Fibered neighborhoods of curves in surfaces

The Journal of Geometric Analysis Volume 13, Number 1, 2003

Fibered Neighborhoods Surfaces

of Curves in

By C. Camacho, H. Movasati, and P Sad

ABSTRACT. The a#n o f this article is to study fibered neighborhoods of compact holomorphic curves

embedded in surfaces. It is shown that when the self-intersection number of the curve is sufficiently negative

the fibration is equivalent to the linear one defined in the normal bundle to the curve. The obstructions to

equivalence in the general case are described.

This article is devoted to the study of the simplest possible type of foliations on surfaces. We consider a compact, smooth, holomorphic curve inside a (holomorphic) surface and a holomorphic foliation by discs transverse to it; the general problem is to classify such objects. These foliations arise naturally when suspensions of groups of diffeomorphisms are constructed. In our specific case, we wish to know whether the foliation (or the fibration over the curve) is equivalent to the foliation by lines on the normal bundle to the curve. This is true when the curve has a sufficiently negative self-intersection number, but we are able to describe the obstructions that appear in the other cases.

Our study is related to the systematic study of neighborhoods of analytic varieties started by H. Grauert in his celebrated article [5]. In this article, negatively embedded submanifolds of codimension 1 are considered. There is a geometric aspect, where formal equivalences of neighborhoods are proven to be in fact holomorphic equivalences; the main point is the vanishing of some special cohomology groups due to the existence of holomorphically convex neighborhoods of the submanifold. As for the formal side, the notion of n-neighborhood, for n E N is introduced and the obstruction to extend isomorphisms between n-neighborhoods to (n + 1)-neighborhoods is described; it lies also in certain cohomology groups which depends on the normal bundle to the submanifold. The Kodaira Vanishing Theorem implies that for n c N sufficiently large this group vanishes, so that the germ of a complex manifold along a negatively embedded submanifold depends only on a finite neighborhood. In [7] the special case of curves is considered; the use of Serre's duality allows simpler proofs. A careful analysis leads to the following linearization result: a negatively embedded curve with self-intersection number smaller than 4 - 4g (g 6 N is the genus of the curve) always has a neighborhood equivalent to a neighborhood of the zero section of the normal bundle.

Our main result states that, when the self-intersection number of the curve is negative and

Math Subject Classifications. 32F75, 32E05, 32L20. Key Words and Phrases. holornorpbic fibrations, Serre's duality~ holomorphically convex neighborhoods of curves.

�9 2003 The lournal of Geometric Analysis ISSN [050-6926

56 c. Camacho, H. Movasati, and P. Sad

smaller than 2 - 2g, the fibration is equivalent to the linear one that exists in the normal bundle. We use the same cohomological property of a holomorphically convex neighborhood as in [5] to solve a Cousin problem. We remark that the linearization result for neighborhoods, previously mentioned, does not imply that a fibration is equivalent to a linear one, even if the self-intersection number is smaller than 4 - 4 g . As for the obstructions to linearization, the existence of the fibration allows us to read them in the first cohomology group of the sheaf of 1-forms along the curve that take values in the normal bundle. We know then exactly when it is possible to linearize a fibered embedding; it remains to be developed the classification of all fibered embeddings with a given normal bundle. In the negative case, it is likely that (2g - 1)-neighborhoods determine the embedding.

This article is organized as follows. In Section 1 we state the results and give some examples. Sections 2 and 3 are devoted to the formal analysis of neighborhoods, and Sections 4 and 5 to the convergent aspects. Section 6 presents a simple technique to generate examples of fibered embeddings which are not linearizable. Finally, there is an Appendix with an application to holomorphic foliations.

1. Statement of results and examples

Let S be a compact, smooth, holomorphic curve embedded in some complex surface M in such a way that it has a neighborhood holomorphically fibered by a family ~ of transversal l-dimensional discs (we will say that S isfibered embedded in M). This means that there exist a covering H of S by open sets {U,~} C M and charts

~p~ = (x~, y . ) : U~ > C 2

such that

(1) S n Ua = ya- l (0 )

(2) Whenever Ua N U~ # 0 , the change of coordinates Fa/~ = gta o gt~ -1 is given by

(3)

x,, = qS~(x~) (1.1) Yc~ = Ocq~(x/~,Y~),

where q ~ and ~7~ are holomorphic functions. We remark that Ft~ v o F• o F~t~ = Id when Uu n U~ n Uy # 0 and that F~u = ld for any or.

