A descent problem of vector bundles and its applications

J. Math. Kyoto Univ. (JMKYAZ) 23-1 (1983) 73-83

A descent problem of vector bundles and its applications

By

Sadao ISHIMURA

(Communicated by Prof. Nagata, Jan. 12, 1981; Revised, N ov. 13, 1981)

§ 0 . Introduction

The descent problem of this paper is the following: if ^{7 i :} 2^—>x is the monoidal transformation of a smooth projective variety X with smooth center Y and if I i s a locally free sheaf on ^{g ,} when is 7r d^; a locally free sheaf? By H ironaka-K leim an's theorem, it is known that every vector bundle o n X is obtained by "extension + descent" ([8], Introduction). Therefore, this descent problem is a n essential part of the construction of vector bundles. In this paper, we shall give a necessary and sufficient condition as follows:

Theorem 1. L et 6^3- be a rank r vector bundle on g . T hen there ex ists a rank r v ector bundle g on X such that I is isom orphic to 7r*S if and only if for ev ery point p of Y ,

ibe„-t(p)

is isom orphic to Q e „,^{( p )}.

As applications of Theorem 1, we shall construct rank r vector bundles o n X from subschemes of codimension r (r=2 , 3). In th e c a s e o f r=2, som e proofs are k n o w n ([2 ], [5 ]). O ur methods is a refinement of that of B arth-Van de Ven. O n the other hand, Griffiths and Harris ([3]) present a problem of vector bundles re- lating with residues and zerocycles. Theorem 5 gives a partial answer for it in the case of threefolds.

In this paper, the ground field k is a n algebraically closed field of arbitrary characteristic a n d X is a smooth irreducible projective algebraic variety over k.

By a vector bundle on X , we always mean a locally free Ox-m o d u le . We will use the following notation: if S is a vector bundle on X , then S^- is the dual bundle of S ; (s), is the scheme of zeros of s for a global section s o f ^{S; ci(S )=} Y means the i-th Chern class of 8 represented by the closed subscheme Y of codimension i of X ;

9x (l) is a very ample line bundle on X .

T h e a u th e r expresses his hearty thanks to P rofessors K. W atanabe a n d H.

Tango for their valuable suggestions and kind advices.

Sadao Ishimura

§ 1. A descent problem of vector bundles

Let 7r: X be the monoidal transformation with smooth center Y and let be a rank r vector bundle o n f e . O u r descent problem is to know a necessary and sufficient condition for the direct im age ir,I to b e a rank r vector bundle o n X . The answer is given as follows;

Theorem 1. T h e re e x ists a rank r v ector bundle

e

on X such that f 4.'n*(6°) if and only if for every point p of Y, 6^3-'0 0 „-i( p ) -. 0„- 1( p).

Remark 1. If i n * ( S ) , then 6' r *( i') and i t ( i ) is a ra n k r vector bundle on X .

Before proving Theorem 1, let us prepare the following elementary lemmas.

Lemma 1. L e t V b e a sm ooth af f ine v ariety ov er k . L e t n : '1⁷–> b e the m onoidal transf orm ation w ith sm ooth center W and let E be the exceptional divisor o n P T hen for ev ery integer q._0 and p>0, HP(E, 9r( – q E ) 0 0 ,) =0 .

P ro o f . There is the following exact sequence on V;

0 O W - q – 1)E) O p ( qE) - - > 174 – q E ) 0 EO . This gives the cohomology exact sequence,

HP(I7, Op( – qE)) — > H P(E , v( – q E )0 E) — > HP^{+ 1}( F , 0 9 ( ( - q – 1)E)) . Since 12'n*Ov( – q E ) =0 fo r all q O and i > 0 , th e sp e c tra l sequence degenerates.

Hence, we see HP( P, op( — HP(V, qE))=0 because V is affine. There-

fore we conclude HP(E, (9p( – qE)(DCE)= O. q. e. d.

Lemma 2. L et V , W be algebraic varieties over k,f: V –>W a morphism and let be a coherent C^y-m o d u le . For every point p of W , if the restriction .F I f -i(,)

is generated by its global sections f or som e open neighborhood U o f p, then the natural m orphism f *f,..F–>.0^{9 4}" is surjective.

