Kunihiko Kodaira: Collected Works, Volume II

Kunihiko Kodaira
Pages: 508
https://www.jstor.org/stable/j.ctt13x1310

1. Front Matter
(pp. i-iv)
(pp. v-x)
3. 31 ON ARITHMETIC GENERA OF ALGEBRAIC VARIETIES
(pp. 648-656)
K. Kodaira and D. C. Spencer

1.Introduction.—There are two distinct ways in which arithmetic genera of algebraic varieties may be defined. Let Mnbe an irreducible non-singular algebraic variety of dimensionnimbedded in a complex pro jective spaceSand letEbe a general hyperplane section of Mn. Then, for an arbitrary divisorDon Mn, there exists a polynomial v(h; D) inhof degreensuch that

dim |D + hE| = v(h, D), for largeh,

where |D + hE| denotes the complete linear system on Mnconsisting of all effective divisors which are linearly equivalent to D +...

4. 32 ON COHOMOLOGY GROUPS OF COMPACT ANALYTIC VARIETIES WITH COEFFICIENTS IN SOME ANALYTIC FAISCEAUX
(pp. 657-660)
K. Kodaira

LetVbe a compact complex analytic variety of complex dimensionnand letFbe an analytic fibre bundle overVwhose fibre is a complex line, i.e., the complex number fieldCand whose structure group is the multiplicative group C* of complex numbers acting onC.Moreover let${\Omega ^v}\left( F \right)$be the faisceau¹ overVof germs of holomorphicp- forms with coefficients inF.The faisceau${\Omega ^v}\left( F \right)$introduced recently by D. C. Spencer and, independently by J.-P. Serre turned out to be of importance to applications of faisceaux to the theory of compact analytic varieties. However, for...

5. 33 GROUPS OF COMPLEX LINE BUNDLES OVER COMPACT KÄHLER VARIETIES
(pp. 661-665)
K. Kodaira and D. C. Spencer

1.Introduction.—This is a preliminary report of a generalization of divisor class groups on algebraic varieties to groups of complex line bundles over compact Kahler varieties. In the case of algebraic varieties, a divisor class defines a complex line bundle whose structure group is the multiplicative group of complex numbers. Therefore, on Kahler varieties, we replace the notion of divisor classes by that of complex line bundles, and determine the structure of the group of such complex line bundles. In this way we are led to the definition of Picard varieties attached to Kahler varieties.

2.Groups of Complex...

6. 34 DIVISOR CLASS GROUPS ON ALGEBRAIC VARIETIES
(pp. 665-670)
K. Kodaira and D. C. Spencer

In a previous paper¹ we have determined the structure of the group of complex line bundles over a compact Kahler variety. In this paper we show that for an algebraic variety this group is isomorphic to the divisor class group, and we obtain, in particular, a new proof of the Lefschetz-Hodge theorem concerning algebraic cycles. Moreover, we prove Igusa’s first and second duality theorems. We continue with the notation of the previous paper.

LetVbe a non-singular algebraic variety of dimensionnimbedded in a projective space. As was shown in Section 2 of the previous paper, every divisor...

7. 35 ON A DIFFERENTIAL-GEOMETRIC METHOD IN THE THEORY OF ANALYTIC STACKS
(pp. 671-676)
K. Kodaira

1.Introduction–Let V be a compact Kähler variety of complex dimensionnand letFbe a complex line bundle¹ over V whose structure group is the multiplicative group of complex numbers. Moreover let Ω˚(F) be the stack (faisceau) over V of germs of holomorphicP-forms with coefficients in F and let H˚(V; Ωp(F)) be the qth cohomology group of V with coefficients in It is important for applications to determine the circumstances under which the cohomology group Hq(V; Ωp(F)) vanishes. In the present note we shall prove by a differential-geometric method due to Bochner² some sufficient conditions for...

8. 36 ON A THEOREM OF LEFSCHETZ AND THE LEMMA OF ENRIQUES-SEVERI-ZARISKI
(pp. 677-682)
K. Kodaira and D. C. Spencer

1. Introduction.—The present note is mainly concerned with a generalization 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 Kähler 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.—LetVbe a compact Kähler variety of complex dimensionnand letSbe a non-singular analytic subvariety ofVof complex dimensionn— 1. Take a sufficiently...

9. 37 ON KÄHLER VARIETIES OF RESTRICTED TYPE
(pp. 683-686)
K. Kodaira

1. This is a preliminary report of a paper¹ concerning Kähler varieties of restricted type. A compact complex analytic varietyVof complex dimensionn≥ 2 is called aKähler variety oj restricted type² or aHodge variety³ ifVcarries a Kähler metric ds² = 2 Σ gαβ (dzαβ) such that the associated exterior form ω = i Σ gαβ dzαβbelongs to the cohomology class of anintegral2-cocycle onV. In what follows, such a metric will be called aHodge metric.It is well known that every non-singular algebraic variety in a...