In each set Uu the fibration G over S is given by dxc, = O.

The normal bundle N to S, defined by the transition maps (x~, ya) = L,~(x~, y~) given by

x~ = ~ba~(x/~) (1.2) ya = A~t~(x~)yt~

for A~(xr := O0~/Oy~(xr 0), is in a natural way fibered by the lines dx~ = 0. Let us call Gp the fiber of ~ and Np the linear fiber in N both passing through p E S.

The problem we study in this article is the existence of a fibered equivalence between the embeddings of S in M and N. That is, whether there exists a holomorphic diffeomorphism H, defined between neighborhoods of S in M and N, that sends S to S and G p into Np for all p ~ S (we say that the embedding is linearizable).

The following examples illustrate the situation.

Fibered Neighborhoods of Curves in Surfaces 57

Example 1.1. Take S as a rational curve with self-intersection number S . S = - 1 inside M. We may blow down a neighborhood of S and get near (0, 0) 6 C 2 a foliation with a singularity of radial type (obtained from blowing down the fibration G). A theorem of Poincar6 (see [1]) allows us to define a diffeomorphism that sends the foliation to the one given by y dx - x dy = 0; such a diffeomorphism can be lifted to produce the equivalence with the linear fibration of N.

Example 1.2 ([11). The curve S will be in this example a complex toms embedded with self-intersection number 0. Let

(z ,w) ~ (~ ( etz')~w+ ~--~zk(z)w~)k>_2

define an embedding of the open set {1 - e < Izl < 1 + e} • II3; here 0 < loci < 1, ~. ~ C* and 0 < e < < [cr The coordinates (z, w), (z r, w') for the surface M are settled as follows:

1. (1 +2E)lotl < Izl < I + E , Iwl < 1

2. (1 ~)1~1 < Iz'l < 1 - 2~, w' - ~ C

W t 3. The change of coordinates is given by z = otz, = O(z, w) for 1 - e < Izl < 1 + E andz = z , w = w for Ic~l(1 + 2 e ) < [z[ < 1 - 2 e .

As for the normal bundle, we have the coordinates (z, w) and (z', w') ; z and z' are related as t

before, and w = ~.w.

Now we look for a diffeomorphism H(z, w) = (z, h(z, w)), defined for (1 - E)I~I < t t

[zl < 1 + e and Iwl small, that satisfies Oh/Ow(z, 0) = 1 and H ( z , w ) = (otz,)~h(z, w)), or h(otz, rl(z, w)) = )~h(z, w). There are formal obstructions to finding such an h. Let us write h(z, w) = w + ~k>_2 hk(z) wk and try to find h2; it has to satisfy the equation

A2(z) + )~2h2(otz) = )~h2(z) �9

= Y~_oolnznin(1-E) la l < [zl < l + e Expanding as power series both the functions hz(z) +o~

and Az(z) + ~ : Y~-oc an zn in a neighborhood of ~1, we get

)dn ( 1 - )~otn) = an .

We demand [~.ot n - II 7 ~ 0 for all n ~ Z (since loci < 1, this is a single condition; it is easy to see that once it is satisfied, the series defining h2 converges). Going further in the process of computing hk for k > 3, we see that to get at least a formal map h we need ])~mctn - 1[ 7~ 0 for all n ~ ~ and m ~ N*.

In [1], it is also presented a small denominator condition in order to ensure the convergence of h; in the absence of this condition, we may have divergence (see [2]).

Example 1.3. There are embedded curves without fibration by discs in a neighborhood. For example, if the curve is plane of degree greater or equal to 2, any fibration would be the restriction of a globally defined foliation in ~(2), by Levi 's extension theorem; but such a foliation has necessarily tangency points with the curve.

Our main result in this article is the following

T h e o r e m 1.4. Let S be a compact Riemann surface of genus g E N fibered embedded in the surface M. Assume that S . S < 0, Then, i f S . S < 2 - 2g, the embedding is fibered equivalent to the embedding orS in the normal bundle N.