P ro o f . It is a local property to show that f *f,.:¹/⁷ --*.F is surjective. Therefore we may assume that0,,m–>..F^. i s surjective for som e integer m >O . Then there is the following commutative diagram;

f *f *(47^{3 -}

f * f *^,F

Since (9Pn–>g^- is surjective, it is only enough to show that f *f*CVm–>CTm is sur-3

jective. F rom th e id en tity m ap Om –> f *f*CPm –> , w e h a v e th a t f *f*OP'n–>

(1,¹³'m is surjective and the proof is complete. q. e. d.

A descent problem of vector bundles 75 P ro o f o f T heorem 1. T h e only if p a rt is obvious. Therefore it remains to prove th e if p a rt. O u r proof consists of two steps.

S tep ^1. We shall show that n ,i' is a rank r vector bundle on X . Let E be the exceptional divisor on a n d l e t cp: E--*Y be the restriction of n. From th e as- sumption and the base change theorem ([9 ], Corollary 2), there exists a rank r vector bundle ^.F on Y such that iC )0Effyo*,.F. Since the problem is local, we may assume that X is a smooth affine variety and iC )0E C ) 0E.

Let X^-- and X^ be the formal completions of :g and X along E and Y respectively, 5e^- ->x^- t h e induced ^morphism a n d le t ..or be th e sheaf o f ideals defining Y.

P u t i® Oxisrn±lex = i n a n d le t i " be the inverse limit of { in}. By th e Found- amental theorem o f G ro th e n d ie c k ([4], Théorèm 2), t h e m orphism 41)"-->

iq ,(i'" ) i s a n is o m o r p h is m . I f f o r every n , th e r e is a n isomorphism 9„:

6 01,15-.+10,¹ such that the diagram

(560g15^-n+20g

i n( ) ., 6ogi jr n + l ox

is commutative, then n*i is a rank r vector bundle o n X from the above theorem.

By the induction on n, we shall show o ur claim . F o r n=0, there is nothing to do.

Suppose that the result is true for each integer smaller than n + ^1. There exists the following exact sequence;

0f "^{- "}O g i l f " ⁺²0 5 6^{°- - -} > — >

o.

Taking cohomology groups, we obtain the exact sequence

H°(E, I r (E , i ⁿ) 111(E, ^{J r n + lox}il J r n + 2 0x

1)

,

On the other hand, since

ir n - "

O g i l i t "⁺²0 g i O g ( ( ^— n-1)E )i(D OE

6 0

5

((—n—

1 )E )® 9 . Hence, cc is surjective b y L em m a I. If is a basis o f H°(E, i n) and

are the sections of H°(E, i n+ i) such that ^{cx(0 =} fo r each i and if is a subsheaf of

i

n +

,

generateted by Z1,..., Zr, then we have

..#+.5^{1 -}i l in +,

Then Nakayama's lemma yields .4' ^'L'ii1+ 1. Since ^!, are linearly independent, is isomorphic to eexpan +26g. Therefore we have e n + 1 - xp r.+ 2 6 x ^co and these isomorphisms satisfy the commutativity condition.

Step ^2. Now we shall show Since n * n d and i have the same rank, it is enough to show that the natural morphism e n *i - - 4 is surjective. B ut

Sadao

this follows from Lemma 2. Therefore we can complete the proof. q. e. d.

R em ark 2. In his paper ([10], Theorem 5), Schwarzenberger proved the above theorem when Y is a point.

§ 2. A relation between rank 2 vector bundles and subschemes of codimension 2 It has been known that there exists a close relationship between rank 2 vector bundles and subschemes of codim ension 2 ([2], [5]). In this section, using Theorem 1, we shall construct rank 2 vector bundles on X from smooth subschemes of codi- mension 2 on X . O ur method is a refinement of that of Barth-Van de Ven. They proved Theorem 2 in [2 ] (Proposition 6.1). For the sake of completeness, we shall prove the following;

Theorem 2. L et Y be a smooth codimens ion 2 subscheme of X and let H be a divisor on X . A ssume the following conditions;

2

(1) A Xy / x e x(H )0 0y, where .Aty l x is the norm al bundle of Y in X .

(2) H²(X , CA— = 0.

T hen there ex ist a rank 2 v ector bundle f on X and a global section s o f S such t h at ci(X )= H , c2( e ) = Y an d (s)0= Y. M o re o v e r i f 11¹-(X, ex(— H ))= 0 , é° is unique up to isomorphism.

