On complex vector bundles on projective threefolds

Invent. math. 88, 427-438 (1987)

I~ vsYltio~ls$

mathematicae 9 Springer-Verlag 1987

On complex vector bundles on projective threefolds Constantiu BSnic~ and Mihai Putinar Department of Mathematics-INCREST, B-dul P~cii 220, Bucharest, Romania

Introduction

One of the first questions which arises in the study of (complex) vector bundles on complex projective manifolds is to determine those topological bundles which admit algebraic, or equivalently, holomorphic structures. In the case of curves, every topological vector bundle has an algebraic structure. Schwarzenberger [15] has shown for surfaces that a topological vector bundle admits an algebraic structure if and only if its first Chern class is algebraic. From the papers of Atiyah-Rees [3], Horrocks [10] and Vogelaar [18] it follows that every topological vector bundle on the projective space IP 3 has an algebraic structure. The same conclusion holds on a homogeneous rational threefold, see [5] and [6]. What about the general case of (complex, smooth) projective threefolds? One can expect that in this case a topological vector bundle admits an algebraic structure if and only if its Chern classes are algebraic. The "only if" part is clear, and for the proof of the "if" part only rank 2 and 3 vector bundles should taken into account. First, the topological classification is needed. This is discussed in the second section of the present paper for any compact connected three-dimensional complex manifold. The rank 3 bundles are parametrized, via Chern classes, by the cohomology classes (cl,Cz,C3) subject to an integrality condition; the rank 2 vector bundles are determined, roughly speaking, by the Chern classes (Cl, C2) and by two other "finite" invariants fl(c 1, c2) and ~(c 1, c 2, fl(Cl, c2))' see Theorem 1 for the precise statement. In the third section we answer into affirmative the above question in stable rank, i.e. >3, by proving the following: Theorem 2. A topological vector bundle of rank 3 on a projective 3-fold admits an algebraic structure if and only if its Chern classes are algebraic.

The remaining case of rank 2 vector bundles is treated in the fourth section, where we prove a weaker result:

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C. B~mic~tand M. Putinar

Theorem 3. For every rank 2 topological vector bundle on a projective 3-fold X with algebraic Chern classes (cl, c2) there exists an algebraic vector bundle with the same Chern classes. Moreover, if c 1 = cl(Kx) , then this algebraic bundle can be chosen to be K xsymplectic, with a prescribed arbitrary value in 7],, 2 for the Atiyah-Rees invariant.

Recall that if E is a holomorphic Kx-symplectic bundle, then the (analytical) Atiyah-Rees invariant [3] is ~(E)=h~

(mod 2).

In Theorems 2 and 3 above one constructs algebraic bundles by the method of extensions [16], in the case of rank 3 bundles by a variant given in [5]. The fact that 1-dimensional cycles are smoothable and that they are very movable, due to the residual intersection device, plays an important role in these constructions. Non-reduced curves (of smooth support) are also intensively used. Accordingly, a key point in the proof of Theorem 3 is played by a method of "tripling" curves, due to Forster and the first author [-4]: it was very interesting to us to see how the triple structures are connected with the integrality condition (c 1 + c l ( X ) ) c 2 - 0 (mod 2) and how they relate the AtiyahRees invariant with 0-characteristics. In a final section we combine the previous results in order to prove the next: Theorem 4. Every topological vector bundle an a projective rational threefold admits algebraic structures.

In this case an essential step in the proof is the fact that the analytical Atiyah-Rees invariant for Kx-symplectic bundles corresponds to the topological ones [3]. We were not able to prove the result for rank 2 vector bundles in its full generality, mainly due to the fact that we failed to understand the intricate question whether the second topological invariant fl has an analytic counterpart. The first section contains preliminaries concerning extensions, double and triple structures, cycles of dimension 1 and a statement derived from the Atiyah-Hirzebruch spectral sequence in K-theory. We adopt the following conventions. A 3-fold means in the sequel a complex, connected, smooth, projective manifold of dimension 3. We abbreviate 1.c.i. for locally complete intersection. We simply say bundle for a topological complex vector bundle, and if the context is clear, for an algebraic vector bundle. Sometimes we denote by the same letter an integer cohomology class as well as its image by the reduction 7/-~7/2.