10. 38 ON KÄHLER VARIETIES OF RESTRICTED TYPE (AN INTRINSIC CHARACTERIZATION OF ALGEBRAIC VARIETIES)
(pp. 687-707)
K. Kodaira

A compact complex analytic varietyVis called aKähler variety of restricted type¹ or aHodge variety² ifVcarries a Kähler metric ds² = 2 Σgαβ(dzαβ) such that the associated exterior form ω = i Σgαβdzαβbelongs to the cohomology class of anintegral2-cocycle onV. In what follows such a metric will be called aHodge metric.The main purpose of the present paper is to prove thatevery Hodge variety is (bi-regularly equivalent to) a¹ non-singular algebraic variety imbedded in a projective space.³ Since every non-singular algebraic variety in a projective llpace carries a Hodge...

11. 39 SOME RESULTS IN THE TRANSCENDENTAL THEORY OF ALGEBRAIC VARIETIES.
(pp. 708-714)
Kunihiko Kodaira

1.Introduction.Recently the theory of analytic sheaves, developed mainly by H. Cartan¹, has been successfully applied to algebraic geometry by J.-P. Serre². This theory is independent of potential theory. On the other hand it is possible to develope a theory of analytic sheaves on compact complex manifolds by a potential theoretic method. The present note is a survey of some results in this direction and their applications to algebraic geometry.

2.Complex line bundles³. LetMbe a compact complex manifold of complex dimensionn. By acomplex line bundle FoverMwe shall mean an analytic fibre...

12. 40 CHARACTERISTIC LINEAR SYSTEMS OF COMPLETE CONTINUOUS SYSTEMS.
(pp. 715-743)
K. Kodaira

LetVbe a non-singular projective variety and letmbe a complete continuons system i.e·, a maximal algebraic system of effective divisorsDonV. Then, lettingMbe the parameter variety ofm, we have a one-to-one algebraic correspondence A → D = DλbetweenMandm. Suppose that a member C = Doofmis a non-singular prime divisor and that the corresponding pointоis a simple point ofM. Then the characteristic linear system ofmonCcan. be defined in a well known wanner.¹ Roughly speaking. the characteristic linear system of...

13. 41 On the complex projective spaces;
(pp. 744-759)
F. HIRZEBRUCH and K. KODAIRA

Introduction. — The principal purpose of this paper is to prove the following theorem announced earlier in [2] and [4].

Theorem. —LetXbe an n-dimensional compact Kähler manifold which is${c^\infty }$-differentiably homeomorphic to the complex projectire space${p_n}(C)$with its usual differentiable structure. in case the dimension n ofXis odd,Xis complex-analytically homeomorphic toPn(C),i.e. there exists a bi-regular map ofX onto Pn(c).Let g be the generator of the second cohomology groupH2(X,Z) ($\cong z$)whose sign is chosen in such a way that g is represented as the fundamental class of a Kähler metric...

14. 42 On the Variation of Almost-Complex Structure
(pp. 760-771)
K. Kodaira and D. C. Spencer

LetMbe an almost-complex manifold of class$C\infty$, and let φ be the exterior algebra of the complex-valued differential forms of class$C\infty$on M. We denote by ∂ the anti-derivation of degree + 1 of φ which maps elements of φ0(functions) into forms of type (0,1) and which satisfies the commutativity relation d∂ + ∂d = 0, where d is the exterior differential of φ. Given an arbitrary Hermitian metric on M, let b be the adjoint of ∂ i`n the sense that (∂φ, ψ) = (φ, ƌψ) for forms rp, φ, ψ ε φ with compact...

15. 43 ON DEFORMATIONS OF COMPLEX ANALYTIC STRUCTURES, I
(pp. 772-845)
K. Kodaira and D. C. Spencer

Deformation of the complex structure of a Riemann surface is an idea which goes back to Riemann who, in his famous memoir on abelian functions published in 1857, calculated the number of independent parameters on which the deformation depends and called these parameters “moduli”. Riemann’s well known formula states that the number of moduli of a compact Riemann surface of genuspis equal to the dimension of the complex Lie group of complex analytic automorphisms of the surface plus the number3p-3. This formula has been generalized by Klein to the case of a Riemann surface with boundary. During...

16. 43 ON DEFORMATIONS OF COMPLEX ANALYTIC STRUCTURES, II
(pp. 846-909)
K. Kodaira and D. C. Spencer

In this section we examine complex analytic families of some simple types of compact complex manifolds, e.g. hypersurfaces, complex tori, ... , and compute the numbers of moduli of these manifolds.

