Then the adjoint operator b of d" with respect to < > is given by

Transcription

1 266 [vol. 30, 56. Note on Kodaira-Spencer's Proof o f Lefschetz Theorems By Yasuo AKIzuK1 and Shigeo NAKAN01' Mathematical Institute, Kyoto University (Comm. by Z. SUETUNA, M.J.A., April 12, 1954) It has been for a long time expected to have a rigorous proof of the celebrated theorem of Lefschetz on the homomorphism between homology groups of an algebraic variety V(n2) and those of its generic hyperplane section S. Recently Kodaira and Spencer have succeeded through their deep investigations in proving not only this theorem but also the lemma of Enriques-Severi-Zariski at the same time. By a differential-geometric method Kodaira gained the fundamental lemma on the cohomology groups with coefficients in the stack of germs of holomorphic forms whose coefficients lie in a complex line bundle over V. Standing on this lemma he succeeded also in obtaining the decisive result : the Kahler variety of Hodge's type is equivalent to a projective variety (namely an algebraic variety). The differential-geometric considerations were necessary indeed for the discovery of the lemma, but not necessary, as we shall show in this note, for the proof itself. But by this simplification we can somewhat improve the lemma and establish the above Lefschetz theorem in the original form.2' Further we shall add a remark that the Lefschetz-Hodge's theorem3' may be deduced from the fundamental lemma on the same principle. 1. Let V be a compact Kahler variety of complex dim. n, u= { U } a simple covering of V, N its nerve, a complex line bundle defined by {f.,,} referred to the covering U, where f.,,(z) is a non vanishing holomorphic function in UJ m Uk and f 1, f13= 1. We shall redefine the characteristic class c(b) as follows : 4~ cl,, = hjk + h,, + h1, h,,= logf,,. (1) Then, if 8 is defined by the divisor D whose local equation is f (z) = 0 and f it f k = f;,,, considering D as a current we obtain easily a Weil's chain of double cochains5' (coelement) D~ 2-1d 1 lobrgfjf-~ 1/-1 2 togf k (c tjk)~ ~r thus D is cohomologous to c({d} ),6)

2 J No. 4] Note on Kodaira-Spencer's Proof of Lefschetz Theorems 267 Returning to the general case, consider the cochain (ta) = (log I f ~k 12) of double degree (0,1), then as SN(t Jk) = 0, it is (t) ~k=&x, X=(a5), where a5 is real c in U~. Putting e-ai=a3, then I f 5 k 12=aJlak and log f Jk + log f Jk =: log a3-log ak d' log a5- d' log ak = d' log fk = d log fk." Therefore the following chain of double cochains is also the Weil's chain. 1 1/-1 1/-1 l og f;k (~~ yk) = c()~ where. p3 = - d' log aj, x ==V -1 d"p5 = 1/ -1 d'd" log a3. (2) Thus c(3) is cohomologous to the real quadratic form 1 x. 2!Tr 2. Let 1 be the module of differential forms p whose coefficients lie in = 3f}, Jk namely q = q J in U5 and ~P J= f JkCpk in UJ fm Uk. Specially when 3 =(0) (trivial bundle), then P is the module of forms on V. We define as the inner product (q, Jr) or <yo, J'> of two forms gyp, fr E (o or ( respectively as follows : V IqJP.J Then the adjoint operator b of d" with respect to < > is given by V a'.7 b~ TZ =-*a5d' J 1 =6"_ *(pj5*~3). J (3) Let i (x) be the adjoint operator of e (x) (the exterior multiplication of real quadratic form x), then it is easy to see i(x)=(-1)'*e(x)* =(`1)r*e(x)* (for (P E P ) (4) Consider now the involutory mapping 7r: P -* (_ such that q- TJ q'3tj a' By easy calculations we get for d" q =(-1)' +r* 1 r/~ ajb~"~ 5 TJ ~ 7J, TJ aj a, '= - 1 *d'(at.*~~ ')=(-1)r 1 *d~~cpj If p is fl-harmonic in P, then since d" cp = b~d = 0 the above relations show that ire = ~p' is also fl-harmonic in (P_s, and conversely. Therefore, if Hp'q(8) denotes the module of 0-harmonic forms of type (p, q) of the bundle B, it is p,q( ) - H'~-p' n-q( - S3). (5)