1. Preliminaries

1.1. Let us first recall the construction of rank 2 bundles by extensions ([16], [14] Ch. I. w5). Denote by X a 3-fold, Y a 1.c.i. curve in X and by L a, L 2 line bundles on X. Assume that det(Nr)~-L-~l| and that HZ(X, LI|

On complex vector bundles on projective threefolds

429

Then there exists a b u n d l e E of r a n k 2, given by an extension

O --. LI -. E -~ ~gyL2 --. O. O n e has d e t ( E ) ~ - L l | true:

a n d the next formulas for the Chern classes are C 1 (E)

= c I (L 0 + c 1(L2) ,

c 2 (E) =

c 1

( g l ) c 1(L2) -t- [ Y].

W e shall consider for o u r p u r p o s e s L~ to be sufficiently positive a n d L 2 to be sufficiently negative. T h e n the c o n d i t i o n H2(X, LI| is satisfied, and, we notice that in this case the equality

h~

+ h2(E)=h~

+ hl(L2l Y)

holds, a r e l a t i o n which will be needed in the c o m p u t a t i o n of the A t i y a h - R e e s invariant. D u e to the c o n d i t i o n on det, the difficulty in p e r f o r m i n g this construction lies in finding the right curves: we shall consider convenient disjoint unions of d o u b l e and triple structures. In o r d e r to o b t a i n bundles of r a n k 3 we need the following variant of the a b o v e c o n s t r u c t i o n [5]. Let Y c X be as above, let F be a r a n k 2 bundle on X a n d L be a line b u n d l e on X. Assume that H2(X,F)=O a n d that d e t N r | (regarded as a 2 bundle on Y) has a nowhere vanishing global section. T h e n there exists a bundle E of r a n k 3 on X, given by an extension 0~F~E~Jr~0. The C h e r n classes of E are given by the f o r m u l a s : c 1(E) = c 1(F), c2(E ) = c2(F ) -t- [Y], c 3 (E) = (c 1(F) + c, (X)) [ Y ] - 2Z(Cr). 1.2. F e r r a n d ' s m e t h o d of " d o u b l i n g " a curve is as follows, [7]. Again X is a 3fold and Y c X is a 1.c.i. curve. A s s u m e that one has an e p i m o r p h i s m N* =,,r where L is a line b u n d l e on Y. C o n s i d e r the closed subscheme Z of X given by the ideal J z = ker (Yy ~ Y y / Y ] ~

L).

Then Z is a 1.c.i. Z has the s a m e s u p p o r t as Y, and (det Nz) [ Y = ( d e t N r ) | A t the level of c o h o m o l o g y classes one has [ Z ] = 2 [ Y ] . The exact sequence

O--,

(;y--,O

gives rise to the exact sequence H 1(Y, L)-~ Pic Z ~ Pic Y--* 1.

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c. BSnic~ and M. Putinar

Notice also that, if L is sufficiently positive, then there exist such epimorphisms e. In that case, P i c Z ~ Pic Y. Next we briefly recall the "tripling" method [4]. One starts with a doubled curve as above. The inclusions of ideals yield the exact sequence o--,~/,.~

s~--,,~/,~~z--'~;~/~--'~/,;~~

o,

and the isomorphisms J r / J z ~ L,

"r162 J z = L2,

J z / J r ,& ~ N* I Y.

Using these, one finds an exact sequence of bundles on Y:

0 --+L2~N]lY~(det N*)|

-I --+0.

Suppose that this sequence splits and let z be a retract of i. Take the subscheme T of X associated to the ideal ~

=

ker (Jz ~ J z / J r J z

~' L 2 )

9

Then T is a 1.c.i. curve with the same support as Y, verifying IT] = 3 [ Y ] and det Nrl Y= (det Ny)| 2. The exact sequences 0--.L ~ Oz ~6)r--.0,

