Math. Z. 2 l l , 87 92 (1992) Mathematische

Zeitschrift f(. Springer-Verlag 1992

Deformations of multiple fibres

Fernando Serrano Departament d'A.lgebra i Geometria, Facultat de Matem;ttiques, Universitat de Barcelona, Gran Via 585, E-08007 Barcelona, Spain

Received February 21, 1991; ~rl final form December 11, 1991

0 Introduction

The aim of this note is to prove that the multiplicities of the fibres of a morphism onto a curve are invariant under small deformation of the map. This completes partial results from E7].

The precise setting is as follows. Let R and C be compact complex manifolds with dim C = 1, and let ~p: R --+ C be a surjective morphism with connected fibres. If ~p l ( p ) = ~n~Bi denotes the analytic fibre of ~p over P e C , the B~'s being the irreducible components and the n~'s their corresponding multiplicities, then the greatest common divisor of the ni's is an integer m called the multiplicity of qg- 1 (P). We say that q~-l(p) is a multiple fibre whenever m > 2. Assume ~p has exactly k multiple fibres with multiplicities {ml, . . . , rag}, (so that some me may appear more than once). It was proved in [-7] that the group ( ~ = ~ 71lintY. is invariant under deformation of (p. In particular, the product of all m~'s is an invariant too. This result will be strengthen in the present paper under the extra assumption that R be a projective variety. Namely, we will prove that the set {ml . . . . . ink} itself is invariant under small deformation (cf. Theorem 1). More precisely, it turns out that a fibre of multiplicity m > 2 "deforms" to a fibre with the same multiplicity in all neighbouring fibrations. In the very particular situation of R being an elliptic surface and ~p the elliptic fibration of R, the invariance of {ml . . . . . m~} was already shown by Iitaka in [6]. His proof was based on the well-known relation- ship between the canonical bundle of R and the multiple fibres. For general fibre genus or in higher dimensions no such relationship exists, and so we have to rely on another "global" invariant. This is provided by the above-mentioned result in [7]. With this at hand, the proof of Theorem 1 depends only on a local analysis, which is performed through suitable base-changes with the help of some results of Horikawa I-4, 5].

1 Invariance of the multiplicities

We are working in the analytic category. A complex manifold is a connected, smooth analytic variety. We reserve the name Jibration for a proper, surjective,

88 F. Serrano

holomorphic map from a compact complex manifold onto a smooth curve, whose fibres are connected, g(C) denotes the genus of a curve C. Let X, Y, M denote complex manifolds (not necessarily compact), and let f: X - , Y, g: Y---, M be surjective, proper, flat, holomorphic maps with connected fibres such that g and h := g o f a r e smooth, and g has fibre dimension 1. If fibres of g and h over t ~ M are written Yt, Xt respectively, then f : X, --, Yt is a fibration as defined before. We call X ~ Y ~ M a family of deformations ofJo: Xo--* Yo, for any 0 ~ M not in the boundary of M. Also, h: X ~ M, g: Y--* M are referred to as families of defor- mations of Xo, Yo respectively.

This paper is devoted to the proof of the following result:

Theorem 1 Let qo: R ~ C be a morphism with connected .fibres from a smooth complex projective variety R onto a smooth projective curve. Suppose ~p has exactly k multiple fibres, and let {ml . . . . . ink} denote their multiplicities. Let f: X ~ Y be any family of deformations of (o over M with X o := R, Yo = C, Jr := ~o, O~M. Then there exists a neighbourhood M c_ M of O with the followino property: if (o has a fibre of multiplicity m > 2 over P c C, then there is a neighbourhood U of P in Y such that for all t e M, f : Xt ~ Yt has exactly one multiple fibre over points in U cn Yz, and with multiplicity equal to m. In particular, f : X~ ~ Yt has exactly k multiple fibres for t ~ M, and their multiplicities are {ml . . . . . mk }.

We say for short that the set {ml . . . . . ink} is invariant under small deforma- tion of rp. In the sequel we shall keep the notation introduced in Theorem 1. When necessary, we will replace M by a smaller neighbourhood of 0 e M, and restrict the family of deformations accordingly. In spite of this, we shall always keep the symbol M for simplicity.

Remark 2 The hypothesis on R being projective algebraic is only used in order to guarantee the vanishing of H~ (R17",d~s)| To) in Lemma 5 below. Note that the theorem asserts the invariance of {m~ . . . . . mk} under any deformation of qo by projective morphisms.

Theorem 1 generalizes a previous result, where no assumption of projectivity was required (cf. Introduction):

Proposition 3 [7] In the situation of Theorem 1, the fibres of any ji: X , -~ Yt have multiplicities whose product equals 1-I~= 1 ml.

