# Kunihiko Kodaira: Collected Works, Volume III

Kunihiko Kodaira
Pages: 480
https://www.jstor.org/stable/j.ctt13x1dxs

1. Front Matter
(pp. i-iv)
(pp. v-x)
3. Vol. III
• On Compact Analytic Surfaces
(pp. 1142-1156)
Kunihiko Kodaira

The present note is a preliminary report on a study of structures of compact analytic surfaces.

1. LetVbe a compact analytic surface, i.e. a compact complex manifold of complex dimension 2. Let$\mathcal{M}(V)$be the field of all meromorphic functions onVand denote by$\dim\mathcal{M}(V)$the degree of transcendency of$\mathcal{M}(V)$over the field C of all complex numbers. By a result due to Chow [2] and Siegel [9], we have dim$\dim\mathcal{M}(V)\leqq 2.$

We denote bypg(V) thegeometric genusofV, byc1the first Chern class ofVand by$c_{1}^{2}(V)$the value of$c_{1}^{2}={{c}_{1}}\cdot {{c}_{1}}$on...

• ON COMPACT COMPLEX ANALYTIC SURFACES, I
(pp. 1157-1198)
K. Kodaira

The present paper is the first part of a study of structures of compact complex analytic surfaces. The main results of this study were announced in a short note¹ which will serve as an introduction to this paper. Naturally that note, written two years ago, does not cover some recent results among which the following would be worth mentioning here:Every compact Kahler surface is a deformation of an algebraic surface. This result was conjectured earlier by Sir William Hodge.²

First we fix our notations. We denote byVa compact complex analytic surface. By acomplex line bundle F...

• A THEOREM OF COMPLETENESS FOR ANALYTIC SYSTEMS OF SURFACES, WITH ORDINARY SINGULARITIES
(pp. 1199-1235)
K. Kodaira

Recently D.C. Spencer and the author proved a theorem of completeness of characteristic systems of complete continuous systems of nonsingular submanifolds of co-dimension 1 of ambient spaces, under the assumption that the submanifolds are semi-regular (see Kodaira and Spencer [2]). The purpose of the present paper is to derive an analogous result for continuous systems of surfaces with ordinary singularities in ambient spaces of dimension 3. Our main theorem may be stated as follows:Let V be a surface, with ordinary singularities only, imbedded in a3-dimensional compact complex manifold W, and letΨbe the sheaf of infinitesimal displacements...

• A THEOREM OF COMPLETENESS OF CHARACTERISTIC SYSTEMS FOR ANALYTIC FAMILIES OF COMPACT SUBMANIFOLDS OF COMPLEX MANIFOLDS
(pp. 1236-1252)
K. Kodalra

Recently D. C. Spencer and the author proved a theorem of completeness of characteristic systems of complete continuous systems of compact submanifolds of co-dimension 1 of ambient spaces.¹ The purpose of the present note is to prove a similar theorem of completeness² for analytic families of compact submanifolds of an arbitrary co-dimension of ambient spaces. A formal aspect of the theorem has been studied earlier by A. Haefliger.³

In what follows, we assume that all manifolds under consideration are paracompact and connected. LetWbe a complex manifold of (complex) dimensiond+r. We denote a point inW...

• ON STABILITY OF COMPACT SUBMANIFOLDS OF COMPLEX MANIFOLDS.
(pp. 1253-1268)
K. Kodaira

There are two aspects of stability of compact submanifolds of complex manifolds. LetVbe a compact complex analytic submanifold of a complex manifoldW. We shall say thatVis astrongly stableor arigidsubmanifold ofWif no small deformation of the complex structure ofWchanges the complex structure of aneighborhood of V in W. Alternatively we callV a weakly stableor simply astablesubmanifold ofWif no small deformation of the Complex structure ofWmakesVdisappear (see Definition 1 below). The rigidity of compact submanifolds of complex manifolds...

• ON COMPACT ANALYTIC SURFACES; II–III
(pp. 1269-1372)
K. Kodaira

This paper is the second part of a study of structures of compact analytic surfaces of which the main results were announced in a short note.¹ In the first part² we have shown that any compact analytic surface of algebraic dimension 1 is a fibre space of elliptic curves over a nonsingular algebraic curve. The present paper deals with the structure of fibre spaces of elliptic curves over non-singular algebraic curves.

LetVdenote an analytic surface. A curve Θ onVis called anexceptional curve³ if Θ is a non-singular rational curve with (Θ2) = −1. The following...

• ON THE STRUCTURE OF COMPACT COMPLEX ANALYTIC SURFACES
(pp. 1373-1376)
K. Kodaira

The purpose of this note is to outline our recent results on the structure of compact complex analytic surfaces. Details will be published elsewhere. Our proof of the results is based on the Riemann-Roch theorem of which the complete form has been established recently by M. F. Atiyah and I. M. Singer.¹

1. We denote byZthe ring of rational integers and by C the field of complex numbers. By asurfacewe shall mean a compact complex analytic surface free from singularities. We consider a surfaceSand denotebνtheν-th Betti number ofSand bycv...

