(Y). B y using th e sameargum ent as th a t in B om bieri [4], Theorem 5 , w e can show tha t, if D is notnumerically 1-connected, S has the structure of a fibre space of curve of genus 2 asa general fibre. Therefore, by Lemma 2.1, S = C , x C 2 g(C2 ) = 2. This provesthe first statement in (A).

The next propotion proves the second statement in (A).

Proposition 2.6. Let S be a surface of general type with condition (A), and theAlbanese torus Alb (S) of S be a simple abelian variety. Then, 012K51 is embedding.

Proo f . Notation is the same as a b o v e . A proof of Proposition 2.6 consistsof two steps.

Step 1: 0 12„si is a one to one mapping.L e t x , y b e p o in ts s u c h th a t 6 (x)12Ksi , — 2ics i(.1). T h e n w e c a n show

lit* K s — 2E, — 2E3,1 0 Ø b y th e sam e m ethod that o f C laim 2.4. Hence, theproblem is reduced to show connectedness of members of In* K s — 2E x — 2Ey l. Ifthere exists a connected member of In*K s — 2E x — 2E3,1, by Theorem 1.11, we canprove 6 t x ) 0 6 tri2x s1,-, • ri2K ,H,v) T h i s contradicts t h e assum ption 0 12 ,, (x) _ -0-12Ksi(Y).Hence, any member o f n* K s — 2E„ — 2Ey 1 is not connected. Let C be a membero f 1Ks — 2x — 2k a n d C = C , + C 2 be decom potion in to tw o positive divisors.By using th e sam e argum ent as th a t in B o m b ie ri [4 ] Lem m a 1 , if K , C , 3(i = 1, 2), then C is numerically 1-connected. So, for at least one C1, say C ,, theinequality K s C, m u s t h o ld . B y our assum ption , K s i s a m p le . Hence,K s C, = 1 or K s C , = 2:(i) K s C , = 1By the algebraic index theorem, we have Cf < O . S in c e K s C , + CI

. is even, C f is

odd. Therefore, C f = — 1 . Since Ks i s ample, C , is irreducible. Since there isno rational curve in S , C , is a non-singular elliptic curve. This elliptic curve isimmersed in Alb (S) by the Albanese mapping of S. This contradicts the assump-tion that Alb(S) is simple.(ii) K ,C , = 2In this case, C? = O. If C , is reducible, C , = C ' + C " . K s C' = K s C" = 1. By(i), such curve does not exist. Hence, C , is an irreducible curve with virtual genus2. Possible cases of C , are as follows:a) an elliptic curve with one node or cusp,b) a non-singular curve of genus 2.

36 Hiro-o Tokunaga

Both cases, by the same argument as that in (i), we can show the existence of anabelian subvariety of A lb (S). This contradicts the assumption.

S tep 2: 0 12„,1 is immersion.Let x be an arbitrary point on S, (9s , x b e its local ring a t x , and X x b e the

maximal ideal of es , x . As well-known, there exists the isomorphism:

es,././111 C Tx* (S)

where Tx* (S ) is co tan g en t sp ace a t x . N ow , consider the exac t sequence ofsheaves,

0 —> 05 (2.1(5 — 2x) —> C5 (21(0 r * 6 9 5,x/X 1 0 Cs (21<5 ) —0 O.

From this exact sequence, we obtain the cohomology exact sequence:

• • H ° V95 (21<s )) * ce Tx*(S) —oH 1(0,(2K , — 2x)) — > • • •

and the problem is reduced to show surjectivity of r*. A proof of surjectivity ofr* consists of two parts.

C ase 1 : The 1-forms coa , co, have simple zero at x.L et co,, co2 be holomorphic 1-forms which d o n o t have zero a t x . S in ce

Alb (S) is simple and by L em m a 1.14. Wa A W 2 , (Ob A (0 2 , and co, A c o , are allnon-trivial holom orphic 2-form s a n d th e y a r e linearly independent. There-fore, (coa A w 2 )(co1 A w2 ), (Wb A W 2)(W 1 A ( 0 2 ) a n d (co, A w2 )2 a r e linearly in-dependent 2-ple 2-forms, and have a t m ost simple zero a t x . This shows r* issurjective.