0~L2 ~6'r~(~z~0

allow to connect Pic T to Pic Y. They also are needed in the computation of the Atiyah-Rees invariant. The obstruction of the above splitting lies in Hi(Y, det Nr | hence, if L is sufficiently positive, the above construction may be performed. As we have seen, by doubling and tripling procedures one can vary the cohomology classes of curves. More important is to remark that, due to the degree of freedom allowed by the choice of the line bundle L, one can also vary the determinant of the normal bundle. 1.3. We shall need the residual intersection device and the smoothness of 1dimensional cycles, see ([9], w7) for an account of these topics. We explain here for later use the meaning of the statement that, on 3-folds, the (smooth) 1cycles are very movable. We rely on the following form of Bertini's theorem: if L is a line bundle on a quasi-projective manifold X and V is a finite dimensional subspace of H~ L) which generates L, then for s general in V the scheme of zeroes of s is non-singular, and connected if dim X > 2. Let Y be a non-singular curve in Y, not necessarily connected. Take k > 0 , so that J r | k is globally generated. Then the general section s in H~ Jr| k) defines a non-singular, connected surface H~ passing through Y (one uses Bertini's theorem on X \ Y and the fact that the image in H~174 k] Y) of the general section has no zeroes). Now, if one considers another general section t in H~ J r| then H t is still non-singular and connected, and the ideal-theoretically equality H ~ H , = Y w Z holds, where Z is a second non-singular curve (we choose t with the additional condition that


431

it smoothly vanishes in the quotient bundle of N* | by the sub-bundle generated by s; the points of Yc~Z are exactly these zeroes). Moreover, we can choose s and t such that H s and H t avoid a given finite subset of X \ Y and Z avoid a given finite subset of Y. In particular Z may avoid a given curve which has no c o m m o n irreducible c o m p o n e n t with Y. Let denote h=cl(L ). One has [Hs]=[Ht]=k. h and consequently k2h 2

= [ Y] + [z]. Notice also that if H ' and H" are general in IH~ then H':~H" is a smooth connected curve and hZ=[H'c~H"]. Here one can again choose a representative curve for h 2, which is disjoint of a given curve. Let assume now that r is a 1-dimensional cycle on X. By a result of H i r o n a k a and Kleiman, ~ is smoothable in its rational equivalence class. In particular, passing to the associated c o h o m o l o g y classes in H4(X, 71) one gets k

[ 4 ] = Y, ~[~3,

n~71,

i=I

where Y/ are s m o o t h and connected curves. As we have already seen one can write [Y1]=kZlh2-[Z1] with kl>>0 and Z 1 disjoint of Yz, Y3 ..... Yk" By repeating this procedure we can finally assume that ~ may be expressed as above, with new curves and coefficients, such that the curves Y~ are pairwise disjoint. This fact, and some information on the parity of n i, is what we shall use in the sequel. 1.4. Let Y be a finite polyhedron and let dzk+l denote the (possible) non-trivial coboundaries in the Atiyah-Hirzebruch spectral sequence for K(Y), associated to the simplicial decomposition of Y. The following assertion is valid ([2] or [8], T h e o r e m I in Chap. VIII): Fix an integer q > l . If ~eHeq(Y, 71) satisfies dzk+l~=0 for all k > l , then there exists a class ~ in K(Y), such that r is trivial on the skeleton yZo-1, and cq(~) = (q - 1)! c~.

2. Toward the topological classification 2.1. Proposition. Let Y be a finite polyhedron of dimension less or equal to 7 and assume HV (Y, 7Z) has no 2-torsion. The cohomology classes (cl, c2, c3) are the Chern classes of a rank 3 vector bundle on Y iff

c3:~(Cl +SqZ)c2

in

H 6 ( y , 7Z2).

Proof The necessity of the condition follows by pulling-back a formula of Wu, see [13], which asserts that the Stiefel-Whitney classes of the tautological bundle on BU(3) satisfy W6 = (W 2 -[- S q2) w4"

Conversely, assume that the congruence condition holds for a triple (c 1, c2, c3). Every class ~eK(Y) decomposes into ~ = [ E ] + k [ l l ] , where E is a r a n k 3 bundle on Y, k e Z and 11 the trivial line bundle, see [11]. Consequently, it suffices to find a ~eK(Y), with ci(~) =c i, i= 1, 2, 3.