P ro o f . Let m: g^.—>X^. b e the monoidal transformation of X with center Y and let E be the exceptional divisor on g . There is the following exact sequence;

2

(2.1) 0 —> - 1 ) .11 y 9*( A X y / x)(:) COE( 1) --+ 0,

where 9 is the restriction of ir to E and CE(1) is the tautological line bundle. From the assumption (1), both line bundles 6E( — 1) and 9*( A X2 y / x)(:)0E(1) are extendable t o Ox( E ) a n d n*(0x(H ))0 0x(— E). M oreover t h e e x te n s io n s o f C A E ) b y n*(0_x(H ))0 0₅₁(— E ) a r e classified by H I (g , x ( — M K ) 2(2E)). N ow , there exists the following exact sequence;

0 7r* ^{( ( 9} x( — H))®(9 (E) it* (0 x( — H ))® g(2E)

- n* (0 x( — H))0 — 2) 0 and taking cohomology groups, we have the following exact sequence

11¹(1, n*(ex(— H ) ) ® x(2E)) H ¹(E , 7t* ( xx(( — TM® 0 ( — 2)) - H²a n * (0x(— H))(:)0x(E)).

Therefore, if f3 is surjective, in particular if H²(g, n*(0x(— H ))0 0x(E ))= 0 , then there exists an extension I on g whose restriction to E is 9*.Aty / x. On the other hand, there are the following exact sequences;

A descent problem of vector bundles 0 * (g (_ H ))H )) --+ n *(0 ,(— H ))® g(E)

7r^* (⁰ x( 1) 0

77

and

H²(g, n*(ex( —H))^{H 2 ( ,}, n* (0 x( — H ))() g(E))

H² (E , 7c* (0 x( — H)) (9 E( — 1)) .

B y t h e assumption ^{(2 ),} w e h a v e H²(1^{, re*(0}x(— H )) H²(X , ex( — H ))= 0 and H²(E, n*(0x(— H ))® E ( - 1 ))= O. H e n c e , w e s e e ^{H 2 ( , * ( e} x(— H))00 f(E))= 0 an d fi is surjective. Therefore there exists an extension I o n g such that ^{i ®} CE--' 9^*./ry/x. By Theorem ^1, there exists a rank 2 vector bundle s o n X such that n aⁱL 2 S . Taking the direct image of the exact sequence

0 eg(E) n*(0x(H))065( — E) ---> 0

and noticing the fact ^ir*O_g(E )---' _x, n_*O_g(E)= 0 a n d n 9 (— E ) J , we get the exact sequence;

(2.2) 0 ---> Ox -- > 45' x(H )0 O.

From (2.2), we have ci(S )= H , c2(S )= Y and ^(s)0= Y, where ^s is the global section o f S corresponding to the injection of (2.2). Moreover i f ^{Ilt(X ,}

e

x

(

^{— H))=0,}

fi is injective by the similar a rg u m e n t. Therefore ^g is uniquely determined by ^#.

q. e. d.

§ 3 . A relation between rank ³ vector bundles and subschemes of codimension 3 3 .1 In this section, we shall study a relationship between rank 3 vector bundles on X an d subschetnes o f codimension 3 o n X . First, we give a criterion for the scheme of zeros of a section to be smooth.

Lemma ^{3 ([5],} Proposition 1.4). L et g be a rank r v ector bundle on X and let ex(1) be a v ery am ple line bundle on X.

(1) I f g ( - 1 ) is g e n e rate d b y its global sections, then f o r all g e n e ral s e H°(X, g), (s) w ill be sm ooth and codimension r.

(2) I f ch (k)---0, th e sam e is tru e u n d e r th e w eak er hy pothesis that g is generated by its global sections.

Now, let S be a rank ³ vector bundle on X . By the above lemma, we assume that there exists a global section si o f (1 such that ^(.500 is a smooth codimension 3 subscheme of X . If we put Z = (s1)0, then ^Z represents the third ^Chern class ^c3(S).

Taking the dual o f si : Ox—>S, we get a homomorphism g : 6^- - ) ex whose image g ( g ') is the id e a l sh e a f f o f Z. Now, tensoring the surjective morphism S^- ->

Jz with (9z, we get SvOCz 'L' irz(De_z '.4^{7 -}i'_{i x} . Hence we have

Proposition 1. L et g be a rank 3 v ector bundle on X . A ssum e th at g ( - 1 )

Sadao Ishimura

is g e n e rate d b y its g lo b al se c tio n s. T hen there ex ists a sm ooth codimension 3 subscheme Z on X and a divisor H on X such that ci(e )= H , c3(S )= Z and O5(H )0

3

(92 =² A A^tzix.