58 C. Camacho, H. Movasati, and P. Sad

This statement can be seen as a generalization of Example 1.1. In order to put Example 1.2 in a general perspective, we have to introduce some notations and definitions. Let 0 (1'~ (N k) be the sheaf of holomorphic 1-forms of S with coefficients in the fiber bundle N k, for k E N (when k = 0, we use O (l'~ for simplicity). We refer to the notation in the beginning of the section.

Definition 1.5. Let m c N*. A fibered embedding of S into M is m-linearizable when there are

adapted charts with changes of coordinates F,~ = (~b~, 0a~) satisfying oJoa~/OyJ~(xl~, O) = 0 for all 2 < j < m (whenever Ua M U~ # 0 ). The embedding is formally linearizable if it is m-linearizable for all m 6 N*.

We want to state a condition to guarantee that a fibered embedding is (m + 1)-linearizable once it is already m-linearizable. So let us assume

x~ = 4,,~(x~)

ya = y~{Aa~(x~) +Ca~(x~)y~ + . . . ] . (1.3)

We associate to co 6 H~ o( l 'O) (Nm)) , represented as the collection {cod} in the covering/4, the 1-cochain

-1 (1.4) Oa~ := Ca~A~ co~ .

The cocycle relations satisfied by the change of coordinates {Fa/~} imply immediately that

{Ca~A~A} E Hi( /4 , N-m). It The lemma below follows.

L e m m a 1.6. {0,~} e HI(/A ', (..9(1'~

As a consequence of Serre duality, as we will explain in Section 2, any element of H 1 (/4, O (1'~ is a 1-coboundary once we allow poles in the open sets Ua. (In fact, the poles can be selected independently of the holomorphic 1-cocycle and outside all possible intersections of type Uu fq U~.) We have then a collection of meromorphic 1-forms {0a} associated to co such that O~ - Ot~ = 0 ~ whenever U~ n U~ ~ 0.

Definition 1.7. Let co ~ H~ o(l'~ Then r(co) = ~ u 7"~es(Oa).

It can be proven that these numbers do not depend either on the choice of the adapted charts or on the choice of the meromorphic l -forms involved. Also, the 1-cocycle {0,~/~ } defined in (1.4) represents an element of H 1 (S, O 0'~ in the covering L/.

Theorem 1.8. Consider a fibered embedding orS into M which is m-linearizable, fir(co) = 0 for all co E H~ (-9(l'~ then the embedding is also (m + l)-linearizable.

All vector spaces H~ 69 (1,~ (Nk)) are finite dimensional, so that we have to check a finite number of conditions in order to apply the above theorem.

Since H~ 0 (1,~ (Nk)) = {0} if the Chem class of (.9 (1,~ (N ~) is negative, or equivalently (S �9 S)k < 2 - 2g, it follows that when S �9 S < 0 only a finite number of conditions have to be verified in order to ensure that the fibered embedding is formally linearizable. Furthermore, we are able to state: I f S �9 S < 0 and the fibered embedding is formally linear&able, then it is also holomorphically linear&able.

Let us go back to Example 1.2 and use the tools involved in the formulation of Theorem 1.8.


E x a m p l e 1.2 his Let us take )~ = 1 for simplicity; the normal bundle to the elliptic curve is therefore trivial. We wish to see whether the embedding is 2-1inearizable. The vector space H~ O(1,~ (N)) is simply H~ O(1'~ which is 1-dimensional; we choose the I-form z -1 dz as its generator. Take the covering/4 corresponding to the open sets { (1 - �9 < I z'[ < 1 - 2�9 and {(1 + 2 � 9 < Izl < 1 +e} and the holomorphic 1-cocycle 012 defined as 012 = A2(z)z -1 dz in V' = {1 - E < Izl < 1 + E} and 012 = 0 in V = {(1 + 2 O l a l < Izl < 1 - 2e}. Now we look for meromorphic 1-forms 01 and 02 such that 01 - 02 = 012.