C ase 2 : One of the 1-forms coa , cob has always 2-fold zero at x.Consider the holomorphic 2-form coa A W . It has 3-fold zero a t x . Now, let

7tx : - - + S be blowing-up at x and Ex b e the corresponding exceptional divisor ofthe first kind. Then In* K s — 3Ex l 0 0 , and, since K s

2 1 2 , w e have

( * K — 3 E ) 2 3 > O .

So, if there exists a numerically 1-connected member of In*K s — 3Ex 1, by Theorem1.11,

hi (Cs ( — (it* K s — 3 E ) ) ) = ( (9 s (27r* K s — 2E )) = O.

Hence r* is surjective, and S tep 2 fo llow s. F o r connectedness o f a member ofrc*Ks — 3Ex 1, we have the following claim.

Claim 2.7. Under the assumption of Proposition 2.6, there exists a numerically1-connected member in 17E* K s — 3Ex l.

Proof o f Claim 2.7. L et s be a n element o f H°(es (K s — 3x)) and C = (s).

Then

(ir x )* C = D + 3Ex

Algebraic surfaces and threefolds 37

where (irx )*C is the total transformation of C . Assume that D = D, + D, whereDi (i = 1, 2) is a positive divisor and put

zl, = Di + (D,Ex )E„.

Then there exist effective divisors C 1, C 2 on S such that (r)*C , = zl i . For theseC 1 , C 2 we have

D, D2 = C1C2 — (Di Ex )(D2 Ex )1D,E x + D2 E„ = 3 .

Hence, if C, > 0 , C 2 > 0 , then by Step 1 and the assumption, we have C, C 2 3.Hence D i D2 1. If one of Ci (i = 1, 2) say C , is a zero divisor, we have

Di D 2 = —(D, Ex )(D2 Ex ).

Since Di i s a positive divisor, D 1E < 0, So, D1 D2 3 . Therefore, Claim 2.7 isproved.

By Step 1 and Step 2, Proposition 2.6 is proved. 0

§ 3 Proof of Theorem (B)

Our proof of Theorem (B) consists of two steps:

Step 1: If 013141 is not birational, its restriction 013K 5 to a general member Sof IK„1 is not also birational.

Step 2: Let D be K v I s where S is a general member of 1Kv l.Then, chi K o .DI is birational.I n o u r case , K y i s am ple, s o b y th e Kodaira vanishing theorem , the

homomorphism

H°((9y(3Ky)) r > H° (S, (95 (K + D))

is surjective. Hence ch,131411s = OlK s + D I, therefore, Theorem (B) follows.Step 1 is proved by Matsuki [7].

Lemma 3.1 (Matsuki [7]). L et V be a threshold which satisfies the assumptiono f Theorem (B) and S be a general member of K . I f 013 ,4 1 is not birational, thenOoKols = Oixs+0 is not also birational.

For a proof, see Matsuki [7].Step 2: If 013„, i is not birational, there exists a Zariski open set V ' which has

the following property:For any point x on V ', there exists a point y on V ' such that

013141(X) — 013K,I(Y) •

The next lemma is crucial.

38 Hiro-o Tokunaga

Lemma 3.2. Let notation be the same as ab o v e . The inequality

dim c H°(6,(1(,, — 2x — 2y)) 2

holds.

Pro o f . By the assumption q 1 (V) 5, there exists at least two holomorphic1-forms with w a (x) = cob (x) = 0, wa (y) = w,,(y) = O. The proof of lemma consistsof three parts.

C ase 1 : For any three linearly independent holomorphic 1-forms co,, w2 , w 3 ,co, A W2 A W 3 * O.

In this case, the proof is easy. Let co, c7) be holomorphic 1-forms such that co,65, w „, oh, are lienarly independent. Then, coa A cob A W # 0, w a A Wb A 6)- # O.Moreover these 3-forms are linearly independent. Hence we have the desiredinequality.

C ase 2 : There exist two linearly independent holomorphic 1-forms 0) 1 , w 2

such that w, A w 2 0 .In this case, by Lemma 1.14, there exists a surjective morphism from V to C,

a curve of genus 2 , and co,, 0)2 are the pullback of holomorphic 1-forms q),, (p2

on C . By Lemma 1.4, we have q 1 (V) g + 3, hence we can choose wa , o4 insuch a w a y th a t wa i s the pullback o f a 1-form on C and c o , is n o t thepullback. By local calculation, we obtain the following claim.

Claim 3.3. For w„, o4 as above, Wa A W b * O.