3 . 2 To obtain further results, we shall study subschemes o f codimension 2 on X corresponding to the second Chern class of S . W e assume that e is generated by its global sections and ch ( k ) = 0 . Let ^77: I-4.X be the ^monoidal transformation of X with smooth center Z = c3(S ) an d let E b e the exceptional divisor on Since ^E is isomorphic to the ^P2-bundle _P ( . . " 1 ) , there is the exact sequence;

(3.1) 0 CE(— 1) — > (pt./1¹'m —> 12si"/ 2( —1) —> 0,

where 9=nIE an d C2E l z is the relative differential sheaf. By Hironaka-Kleiman's theorem ([7]), there exists the following exact sequence;

(3.2) 0 Ox(E) — > 77^*

e

² 0

where 2 is a rank 2 vector bundle on From ^(3.2) and [6] (Theorem 8.13), we have the following commutative diagram;

0 Cg(E)OCE rc*SOCE 200E 0

0 C E (-1 ) 9*../rzix ( 2 l z ( - 1 ) - - + 0

Therefore, we have ^{..20 6}E S2'i/ z( — 1). From the assumption on e, .2 is generated by its global sections. Hence if ^s2 is a general global section of ^2, then, ^{Y, =(s}2)0

is a smooth codimension 2 subscheme o f g^. a n d Y, = c2(.2). Since .20 0EL:' we see that the restriction s2IE o f s2 to E is a section of f4/,(— 1) and

n

E=(s21E)o. On the other hand from ^(3.1) and ^{H ° (E , E(- 1 ) ) = Hl(E, 0}E( - 1 ) )

=0, we have H°(E, f4/ 2( - 1 ) ) .'tzH°(E, (ptirzix)'_=_H^°(Z, ,Kz i x). Thus, we see that there exists a section s3 of Arm such that ^s3 corresponds to ^s21E. Now we assume dim X 5 . T h en w e see th at d im Z<rank.irm a n d s3 is nowhere vanishing, because ^s3 is general. Therefore, we find that (s21E)₀= Y¹

n

^E contains no fibres of 9 . Since Y,

n

E is isomorphic to Z, there exists a smooth subscheme Y of codi- mension 2 on ^X such that ^Y, is the strict transform of Y , Y , Y and ^{Z c} Y. N o w from (3.2), w e have n*(c i(S))= c i(0 x(E))+ c1(.2) an d rc* (c 2( ‘ ) ) c 1(0 g(E))c 1(2) + c2(2). Since c1(2)=7r*(c¹(g))— ci(Og (E)), c2(2 )= Y , and Y, Y, we have c1(2 )=

H1 and c2(S )= Y, where H , is the strict transform of H . By the similar argument

2 2

as in Proposition ^{1 ,} w e h a v e ^{A.iry i i x '-'051(H ,)0 0 ,i.} S in c e A.4¹',1 1 g @ y ip

2 2

coy 0coI and ^n*(cox) 0 02

,(2 E )- cog, we have A./VI/a l l c oy i0 0x(— 2E) On*coXcL-'i*( A d r2 y / x)(:)^,CI ( —2E), where i is the isomorphism Y, Y. T h erefo re

2 2

we have ^i*( A .Arypc) *

(49x(H)) 0 1(E)00 ^yi. Then we conclude A d iÇ y lx : 0 x(H)

®Cy( Z ) . Hence we have

Theorem 3. L et X be a sm ooth projective v ariety ov er k w ith dim and let e be a rank 3 v ector bundle on X . A ssum e that 6° is generated by its global

A descent problem of vector bundles 79 sections and ch (k)= O. T hen there ex ist a sm ooth subschem e Z of codim ension 3, a sm ooth subschem e Y of codim ension 2 , Y Z a n d a div isor H o n X such that

2 C i ( S ) = H , c2(S )= Y, c3( S ) =Z a n d x(H)C)Cy(Z )-- A dry/x.

2

A . / r y / x 0 . / r z i y , we see ./V'Yi2

y O z. But, recently, H. Tango has proved the fol- lowing Theorem;

Theorem 4 . L et X be a smooth projective v ariety w ith dim X d , o ' a rank s v ector bundle on X and let Ox(1) be a v ery am ple line bundle on X . W e assume that S (-1) is generated by its g lo b al s e c tio n s an d s <d <2 s - 1 . T hen there ex ist sm ooth algebraic s e t s Y , Z o f X s u c h th at Y D Z , c^s_1(1 )=Y , c⁵( e ) = Z and

drZ /Y ^{2 2}-

°Z.