We are going to replace ~o by a new fibration in such a way that every m~ appears a suitable number of times as multiplicity. Suppose first of all that g(C) _> 1. Set p := least common multiple of { m l , . . . , m~}. Consider any 6tale cover D -~ C of degree p, and set S := R x D. Every multiple fibre with multiplicity

C m~ of ~0: R --* C gives rise to p identical multiple fibres of the induced map 7:: S -~ D, with multiplicity ml. On the other hand, Y--, M is differentiably locally trivial [2], so that Y becomes diffeomorphic to C x M after making M and Y suitably smaller. Since D x M is a topological covering space of Y, we can lift to D x M the analytic structure of Y. This defines a complex manifold W with underlying topological space D x M, so that W ~ M is a family of deformations of D, and W --, Y is 6tale of degree p. Define V:= X x W. Then V--* Wis a family of deformations of 7' over M,

Y and every fibre f - ~ (P) o f f : Xt ~ Y,, t e M, lifts to / t identical fibres of Vt --* Wt with the same multiplicity as f -~ (P ) . In case C ~-IP ~ we may consider a suitable finite map D -4 C of degree p with g(D) > 2, and proceed as in the last part of

Deformations of multiple fibres 89

Theorem 2.3 in [7]. We also get a new fibration 7~: S--, D and a deformation V--, W o f 7 j with the properties stated above. Note that in all cases, Theorem 1 will be proved for q) and the family f" X ~ Yas long as it is proved for ~P and V ~ W. In the sequel we shall work with this last family, but for simplicity we call itJ~ X --, Y again. We remark it for emphasis:

Claim 4 We may assume from now on that g(C) > 1, and each mi appears in {m~ . . . . . m~} a number of times which is a multiple of p : = lcm(ml . . . . . m0. Furthermore, if P1 . . . . . P, , e C are the points for which q~-l(Pi) has multiplicity m->_ 2, suitably ordered, then there exist ne ighbourhoods U t . . . . U~ u in Y of P1 . . . . . P , , respectively, such that for any q = 0 . . . . . r - 1 , the maps f - ~(Ui) --, Ui are pairwise analytically equivalent for all i = q# + 1 . . . . . (q + 1)kt.

Lemma 5 Let D be a smooth irreducible curve with g(D) > 2, 6: D ~ C a finite morphism, and S a desingularization of R x D. Suppose the induced morphism ?: S ~ R

c is btaIe. Then, up to shrinking M if necessary, there exist a family of deformations V--* W ~ M of the natural map tp: S--*D, with ip beino Vo ~ Wo, and finite holomorphic maps V--* X, W--* Y, such that the following diagram commutes:

V --* X

W ~ Y

"-~ M ~ .

Furthermore, over 0 ~ M this diagram coincides with 7

S ~ R

D ~ C .

Proof Since X --* M is differentiably locally trivial [2], we can make M smaller and shrink X accordingly so that X is diffeomorphic to R x M . The topological covering space S x M of X inherits an analytic structure from X, thus becoming a complex manifold V endowed with an 6tale map V--, X. V ~ M is now a family of deformations of S.

Claim 5.1 Up to shrinking M if necessary, there exists a family of deformations V ~ W o f ~ P : S ~ D o v e r M .

Proof of Claim 5.1 Let TD stand for the tangent bundle of D. By [5, Theorem 8.1] it suffices to show that the canonical homomorph i sms Hi(D, T o ) ~ Hi(S, ~P*(TD)) are surjective for i = 1 and injective if i = 2. The second requirement is clear because dim D = 1. As regards to the first, we make the following simplification. Take a sufficiently ample and general hyperplane section L of S. If dim S > 2, by the vanishing theorem of Kodai ra we have H 1 (S, !P*( To) | L - 1 ) = H 2(S, 7~,( To) | L - t) = 0. The cohomology of the exact sequence

(0 ~ L - ~ ~ CSs ~ C~L - , o) | ( ~ * ( T D ) )

yields H l(S, T*(TD)) = H ~(L, ~*(TD)IL). Hence, by taking successive hyperplane sections in this manner it follows that the surjectivity of Hi(D, T D ) ~ HI(S, 7J*(TD)) will be proved as soon as it is proved for d imS = 2.

90 F. Serrano

We will then assume S to be a surface. The Leray spectral sequence for ~u provides an exact sequence

0--~ Hi (D, tP. ~* (To) ) ---*H'(S, ~U*(TD) )

H~ R ~ ~g, ~u*(To)) --* 0 .