Claim 3.4. Let w a , cab be as above. Then there exist holomorphic 1-forms w,6) on V with following three conditions.(i) wa , 04, to, C13 are linearly independent.(ii) wa A W b A 0.) * 0, W a A Wb A 6.-.) * O.

(iii) w a A w b A W, wa A oh, A 6 are linearly independent holomorphic 3-forms.

Proo f o f Claim 3.4. S in c e coa A W b * 0 , and Q1, is genera ted by globalsections, the re ex ists a t least one holomorphic 1-form with W a A W b A W O.Assume Wa A W b A (1) 0 for any other holomorphic 1-forms cu. I n this case, bylocal calculation, the restriction of co, A to F, a general fibre of f: V — >C, isidentically zero. Let cob , co,(= w), w 2 . . . . . W k b e a part of a basis of H

°(11,)

which is not in f* H°(,(4 ) . Since g21, is generated by cob , c o , if Wa A W b A

W i * 0 (i = 2, ..., k ), F becomes an elliptic surface. This contradicts the assump-tion K(V) = 3. Therefore, the conditions (i) and (ii) fo llow . For the condition(iii), a proof is similar to that of the condition (ii).

By Claim 3.3, 3.4, we obtain desired inequality for Case 2.C ase 3: For any two linearly independent holom orphic 1-form s co, co',

co A ol O . But there exist three linearly independent holomorphic 1-forms w1 ,0 25 (03 with (01 A (0 2 A W 3 * O.

B y Lem m a 1.14, there exist a ra tiona l m ap ço: V --+S o f V t o a surface ofgeneral type and holom orphic 1-forms 1,, ce3 on S with co, = cp*a i . Consider

A lgebraic surfaces and threefolds 39

the following commutative diagram:

V ------ —>S

where E is a composition of blowing-ups at non-singular centers such thates'o =g 0 9 is a morphism.

L et E b e a n exceptional locus of e a n d E' = e * E . Since irreducible com-ponents of E are all ruled surfaces or 132 and S is a surface of general type, 9 (E) isa curve. Therefore the following commutative diagram holds:

V\E S\O(E).(ply\E

This shows that, generically, 9 and 9 are the sam e fibrations. In the follow-ing argument, we assume tha t the Zariski open set in Lemma 3.2 satisfies V' cV\E', and at least one of 1-forms oh, and W contained in 9*H

°(f2k).

i) The case in which both 1-forms oh, and oh, are contained in 9*Fr(flk).B y th e assumption, w a A co, 0 a n d these a re linearly independent. Let

w , Co" be linearly independent holomorphic 1-forms which is not contained in9*1

-1°(Q , ) . It is c lea r tha t co a A co, A w # 0 and co a A cob A 6) O . S o , it is

enough to show tha t w e can take co, (7) in such a w ay tha t COa A Wb A co andwa A wb A di are linearly independent. Now, fix co(e 9* H

°( 4 ) ) . Let (75 be anyother holomorphic 1-form which is not contained in co*I-1° (Q ) . I f wa A 04 A CO

a n d (Oa A W b A (7) a re linearly dependent fo r any 1-form 6 , th is shows th a t ageneral fib re of 9 i s a n elliptic curve. T his contradicts th e fac t tha t K(17

) =K(v) = 3.

ii) The case in which wa is contained in co*I-P(S2) but co„ is not.By the assumption, we have co„ A oh, # O.ii-a) The case in which there exists a holomorphic 1-form w such that wa ,

cob , w are linearly independent and oh, A 0 4 A CO —= O.By taking another fibration, this case is reduced to the case i).ii-b) There exist no holomorphic 1-forms as those in ii-a).This case is clear.By i) and ii), w e obtain the desired inequality in c a s e 3 . By Case 1, 2, 3,

Lemma 3.2 follows.

Corollary 3.5. Let S, x, y be the same as abov e . Let 7E: S —> S be a composi-tion of blowing-ups at x and y. Then, In*D — 2Ex — 2Ey 1 0 where D = K v1 5 ,and Ex , Ey are the exceptional divisors of the f irst kind appearing in the blowing-ups,respectively.

40 Hiro-o Tokunaga

Pro o f . Consider the exact sequence of sheaves,

0 —> —> & ( K ) C s (D) 0 .

Taking cohomology sequence, and by Lemma 3.2, Corollary 3.5 follows.

Proposition 3.6. 0 1„,,D1 is birational.

