m22 are rational integers having no common divisor and let e = Zmej. Then, using c2 = 24, b2 = 22, b+= 3, b= 19. (8)

Transcription

1 ON THE STRUCTURE OF COMPACT COMPLEX ANALYTIC SURFACES, II* BY K. KODAIRA DEPARTMENT OF MATHEMATICS, JOHNS HOPKINS UNIVERSITY Communicated by D. C. Spencer, April 13, 1964 This note is a continuation of our previous report' on the structure of compact complex analytic surfaces. 5. We shall employ the notation of our previous report. Thus we denote by S a surface and by b,, CP, pa, q,..., respectively, the vth Betti number, the Ath Chern class, the geometric genus, the irregularity,... of S. Any complex line bundle over a regular surface is determined uniquely by its Chern class. Hence, a regular surface is a K3 surface if and only if its first Chern class vanishes. It follows that any deformation of a K3 surface is a K3 surface. In this section we shall outline a proof of the following theorem which has been conjectured earlier by A. Weil and independently by A. Andreotti.' THEOREM 9. Every K3 surface is a deformation of a nonsingular quartic surface in a projective 3-space. (i) Assume that S is a K3 surface. Then, using Theorem 1 and the formula (6), we get c2 = 24, b2 = 22, b+= 3, b= 19. (8) Let { rf,..., F,,..., r22} be a Betti base of 2-cycles on S and, for any cohomology class c C H2(S, Z), let c(ru) denote the value of c on ri. We choose ej C H2(S, Z) such that ej(rk) = a for j, k = 1, 2,..., 22 and define A (x, y) = ajkxjyk, aik = ejek. There exists on S a nonvanishing holomorphic 2-form 4w. Following Weil and Andreotti we associate with S the point X X(S) (Xi, = =.... X,... * *22), is = frat in a projective space p21 of dimension 21. We have A(XX) fs = A,t = 0. Thus, X is on the hypersurface M in p21 defined by the quadric equation: A(x, x) = 0. This result is due to Weil and Andreotti.' LEMMA 4. A cohomology class c C H2(S, Z) is the Chern class of a complex line bundle over S if and only if the point m = (ml,...,im,,...m22), mj = c(rj), satisfies the linear equation: A (X, m) = 0, X = X(S). LEMMA 5. If the linear equation: A (X, m) = 0, X = X(S), admits one and only one rational solution m E P21 and if, moreover, that solution m satisfies the quadric equation: A (m, m) = 0, then S is an analytic fiber space of elliptic curves over a projective line A of which the singular fibers4 are either of type I, or of type II. Proof: We choose homogeneous coordinates mj of m such that ml,..., mi,, m22 are rational integers having no common divisor and let e = Zmej. Then, using Lemma 4, we infer from the uniqueness of the rational solution m, that, for any 1100

2 VOL. 51, 1964 MATHEMATICS: K. KODAIRA 1101 complex line bundle F over S, there exists an integer n such that c(f)-ne is an element of a finite order. Hence, we get (F2) = n2a (m, m) = 0. (9) This proves that S is nonalgebraic. By Lemma 4 there exists a complex line bundle E over S with c(e) = e. Using the Riemann-Roch inequality (7) we obtain dim H0(S, 0(E)) + dim H0(S, o(-e)) _ 2, while E is nontrivial. Hence, we infer the existence of nonconstant meromorphic functions on S. Consequently, S is an analytic fiber space of elliptic curves over a nonsingular algebraic curve A. Since h1,0 vanishes, A is rational. Since the canonical bundle of S is trivial, S has no multiple singular fiber. Moreover, by (9), S contains no curve C with (C2) 5 0. Therefore the singular fibers of S are either of type I, or of type II, q.e.d. (ii) In this subsection we assume that S is an elliptic K3 surface of which the singular fibers are either of type I, or of type II. The base curve A of S is a projective line. We indicate a point on A by its nonhomogeneous coordinate u. We denote by 0(u) the fiber of S over u and by <g = cg(u) the functional invariant5 of S. Suppose that S has j singular fibers C(a1),..., C(aj) of type I, and r singular fibers Then we have6 C(,r*),.C('r) of type II. j + 2r =- C2 = 24. There are certain relations7 between the type of C(u) and the behavior of <g at u. We infer from those relations that r is not greater than 8, that al,..., aj are simple poles of 9 and that r Q/(9-1) = a H (u-tv) II (u-r,)3/ HI (U-,.)22 (10) v = 1 v=r+1r + 1 where a 5 O. T,i,# v;} andr,pv Tfor r) v < X < r. S belongs to the family8 5(g, G), where G denotes the homological invariant of S. Let B be the basic member of 5({g, G). The cohomology group H2(A, G) vanishes. Hence, (9J, G) consists of the elliptic surfaces9 Bh*(8), s E H'(A, 9(f)). Consequently, S is a deformation of B. (iii) Let p2 denote a projective plane on which a system of homogeneous coordinates (x,y,z) is fixed. Take two copies p2 X Co and p2 X C1 of p2 X C and form their union W = p2 X Co U P2 X C1 by identifying (x,y,z,u) E p2 X Co with (xi, yi, z1, u1) E p2 X C1 if and only if UUa = 1, U4X1 = x, u6y, = y, zi = z. Note that W is a complex analytic fiber bundle of projective planes over the projective line A = Co U C1. For any point r = (To, T1,..., T8, 01,..., UJ12) in the space C21 of 21 complex variables, we set 8 12 g(u) = rdfl(u - Tv), h(u) = H(u - t = I(U)3/ [(u)3-27h1(2 2], af1 vbb and define a subvariety BT of W by the equations.