Due to vanishing of 012 in V, we see that 01 ---- 02 in that set. We conclude that there exists a global meromorphic 1-form 0 = f (z) dz defined in a neighborhood of the annulus which satisfies the relation

f(az)ot dz - f ( z ) dz = A2(z)z -1 dz

for z ~ V'. Since

7~es(O)=fz l=lO- fz l=la lO

and taking into account the above relation, we arrive at

T~es(O) = - f A2(Z)Z -1 dz = -ao A

JIz I=1

Finally, we observe that S �9 S = 0, since N is trivial. Once again we arrive at the conclusion that a0 = 0 is the condition for the embedding to be 2-1ineafizable.

We observe that H~ O(1'~ = {0} when N is not trivial.

2. The residues

In this section we make two remarks about Definition 1.7 (notation as in Section 1).

First of all, we used that the 1-cocycle {0cr } is a coboundary if we use meromorphic 1-forms. To see this, consider {0~} ~ Hi(/4, O(1'~ clearly we have also {0~Z} ~ H I ( u , O(I '~ for any effective divisor D. But H 1 (/4, 0 (1,~ (D)) and H~ O ( - D ) ) have the same dimension, by Serre Duality ([6], p. 76). It follows then that H 1 (/4, 0 (1'~ (D)) = {0}, and {0~ } is a coboundary for the cohomology with coefficients in O (1'0) (D).

The second remark is a proof that the residues r(w) of Definition 1.7 do not depend on the way we write the cocycle {0~} as a coboundary (see Remark 1). Let us take adapted coordinates {(xu, y~)} and {(xa, z~)} for a m-linearizable embedding, as in (1.3); we assume

Ya = Acr + Ccr ~+l + . . .

' m + l Z~ = Aa~(x~)z~ + Cafl(xfi)z ~ + " "

z~ = y~ + ( . . . ) + C ~ ( x a ) y m+l + . . . �9

It is easy to see that

- 1 ' Aa~C~ = A~Ca~ + ArniCa - C~ . (2.1)

Given {w~ } c H~ O (~ 1)(Nm)), w e choose appropriate meromorphic 1-forms as to have

O~ - O fl = A-~ C ~ w ~ (2.2)

60 c. Camacho, 14. Movasati, and P. Sad

Combining (2.1) and (2.2) gives ! t

0 a - 0~ = 0a - 0~ + Caoga - C~ o9~.

Therefore, ! ! 0a - 0a - Cawa = 0~ - 0~ - C~Og~

and the meromorphic 1-form ( given as (a = 0'a - 0a - Caoga in each Ua is globally defined.

We apply the Residue theorem to conclude that ~ a Res(Oa) = ~-~a Res(O'a)"

3 . P r o o f o f T h e o r e m 1 . 8

Let us consider adapted coordinates as in (1.3) for the m-linearizable embedding of S into M, and replace the coordinates (xa, Ya) by new coordinates (xa, za):

za = Ya - ca(xa)Y m+l (3.1)

where ca is a holomorphic function for each or. The new changes of coordinates are given by the maps

( , ) F~'t~ xt~, z~) = (x~, ~/a~(x~, z~)

where rl~(x~, z~) = Aa~(x~)z# + C ~ ( x ~ ) z ~ +1 + . . .

and Aa~-I Ca = a ~ c a ~ + c~ - aam~ca .

In order to prove that the embedding of S in M is (m + 1)-linearizable, it is enough then to verify that { AJCa~} c HI(L/, N -m) is a co-boundary. Let us then start by writing

A J CaB = Aam~ta - t~

where ta ~ C~(Ua) . The collection of 1-forms Va = 0ta clearly defines an element v HO(s, E(0 ' I ) (N-m)) , a C a ( 0 , 1)-form with coefficients in N -m. By Serre duality (see [6], p. 76), if we are able to prove that

s V A O 9 = 0

for all o9 c H~ o(l"~ then v = 0h for h ~ H~ E~~176 Thus, each ta - ha is a holomorphic function and

A ~ C a ~ = a a ~ ( t a - h a ) - (t~ - h E ) .

Let us then fix o9 = {oga} 6 H~ (Q(I'~ From the existence of meromorphic 1-forms {0a} such that

we get

0 a - - 0/~ ( m - t ~ ) = Aa~ta o9~ = taoga - t~og~ .