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C. B~nic5 and M. P u t i n a r

First, we choose a line bundle L with c a ( L ) = q , and we put ~1=[L1, so that Ca(~l)=Cl, c 2 ( ~ 0 = 0 and c3(~0=0. The only possible non-trivial coboundary in the Atiyah-Hirzebruch spectral sequence, starting at E r4, q, is d3: H4(Y, T Z ) ~ H 7 ( y , I ) . As the values of d 3 lie in the 2-torsion part of HV(y, 7/) one gets d3=0. Therefore there exists by 1.4 a class ~2~K(Y), such that ca(~2) ~-0, ez(~2)=e2; let us denote C3(~2)=T. Similarly one finds ~36K(Y) with Ca(~3)=0 , C2(~3)=0 and e3(~3)=2o, where cr is a given, arbitrary, class in H6(y,z). Take ( = ~ a + ~ 2 + ~ 3 . Then c1(r c2(~)=c2, c3(~)=cac2+z+2a. Since r=-Sq2c2 and c3=-CaC2+Sq2c2 in H6(X, Z2), one can find a cohomology class a such that C3(~)=C 3. 2.2. Theorem 1. Let X be a compact connected 3-dimensional complex manifold. (1) The cohomology classes (q,e2,c3) are the Chern classes of a rank 3 topological vector bundle on X iff (c I + c a ( X ) ) e 2 - c 3 (rood2). Moreover, the triple (Cl, c2, c3) determines the bundle up to an isomorphism. (2) The pair (cl, c2) corresponds to the Chern classes of a rank 2 topological vector bundle on X iff (c 1 +cl (X))c 2 =-0 (mod 2). Moreover, VecttZop(X) ~ {(q, c2,/~(ca, c2), ~(ct, e2, P(ca, c2)))},

where (ca, c2) are as above, ~ is an arbitrary element in the quotient H 5(X, Z2)/(c I + c a (X)) H 3(X, Z) + c 2 H a (X, 7~),

and ~ is an arbitrary element in some quotient of

H6(X, 712)/(c 1 -t- e 1 (X))Ha(X, Z). (Notice the abuse of notation in writing the denominators.) For (1) see [61, Theorem l, or the above proposition. The statement (2) follows by analysing the first two steps of the Postnikov tower of the map B U ( 2 ) ~ B U , as described in [171. For the first step E7---~BU one uses [12], namely Formula 1.5 written for Chern classes, and further one applies the above proposition to the space Y - S X , the suspension of X. By remarking that the cup product H e x H4--+H 6 is trivial on SX, one finds that the set of homotopy classes of liftings X ~ E 7 of a given class ~e[X, BU], with c3(~)=0, is an affine space over HS(X, ~2)/A(~), where A(~) is the range of the map

Ha(X, Z) • H3(X, 7Z) --. H s ( x , 7/.2), defined by (z 1, "C3)V"~'C2(~)'Ca +(q(~)+Sq2)r3 9 Furthermore IX, B U(2)] ~ [ X , Esl, and the enumeration of the liftings relative to E 8 ---~E7 is done in [17], p. 101. All these facts imply (2). It turns out that the right interpretation of the parametrization in (2) is given by actions of the groups H6(X, TZ2) and HS(X, Z2) on the fibres in the decomposition [X, B U(2)] - [X, Es] ~ IX, E73 ~ IX, BU].


433

The actions are transitive and those corresponding to the second arrow have precise isotropy groups. We do not know the isotropy groups for the first arrow, or equivalently, whether the parameter space for ct is exactly

H 6 ( X , 712)/(c 1 + c I (X))H*(X, Z). The question is related, via Kx-symplectic K-theory [3], to the problem of lifting (up to h o m o t o p y ) the determinants of automorphisms of the bundles, from the skeleton X 5 to X (a problem which "lies" in H ~ ( X , Z ) , in fact in H 1(X, Z2) ). Let us assume in addition that H ~ ( X , Z ) = 0 . Then one has no 3 in the parametrization of rank 2 bundles, and, if cl ~ c~ (X) (mod 2), then c~ disappears too. In the Kx-symplectic case, i.e. when c 1 - c I(X) (mod 2), the space of values for ~ is precisely H 6 ( X , ~ 2 ) = 7 ] 2 , see [6] or the above remark. As we have already mentioned in the introduction, in this case the analytical invariant c~ distinguishes the topological type; more precisely, if E~, E z are holomorphic Kx-symplectic bundles with e(E1)+c~(Ez), then E 1 is not topologically isomorphic with E2, see [-3]. Finally, let us remark that the statements (1) and (2) in the theorem are valid (replacing q ( X ) by Sq 2) for an arbitrary finite polyhedron X of dimension less or equal to 6 and without 2-torsion in H6(X, Z).

