Math. Proc. Camb. Phil. Soc. (1999), 125,83 Printed in the United Kingdom 1999 Cambridge Philosophical Society

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1 Math. Proc. Camb. Phil. Soc. ( ,83 Printed in the United Kingdom 1999 Cambridge Philosophical Society 83 Irregularity of canonical pencils for a threefold of general type* BY MENG CHEN Department of Applied Mathematics, Tongji University, Shanghai, China AND ZHIJIE J. CHEN Department of Mathematics, East China Normal University, Shanghai, China (Received 13 January 1997; revised 2 April 1997) 1. Introduction Let X be a complex nonsingular projective threefold of general type. Suppose the canonical system of X is composed of a pencil, i.e. dimφ K X (X) 1. It is often important to understand birational invariants of X such as (Xq(Xh (O X ) and χ(o X ) etc. In this paper, we mainly study the irregularity of X. We may suppose that K X is free of base points. There is a natural fibration f: X C onto a nonsingular curve after the Stein factorization of Φ K X. Let F be a general fibre of f, then we know that F is a nonsingular projective surface of general type. Set b g(c) and (F q(f) for the respective invariants of F. The main result is the following theorem. THEOREM 1. Let X be a complex nonsingular projective threefold of general type. Suppose the canonical system is composed of pencils. Then b q(x) b q(f). The upper bound for q(x) is exact. Furthermore one of the following holds: (i) b 0 and (F) 1; (ii) b 1 and (F) 1; (iii) b (F) (X) 2; (iv) 2 b (X) 1 and (F) 1. Under the same assumptions for an algebraic surface S, we know from [7] that q(s) 2. But in higher dimensional cases, a trivial example shows that q(x) can be arbitrarily large. Therefore it is interesting to find the best upper bound of q(x). 2. Proof of main theorem 2 1. Basic facts. Let X be a complex nonsingular projective threefold and f: X C a fibration onto a nonsingular curve. Then f w X/C is a semi-positive vector bundle on C and for each i 0, Rif ω X/C is a semi-positive vector bundle on C [4 6]. From the spectral sequence Ep,q Hp(C,Rqf ω X ) En Hn(X,ω X we get by direct calculation that h (O X ) h (C,f ω X ) h (C,R f ω X ) q(x) h (O X ) b h (C,R f ω X ). * Project partially supported by NNSFC.

2 84 M. CHEN AND Z. J. CHEN Therefore we obtain χ(o X ) χ(o F )χ(o C ), where we set deg f ω X/C and degr f ω X/C. We can also refer to corollary 3 2 of[6] for the above formula. We also see that i 0 for i 1, Proof of Theorem 1. We can suppose that K X is free of base points, i.e. Φ K X : X Φ K X (X) (X) is a morphism. Let X f C s W, where W Φ K X (X is the Stein factorization of the canonical map. Set b g(c). Let F be a general fibre of f. Under the assumption of the theorem, we see that F is a nonsingular projective surface of general type. Therefore q(f) (F). Let K X lin M Z, where M is the moving part and Z the fixed part. We have M num af with a ( (X) 1)degs. From the exact sequence 0 O X (K X F) O X (K X ) O F (K F ) 0, because H (K X F) H (K X ) is an inclusion, we have (F) h (K F ) 1. If b 0 or 1, this is just (i) and (ii). Suppose b 1, we show that (iii) or (iv) holds. Case 1. (F) 1. Here, f ω X is locally free of rank one, i.e. it is invertible. Since f ω X/C is semi-positive, deg f ω X/C 0, i.e. we know that deg f ω X 2b 2. By the Riemann Roch theorem, we have (X) h (f ω X ) h (f ω X ) deg f ω X b 1. Therefore (X) b 1; hence b (X) 1. Case 2. (F) 2. We know that f ω X is a vector bundle of rank (F because C is a nonsingular curve. Let be the saturated subbundle of f ω X generated by global sections in H (C,f ω X ). Since K X is composed of pencils and Φ K X factors through f, must be of rank 1, i.e. it is an invertible sheaf. We have the exact sequence 0 f ω X 0, which implies f ω X/C ω C 0. Since f ω is semi-positive, the quotient sheaf X/C ω C of f ω X/C satisfies deg ω C 0, i.e. deg (p (F) 1) (2b 2). g By the Riemann Roch theorem, we have therefore ) ( (F) 1)(b 1). From the long exact sequence ( (F) 1)(1 b); 0 H ( ) H (f ω X ) H ( ) H ( ), we obtain ) ( (F) 1)(b 1) 0. By Clifford s theorem, we have deg 2 ) 2 and by the Riemann Roch theorem, we have Therefore ) b 1 and b 1. ( (F) 1)(b 1) b 1. (2 1) (2 2) (2 3)