Finally, an application of Stokes' theorem to the 1-form ~ defined in each Ua as 0a - tao9a gives

r(og) = ZTges (Oa) = (2izr) -1 ]~ v/x O 9 . d b a

But r(o9) = 0 by hypothesis. The proof is complete.


4. An adapted Cousin problem

Let us go back to the setting introduced in Section 1. We assume hereafter that S �9 S < O.

We define an (almost) global system of coordinates for the normal bundle N to S as follows. The first coordinate is the projection over S. As for the second one, consider the coordinates given by (1.2) and a meromorphic section s = {s~(x~)}; clearly IN = {lc~(x~) = yas~ 1 (xa)} is a well-defined meromorphic function of N. We may consider this function as giving the second coordinate for the points of N, with the exception of the fibers which pass through the zeroes and poles of S. If the divisor associated to s is Y~/k_l niPi (where ni E Z and pi E S, i = 1 . . . . . k),

then 1N has divisor S - ~ k 1 niNpi. We want a similar structure for M and its fibration ~ over S (in fact, in some neighborhood of S in M; this will be our understanding from now on). Let Gp be the fiber of ~ through the point p 6 S.

Theorem 4.1. Assume S �9 S < O. I f the embedding o f S into M is m-linearizable for some m > 2g - 2, then the divisor S - ~ - 1 ni Gpi is a principal divisor o f M.

The proof of this theorem relies heavily on some results of Complex Analysis, which we now recall. First of all, the curve S has a fundamental system of strong Levi-pseudoconvex neighborhoods (see [7], Theorem 4.9). Let us consider the following.

(1) A neighborhood U C M of S which is a strong Levi-pseudoconvex domain; OF is its structural sheaf,

(2) The subsheaf 3/[ C OF of germs of holomorphic functions on U which vanish along S.

We may state the crucial property ([5], p. 357):

Theorem 4.2. H i ( U , ./~q) = {0} fora l lq > max{0, 2g - 2}.

We may now give the proof of Theorem 4.1. Let us write the divisors S - E L 1 ni Gpi and

S - ~-~k_ 1 niNpi in the coordinate systems given by (1.3) and (1.2) as y ix~ ni = O, 1 < i < k. --nj

We define the multiplicative cocycle {yix~ ni/(yjxj ) o F/) -1 } associated to the first divisor; our aim is to prove it is multiplicatively trivial. We have then:

( - n j ) L~jl yixT"~ y i x ~ i y j x j o

_ o _ o _ on,, (y,x;-,,,),# (y,d-"')n, Since S - ~ k 1 niNpi is principal (in the surface N), the multiplicative cocycle

l is trivial: there exist nonvanishing functions {ti } such that

yix : , , , / = t i / .

It follows that

(n,) yixZ ni ti tj o Fij 1 y j x j o Lij 1

( y , x ; . n ' ) - - o - o n , ' ' ; o ' # ( y ; d - " ' ) n , '


As for the cocycle {(tj Fi~l/ t j L~jl) �9 -n~ -1 -nj o o ( ( y j x j ) o L i j / ( y j x j ) o Fi;1)}, let us define uij in order to have

t j o F i ; 1 ( y j x ; n J ) ~ 1

1 q - u i j : - - ' j ~ ( y j x ; n J ) oF i : 1

and vij := log(1 + Uij) (log 1 -- 0); clearly {vij} belongs (via the system of coordinates of M) to H 1 (U, Aim) . Theorem 4.2 implies that this additive cocycle is trivial, and consequently 1 + uij = e vij = Ci/(Cj) o F/j I for nonvanishing functions {ci}. Finally, we may write

y i x ? ni tici ( n 0 y jx j o Fij 1 ( t jc j) o Fi~ 1 '

--nj so that {yixZ n i / ( y jx j ) o Fij 1} is trivial.

5. Consequences

Let us present now some consequences of Theorem 4.1. We continue with the hypothesis S �9 S < 0 and keep the notation of the last section.

T h e o r e m 5.1. Assume the embedding o f S &to M is m-linearizable for some m > 2g - 2. Then the embedding is linearizable.

