VOL. 39, 1953 MA THEMA TICS: KODAIRA AND SPENCER 1273 This result reduces to a special case of Serre's duality theorem. 6 See Garabedian, P. R., and Spencer, D. C., "A Complex Tensor Calculus for Kahler Manifolds," ACta Math., 89, 279-331 (1953). 7 In view of de Rham's theorem, any cohomology class may be regarded as a class of d-closed forms. By a d-closed form we shall mean a form which is closed under d. 8 Garabedian and Spencer, lOc. cit., p. 290. ON A THEOREM OF LEFSCHETZ AND THE LEMMA OF ENRIQ UES-SE VERI-ZARISKI* By K. KODAIRA AND D. C. SPENCER DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY Communicated by S. Lefschetz, October 8, 1953 1. Introduction.-The present note is mainly concerned with a generali- zation of a theorem of Lefschetz which establishes a relation between the cohomology groups of an algebraic variety and those of its general hyper- plane sections. We generalize the theorem to the case of cohomology groups of Kihler varieties with coefficients in some analytic stacks. Our generalization includes the lemma of Enriques-Severi-Zariski as a special case. 2. Some Exact Sequences of Stacks.-Let V be a compact KShler variety of complex dimension n and let S be a non-singular analytic subvariety of V of complex dimension n - 1. Take a sufficiently fine finite covering UjI of V. Then, in each U1, the subvariety S is defined by a holomorphic equation sj = 0. We associate with S the complex line bundle' {IS determined by the system {Sjk of non-vanishing holomorphic functions Sjk = Sj/Sk defined, respectively, in Uj n Uk. We note that the charac- teristic class2 c(ISI) of {SI is the dual of the homology class of the cycle S. For simplicity's sake, we set c(S) = c({S}). Now, let F be an arbitrary complex line bundle over V and let QP(F) be the stack over V of the germs of holomorphic p-forms with coefficients in F. Take a point p e S n U, and choose a system of local coordinates (, ..., ,z) in Uj such that s z= . Then each germ tj e QP(F) can be written in the form '1.s = E 1.ial...ap dzj' ... dzj,p + E nja...a, dz4 dza2 ... dzj0', l<ai<...<aj, l<a2<...<ap where QP(F)P, is the fiber of QP(F) over the point p. With the help of this expression, we associate with qj two germs , v>of holomorphic forms on S defined, respectively, by lj =E I ('bai ..a,)s dz7l .. 1 <al< ..<as, .dzj ?1j~= I ~ .a,)Sdz7 ... dz7' ;7 dz(jla... apSdj2*dzjp 1 <as< ..<av Downloaded by guest on September 25, 2021 1274 MA THEMA TICS: KODAIRA AND SPENCER PRoC. N. A. S. where ( )s denotes restriction to S. Let {fjk} be the system of non- vanishing holomorphic functions defining the bundle F. Assuming that p e U1 n Uk, we infer readily from the relation 7s = jf that ;= (Cfjk)S (1) and that -1'=JjSj)Sflk + aL ?lk J (2) where Olka = Z (flkaa2 . av)SdZk' ... dzak* 1 <a2< ..<ap ' The equality (1) shows that e QP(Fs8) and that r': , ->. is a homo- morphic mapping of the stack QP(F) onto Q'(Fs), where Fs denotes the restriction of F to S. Letting U'P(F) be the kernel of this homomorphism r', we obtain therefore the exact sequence 0 UfP(F) Q(F)UP - > Q(Fs)P - 0. (3) Now, it follows from (2) that, if v; = 0, the relation 'i = (fis )kSk holds. This shows that 7 e P-'1(Fs - {S}s) for any j e U2P(F). More- over rf :,fj -- I is a homomorphic mapping of UPv(F) onto UP-1(FsT {S)s). As one readily infers, rj = s,qj is holomorphic if and only if . vanishes, while i:rj - j = sj,5 is an isomorphic mapping of QP(F - {S}) into Q'll(F). Thus we see thaat the sequence i rw 0 QP(F - {SI) Qt(F)O" QP-'(Fs - {S}s) -40 *(4) is exact. From (3) and (4) we obtain the corresponding exact cohomology sequences: H V-P-l(V (F)) -- H-1(S, QP(Fs)) Hq(V, Q`"(F)) - ... (5) and H-1(V, R"P(F)) Hq-(StH UP-(Fs- {Sjs)) ' Hq2(V, UP(F' IS}))) >*> 6 Downloaded by guest on September 25, 2021 VOL. 39, 1953 MA THEMA TICS: KODAIRA AND SPENCER 1275 We note that Q°'(F) - Q(F - {S)). Thus, in the extremum case p = 0, the above sequences (5), (6) reduce to a single exact sequence i* r'* ~~~~~~~~~~~~6* . .. - PH-1(V, Q°(F)) - Hg-1(S, QP(Fs)) - Hg(V, °(F - S)).... (7) 3. Generalization of a Theorem of Lefschetz.