3 . 268 Y. AKIZUKI and S. NAKANO 3. Let us now prove E the following : `Sls + {Pjd"(*q )}-(-1)r*(dlfpj)*co 1)r+l*d1f(P;i*~;i). Therefore, by using (4), we get (6): [Vol. 30, (s) (be' +S'b) =- 1 *e(x)*cpj=v-1 i(x)9'j. Next also adopting H. Cartan's notation e(q), 9 have as the fundamental formula for Kahler variety i(q)d"-d" i(q)=s'. If we use Weil's notation L = V --1 e(9), then n its adjoint is A = -- i/ -1 i(2),6 ' and the above formula will be writte r't n in the following form, hence ~' is also a linear operator on P. d" A-Ad"=V f '. Let now q be El-harmonic, then since b9 = d" ~ = 0, as <9', a~ > = a < y~, +>. < Arp, bt'ep > = (dcp, (ba' + S'b)co > = A9', T/-1 i(x) ~> = -VT A9', i(x)p> On the other hand, it will be <A91, bs'~p > = <d" X19', ('9'> = < (d"a -Ad")9', c'9'> =i/ T <'9','(P. S~ > by (7). Hence we have the important formula for El -harmonic forms - <A9', i(x)9' > = S'rp, S'9'> >o (g) 4. Let x =1/-1 xardzxdz~ and ass ume that = CJaf~a, We (7) it will be (b6'-+- S'b)cj = V'i i(x)q j In fact by (3) - * ~PJ( * p~j)=se's' - (--1)' *{ PJd»(*9'3) }~~ 3'b9'3='"9'- SSS'(*Pj*9'i)=(SIS"~p3+(- et] form is xa everywhere (ta, t ~) (ta, t ~) : symmetric posiwhere positive definite.) produ tive definite. (Then we say that x is every In this case we may define the Kahler m etric on Y taking this form as the fundamental form. Accordi ugly we can see by (8) that, as we may consider in this case A= i(x), by (s) the quadratic Therefore if there exists a El-harmonic form 9'O of type (p, q) with p + q>_n +1, then by a well-known Hodg 's theorem on effective e form we see Ap~0 and (d9', A9> >0. Th is contradicts s with the

4 No. 4] Note on Kodaira-Spencer's Proof of Lefschetz Theorems 269 above inequality, so it must be qp,4= 0, when L1g1= 0 and p + q>_n +1. Thus we get Kodaira's lemmas : Theorem 1. If the characteristic class c(13) contains an everywhere positive definite quadratic form, namely c(73) >0, then HPe(S}' )...0 for p+q>_n+1, where n is the complex dim. o f the underlying space V of 8. By the involutory mapping 7r: -~ &, we have also Theorem 1'. I f the characteristic class c(23) contains an everywhere negative definite form, namely c(.8) < 0, then it is i1p (3)0 for p+q<n--1. (9) Further let QP(t3) be the stack of germs of holomorphic p-forms whose coefficients in. Then by the Dolbeault's theorem where HJ(Q20(!3)) = II ~(V, Q2(3)). Hence Theorem 1". I f c(3) contains an everywhere positive or negative definite form, then it is respectively H(9(3))0 ~^for p + q>_ n+1 or <n -1. (10) 5. Let S'' be a non-singular analytic subvariety of V n, s the restriction of the bundle 3 on S, and the sequence be exact. 0 -~ Q'p()3) 4. QP(?3) - QP(8) - 0 (11) And let i E Q'p(8) _ al ap - ~a 1...ap z... z, a 1<... <o p where we assume that z1= 0 the local equation of S. Then it is clear, that for p_>1 ~' = 1<a2<... <ap (m2...p)sdz2.. aaa. dza belongs to Sip-1{(C7- {s})s}. If we denote by r the mapping Q''(3) -~ Qp-1 [(B- {s}) s} such that r: E Q'p(3) -~ E Qp-1 ((8 - ~s} )s} then we can easily prove that the sequence 0 -'l.?gp(^u - {s}) - Q''(^8) - 9p-1 [(8 - {s})s}-*0 (12) is exact. By taking the sequence of cohomology groups corresponding to (12), we obtain the exact sequence --} H~(Qp(3 - {s})) -* H~(SYp(i3)) -* Hq(Qp-1 (( - {s}) s}) -~. (13) Now let us assume that S is so ample that c(3 - {s}) contains an everywhere negative definite form, then we see by theorem 1" that Hq(2p(3-- {s}))^'0 for p+q<n-1,