3. The proof of Theorem 2 3.1. We follow the general framework of [5]. We have to prove that if c 1, c 2, c a are algebraic c o h o m o l o g y classes, satisfying (c 1 + c l(X))c z - c a (mod 2), then there exists a rank 3 algebraic bundle with these Chern classes. It is useful to write d o w n the transformation formulas for the Chern classes of a rank 3 bundle G, tensorized with a line bundle P : C1 (G @ P ) = c 1 (G) -[- 3 c 1 (P),

c 2 (G @ P) = c 2 (G) + 2c 1(G) c 1(P) + 3 c 1(p)2, c 3 (G | P) = c 3 (G) + c 2 (G) c 1(P) + 171(G) c 1(p)2 + cl (p)3. We can tensorize the topological bundle having Chern classes (ca,c2,ca) with a sufficiently positive algebraic line bundle, such that c I is the Chern class of an ample line bundle. By tensorizing again with a sufficiently high power of this line bundle, and by using 1.3, we can assume in addition that c z = ~ k i [ Y i ] , where ki>O and Y/are pairwise disjoint s m o o t h curves. The ample line bundle L appearing in the sequel satisfies c l ( L ) = c 1. Put h = q . Let us consider general elements H 1 and H 2 in IH~ L)I and denote C = H 1 t~H2, so that [ C ] = h 2. By the exact sequence 0 -~ C ( - H a - H 2 ) ~ C ( - H 1 ) O

C(-Hz) -*Jc ~0,

(1)

we get N c ~ C ( H O O C ( H 2 ) ] Y = 2 L I Y . Accordingly, for every k > 0 , one finds epimorphisms Jc/Jc2 -, det Nc | L2k- ' [ C --*O,

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a n d c o n s e q u e n t l y one can use F e r r a n d ' s m e t h o d . Since Hi(C, det N c @ g 2 k - l l C ) = 0 for large values of k (but i n d e p e n d e n t of the r e a l i z a t i o n C of h2), one gets 1.c.i. curves C' s u p p o r t e d by C, so that [C']=2[C] a n d detNc,~-LI-2klC '. Using this fact a n d the m e t h o d of Serre, one can find integers - a n d we fix such a value, say k 0 - so that the next s t a t e m e n t holds: for an a r b i t r a r y disjoint u n i o n Y of such d o u b l e structures (with k = k o ) there exists an extension O-+Lk~ ---rJ r L1 - k~---~0, (2) with F locally free. W e shall consider only curves Y which are disjoint of the curves Y/which a p p e a r in the expression of c 2. O n e has c 1 (F) = c I (L) = h = q , c 2 (F) = (k o (1 - ko) + 2d) h 2, where d is the n u m b e r of d o u b l e curves c o n s i d e r e d in Y. The integer d__> 1 is a r b i t r a r y a n d it will be c o n v e n i e n t l y choosen at the end of the proof. 3.2. Start with an F as above. W e seek for integers t>>0 a n d curves Z such that there exist extensions of the form

O-* F @L3'--* E@ L2t-+ J z ~ 0 ,

(3)

with E locally free of r a n k 3, a n d such that the C h e r n classes of E are the given q , c2, c 3. A c c o r d i n g to the c o n d i t i o n s in Serre's c o n s t r u c t i o n one has to verify firstly that H 2(X, F | L3t) = 0 a n d that det N z | F | L3' has a section without zeroes on Z. There is a to, such that for every t > t o one has H2(X,F| for every F as above (use (1), the exact sequence 0 --* Lk~ 1 | det N c ~ COc, ~ (tic --* 0, a n d (2)). If we can find an extension (3) with E locally free, then c 1( E ) = c 1( F ) = q . F o r the second C h e r n class o n e finds