-We say that a complex line bundle F is ample if the characteristic class c(F) of F contains a real d-closed (1, 1)-form3 -y > 0. THEOREM 1. Assume the bundle {S} to be sufficiently ample compared with F in the sense that each cohomology class c(m{S} - F), m = 1, 2, .... n, contains, respectively, a real d-closed (1, 1)-form ym > Ofor which OP (,y,, u, z), 1 _ p, q, < n, are positive definite at each point z on V, where OP "(,y U, z) are the Hermitian forms in u introduced in the previous note.4 Then the restriction map r'*:HQ(V, UP(F)) -*H6(S, QP(Fs)) is an isomorphism into or onto according as p + q = n - 1 or p + q < n-1. Proof: Set Fm = F - m{S}. Then, by virtue of Theorem 1 of the previous note,5 the cohomology group H"(V, f2P(Fm)) vanishes for q _ n - 1, 1 _ m .< n. Hence, replacing F by Fm, we infer from (5) the iso- morphism Hq(S, UP(Fis)) f-Hq+l( V, U'P(Fm)), for q < n - 2, 1 m _ n, (8) where Fms denotes the restriction (Fm)s of Fm to S. Similarly, setting F = Fm-, in (6), we get Hq(V, QZP(Fmi-)) H~g(S UP-'(Fms)), forq_ n - 2, 1 < m n, (9) provided that p > 1 (we assume always p . n - 1). Combining (8) with (9), we obtain Hq(V, O"'P(F)) _HP+-q(V, Q"0(Fp)), forp + q< n - 1, p _ 1, (10) while we have Q°"(Fp) _ QO(Fp+1). By Theorem 1 of the previous note,6 we have Hq(V, UIP(F)) HP+q(V, EP(Fp+1)) = 0, for p + q < n-1. (11) This clearly holds also in the case p = 0. Now, combining (5) with (11), we infer readily that r'* :Hq( V, iP(F)) -- Hq(S, QP(Fs)) is an isomorphism into or onto according as p + q = n - 1 or p + q < n - 1, q. e. d. The above theorem may be regarded as a generalization of a theorem of Lefschetz concerning the homology of hyperplane sections of algebraic Downloaded by guest on September 25, 2021 1276 MA THEMA TICS: KODAIRA AND SPENCER PRoc. N. A. S. varieties. In fact, letting C be the field of complex numbers, we have the commutative diagram7 H8(V,C) E Hq(V,UP) r*I P+q= s Ir* Hs(S, C) _ E Hq(S, UP), p+q=s where UP = QP(O) is the stack over V of germs of holomorphic p-forms (without coefficients in a bundle), PP = UP(Os) the corresponding stack over S and where r* is the canonical restriction map. We therefore infer from Theorem 1 the following: THEOREM 2.8 If the bundle {S} is sufficiently ample in the sense that the cohomology class c(S) contains a real d-closed (1, 1)-form y > 0 for which the Hermitian forms O, q(-y, u, z), 1 < p, q _ n, are positive definite at each point z on V, then the restriction map r*:HR(V, C) -- H(S, C) is an iso- morphism into or onto according as s = n - 1 or s < n - 1. 4. The Lemma of Enrigues-Severi-Zariski.9-The lemma of Enriques- Severi-Zariski may be formulated as follows: Let V be a non-singular algebraic variety imbedded in a projective space and let Sh be the section of V cut out by a general hypersurface of order h. Given a divisor D on V, there exists an integer ho(D), depending on D, such that the isomorphism r'* HO(V, Q°({D})) - HO(Sh, Q°({D}sh)) (12) holds for all h > ho(D). It is clear that this lemma reduces to a special case of our Theorem 1 above. In fact, since the cohomology class c(Sh) contains the real d-closed (1, 1)-form h.w, w = i Ei g,AdzGd2J > 0 being the fundamental form associated with the "standard" K.hler metric on V, we can find an integer ho = ho(D) such that {Sh} is sufficiently ample compared with {D in the sense of the hypothesis of Theorem 1 for all h, ho(D). 5. Characteristic Deficiencies.-In analogy with the case of algebraic varieties,10 we define the deficiency def(F/S) of the bundle F over S by def(F/S) = dim HO(S, fP(Fs)) -dim r'* HO(V, SP(P)). The deficiency def(I{S1S) of {S) over S may be called the characteristic deficiency of S, since def({S}/S) is equal to the deficiency of the charac- teristic system on S of the complete linear system si .