3 Irregularity of canonical pencils for threefolds 85 By assumption, we must have (F) 2 and ) b 1 0. (2 4) Therefore (X 2by(2 2) and (2 3). Furthermore, ) 0 and deg 2b 2by(2 1). Since deg f ω X/C deg deg 2degω C 4 2b 0, we get b 2, which is item (iii). Next, we begin to show the inequality for q(x). (i) If b 0, then R f ω X/C is a locally free sheaf of rank q(f) and so is R f ω X. Note that any vector bundle on is decomposible and we can write R f ω X as O (n ) O (n ) O (n r where r q(f) and n i 2 for i 1,,r. It is obvious that q(x) b h (R f ω X ) q(f). (ii) If b 1, we prove by mathematical induction on the rank r q(f). For r 1, it is obvious that h (R f ω X ) 1. Suppose, for r k, we have q(x) 1 k, then, for r k 1, we want to show the same result. R f ω X. Actually can be an arbitrary vector bundle on C providing that ω is semi-positive. Using C Riemann Roch, we get h ( ) deg h ( ). If h ( ) 0 then h ( ) 0. Choosing a global section s H ( let be a saturated invertible subbundle of generated by s. We obviously have deg 0. Therefore we obtain the following extension of 0 0. We have h ( ) 1. Because ω C O C, is semi-positive. Therefore, as a quotient bundle of, is also semi-positive. Thus we get by our assumption that h ( ) h ( ) h ( ) 1 k. Therefore q(x) 1 q(f). (iii) If b (X) (F) 2, we see q(f) (F) 2. If q(f) 1 then R f ω X is zero or an invertible sheaf and the inequality for q(x) is obvious. In what follows, we always suppose q(f) 2, i.e. is a rank two vector bundle and ω is semipositive. We have deg 4. Using Riemann Roch theorem, we C have h ( ) h ( ) deg 2(1 b) 2. Similarly we can choose a section s H ( ) and let be the saturated invertible subbundle of generated by s. We have deg 0. We have the exact sequence 0 0, which implies another exact sequence ω C ω C 0. We see that ω C is also semi-positive and deg 2. If deg 2, then h ( ) h ( ) h ( ) 2. If deg 1, then deg 3 and then h ( ) 0. We have h ( ) h ( ) by Riemann Roch. If h ( ) 0, then