P ro o f . From the assumption, 0° gives a twisted embedding o f X into the Grassmann variety Gr (r —1, r — s — 1) such that the pull back o f th e universal quotient bundle .2 is g , where r = dim H°(X , 0') ([7 ], Remark 3.2 (iii)). Using a theorem of Kleiman ([7 ], Theorem 3.3), we get three dense open sets 14(1, ^W2 and W3; (1) W, is a dense open set of Gr (r —1, ^{P r - 1} such that for every z e IV,, X n i( L ) is smooth of dimension d — s, where Lz is the 0-dimensional linear sub- space of p^{r - i} which is represented by z.

(2) W 2 is a dense open set of Gr (r — 1, 1) such that for every y ^{E W 2,} X n 62(4)=0 where Ly is the 1-dimensional linear subspace of Pr^- i which is represented by y.

( 3 ) W 3 is a dense open set of Gr ( r- 1 , 1 ) such that for every y e W 3 , X n 0-1(4) — 0-2(Ly)) is smooth of dimension d— s +1.

Let us consider the following diagram;

Fl (r — 1, 1, 0)

Gr(r — 1, 1) p r- i

, where Fl (r — 1, 1, 0) is the flag variety {(y, z) e Gr (r — 1, 1) x

I

Ly 3 z). Then cc '(W2) n rx^- '( W3) n fl^{- 1} ( W1) is a dense open set of Fl (r — 1, 1, 0). F o r a fixed point (y0, zo) of ce^{- 1}( W2) n oc¹( w3) n [3^{- 1}( W1) , set Z= X n 0^-1(Lz 0)= {X E X I L D zo}

an d Y=x n 0^-1(4 ⁰) =fx e^X

I

^{L„ n 400 0 . Then Z} an d Y are smooth algebraic sets of dimension d — s and d — s + 1 respectively from (1), (2) and (3). Moreover, since cs_ i(..2) = i(Ly o), cs( 2 ) = i(Lz o) and L 03z0, we have Y= c5 1„(S), Z = 4 1 )(0'), a n d Y Z . Now let C be 4 ,0 n IV, and let 2 b e Fl (r —1, r — s —1, 0) n xx c=

z ) L^x z , x e X , z e C } . Then 2 is an algebraic set of Y x C and there is the following diagram;

2 3 3

Remark 3. From 9x(H)0(9y(Z)'_•-.' A X y / x, (9x(H )0 0z . A.APzix a n d A .4⁷'z / x

Sadao

Y

, where 9 and Li are projections. For every z e C, i/JV (z) is isomorphic to 90(tliV (z ))

= { x E X I Lx3z}=cr1( L ) n x a n d dim 00- 1 (z)=d— s= dim Y - 1 , because z e and (1). Therefore,

2

is a divisor of Yx C . If z e C and z zo, we have 90(06¹(z0))

=Z and 90( 0 ( z ) )

n z {x

^E X I Lx z and Lx 3 zo}. Since Ly 0 contains z and zo, w e g e t 90(00- '(z))

n z =

Ix e X Ex D Ly.} = X n 0^-2(Ly 0) = 0 . T h e r e f o r e w e have

Zi,Z = 0 and .irz i y O z. q. e. d.

3 . 3 In the rest o f th is section, we shall construct rank 3 vector bundles on X from subschemes of codimension 3.

Theorem 5. Let X be a smooth projective variety over k, H an effective divisor on X and let Y an d Z be sm ooth subschemes of codimension 2 an d 3 o n X , re- spectively, with the relation Y D Z . A ssume the following conditions;

(1) ./ rziy (92.

(2) L-2. x( H ) 0 y(Z) (3) H²(X, (9x( — H ))= 0 (4) H²(X , e ) = e1)=0.

Then there exists a rank 3 vector bundle on X such that H , Y and Z correspond to the i-th Chern classes ci( d ) ( i = 1, 2 and 3, respectively).