c2(E @ L2t)=c2(F @ L3t)-I- I-Z]. As we claim that c2(E)=Cz=~_ki[Yi], one derives that Z has the associated c o h o m o l o g y class [ Z ] = ~ ki[Yi] + (t(3t + l ) - k o ( k o - 1 ) - 2d)h 2. T a k e t>=to such t h a t t ( 3 t + l ) - k o ( k o - l ) - 2 d > l . Let us explain how such curves Z m a y be chosen so t h a t they satisfy for large values of t the c o n d i t i o n on det N z @ F | 3t, keeping in m i n d that we have to take care also of c 3. Before d o i n g this, we do the following observation. Let C be a n o n - s i n g u l a r curve, C = H lr~H2, as above. If P ' is a line b u n d l e a n d if the degree of P' exceeds a suitable integer, which d e p e n d s only on c~ (X) a n d h, then there exist e p i m o r p h i s m s J c / J 2 ~ P ' ~ O . Let us fix such a degree. A n e p i m o r p h i s m as a b o v e d e t e r m i n e s a d o u b l e structure C', a n d let us define e=Z((2c, ). F o r the sake of simplicity we call such a curve an e-curve. If e p i m o r p h i s m s J c / J c 2 --* P"--* 0 are t a k e n into account, with deg P " = deg P ' + 1, then we o b t a i n curves C' with z((gc,)=e+ 1. T h e y are s i m p l y called ( e + 1)-curves.


435

Let us come back to the construction of Z. We can assume that ~ki[Y~] = [T], where T is a 1.c.i. curve supported by the union of Y~ (choose smooth surfaces S i which contain correspondingly Y~, then replace Yi by its (k i - 1)infinitesimal n e i g h b o u r h o o d in Si, and take the union). Fix such a T. Take the arbitrary decomposition t(3t+l)-ko(ko-1)-2d=2a+2fl,

a > 0 , fl_>-0.

(4)

The desired curve Z will be a disjoint union of T with a copies of curves of type e and f copies of type e + 1. Moreover, we may assume that Z is disjoint of the curve Y which appears in (2). There is an integer t~ such that for every t>t~ the bundle d e t N z | 1 7 4 3t possesses a nonvanishing section: for this purpose restrict (2) to any c o m p o n e n t of Z and so on .... All in all, we can find an integer t 2, so that for t > t 2 there exist extensions (3) with E locally free and Cl(E)=cl, c 2 ( E ) = c 2 , for every curve Z as above. The integer t 2 depends only on c l ( X ), h = c 1, ko, the decomposition c 2 = ~ k ~ E Y ~ J = [ T ] and degP'. 3.3. Let us take an E as above. By the variation of a and f in (4) we would like to obtain c3(E)=c 3. One has: c 3 (E | L2t) = (c I (F | L 3t) + c 1(X)) [ Z ]

--

2Z(Cz).

hence 8h 3t 3 + 4 h 3t 2 + 2 h c z t + c 3 ( E ) = (6h t + h + c 1(X))(c z + 2(a + f ) h 2) - 2(Z(6'T) + ae + f ( e + 1)). By requiring c 3 ( E ) = c 3, one finds an equation in a and f, which must be solved. D u e to the integrality condition ( h - l - c l ( X ) ) c 2 - c 3 (mod2), we can divide by 2 and write this equation as P(t) = aA(t) + f ( A ( t ) - 1), where A ( t ) = 6 h 3 t + h 3 + h 2 c l ( X ) - ~ and P ( t ) = 4 h 3 t 3 + lower terms (which depend only on cl(X), c 1, c 2, c 3 and the fixed curve T). F r o m now on, we reverse the way of the argument. Thus, A(t) and A ( t ) - 1 are relatively prime numbers, hence for t>>0 it is possible to find a decomposition P(t) = a,A(t) + fl,(A(t) - 1), where at, flit are positive integers. Since P (t) = (a t + flit)A (t) - B (t), we asymptotically find a t + f t ~ t 2. Thus t ( 3 t + 1 ) - k o ( k o - 1 ) - 2 ( a + f ) ~ t 2 for t ~ 0 . Consequently, there exist integers t > t 2 , sufficiently large such that P(t) can be expressed as above and the n u m b e r t ( 3 t + l ) - k o ( k o - 1 ) - 2 ( a t + f l t ) is positive. Take d to be half of this number. Choose Y, construct F, then construct Z associated to a t and flit and finally one obtains the desired bundle E.