Pro o f . Assume 0 1„5 , D1 is n o t b ira tio n a l. S in c e c 6rimcvils = 013141 isno t a lso b ira tiona l. L e t x , y b e points with O IK , „ 1(x) = O lK s .„0 (y). Then, byCorollary 3.5, In*D — 2Ex — 2Ey 1 0 Ø . W e m a y assume that a Zariski open setS n V ' is contained in SVC w here is defined as follows:

= {C is an irreducible curve on S with DC < 2D 2 , C 2 < 0}

By Proposition 1.10, is a finite set. By Lemma 1.5, we have

(n*D — 2Ex — 2Ey )2 12 — 8 = 4 > 0 .

Hence the problem is reduced to show connectedness o f a member of In*D —2Ex — B y the assumption, a numerically connected member does not exist.Let C be an effective divisor linearly equivalent to D and a defining section s of Cis contained in H

°(e s (D — 2x — 2y)). Put

1r* C = A + 2E x + E y( n * C is a total transform)

and let A = A , + A2 be a decomposition into positive d iv iso rs . Put

A i = A i + (A i Ex )Ex + (A i Ey )Ey( i = 1, 2)

Then there exist effective divisors Ci (i = 1, 2) on S with n*Ci = A , C 1 + C2 = C.Therefore, we have

, A 2 = C1 C2 — (A 1 Ex )(A 2 E,c ) — (A 1 Ey )(A 2 Ey )tA ,E x + A 2 Ex = 2, A,Ey + A 2 Ey = 2 .

Therefore, the inequality

A 1 A 2 C 1 C2 —2

holds and the equality holds if and only if

A i Ex = A i E,= 1 (i = 1, 2) .

Moreover, i f C, = 0 , then, since A , i s a positive divisor, we have A ,E x 0 ,A 2 Ex 0 a n d at least one of the inequalities is strictly inequality, hence, A 1 A2

3. Hence both C , and C2 are positive divisors. Moreover, since we can showC1 C2 1, A is not numerically connected if and only if one o f the followingsholds:

(1) CI C2 = 1 A ,Ex = A ,E y = 1 , A ,E x = 2 , A 2 Ey = 0

A ,E„ = A ,E y = 1 , A 2 E„ = 0 , A 2 Ey = 2

LII

A lgebraic surfaces and threefolds 41

A 2 E„ = A 2 Ey = 1 , A ,E „= 2 , A ,E y = 0

A2 Ex = A 2 E y = 1 , A ,E x = 0 , A ,E y = 2

A,E„ = A i Ey = 1 (i = 1, 2)

(2) C, C2 = 2 A,Ex = A i Ey = 1 (i = 1, 2)

Case ( 1 ) : The case in which C1 C2 =1, A 1 Ex =A 1 Ey =1, A 2 E = 2, A 2 Ey =0.This shows that X , y are smooth points on C ,, and x is a double point on

C2. H e n c e C 1 C2 2 , and th is is a contrad ic tion . Similarly, we can show thatother cases in (1) do not occur.

Case ( 2 ) : If D C , 3 (i = 1, 2), we can prove C 1 C23 . H e n c e for at leastone of Ci , say C,, DC, 2.

Case (a): DC, = 1Since we have C , + C2 D, w e have C, C2 = 2 , C,(C, + C 2 ) = 1 and C = 0 ,

and the virtual genus of C, is integer, so this case does not occur.Case (fl): DC, = 2By a similar calculation in the case (a), we have Cf = 0, and the virtual genus

of C , is 3. M oreover, w e can prove that C 1 is irreducible. Therefore, S has astructure of a fibre space with curve of genus 3 as a general fibre.

Claim 3.7. A general member o f 1K ,I does not hav e a structure of a fibrespace with curve of genus 3 as a general fibre.

Proof of Claim 3.7. Assume a general member S o f 1K , has a structure of afibre space p: S —■ C with curve of genus 3 as a general fibre. By Leray's spectralsequence, we have an exact sequence:

0 —+ I-11 (C, 111(S, e s ) —> H° (C, R 1 p,,,(9,) —> 0 .

Hence, 111 ((95 ) = g + 2 o r g + 3 where g is the genus of C . If h i (e s ) = g + 3, wecan show that S = C x F, g(F)= 3. Assume that the following Claim 3.8 holds.