P ro o f . Our proof consists of several steps.

S tep I. L e t n, : X, —>.X be the monoidal transformation of X with center Z and let El be the exceptional divisor on X , . There is the following exact sequence;

(3.3) 0 — l) çor./rzix gridz( ^—1) 0

, where 91 is the restriction of n, to El. Now, we shall show that there exists a rank 2 vector bundle .2, o n Xⁱ s u c h th a t ..2 1 0 0 E 1 Q E - 1 I z ( - 1 ) . F r o m (3.3), we have

1 1 °(Z, .¹¹f z ix ) -^{1 1 °}(E1, (Pid^{1 t}zix) ^{1 1 °}(Ei^, 1 ) ) . O n t h e o th e r h a n d , there exists the following exact sequence;

(3.4) 0 — ) (PtA^çzir 9tAtzix ^{- - 4} " ⁴'s/1(0⁰z) ^{— 4} O.

Now, let U = Spec A be an affine open set of Z and let V= Proj A[xo, xi, x2] be an inverse image of U . Restricting of (3.3) and (3.4) to V , and tensoring with 0,(1), we have the following commutative diagram with exact rows and exact columns;

A descent problem of vector bundles 81

0 0

Cy(I) Cy(1)

0 — * Cy x. Cy(¹)'¹³⁶t r i q u 0 x

x,2I x

x i( M I )^{6 )2}C y ® . , y , n v

0 Cv

, where Y, is the strict transform of Y. F r o m the third column of the above diagram, we have the following exact sequence,

(³.⁵) ⁰( P r . / r z i r g-411z(— I) 4 4 ( X z lx )0⁰E¹(¹) 0 -fr¹nE^{1- 4 0} By our hypothesis (1) and the injection of (3.5), there is a global section s , of

— 1) such that (s1)0= Y, n E , . Thus, if there exist a rank 2 vector bundle .2, on X , and a global section s2 o f .2, such that (s2)0= Y,, then we see

2 3

Q 11 (- 1) b e c a u s e o f A d ry^, n E i/ E, TT( A APz / x⁾ 0 0E 1) >C<) YinEi^, H ⁱ(E i, ⁽Pf

3 3

( A drz/xY0 CE,( — 1)) H²(EI , çor( A ./1⁷'z i xr" 0 CE1(— l))= 0 a n d t h e uniqueness of

2

Theorem 2. B u t f r o m t h e assum ption (2 ) a n d Y1'1' Y, w e h a v e A X y d x , 4 (0 ^x(H ))0(9,,¹(— E¹) 0 0^{y 1}. Here, there exists the following exact sequence;

0 — 4 nr(ex(— H )) Trf(0,(— WOO E^t( ^{1) - >}

, where .9⁹1 = nr(0,( — H ))0 0 ,i(E ,). Taking cohomology groups, there is the fol- lowing exact sequence;

H²(X 5(— H))) --_3 _H²(X - r i )

— > H²(E1, rtf(9x(— H ))0 0_{E 1}(-- 1)) .

B y th e assumption (3 ), w e h a v e 11²(X1, n f(ex( — H2(X, (95( — H ))= 0 and H²(E1, nt(Cx(— H)) 0 0 E l⁽ — 1)) = 0, w e s e e H²( X1, .2⁹) = 0. T herefore, b y Theorem 2, there exist a rank 2 vector bundle .2, on X , and a global section s2 o f .2, such that c1(.21) =7C,K(H)—E1, c2(.21) =. Y, and (s2)0 = Y,.

Step 2. In this step, we shall show that there exists a rank 3 vector bundle gi

o n X , such that _SiOCEI - ."(Pr(-/rzix). N ow , tw o v ecto r b u n d les 0,1( — 1) and frE'i / z( — 1) a r e extendable to t 9 ( E ) a n d .2 , respectively. M o r e o v e r th e ex- tensions of e² 1(E1) b y ..21 a r e classified by I P (X , , .2r® Ox i(E ,)). N o w , there exists the following sequence;

0 — > 05 i(E ) — > OCE i(— 1) — > 0 and taking cohomology groups, we have the following exact sequence;

H '( X ,, .2r0e_{x 1}(E1))--'—' )11¹(E1, ..2₁OCE₁( -1 )) — H ²(X1,

Therefore i f y is surjective, in particular if H²(X1, .20= 0, then there exists an extension Si o n X , whose restriction to E , is cpr.Atz i x . O n the other hand, by the same argument of the proof of Theorem 2, we get the following exact sequence, (3.6) 0 ---> ^{7 0}1(^{2 2}1)0⁶x²(E2) --> _ - 0

, where n2: X2—)X1 is the monoidal transformation of X , with center Y, and E 2 is the exceptional divisor on X 2 . Moreover, there exist following two exact sequences;

O - - > g I (² 1) n i G r i ) 0 0 “ E 2 ) 4 (2 1 )0 0 E 2 ( ^{— 1}) 0 E2)x 2 - - - > 0 E2O .