Proof . Consider a neighborhood U C M of S; a holomorphic diffeomorphism from U to a neighborhood of S in the normal bundle N will be now defined. Let g : U ) S and gN : M > S be the projections associated to the fibrations in U and N; the meromorphic functions introduced in the last section are denoted by l and lN. Take kO(p) E N as the point such that (gN (~(P) ) , IN(~P(P)) = (g(P), l (p)) ; in fact, this map is defined outside U~_ 1Gpi, the support of the divisor of the function l. In order to prove that it extends to a fiber a p i , w e take local coordinates lp and ~ N around Pi such that l o @ = IN o ~N = yx-ni and x (gt (q)) = x 0PN (qJ (q)) (this last condition comes from the fact that ~ preserves the fibrations). It follows immediately that y ( ~ ( q ) ) = y(~N(qJ(q)) . []

A short way of expressing this proof is that we have constructed a holomorphic diffeomorphism qJ that carries the 1-form ldg into lNdgN: ~*( lNdgN) = ldg. In fact, the proof consists in finding local expressions for the map qJ, which turn out to be unique; this property allows us to glue all the local diffeomorphisms.

Theorem 1.4 is a corollary to Theorem 5.1; it is enough to remark that if S. S < 2 - 2g then H~ o(l 'O)(Nm)) = {0} for all m 6 N*; consequently the embedding is m-linearizable for any m 6 N*, due to Theorem 1.8.

Finally, we remark that if the embedding is formally linearizable, it is m-linearizable for any m 6 N* (by Definition 1.5), and again we may apply Theorem 5.1 to get that the embedding is actually linearizable.

R e m a r k 5.2. There is a nice fact about Theorem 4.2 when S. S < 2 - 2g: H ~ (U, .Ad q0) = 0 for some q0 > max{0, 2g -- 2} implies H 1 (U, Adq) = O for all q > max{0, 2g - 2}.

The proof is standard. Let us consider the short exact sequence

0 > . / ~ q + l > .A'~ q > .A/lq/.A/[ q+l ) O.


It is easy to check that J~q/.A/[ q+l is isomorphic to O(N-q). By Serre's duality, we have that H 1 (S, O(N-q)) -~ N0(S , (._9 (1'0) @ Nq). But H~ O (1,~ | N q) = 0 if S . S < 2 - 2g, so that H 1 (S, O(N-q)) = 0 as well. The long exact sequence associated to the above sequence gives

�9 .. > H l (U, .A-4 q+l ) > H 1 (U,.A/[ q) > 0 .

It follows that there is a surjective map HI(U, .All q~ > HI(u, .A/~ q) for any 2 - 2g < q < q0; consequently, H I ( u , .A4q) = 0 in the case q > max{0, 2g - 2}.

6. Examples

In this Section we construct examples of fibered embeddings that are not linearizable. We use the notations of Section 3.

Consider a fibered embedding of a curve S which is m-linearizable; it is easy to prove the converse to Theorem 1.8: if r(w) r 0 for some w 6 H~ O(l'O)(Nm)), then the embedding is not (m + 1)-linearizable.

We start by selecting a non-trivial linear bundle N over a curve S such that H ~ (S, O (l'~ (N)) {0}. Fix a nonzero o9 ~ H~ O(1,~ and take some meromorphic section s of N. The

goal is to construct a fibered embedding of S into some surface which is not 2-1inearizable and which has associated normal bundle N (m = 1 was chosen just for simplicity).

Step 6.1. Let 8 denote the sheaf over S of germs of biholomorphisms of the form (x, y) > (x, g(x, y)) where x belongs to an open subset of S, y ~ C and g(x, O) = O. The elements of HI(S, s correspond to the fibered embeddings of S into surfaces. In order to see this, an element of Hi(S, s is represented (in some covering of S by discs {V~ }) as a collection {G~} of biholomorphisms of the form Ga~ (x, y~) = (x, gat~ (x, y~)), x E V,~ n Vt~ which satisfy

G ~ = Id, Gy~ = G~,~ o G ~

whenever V~ n V~ n V• 7~ 0.

Once we get these maps, we may define a surface M as the quotient of Ua Va • D under the equivalence relation (x~, ya) ~ (x~, Yt~) iff (x~, ya) E V a x II), (x#, y/~) E V/~ x D and (xc~, y~) = (x/~, ga~(x~, y/~)); the curve S is fibered embedded since Gc~#(x#, O) = O. The associated normal bundle is given by the cocycle {L~t~ = Oga~/ay# (x/3, 0)}. The surface M has automatically a system of coordinate charts such that the changes of coordinates are given by the maps in Gate.