5 270 Y. AKIZUKI and S. NAKANO [Vol. 30, Hq(Q 1(B _ {s})s)^'0 for p+q<n-1. Putting these in (13), we have, if p> 1, for p + q <n -1 H~(Q'p(3)) _ 0. (14) Moreover it is also the case even when p=0, because Q10() =Q (3 - s}), H~(Q0(.3 _. {s} ))^r0 for q < n - 1. On the other hand, taking the sequence of cohomology groups of the sequence (11), we get the exact sequence --* Hq(,~~p( )) -* El(QP(J3)) ~ H (QP(8s))- H~+1(Q"2 (8)).. Theorem 2. If the divisor S7 1 is so ample such that c(18-- {s} ) contains an everywhere negative definite form, then for p + q<n --1 there exists the isomorphism H~(DGp(^U)) ~ H~('~~p(~s)) and this is an isomorphism onto or into according as p + q<n - 2 or p+q=n-1. Consider the special case, where V is a projective variety, S a generic hyperplane section of V. Taking as { 0 } (30} the trivial bundle), then clearly {0} - {S} contains an everywhere negative definite form, so the mapping Hq(Qp(0)) -* HH(9p(O8)) for p + q ~ n -1 is isomorphic. But we see by the Dolbeault's theorem H4(9P(O))~Hp,9(v, C), Hq(Qp(OS)),.,H"q(S, C), where C is complex number field. Hence we have the Lefschetz theorem in the classical form: Theorem 3. Let V be an algebraic variety of dim. n without singularities immersed in a projective space, S be a generic hyperplane section of it (consequently S is irreducible# and has no singularities), H(V, C) the cbhomology group of degree r. Then Hr(V, C) is isomorphic to H'($, C) if r_<n--2, and Hn+i(V, C) is isomorphic to a submodule of Hn-1(S, C). 6, we can prove directly by theorem 1" also the following theorem of Kodaira-Spencer, consequently the Lefschetz-Hodge's theorem. Theorem 4. Any complex-line-bundle l3 over a compact projective variety Y is equivalent to a divisor-class D on Y.9)9a) Proof. Let 3 be any given complex-line-bundle over V, S a generic hyperplane section of V, 3,, =l3 +h {S} for a natural number h and K the canonical divisor of V. Consider now the mapping r letting correspond to an n-form with coefficients in h- {K} its Poincare-residue with respect to S. Then the sequence of stacks

6 No. 4] Note on Kodaira-Spencer's Proof of Lefschetz Theorems * Sn( h-1 {K} ) Q( h {K} ) 4 QS-1((!8h-1 {K} )s) -*0 is exact, and to this corresponds the exact cohomology sequence : H (V, Sn(^Uh_- {K})) H (S, SS_1((~h-+1- {K})8)) -* H1(V,Sn(3h-1- [K})) -. If we take h sufficient large, as it can be readily seen, we can assume c(3,_1 ~ - { K }) >0. Hence in the case we have by theorem 1" that H1(V, Sn(^Uh-1- {K}))O and therefore the mapping r* is a homomorphism onto. We shall now prove theorem 4 by induction with respect to dimension n of V. Then there exists a divisor D' of Sn-1 with 1- { K })s = f D' + (h- 2)S' -- K'} where S' is a generic hyperplane section of Sn-1 and K' the canonical divisor of S. If we denote by Sq(D) (or Q '(D')) the stack of germs of meromorphic forms whose divisors are multiples of -D (or -D') on V (or S), then we have dim H (S, Qs-1((h-1- {K} )S)) =dim H (S, Sts-1({D'+(h-2)S'-K'})) =dim H (S, 9s-1(D'+ (h-2)s'-k')) =dim H (S, 9s(D'+ (h-2)s')) =dim ~D'+(h-2)S'Is+1. Since h is sufficient large we can assume this number is greater than one. Moreover r* is an onto-mapping and therefore dim H (V, 9'(3h- {K})=dim H (V, Qo(S3h))>1. Hence there exists a non-trivial cross section in S (!3h), consequently a divisor D of V with 3={D}, q.e.d. Bibliography 1 H. Cartan, Seminaire (1951-'52), I. 2 De Rham and Kodaira, Harmonic Integrals, Princeton lecture, Kodaira and Spencer, Groups of complex line bundles over compact Kahler varieties, P.N.A.S. 39, Kodaira and Spencer, Divisor class groups on algebraic varieties, P.N.A.S. 39, K. Kodaira, On a differential-geometric method in the theory of analytic stacks, P.N.A.S. 39, Kodaira and Spencer, On a theorem of Lefschetz and the lemma of Enriques- Severi-Zariski, P.N.A.S. 39, K. Kodaira, On Kahler varieties of restricted type, Mimeographed prints, Princeton Univ., A. Weil, Sur les theoremes de de Rham, Comm. Math. Helv. 26, 1952.

7 272 Y. AKILUKI and S. NAKANO [Vol. 30, References 1) The authors are very grateful to Profs. Kodaira and Spencer for their kindness to send us their mimeographed prints before publishing them. 2) Compare Kodaira-Spencer, 6, Theor. 2. 3) See Kodaira-Spencer, 4, Theor. 4. 4) This definition is different by sign from that of Kodaira-Spencer, 3. 5) Weil, 8. 6) De Rham-Kodaira, 2, 28, lemma 3. 7) We use here the same notations as in H. Cartan's seminary, 1. 8) Since the product is Hermitian (Lo, rb)=(/ -1 e(2)p, ') = (e(s2)c, -'v'- 1 ~')=(c', -1/ -1 i(s2)c) _ (q', A~b). 9) Compare Kodaira-Spencer, 4, pp a) When I finished this paper, Prof. Weil kindly sent me his lecture-prints in which I found this theorem stated in the case when f2~ is rational function. Cf. A. Weil, Fiber-spaces in algebraic geometry, p. 26, Univ. of Chicago, Winter, 1952.