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4. The proof of Theorem 3 4.1. Let us consider a topological bundle F of rank 2, with algebraic Chern classes c a and c z. We want to construct an algebraic bundle E of rank 2 and with the same Chern classes. We begin as in 3.1. By successive convenient twistings of E with algebraic line bundles, we can assume (and we do) that c a = c l ( L ), where L is an ample line bundle, and c2=2ni[Yi] , with ni>3 and Y/ disjoint connected s m o o t h curves. Let us denote h = q ( L ) . Let Y~I,.... Y~r be those components whose coefficient n i is odd. We choose an even integer m >>0, so that m 2 h 2 = [ Y J + . . . + [ Y i r ] + [ U ] , where U is smooth and disjoint of the other components, see 1.3. We also choose an even integer n>>0, such that n2h 2 =[U]+[Z], with Z s m o o t h and disjoint of all Y/. Accordingly, [ Y J + . . . + [ Y j = ( m 2 -n2)h2+[Z]. Further we tensorize once more F with Lq. One gets c2(F| 2. |f we take q>>0, then the new bundle F | q has c 2 = ~ 2 n i [ Y i ] + 3 [ Z ], with new coefficients and new components Yi, where n~>0, Y~and Z are smooth, connected and pairwisely disjoint. The new c I is still the first Chern class of an ample line bundle; we keep the same notation L for it, and h for c~ (L). After these normalizations of the initial topological data, we can look for extensions. Let us choose, as in w3, an 1.c.i. curve Y supported by the union of the curves 1I// and satisfying [ Y ] = ~ n i [ Y i ] ; it follows that c 2 =2[Y3 +3[Z]. We seek for integers k and for 1.c.i. curves T, which give rise to extensions

O--. Lk ~ E ~ JrLa-k ~O, with E locally free. One has c l ( E ) = c l ( L ) = h = c I and c 2 ( E ) = k ( 1 - k ) h 2 + I T ] . If we show that for k>>0, one can choose T so that there exist extensions with k ( 1 - k ) h 2 + [ T ] = c 2, then the p r o o f will be finished. We take T to be a disjoint union of a convenient double structure Y' on Y with a triple structure Z" on Z and with 89 double structure C' supported on pairwise disjoint curves of the form C=H'c~H", where H ' and H" are general elements of IH~ L)t. First of all, notice that H2(X, LZk-1)=0 for k>>0. We have to accomplish Serre's condition: det N T ~ L I - Z k I T. For an arbitrary 1.c.i. curve S(e.g., for Y or for the previously described curves C) one finds, for large values of k, epimorphisms

3~s/Js2 -, det Ns @ L2k- l -. O,

hence double structures S' with detNs, lS~-L1 2kls. Since for k>>0 the restriction m a p P i c S ' ~ P i c S is bijective, one obtains exactly the desired relation detNs,~-La-2klS'. It remains to prove that (again for k>>0) there are triple structures Z " on Z with detNz,,~-La-2klZ ''. Because detNz~(~z| [ Z ] ---c 2 (mod 2) and deg coz - 0 (mod 2), we get that the degree of det N z | L2k- a ]Z has the same parity as (c1+c1(X))c2, therefore it is even by the integrality condition. Consequently, for every k, one finds a line bundle Pk on Z, so that Pk2 ~ d e t N z | 2k- a IZ. There exists an integer k 0 with the following property: if P is a line bundle on Z and degP__>ko, then there exists an epimorphism J z / J Z - ~ P ~ O and a retract J z , / J z J z , ~ , P 2 for the inclusion O ~ P 2 ~ Jz,/oCzJz,, where Z' is the associated double structure. This retract defines a triple structure Z" on Z. By enlarging ko, if necessary, we have isomorphisms


437

and consequently detNz,,~--L1-2ktz". P = Pk for k >>0 and thus we get the needed curve Z".

PicZ'~PicZ'~PicZ

We

take

4.2. Let F be a rank 2 topological bundle with Chern classes q , c 2, and in addition, Cl-Cl(X)(mod2 ). As described above, we may assume after a normalization that c 1 = c 1(L), with L ample on X and c 2 = ~ 2n i [Y//] + 3 [Z], where ni>O, and Y// and Z are connected, pairwise disjoint smooth curves. By the exponential exact sequence, the condition c~ + c~ ( X ) - 0 ( m o d 2) implies the existence of an algebraic line bundle A on X, with the property L | Kx ~ ~ A 2. We have already seen that extensions like