• ON THE STRUCTURE OF COMPACT COMPLEX ANALYTIC SURFACES, II
(pp. 1377-1381)
K. Kodaira

This note is a continuation of our previous report¹ on the structure of compact complex analytic surfaces.

5. We shall employ the notation of our previous report. Thus we denote bySa surface and bybν, cν, pg, q, …, respectively, theνth Betti number, theνth Chern class, the geometric genus, the irregularity,… ofS. Any complex line bundle over a regular surface is determined uniquely by its Chern class. Hence, a regular surface is aK3surface if and only if its first Chern class vanishes. It follows thatany deformation of a K3surface is a K3surface. In...

• ON THE STRUCTURE OF COMPACT COMPLEX ANALYTIC SURFACES
(pp. 1382-1388)
K. Kodaira

By asurfacewe shall mean a compact complex manifold of complex dimension 2. We fix our notation as follows.

S: a surface,

bν: theν-th Betti number ofS,

cν: theν-th Chern class ofS,

$\mathcal{O}$: the sheaf overSof germs of holomorphic functions,

$q=\dim{{H}^{1}}(S,\mathcal{O})$: the irregularity ofS,

${{p}_{g}}=\dim{{H}^{2}}(S,\mathcal{O})$: the geometric genus ofS.

Note thatc12andc2are (rational) integers.

By a theorem of Grauert [2], any surface is obtained from a surface containing no exceptional curve (of the first kind) by means of a finite number of...

• ON THE STRUCTURE OF COMPACT COMPLEX ANALYTIC SURFACES, I.
(pp. 1389-1436)
K. Kodaira

The present paper is a continuation of a series of our previous papers [8], [9]. The main results of this paper were announced in two short notes [10], [11].

We call a compact complex analytic surface free from singularities simply asurface. In our previous paper [8], we have shown that any surface with non-constant meromorphic functions is either an algebraic surface or an elliptic surface and, in [9], we have examined the structure of elliptic surfaces in detail.

In Section 1 of the present paper we establish certain relations between several numerical characters, e. g., the geometric genus, the...

• ON CHARACTERISTIC SYSTEMS OF FAMILIES OF SURFACES WITH ORDINARY SINGULARITIES IN A PROJECTIVE SPACE.
(pp. 1437-1466)
K. Kodaira

In our previous paper [5] we have proved a theorem of completeness of characteristic systems for analytic families of surfaces with ordinary singularities in ambient threefolds. In this paper we examine the application of the theorem to surfaces with ordinary singularities in a projective 3-space. The theorem asserts that the characteristic systems of a maximal analytic family of surfaces with ordinary singularities in an ambient threefold are complete if the surfaces are semi-regular. We show, in Section 2, that, in a projective 3-space, the semi-regularity coincides with the regularity and that the regularity is equivalent to the linear independence of...

• COMPLEX STRUCTURES ON S1 X S3
(pp. 1467-1470)
K. Kodaira

1. By asurfacewe shall mean a complex manifold of complex dimension 2. Let C2denote the space of two complex variables (z1,z2) and letWbe the domain C2– (0,0). Moreover, letZbe the infinite cyclic group generated by a complex analytic automorphismgofWof the form

$$g:({{z}_{1}},{{z}_{2}})\to ({{z}_{1}}^{\prime },{{z}_{2}}^{\prime })=(s{{z}_{1}}+\lambda {{z}_{2}}^{m},t{{z}_{2}}),$$

wheremis a positive integer ands,t, λ are constants such that

$$0\textless\left| s \right|\leqq \left| t \right|\textless 1,\quad ({{t}^{m}}-s)\lambda =0.$$

Obviously,Zis a properly discontinuous group of complex analytic automorphisms ofWfree from fixed points, and the quotient surfaceW/Zis a compact surface which is topologically homeomorphic to S1...

• ON THE STRUCTURE OF COMPACT COMPLEX ANALYTIC SURFACES, II.
(pp. 1471-1510)
K. Kodaira

In Part I of this paper (reference [5]) we have introduced a classification of surfaces. In this Part II we are mainly concerned with surfaces of class VII0. First, in Section 8, we show that a surface belongs to the class VII if and only if its first Betti number is equal to 1 (Theorem 26). Then, in Section 9, we determine all elliptic surfaces of class VII0(Theorem 27). By a Hopf surface we shall mean a surface of which the universal covering manifold is (complex analytically homeomorphic to)C2O, whereC2is the space of two...