3 1102 MATHEMATICS: K. KODAIRA PROC. N. A. S. y2z - 4x3 + g(u)xz' + h(u)z' = 0, yi Iz1-4x,1+ Ul8g(1/U,)XZ,2 + u112h(1/u,)z,3 = 0. B. is an analytic fiber space of elliptic curves over A of which the functional invariant is gje and the singular fibers are either of type I, or of type II, provided that r satisfies the following conditions: (a) To # 0; (b) if Ad, = r,, u $ 0, then a. # for v $ X; (c) gjt has no multiple pole. Moreover, B, has a global holomorphic section over A defined by the equations: x = z = xi = zi = 0. Clearly, BT is a deformation of Ba, a = (1,1,..., 1,0,..., 0). Now we let r = (3a'3, T1,...., T8, TJ..., Tr, 0r+1b..., T12). Then S, coincides with 9, and therefore B, coincides with the basic member10 B of 3Y(gJ, G). Consequently, B is a deformation of Ba. (iv) Let S be an arbitrary K3 surface. Denoting by 0 the sheaf over S of germs of holomorphic vector fields, we have dim H'(S, 0) = h'l = 20, dim H2(S, 0) = h' 2 = 0. Hence, we infer the existence" of an effectively parametrized complex analytic family of small deformations St, t C U, of S, where U denotes a domain in the space C20 of 20 complex variables. Assuming that the surfaces St, t E U, have one and the same underlying differentiable manifold X, we define X(St) by means of a Betti base { rj} of 2-cycles fixed on X. It has been shown by A. Weil and independently by A. Andreotti that t -- X(St) is a biholomorphic map of U onto an open subset of M. Moreover, by (8), the signature of the quadric form A (x, x) is (3, 19). Hence, we can find a point t E U such that X = X (St) satisfies the hypotheses of Lemma 5. The corresponding surface St is then an elliptic K3 surface whose singular fibers are either of type I, or of type II. Therefore, by the results of (ii) and (iii), St is a deformation of Be. Hence, S is a deformation of Ba. This result implies that any nonsingular quartic surface in a projective 3-space is a deformation of B,. Consequently, every K3 surface is a deformation of a nonsingular quartic surface in a projective 3-space. 6. Let C2 denote the space of two complex variables zi and Z2. THEOREM 10. If the canonical bundle of S is trivial, then S is a K3 surface, a complex torus, or an elliptic surface of the form C2/G, where G is a properly discontinuous nonabelian group of affine transformations without fixed points of C2 which leave invariant the 2-form dz, A dz2. The first Betti number of C2/G is equal to 3. We shall outline a proof of this theorem. By hypothesis the first Chern class cl of S vanishes. Hence, by a result of Atiyah and Hirzebruch,12 the Todd genus of S is even, while the geometric genus p0 of S equals 1. Consequently, by (6), the irregularity q of S is even and 12q - 2b, + b+ b- = 22. In case bi is even, we have, by Theorem 1, b+ = 3, bi = 2q, and therefore 8q + b- = 19. Hence, q is equal to either 0 or 2. If q = 0, then S is a K3 surface. If q = 2, then S is a complex torus. In case bi is odd, we have b+ = 2, 2q = b,+ 1 > 1, and 8q + b- = 18. Hence, q equals 2, and b, equals 3. In view of Theorem 6, S is therefore an elliptic surface. Moreover, we infer in the same manner as in section 3 (ii) that the universal cover-