Since ~P,(Cs) = CD, projection formula gives ~P, 7 '* (TD)= To, R 1 71, 7~'*(To) = (RI~P, Cs) | TD. Relative duality (cf. [1]) yields RI~t ' ,Cs = (7~,(Ks | 7J*T~) )~, where Ks is the canonical bundle of S, and ~ stands for dual as Co-module. We now quote Fujita's theorem in [3]: 7 J , ( K s | 1 7 4 where n = d i m H l ( S , Cs) - g(D) and E is a vector bundle satisfying H~ E ~) = O. Combining all the above one gets H~ R 1 7 ' , 7 ~ * ( T o ) ) = H ~ 1 7 4 (E ~ | To)) = 0, since g(D) > 2 by assumption.

Claim 5.2 There exists a ho lomorphic map W ~ Y for which the diagram of Lemma 5 commutes.

Proof of Claim 5.2 Set e := (po), = go 7 ~. By [4, 7.3] applied to V ~ W and the composite V ~ X ---, Y, it follows that a map W ~ Y with the required conditions exists as long as the homomorph i sms Hi(D, 3*(Tc)) ~ Hi(s , e*(Tc)) are surjective for i = 0, and injective for i = 1. The claim for i = 0 is obvious, and for i = 1 it follows from the Leray spectral sequence associated to 7-': S ~ D. Lemma 5 is now proved. []

Lemma 6 I f the fibre of ~p: R = Xo --* C = Yo over P e C has multiplicity 1, then there is a neighbourhood U for P in Y such that X , -o Yt has fibres of multiplicity 1 over all points in Yt c~ U, for t ~ M close enough to O.

Proof. Consider the set {ml . . . . . ink} decreasingly ordered: m I ~ . . . => mk. Let ~0- 1 (P1) . . . . . tp- 1 (Pk) be the multiple fibres of ~0, with O 1 (Pi) of multiplicity m~. By the hypothesis of Claim 4 we may assume that ml . . . . = my, > mr,+l > �9 �9 . ( r eN) . Choose a line bundle H on C such that H | = (_( ;c (P1 + . . . + P r u ) " These data give rise to a cyclic covering C ' ~ C of degree /~, totally ramified at P1 . . . . . P , , and unramified elsewhere [1]. If R' stands for the normalizat ion of R x C', then R' is smooth (see the proof of [1, I I Ig . l ] , valid in any dimension),

c R' -* R is 6tale, R' --, C' has no multiple fibre over the preimages of P1 . . . . . P , , in C ' (loc. cit.), and all multiple fibres of R' --, C' are preimages of multiple fibres of q) with multiplicities < ml. N o w we can apply the same procedure to R' --, C', and so on. At the end one gets a finite map D --* C which is 6tale away from P 1 , . , . , Pk and such that the points on D mapping to Pi are ramification points with multiplicity m~. The normalizat ion of R x D is, as before, a smooth variety S with

c the natural maps S ~ R being 6tale and 7/: S -4 D free from multiple fibres. Since g(D) > 2 we may consider the family V --, W o f deformations of 7 ~ over M obtained in Lemma 5. F rom the fact that the product of the multiplicities of the fibres is a deformation invariant (cf. Proposi t ion 3) we conclude that V, ~ Wt has no multiple fibre, for any t e M . Moreover, W ~ Y is unramified over some small ne ighbourhood U _~ Y of the given point P. The claim is now obvious. 2~

Lemma 7 Suppose cp: R = Xo --, C = Yo has a fibre o f multiplicity m >-_ 2 over P ~ C. There exists, then, a neighbourhood U o f P in Y such that for all t ~ M near O, X, ~ Yt

Deformations of multiple fibres 91

has at most one multiple f ibre over points in U ~ Y,. Moreover, the multiplicity o f this f ibre is not greater than m.

Proo f In view of Claim 4, there exist points P I : = P, P2 . . . . . P,, in C, and neighbourhoods Ui of Pi in Y, i = 1 , . . . , m, such that the restrictions of the family X ~ Y to the U{s are pairwise isomorphic. In particular, ~p-- 1 (p~) has multiplicity m for all i = 1 . . . . . m. Construct an m-cyclic covering D--. C branched at P1 . . . . , P~, and unrarnified elsewhere, and denote by S the normalizat ion of R • D.

C We know that S is smooth, S ~ R is 6tale and the natural map S ~ D has no multiple fibres over the preimages Q1 . . . . . Q,, ED of P1 . . . . . P,, (cf. proof of the preceding lemma). Applying Lemma 5 one gets a family V--* W of deformations of S ~ D, and for t near 0, all fibres of V~ ~ ~; over points close to Q ~ . . . . . Q,, have multiplicity i (Lemma 6).

