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', lof holomorphic forms on S defined, respectively, by

lj =E I ('bai ..a,)s dz7l .. 1

;7 dz(jla... apSdj2*dzjp 1

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

' 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 . THEOREM 3.11 The characteristic deficiency def( {S}/S) of a non-singular prime divisor S on V is never greater than the number gi of the linearly inde- pendent simple differentials of the first kind attached to V, and the equality Downloaded by guest on September 25, 2021 VOL. 39, 1953 MA THEMA TICS: KODAIRA AND SPENCER 1277 def({S}/S) = gi holds if the bundle {S) is sufficiently ample in the sense that the cohomology class c(S) + cl contains a real d-closed (1, 1)-form 7y> 0, where cl is the first Chern class of V. Proof: Setting F = IS} in (7), we get the exact sequence r'* ,* ... IIH`(V, °({SI)) - I (S, fp({SIs)) ) H1(V,V -P) H1(V, Q°({S})) -- .... (13) By a result of Dolbeault,'2 dim H'(V, Q°) = gi, while Theorem 3 of the previous note asserts that H'(V, Q°({S})) vanishes if c(S) + cl contains 'y> 0. Hence, from (13), we conclude our theorem. 6. Remarks.-The Euler characteristic xv(e) of V with coefficients in a stack Ei over V is defined by Xv(e) = E(-1) dim Hq( V, ) provided that each cohomology group Hq(V, (5) has a finite dimension. Setting XO (F) = Xv(Qv(F)), we infer from (5) and (6) the equalities

XI (F) = xv(Q"p(F)) + XI(Fs), Xv(f2P(F)) = xP(F - {SI) + XP71(Fs - ISIs), and consequently

XP (F) = XO (F - IS)) + XP(Fs) + XP '(Fs - {S}I). (14) In this connection we add here a remark on our note concerning arithmetic genera of algebraic varieties.13 Since, as (2) shows, 7j is not invariantly defined unless tj = 0, the mapping qj - tj + nJ does not give the iso- morphism W2P(F)1"(F - IS)) - Qi(Fs) + £P-1(Fs- ISIS) except for the case p = 0 or p = n. Thus the formula (13) in our note14 cannot be justified, whereas the formula (16) of fundamental importance, which we derived from (13), still holds, since (16) reduces to a special case of our equality (14) above. A similar remark applies to the note.16 * This work was supported by a research project at Princeton University sponsored by the Office of Ordnance Research, U. S. Army. 1 For the relation between divisors and complex line bundles, see K. Kodaira and D. C. Spencer, "Divisor Class Groups on Algebraic Varieties," these PROcEEDINGS. 39, 872-877 (1953). 2 See Kodaira and Spencer, loc. cit. 3By a d-closed (p, q)-form we shall mean a (p + q)-form of type (p, q) which is closed under d. Cf. Kodaira, K., "On a Differential-Geometric Method in the Theory of Analytic Stacks," these PROCEEDINGS, 39, 1268-1273 (1953). Downloaded by guest on September 25, 2021 1278 MA THEMA TICS: PEARSON AND VANDIVER PRoc. N. A. S.

Kodaira, loc. cit. Kodaira, loc. cit. 6 Kodaira, loc. cit. I This follows from a result of Dolbeault, P., "Sur la cohomologie des variete ana- lytiques complexes," Compt. rend., Paris, 236, 175-177 (1953). 8 Lefschetz, S., L'Analysis situs et la g6ometrie algebrique, Gauthier-Villars, Paris, 1924, pp. 88-91. Zariski, O., "Complete Linear Systems on Normal Varieties and a Generalization of a Lemma of Enriques-Severi," Ann. Math., 55, 552-592 (1952). 10 See, for example, Zariski, O., loc. cit. 11 This theorem may be regarded as a generalization of a "fundamental" theorem for algebraic surfaces to the effect that an algebraic surface of irregularity q possesses exactly q independent simple differentials of the first kind. See Zariski, Algebraic Sur- faces, Springer, Berlin, 1935, p. 123. 12 Dolbeault, loc. cit. 13 Kodaira, K., and Spencer, D. C., these PROCEEDINGS, 39, 641-649 (1953). 14 Kodaira and Spencer; loc. cit. in ref. 14, p. 645. 16 Spencer, D. C., "Cohomology and the Riemann-Roch Theorem," these PRO- CEEDINGS, 39, 660-669 (1953).

ON A NEW PROBLEM CONCERNING TRINOMIAL CONGRUENCES INVOLVING RATIONAL INTEGERS By ERNA H. PEARSON AND H. S. VANDIVER

ATLANTA, GEORGIA; AND DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TEXAS Communicated October 5, 1953 As a background for the study of the topic mentioned in the title, we first consider more general problems. Let

CIXJI + ...+ C"8 a= C, (1) where the c's are given elements in a finite field of order pn, p prime, none zero, the m's divide p' - 1 with the x's to be determined in this field, none zero. For c $ 0 and s = 1 it is known that there exist fields of this type where the equation has no solutions. On the other hand,' if s 2 2 with c $ 0 and s > 2 for c = 0, we know that the equation always has at least k solutions in non-zero x's, for k any given positive integer, provided pn exceeds a certain limit. The passage from the case s = 1, c 5 0 to the case s = 2, c $ 0 is a fundamental one. Consideration of the first case for various values of p involves the use of congruences with respect to a prime ideal in an algebraic field and leads into the theory of class fields and laws of reciprocity. In the second case, however, different types of developments have arisen, particularly having Downloaded by guest on September 25, 2021