4 86 M. CHEN AND Z. J. CHEN 1 deg 2h ( ) 2 by Clifford s theorem. Therefore we always have h ( ) h ( ) 1. Hence h ( ) 1. If deg 0, a similar method shows that h ( ) 2 and h ( ) 0; we again have h ( ) 2. In a word, we have q(x) 2 q(f) in item (iii). (iv) In this item with (F) 1, it is obvious to have q(x) b 1, because q(f) Examples. The examples are mainly from [3], therefore only the value of q(x) is given, while detailed explanations are omitted. Example 1. Let S be a nonsingular projective surface of general type with (S) 1 and C a nonsingular curve of genus b 2. X C S. We can see that K X is composed of pencils and Φ K X factors through the projection map p : X C. This is the simplest example. We have (X) b 2 and q(x) b q(f). Example 2. Let S be a minimal nonsingular projective surface of general type with (S) q(s). Suppose S admits a torsion element η Pic S with 2η lin 0. Such that h (K S Set Y S. Let p : Y and p : Y S be two projection maps. Take an effective divisor D on with degd a 3 and take δ (D) (η) and R lin 2δ lin (2D). Thus the pair (δ,r) determines a double covering π: X Y.We can find that the canonical system of X is composed of pencils and Φ K X factors through π and p. Let f p π, then f: X is the derived fibration of the canonical map. It is obvious that X is minimal. We have b 0, (F) (S) 1, q(f) q(s K F 2K S, (X) a 1, q(x) q(f) and K X 3K S (a 1). We know that a surface S of general type with χ(o S ) 1 must satisfy 0 (S) q(s) 4. To realize a specific example, we only have to find a pair (S,η) as above. (a) Take a numerical Godeaux surface with K S 1 and (S) q(s) 0 such that S admits a torsion element of order 2. We can show by [2] that h (K S (b) Take a minimal surface S with K S 2 and (S) q(s) 1. The existence of this surface is well known. For any torsion element η of order 2, we can show according to Horikawa s theorem that h (K S (c) Take a genus 2 curve C, let p and p be two projection maps of product C. Let θ Pic C be a torsion divisor of order two. Take a divisor D in with deg D 3. On the ruled surface C, take δ (D) (θ) and R 2δ (2D then the pair (δ,r) determines a double cover π: S C. S is just a minimal surface with (S) q(s) 2 and K S 8. Denote f i p i π (i 1, 2). We see that the fibration f : S has exactly six multiple fibres with multiplicity 2. Let F 2E and F 2E be two multiple fibres, take η E E, then 2η lin 0 and we can see that h (K S (d) Let C be a nonsingular curve of genus two and E an elliptic curve. Take the product S C E and let q, q be the two projection maps. Take a torsion element θ Pic C of order 2 and P a point on E. Let δ q (θ) q (P) and R 2δ; then (δ,r) determines a double cover π: S S. We can see that S is just a minimal surface with (S) q(s) 3 and K S 8. Take a 2-torsion element τ Pic E and η π*q (τ); we can see that h (K S (e) Let S C C, where C is a nonsingular curve of genus two. It is obvious that (S) q(s) 4 and K S 8. Let q i, i 1,2, be the two projection maps, and take a 2-torsion divisor θ Pic C, η q (θ) q (θ); then we can see that h (K S

5 Irregularity of canonical pencils for threefolds 87 Example 3. Let S be a minimal surface with (S) q(s) 1 and K S 2. We know that the Albanese map of S is just a genus two fibration onto an elliptic curve. It can be constructed from a double cover onto a ruled surface P which is over the elliptic curve with invariant e 1. Furthermore, all the singularities on the branch locus corresponding to this double cover are negligible. Let π : S P be this double cover with covering data (δ ). Note that q(p) 1. Take an elliptic curve E and write T E P. Let p and p be the two projection maps. Take a 2-torsion element η Pic E. Since P is a fibration over an elliptic curve, we can take a 2-torsion element τ Pic P such that π (τ) lin 0 through the double cover π. Let δ (η) (δ ) and R 2δ, then the pair (δ ) determines a double cover Π : Y T. Let φ p Π. Take a divisor A on E with dega a 0. Let δ φ*(a) Π (τ) and R 2δ, then the pair (δ ) determines a smooth double cover Π :X Y. We can see that X is a minimal threefold of general type and the canonical system K X is composed of pencils. Φ K X factors through Π and φ. This example satisfies b 1, (F) 2, q(f) 1, K F 16, (X) a, q(x) b q(f) 2 and K X 12a. REFERENCES [1] BEAUVILLE, A. L application canonique pour les surfaces de type ge ne ral. Invent. Math. 55 ( [2] BOMBIERI, E. Canonical models of surfaces of general type. Publ. Math. Inst. Hautes Etude. Sci. 42 ( [3] CHEN, M. Complex varieties of general type whose canonical systems are composed with pencils. J. Math. Soc. Japan 51 (1999 to appear. [4] KAWAMATA, Y. Kodaira dimension of algebraic fiber spaces over curves. Invent. Math. 66 ( [5] KOLLA R, J. Higher direct images of dualizing sheaves I. Ann. of Math. 123 ( [6] KOLLA R, J. Higher direct images of dualizing sheaves II. Ann. of Math. 124 ( [7] XIAO, G. L irre gularite des surfaces de type ge ne ral dont le syste me canonique est compose d un pinceau. Compositio Math. 56 (