Given {Ga/~} ~ H I ( s , S), we may construct another element of Hi(S, s in the following way: 1) we take a section s = {sa} of N, which can be supposed without zeroes or poles in the intersections Vc~ n V/~; and 2) we consider the collection of local biholomorphisms {G~g } given by

(x~, ya) = Gc~#(x~, y~) = (x~, g,c~(x~, y~)) = (x~, s~ l gc~(x~, s#y~)) .

It is easy to see that {Ga/~} E H l (S, g); but the associated normal bundle L to {Ga~} changes to L | N*. In particular, applying this construction to elements of s which have fixed associated normal bundle N produces embeddings with trivial normal bundles.

We have therefore a way of replacing an embedding of S into a surface M with associated normal bundle N by a different embedding of S into a surface/s with associated trivial normal bundle, and vice-versa.


Step 6.2. Suppose now that we have a holomorphic surface ,Q (where S is fibered embedded with trivial normal bundle) which is not 2-1inearizable. There exists in S a 1-form cr 6 0 d'~ such that T~es(cr) ~ 0; we choose the section s as the quotient og/cr. The covering {Va} for kl can be selected in order that no zeroes or poles of s appear in the intersections Va M V~. Let the

change of coordinates between coordinate charts of M be

(rn) m ~'u/3(x, y/3) = x, y~ + ~[~ (x)y~ . m = 2

We write as {Au~} the transition functions for N. We put ~ = {~u} and w = {~ou} and define M as in Step 6.1 to have change of coordinates

Fu~(x, y~) = Ac~(x)y~ § ~ ~]c~fl(m) (x)y~m m----2

where AuTs~-I O(mfl ) r(m) = sue �9

In particular, Res (w) -1 (2) T~es({A~A (2) (2)0. = TCes({aa~oal~w~} ) = r/,~/~s~tr~}) = ~es({(~/3 /3}) = 7~es(cr) # O, and M is not 2-1inearizable.

Step 6.3. At this point we have just to exhibit examples of fibered embeddings of S with associated trivial normal bundle which are not 2-1inearizable. This is straightforward. We take a suspension of a special representation of the fundamental group of S into Diff0 (C), the group of germs of holomorphic diffeomorphisms of C which fix 0 c C. Suppose the fundamental group is presented with the generators al, bl . . . . . ag, bg (where g E N is the genus of S) and the relation

al .bl .a ~ 1 .b l 1 . . . ag .bg.a g l .b g 1 = 1. We represent al by the local holomorphic diffeomorphism oo n l(t) = t § Zm=2 lnt , a2 by another local diffeomorphism which commutes with l(t), and the

remaining generators by the identity map. Let ,Q be the suspension of such a representation; then M is not 2-1inearizable if 12 7 k O, since this property persists under holomorphic changes of coordinates.

R e m a r k 6.4. Examples of linear bundles over compact Riemann surfaces such that the space of holomorphic 1-forms (with coefficients in the bundle) has positive dimension can be found in [6], p. 111. For example, if the Chern class c of the bundle is positive, the dimension is c § (2g - 1), where g ~ N is the genus of the curve. If c = 2 - 2g then the dimension is 1 when the bundle is the anticanonical one.

7. Appendix: The index theorem for foliations

Let us take a fibered embedding of S into M as before (the notation is taken from Sections 1 and 2). Consider in M a holomorphic line field I. In coordinates, one has a collection of meromorphic functions { I~ (x~, ya) } such that the line field is defined as dyu - Iu (xa, y~) dx~ = 0; we assume that there are no poles in the intersections Uu f3 U#. The compatibility condition when ~ n b'/3 r

lu(xa, Yu) dxa = U~(x~ , y/3)I~(x~, y~) dx~ + Vu~(x/3, y~) dx~ , (7.1)

where U ~ = OOc~/Oy~ and Va/~ = OO~#/Ox~. The function Iv can be regarded as the slope of the line field in the coordinates (x,~, yu). The restriction of the condition above to S gives

Iu(xu, O) dx~ = A~(x/3)I~(x[~, O) dxl~ .