O ~ Lk ~ E ~ JTLI-k ~ o give rise to algebraic bundles E of rank 2 with prescribed Chern classes ct and c 2. Moreover, det E = L, hence for E . . . . : = E | A - 1 one gets det (E . . . . )-~ Kx. We shall show how to vary the extension parameters in order to obtain for the invariant ~(E . . . . ) = h ~ 1 7 4 1 7 4 (mod 2) both values 0 and 1. This will conclude the proof of the Kx-symplectic part of Theorem 3. Let us come back to the construction presented in 4.1. The triple structure Z" on Z involves an integer k>>0 and an epimorphism J z / ~ d --.P ~ 0 , where P is a line bundle on Z with P2---det Nz| Using the extension, the exact sequences

O~P~(gz,~Cz~O

,

O--.PZ~6z,,~Cz,~O

and choosing a sufficiently high value of k (which depends only on the given data Y, Z and A) in order to have vanishing terms in computations, one gets

h~176 ) =h~ @ A - ~) + hl (Ll-k | A - ~l Y') + 89k(k--1)h' (Lt-k | A - t I C') +hl(Ll-k@A '[Z)+hl(P| ). Let us denote Q = P | 1 7 4 Q2 ~ d e t N z | 1 7 4

We have 2)iZ ~det

Nz|

~-Kz,

hence Q is a 0-characteristic on Z. Now, we modify the curve T by changing the triple structure on Z. more precisely we start with an epimorphism J z / J 2 ~ P | where R2~-Cz. If T is the new curve and /~ is the new bundle, then c~(E. . . . )--CX(/~ . . . . ) = h l ( Z , Q ) - h ~ ( Z , Q | 9 Because h 1 (rood2) takes both values 0 and 1 when the 0-characteristic changes (see [13 for the precise result), we can modify the value c~(E. . . . ), and the proof is complete. 5. The case of rational manifolds

We prove Theorem 4. Let X be a complex projective, smooth rational threefold. Since X is simply connected, H I ( X , Z ) = H s ( X , Z ) = O and H z ( x , z ) has no torsion. By ([6], th. 1, or w above) the topological classification of rank 2 bundles is given by the Chern classes and, when q + q ( X ) = O (rood2) (in

438

c. B'Snic~tand M. Putinar

w h i c h case we can n o r m a l i z e so t h a t c 1 = cl(Kx) ), by the A t i y a h - R e e s i n v a r i a n t . M o r e o v e r , the a n a l y t i c a l i n v a r i a n t c~ d i s t i n g u i s h e s b e t w e e n t h e t o p o l o g i c a l types. By u s i n g these facts a n d T h e o r e m s 2 a n d 3, it r e m a i n s to p r o v e t h a t e v e r y c o h o m o l o g y class in H Z ( x , z ) or H4(X,Z) is algebraic. F o r H Z ( X , 7Z) this is c l e a r by t h e e x p o n e n t i a l e x a c t s e q u e n c e . B e c a u s e we h a v e n o t f o u n d a reference for t h e a n a l o g o u s s t a t e m e n t for H'*(X,~E), we s k e t c h b e l o w a p r o o f of it. As it is well k n o w n , t h e r e exists a r a t i o n a l t h r e e f o l d t o g e t h e r w i t h t w o m o r p h i s m s Z ~ X , Z --* IP 3, w h i c h are b o t h c o m p o s i t i o n s of m o n o i d a l t r a n s f o r m a t i o n s w i t h s m o o t h center. T h u s we are r e d u c e d to p r o v e the f o l l o w i n g fact. If X ~ Y is a m o n o i d a l t r a n s f o r m a t i o n o f 3-folds w i t h s m o o t h center, t h e n e v e r y c o h o m o l o gy class in H4(X, Z) is a l g e b r a i c if a n d o n l y if the s a m e p r o p e r t y is t r u e o n Y. F o r t h e proof, if X has the p r o p e r t y , t h e n Y has it too, as . / , f * = i d , in c o h o m o l o g y as well as in t'he C h o w ring. A s s u m e t h a t Y has the p r o p e r t y a n d let be a n e l e m e n t o f H4(X,Z). T h e n ~ - f * f , ~ is of t h e f o r m i,(r/), w h e r e r / e H 2 ( E , Z ) , E d e n o t i n g the e x c e p t i o n a l d i v i s o r a n d i: E ~ X the inclusion. But r/ is a l g e b r a i c in v i r t u e of HZ(E, (9)=0, w h i c h c o n c l u d e s the proof.

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