Taking cohomology groups, there are two exact sequences;

H²(X2, ⁷1(-9⁹1)) --> H²1X2, nI(¹é'1)0⁰x2(E2))

H²(E2, ⁷riG ri)0⁰E2( — 1)), H¹(E2, ⁰E2) H2(X2, ex2(^— E2)) H2(X2, ^2x2).

From the above cohomology sequences and the assumptions (3), (4), we can show H²(X2, TrIG⁹i ) 0 (⁹x2(E2))=¹/²(x2, C ²(- E²))= O. Therefore, since (3.6), w e have H²(X2, 4(..21")) H²(X1, ..2r)=0.

Step 3. Since Si O C E , Ti(.A^Pz ix ), there exists a rank ³ vector bundle ó on X such that gal ''.'7ErS by Theorem 1 and there is the following exact sequence;

0 Oxi(E,) — > .2, - - + 0.

H e n c e , w e h a v e nr(c,(S ))= c1(0 x i(E1))+c,(.21), n t(c2(S ))= c x i(E ,))c 1(.2 1) + c2(.2 1) a n d nr(c³(g ))= c1(Ox¹(E1))c2(² 1). Since c1(.21) = n(H )— E1, c2(² 1)=

Y, a n d Y , Y, we conclude ci(S )=H , c2(S )= Y and c3( )= Z. q. e. d.

Remark 4 . I n [3] (p 504), Griffiths and H arris present a problem o f vector bundles relating with residues and zerocycles. Theorem 5 is a partial answer to it in the case of dim X =3.

Corollary 1. L e t P be a projectiv e space P³, Y a smooth codimension 2 sub- scheme of degree d2 on P and let Z be a smooth codimension 3 subscheme of degree d3 o n P. A ssu m e the follow ing conditions;

2

(1) A . A r y i p Op(di) 0 0y(Z ), w here d, is a positive integer (2) H'(Y, Cy)=0.

Then there exists a rank ³ vector bundle o n P such that c1(1)= d1 (i= 1, 2 and 3).

It is interesting to find indecomposable rank r vector bundles on X (r X ) . In our case, to construct indecomposable rank 3 vector bundles on X , we must find subschemes H , Y and Z satisfing the conditions of Theorem 5. It is difficult to find them in general, even if X P", B ut if Xg_^-2P³, we can find indecomposable rank 3 vector bundles on X as follows:

A descent problem of vector bundles 83 E x a m p le . L et cp: P ' P ³ b e the morphism defined by the composition of a d-uple em bedding P'— *Pd (d__.3) a n d a g e n e ric projection P ' —*P³. S i n c e

2

(" 9 p 3 ( 1 O p i ( d ) a n d A dIrpi/p3 -.. 0,3(4)06p1(-2), we have the following two cases;

2

Case 1 . A . A f p 1 1 p 3 C 9 p 3 (2) 0 0 p 1 (2d — 2).

By Corollary 1, there exists a rank 3 vector bundle S o n P³ s u c h t h a t c,(S)=2, c2(S )=d and c3(S )= 2 d - 2 for all d 3. Since the Chern polynomial cr(e )=1 + 2t+dt²+ (2 d -2 )t³ is irreducible, S is indecomposable.

2

Case 2. A ../r p l 11,3 p 3 ( 1 ) C ) 1 9 p l (3 d -2 ).

By Corollary 1, there exists a rank 3 vector bundle S o n P³ s u c h th a t c ,( ) = 1 , c2(S )=-d and c3(6 ) = 3 d -2 for all d 3. Since the Chern polynomial 46'1=1+

t+d t²+(3 d -2 )t³ is irreducible, S is indecomposable.

DEPARTMENT OF MATHEMATICS, TOKYO METROPOLITAN

UNIVERSITY

References

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