Fibered Neighborhoods of Curves in Smfaces 65

The line field along S defines a meromorphic slope 1-form

Is = { lu (xu, O) dxu }

with coefficients in N.

0 to both sides of (7.1) and restrict to S. We obtain: Let us apply the derivation 0-~

O Iu/Oyc,(xa, O) dxu - O l~/Oy~(x~, O) dx# (7.2) -1 r

= 2Cc~ ( x ~ ) A ~ (x~)l~ (x~, O) dx~ + A~# (x#)Au~ (x~) dx~ .

T h e o r e m 7.1. Suppose the embedding o f S is 2-1inearizable. Then

E ~es(OIc~/Oya(xc~, O) dxu) = S . S O

Ol

in any of the following situations:

(1) The slope 1-form Is is holomorphic.

(2) The coordinate system is adapted (see Del~nition 1.5).

P r o o f 1) Let us suppose that the slope is holomorphic. From Lemma 1.6 we know that -1 {0~ = C ~ Aa# I~ (x~, O) dx~ } belongs to H 1 (/4, O (1,~ we may write then (as in Definition 1.7)

0~ - 0~ = 0a~ and therefore r(Is) = ~-~ Tr We claim that the forms 0,~ may be chosen as holomorphic 1-forms.

In fact, let us replace the coordinates {y~} by the coordinates {z~} given as

za = Yc~ + Bc~y~

where {Ba } are holomorphic functions to be chosen. The changes of coordinates of the embedding are now given as

2 z~ = A,#zfl + ( C ~ - Au~B/3 + aaflB~) z2~ + . . . .

Since the embedding is 2-1inearizable, one can choose the functions {Ba} in order to have A ~ Cu/~ ----- B/~ - A ~ Bu, or

A~-/Cc#~ I/~ (x/~, 0) dx~ = Bt~ I# (x/~, 0) dx~ - Au# Ba Iu (xa, 0) dx= .

It follows that r(Is) = 0. From [6], p. 102 and (7.2) we get that

S. S = Z ~es(OI~/Oy~(x~, O) dxa) - r(Is) Ol

2) In the case of an adapted coordinate system we have already that C~/~ = 0. The proof follows immediately from (7.2). [ ]

The numbers Res (0 Io/Oy,~ (x~, O) dx,~) are the indices for the line field introduced in [3]. We observe that Theorem 7.1 holds true without the hypothesis on the embedding if S is invariant for the line field. The case of foliations on line bundles was treated in [4].

66 C. Camacho, H. Movasati, and P. Sad

References

[1] Arnold, V. Chapitres Suppl~mentaires de la Th~orie des t~,quations DiffgrentieUes Ordinaires, I~ditions Mir, Chapitre 5, (1980).

[2] Arnold, V. Bifurcations of invariant manifolds of differential equations and normal forms in neighborhoods of elliptic curves, Funct. Anal. Appl., 10-4, 249-259, (1977).

[3] Camacho, C. and Sad, P. Invariant varieties through singularities of vector fields, Annals of Math., 115, 579-595, (1982).

[4] Camacho, C. Dicritical singularities ofholomorphic vector fields, Contemporary Mathematics, 269, 39-46, (2001).

[5] Grauert, H. Uber modifikationen und exzeptionelle analytische Mengen, Math. Annalen., 146, 331-368, (1962).

[6] Gunning, R. Lectures on Riemann Surfaces, Princeton University Press, 1966.

[7] Laufer, H. Normal Two-Dimensional Singularities, Princeton University Press, 1971.

Received October 12, 2002

Instituto de Matem~itica Pura e Aplicada, Estrada Dona Castorina, 110. Jardim Bo~nico 22460-320, Rio de Janeiro, Brasil

e-mail: [email protected]

Instituto de Matem~ttica Pura e Aplicada, Estrada Dona Castorina, 110. Jardim Bot~nico 22 460-320, Rio de Janeiro, Brasil

e-mail: hossein @ mpim-bonn.mpg.de

Instituto de Matem~itica Pura e Aplicada, Estrada Dona Castorina, 110. Jardim Botfinico 22 460-320, Rio de Janeiro, Brasil e-mail: [email protected]

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