4 VOL. 51, 1964 MATHEMATICS: K. KODAIRA 1103 ing surface of S is 02, and the covering transformation group G of 02 is generated by the affine transformations (Z1, Z2) -* (Z1 + az2 + p3, Z2 + a,), v = 1,2,3,4, of which the coefficients satisfy the condition that a3 = a4 = 0,?33 # 0. lac2-2al = m34 $ 0, where m is a rational integer. 7. Let S be a surface free from exceptional curve, and let bi, pa, and cl denote, respectively, the first Betti number, the geometric genus, and the first Chern class of S. LEMMA 6. If p, is positive, then c12 is nonnegative. LEMMA 7. If b1 is even, p, > 0, and c1 = 0, then the canonical bundle of S is trivial. LEMMA 8. If p, is positive, C12 = 0, and cl $ 0, then S is an elliptic surface. Proof: If pa > 0 and if cl $ 0, then, in view of Theorem 6, S is either algebraic or elliptic. On the other hand, in case S is algebraic, this lemma is reduced to a classical result of Italian geometers.13 Class bi p9 Cl C12 Structure I Even 0 Algebraic II 0 + =0 0 K3 surfaces III 4 + = 0 0 Complex tori IV Even + $ 0 0 Elliptic V Even + + Algebraic VI Odd + 0 Elliptic VII 1 0? With the aid of these lemmas we derive from Theorems 2, 5, 6, 8, and 10 the following theorem (see above table). Note that an elliptic surface is free from exceptional curve if and only if its Chern number c12 vanishes. THEOREM 11. Surfaces free from exceptional curves can be classified into the following seven classes: I. the class of algebraic surfaces free from exceptional curves with p, = 0; II. the class of K3 surfaces; III. the class of complex tori (of complex dimension 2); IV. the class of elliptic surfaces with b1=- 0(2), p0 > 0, and c12 = 0; V. the class of algebraic surfaces free from exceptional curves with p, > 0 and C12 > 0; VI. the class of elliptic surfaces with b1= 1(2), p, > 0, and c12 = 0; and VII. the class of surfaces free from exceptional curves with bi = 1 and p, = 0. * This work was partly supported by the National Science Foundation grant G7030. l Kodaira, K., "On the structure of compact complex analytic surfaces," these PROCEEDINGS, 50, (1963). 2 Compare Grauert, H., "On the number of moduli of complex structures," in Contributions to the Function Theory (Tata Institute of Fundamental Research, 1960). 3 Ibid. 4 Kodaira, K., "On compact analytic surfaces II-III," Ann. of Math., 77, ; 78, 1-40 (1963), Theorem Ibid., section 7. 6 Ibid., Theorem Ibid., section 9, Table 1.

5 1104 BIOCHEMISTRY: H. T. MILES PROC. N. A. S. 8Ibid., Definition 8.1. 'Ibid., section Ibid., Theorem '1 Kodaira, 'K., L. Nirenberg, and D. C. Spencer, "On the existence of deformations of complex analytic structures," Ann. of Math., 68, (1958). 12 Atiyah, It. F., and F. Hirzebruch, "Riemann-Roch theorems for differentiable manifolds," Bull. Amer. Math. Soc., 65, (1959). 13 Enriques, F., Le Superficie Algebriche (Bologna, 1949). THE STRUCTURE OF THE THREE-STRANDED HELiX. POLY (A + 2U)* BY H. TODD MILES NATIONAL INSTITUTE OF ARTHRITIS AND METABOLIC DISEASES, NATIONAL INSTITUTES OF HEALTH, BETHESDA, MARYLAND Communicated by Norman Davidson, April 2, 1964 The interaction of polynucleotides to form helical structures results in characteristic changes in the infrared spectra1-4 in aqueous (D20) solution. Although the origin of these spectral changes is complex and not at present understood in detail,1' 5 5a it appears from all observations to date that when a carbonyl group in a nucleotide base is hydrogen-bonded to another polynucleotide chain in a helix, there is an increase in the vibrational frequency of that carbonyl group. We now propose this frequency shift as an empirical criterion of interbase hydrogen bonding in a helix, and apply the criterion in support of the structure proposed for poly (A + 2U) based on the hydrogen bonding scheme I. A stereochemically satisfactory threedimensional model of the proposed structure has been constructed. The essential conclusion which we derive from the infrared spectrum of the threestranded helix (Fig. 1) is that the uracils in the two poly U strands are hydrogenbonded to the adenine in the poly A strand by different oxygen atoms. This conclusion requires only that the 1691 cm-1 band and the 1657 cm-' band of uridine involve the vibrations of different oxygen atoms and is independent of which assignment is made of the bands. A correct assignment is important, however, for the two-stranded helix, poly (A + U). From synthesis and spectroscopic observation of uridine-4-o8 we have obtained evidence that the two bands in question do have predominant contributions from the vibrations of different oxygens and that the higher frequency band has a major contribution from a C2-O vibration and the 1657 cm-1 band from a C4-O vibration. Materials and Methods.-The infrared spectra of D20 solutions were measured with a Beckman IR-7 spectrophotometer using matched CaF2 cells of fixed path length. 14 The spectra were digitized at 1.25 cm-' intervals with a Gerber X-Y reader and normalized to an extinction coefficient basis with a Honeywell-800 computer, as described in a recent communication. Uridine-4-O'8 was prepared by nitrous acid deamination of cytidine6 in 91 per cent D2018. Because of isotope dilution by exchange between solvent and the nitrous acid oxygens the 018 content of the uridine was estimated to be -82 per cent. Paper chromatography in two solvent systems (2-propanol-HCl and 1-butanol-