• A CERTAIN TYPE OF IRREGULAR ALGEBRAIC SURFACES
(pp. 1511-1519)
K. KODAIRA

By acomplex4-manifoldwe shall mean a complex manifold oftopologicaldimension 4. For any complex 4-manifoldMwe letcν(M) denote theνthChern class ofM. By the Hirzebruch index theorem, the index ofMis equal to

$$\tau (M)=\frac{1}{3}[c_{1}^{2}(M)-2{{c}_{2}}(M)].$$

A. Van de Ven has pointed out in connection with his recent results(1) that there are not many known examples of compactconnectedcomplex 4-manifolds with positive indices. The purpose of this note is to exhibit a series of compact connected complex 4-manifoldsMn,m, n, m= 2, 3, 4, …, with positive indices. Each complex 4-manifoldMn,m...

• PLURICANONICAL SYSTEMS ON ALGEBRAIC SURFACES OF GENERAL TYPE
(pp. 1520-1524)
K. Kodaira

1. By a minimal nonsingular algebraic surface of general type we shall mean a nonsingular algebraic surface free from exceptional curves (of the first kind) of which the bigenusP2and the Chern numberc12are both positive, wherec1denotes the first Chern class of the surface. We remark that if either the bigenusP2or the Chern numberc12of a nonsingular algebraic surface free from exceptional curves is nonpositive, then the surface is one of the following five types of surfaces: projective plane, ruled surface,K3 surface, abelian variety, elliptic surface.¹ In this note, we letSdenote...

• ON THE STRUCTURE OF COMPACT COMPLEX ANALYTIC SURFACES, III.
(pp. 1525-1553)
K. Kodaira

In this paper, the third in a series (see [6]) under the same title, we study Hopf surfaces. First, in Section 13, we prove that a surfaceSis a Hopf surface if the second Betti number ofSvanishes and if the fundamental group ofScontains an infinite cyclic subgroup of finite index (Theorem 41). This gives a topological characterization of Hopf surfaces.

In Section 14, we consider degenerate forms of Hopf surfaces. LetR(m)denote a projective line bundle over a projective line which is a compactification of a complex line bundle over a projective line, where...

• Pluricanonical systems on algebraic surfaces of general type
(pp. 1554-1576)
Kunihiko Kodaira

By a minimal non-singular algebraic surface of general type we shall mean a non-singular algebraic surface free from exceptional curves (of the first kind) of which the bigenusP2and the Chern number$c_{1}^{2}$are both positive, wherec1, denote the first Chern class of the surface (see § 3). LetSdenote a minimal non-singular algebraic surface of general type defined over the field of complex numbers and letKbe a canonical divisor onS. The number of non-singular rational curvesEonSsatisfying the equation:KE= 0 is smaller than the second Betti number of,...

• ON THE STRUCTURE OF COMPLEX ANALYTIC SURFACES, IV.
(pp. 1577-1595)
K. Kodaira

In Part I of this paper we have introduced a classification of surfaces (see [9]). In this part we incorporate with this classification some classical results on algebraic surfaces due to Italian geometers such as Enriques’ criterion for ruled surfaces and Castelnuovo’s criterion of rationality (see Enriques [5]). We formulate the classical results with which we are concerned in Theorems 48-53 in Section 16. Rigorous proofs of these theorems are found in a recent monograph (Šafarevič [12]) in which a group of Russian mathematicians has expounded the theory of classification of algebraic surfaces. Most of their proofs are based on...

• On Homotopy K 3 Surfaces
(pp. 1596-1607)
Kunihiko Kodaira

By asurfacewe shall mean a compact complex manifold of complex dimension 2. A surface is said to be regular if its first Betti number vanishes. AK3surfaceis defined to be a regular surface of which the first Chern class vanishes. EveryK3 surface is diffeomorphic to a nonsingular quartic surface in a complex projective 3-space (see [1], Theorem 13). Thus there is a unique diffeomorphic type ofK3 surface.By a homotopy K3surface we mean a surface of the oriented homotopy type of K3surface. The purpose of this paper...

• HOLOMORPHIC MAPPINGS OF POLYDISCS INTO COMPACT COMPLEX MANIFOLDS
(pp. 1608-1622)
K. KODAIRA

In this paper we prove an inequality in the manner of the Nevanlinna theory expressing certain properties of holomorphic mappings ofn-dimensional polydiscs into compact complex manifolds of the same dimension and discuss some of its applications.

1. LetWbe a compact complex manifold of dimensionn. For a pointwinW, we denote a local coordinate ofwby (w1,w2, …,wn). Take a complex line bundleLoverW. By a theorem of de Rham, the Chern classc(L) ofLcan be regarded as ad-cohomologyclassofd-closed 2-forms onW. We say that...

4. Back Matter
(pp. 1623-1623)