Claim 3.8. L et <99 = {S,}, c 4 , 4 = { t e CIIt < e l be a fam ily o f general mem-ber o f IK,,1 with So = S. By a theorem of Lefshetz, holomorphic 1-forms of So areobtained by restriction of holomorphic 1-forms on V to So . L et coi ls ° , . ,Wg+ 2150 w 9 + 31s0 (o r (0 1 iso , . . . , wslso W 9+1150 , W g + 2 0 be holomorphic 1-forms and, 1

W11S0' • • • , Woo b e the pullback o f holomorphic 1-forms o n C = Co . Then, coils ,(i = 1, g ) are also the pullback of holomorphic 1-forms on C, where C, is curve ofgenus g.

By Claim 3.8, we have co, A coi l s , 0 ( i,j = 1, g). Since 0 1„ 1 i s a finitemorphism, w A co; 0 o n V. By Lemma 1.13, there exists a morphism f : V Cwhere C is a curve o f genus g. Therefore, by Lemma 1.4, we have h i (e ,)>g + 2. I n c a s e h1 V 9) = g + 3, a general S is the product C x F of curves C andF with g(F) = 3. S o the similar facts in C laim 3.8 hold for wa + ,, 0 g + 2 , cog + 3 .Hence, we have w i A co; 0 (i,j = I, ..., g) a n d coo , A w g + k 0 (h, k = 1, 2, 3).Since Q1, is generated by global sections, this is a contradiction.

42 Hiro-o Tokunaga

Proof o f Claim 3.8. F irs t , w e sh o w th a t co n d itio n o f Theorem 1.12is s a t is f ie d . F o r p: S C, H2(C, 0c ) —> H2 (S, p*Coc ) is injective. becauseH 2 (C, Oc ) = O. N e x t, w e sh a ll p ro v e pf: 11 1(C, Oc ) (S, p*Oc ) is surjective.By Leray's spectral sequence, we have an exact sequence:

I-11 (C, 111(S, p*Oc ) —> W(C, R 1 p p*Oc ) _ o .

Therefore, it is enough to show H ° (C, 12 1 p* p* 9c ) = 0. By the projection formula,R 1 p* p* Oc Z R 1 p ( 9 0 O c . In case h

°(C, R 1 p &5) = 2, by Theorem 1.9, we have

R 1 N e s 0 Oc Z(13*(0 51 )'1 O c Z (Cc Ci Cc CD S v ) 0 e c

where e is an invertible sheaf with 111(C, e [Kc ]) = O. S in c e g(C) 2, we haveW (C , p* Os e c ) = O.

In case h° (C, R 1 p & ) = 3, by Theorem 1.9, we have

R 1 p (9 ® Oc Z Oc 0 ec 0 Oc

Hence, we have H ° (C, p * (9 s 0 O c ) = O.Therefore, pr is surjective. By Theorem 1.12, there exists a family of curves4 ' = f t e C t < e' < el w ith Co = C, and a holomorphic map with Olso =

p which satisfies the following diagram:

ois. P

4'

F or families 9°, and (6, there exist families of Albanese tori dey (91 and f (e)such that the following diagram is commutative:

„glee G.99 )\

I)44 0)

/1' A'

Let S, be a general member of 1K, 1, so, we have

we (9°) = Alb (S0 ) x 4 ' = Alb (V ) x 4 '.

Since there exists no continuous family of abelian subvarieties in a fixed abelianvariety, we have / 0 ) = i(co ) x S . This proves Claim 3.8.

By the argum ent as above, there exists a numerically 1-connected member inD — 2E„ — 2Ey l. Therefore, Proposition 3.5 follows.

DEPARTMENT OF MATHEMATICSKYOTO UNIVERSITY

A lgebraic surfaces and threefolds 43

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[ 3 A. Beauville, Complex Algebraic Surfaces, London Math. Soc. L.N.S. 68.[ 4 ]

E. Bombieri, Canonical models of surfaces of general type, Publ. Math. Inst. Hautes Etud. Sci.,42 (1973), 171-219.

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[ 8 ] Z . Ran, On subvarieties of Abelian Varieties, Inv. math., 62 (1981), 459-479.[ 9 ] Z . Ran, The structure of Gauss-like maps, Compo. Math., 52 (1984), 171-177.[ 1 0 ] K . U e n o , Classification Theory of Algebraic Varieties and Compact Complex Spaces, Springer

L.N.M. 439 (1975)

Author's present addressDepartment of MathematicsFaculty of ScienceKochi UniversityKochi, 780 JAPAN