Fix a point I e M near 0, and let us denote X, ~ Y~, V, --+ Wt by q)': R' ~ C', ~ ' : S' ~ D' respectively. Write r~ . . . . , rd for the multiplicities of the multiple fibres of q~' near P1 (over points Q(~,l), �9 . . , Q(~,d) in U1 c~ C ' respectively). By the assumptions of Claim 4, rl . . . . . rd are also the multiplicities of the multiple fibres of ~o" near P~, i = 2 . . . . . m (over points Q(i, 1) . . . . . Q,,d) ~ U~ c~ C'). Let e~i,h~, s = 1 . . . . . z(i, h), denote the ramification indices of D ' - ~ C' over Q,,*I. One observes that S' ~ R ' is etale of degree m and S' ~ D' has no multiple fibre over points on D' mapping to Q(~,h)a C'. The commutat ivi ty of the diagram

S' ~ R '

D' ~ C '

implies e~i,h) > rh for all f, h, s, and m > deg(D' --. C ' ) > deg(D ~ C) = m, so that ~- :,~(i, h)

deg(D' ~ C') = m. Consequently ~(i, h)rh < e:i,h) +.. �9 + ~,,h) = m. Hurwitz for-

mula for the map D ~ C yields

(.) 2g(D) - 2 = m(2q(C) - 2) + (m - 1)m.

The same formula on D' ~ C' gives

(**) 2g(D') - 2 > m(29(C' ) - 2) + d a(i,h)

Y~ Z (e;,,n,- 1) i = 1 h ~ l s = l

d

= m ( 2 9 ( C ' ) - 2 ) + ~ ~ ( m - ~ ( i , h ) ) i = 1 h = l

d

> m(29(C') - 2) + ~ m(m - m/ra)�9 h = l

Inasmuch as D' is a smooth deformation of D we have g(D') = g(D). Likewise g(C') = g(C). Combining with (.) and (**) it implies

d

1 > I - 1 /m > y~ (1 - l /r~) > a l l2 . h = l

One concludes that d = 1 and rl < m, as desired. []

92 F. Serrano

Proof of Theorem I Let q~-l(P1) . . . . . (o-~(Pk) denote the mul t ip le fibres of q0, with mult ipl ici t ies ml . . . . ,mk respectively. Lemmas 6 and 7 combined with the fact that C is compac t implies that for t ~ M close enough to 0, X, --, I1, has at most k mul t ip le fibres, each one of them near a co r respond ing ( p - l ( P i ) and with mult ipl ic i ty not grea ter than mi. On the other hand, the p roduc t of the mult ipl ic- ities of all fibres is a deformat ion invar iant (Propos i t ion 3). It turns out that {ml . . . . . ink} are precisely the mult ipl ici t ies of the k mul t ip le fibres of X~--* Y~. []

One can also show tha t fibres which are mul t ip les of a manifo ld can only deform to mul t ip le manifo ld fibres. In other words:

Corol la ry 8 With the hypothesis of Lemma 7, !f ~o-I(P) is a multiple oJ'a smooth irreducible divisor, then the only multiple fibre of X, ~ Y, over U n Yt has a reduced structure which is a smooth divisor.

Proof Proceed as in the p roo f of L e m m a 6. We obta in a commuta t ive d iag ram of de format ions

V ~ X

W ~ Y

such that the fibre of Vo ~ Wo over any poin t Q e Wo ma pp ing to P ~ Yo is smooth. Consequent ly , V, ~ Wt has smoo th fibres over U • W , for some small ne ighbour- hood of Q in W. Since V--, X is ~tale the claim follows.

References

1. Barth, W., Peters, C., van de Ven, A.: Compact Complex Surfaces. Berlin Heidelberg New York: Springer 1984

2. Ehresmann, C.: Sur les espaces fibr6s diff&entiables. C.R. Acad. Sci., Paris 224, 1611 1612 (1947)

3. Fujita, T.: On K/ihler fiber spaces over curves. J. Math. Soc. Japan 30, 779-794 (1978) 4. Horikawa, E.: On deformations of holomorphic maps II. J. Math. Soc. Japan 26, 647 667

(1974) 5. Horikawa, E.: On deformations of holomorphic maps III. Math. Ann. 222, 275 282 (1976) 6. Iitaka, S.: Deformations of compact complex surfaces I1. J. Math. Soc. Japan 22, 247 261

(1970) 7. Serrano, F.: Multiple fibres of a morphism. Comment. Math. Helv. 65, 287 298 (1990)