(α)Projective space.Let Vo= Pn(C) be a complex projective space. We have dimH1 (Vo, Θo)= 0 (Bott [7]; see Lemma 14.1 below) and therefore, by Theorem 6.3, the family {Vo} consisting of the single memberVois complete. Hence the number of modulim(V<,sub>o)equals 0 and the equalitym(Vo)= dim H1(Vo, Θo) holds.

(β)Hypersurjaces in projective spaces.LetPn+1=Pn+1(C) be...

17. 44 ON THE EXISTENCE OF DEFORMATIONS OF COMPLEX ANALYTIC STRUCTURES
(pp. 910-919)
K. Kodaira, L. Nirenberg and D. C. Spencer

LetM= {t |t| < 1} be a spherical domain on the space ofmcomplex variablest= (t1, ... , tλ,..., tm,), where |t|² = Σ |tλ|², and letV= {Vt|tεM| be a complex analytic family of compact complex manifoldsVtoverM(see Kodaira and Spencer [4], § 1).Vmay be defined as a family of complex structures Vtdefined on one and the same differentiable manifoldXin the following manner:

Let{Uj}be a finite covering of X by sufficiently small neighborhoods UJ.

(i)There exists on each Uj...

18. 45 A THEOREM OF COMPLETENESS FOR COMPLEX ANALYTIC FIBRE SPACES
(pp. 920-933)
K. Kodaira and D. C. Spencer

We begin by recalling several definitions, introduced in the authors’ paper [3], concerning complex analytic families of complex manifolds.

By a complex analytic fibre space we mean a triple (ϑ, ῶ,M) of connected complex manifolds ϑ, M and a holomorphic map ῶ of ϑ ontoM. A fibre${\varpi ^{ - 1}}\left( t \right),t\varepsilon M$of the fibre space issingularif there exists a point$p\varepsilon {\varpi ^{ - 1}}\left( t \right)$such that the rank of the jacobian matrix of the map ῶ atpis less than the dimension ofM.

DEFINITION 1. We say that ϑ→M is a complex analytic family of compact, complex manifolds if (ϑ,...

19. 46 EXISTENCE OF COMPLEX STRUCTURE ON A DIFFERENTIABLE FAMILY OF DEFORMATIONS OF COMPACT COMPLEX MANIFOLDS
(pp. 934-955)
K. Kodaira and D. C. Spencer

The concept of complex analytic family of compact complex manifolds can be regarded as a collection {Vt} of compact complex manifolds {Vt} depending on a parameter t ranging over a complex manifoldMprovided that the union$\upsilon = {U_t}{V_t}$of the Vthas a structure of complex manifold compatible with the complex structures of the Vtand ofM(see Definition 2 below). We begin by recalling the corresponding definition of a differentiable family of compact complex manifolds which was introduced in the authors’ paper [3] (to which we refer the reader for detailed explanations).

By “differentiable” we shall always mean “differentiable...

20. 47 A THEOREM OF COMPLETENESS OF CHARACTERISTIC SYSTEMS OF COMPLETE CONTINUOUS SYSTEMS
(pp. 956-979)
K. Kodaira and D. C. Spencer

The theorem of completeness of characteristic systems of complete continuous systems of curves on algebraic surfaces, conceived by Italian geometers around 1900 (see [8], no. 4, pp. 39-42), was first proved in 1921 by Severi [6J under the assumption that the curves are arithmetically effective. His proof is based on a theorem of fundamental importance due to Poincare [9J, [10J (see also Zariski [12]). Later, Severi [7] removed the assumption of arithmetical effectiveness and proved the theorem of completeness for semi-regular curves. We remark that a curve0on a surface is called semi-regular by Severi if the canonical linear...

21. 48 ON DEFORMATIONS OF COMPLEX ANALYTIC STRUCTURES, III. STABILITY THEOREMS FOR COMPLEX STRUCTURES
(pp. 980-1013)
K. Kodaira and D. C. Spencer

In this paper, the third in a series under similar title, the authors apply the theory of elliptic differential equations to questions concerning the deformation of complex analytic structures and derive the fundamental theorem stated without detailed proof in Section 2 of the first paper, on which many of the results in the first and the second papers (reference [8] of the Bibliography) depend. The authors prove also various stability theorems, some of which are stated in the first paper.

A knowledge of the first and the second papers is not assumed in this paper though it is a sequel....

22. 49 ON DEFORMATIONS OF SOME COMPLEX PSEUDO-GROUP STRUCTURES
(pp. 1014-1092)
K. Kodaira

A pseudo-group of transformations in the space of several real or complex variables is a collection of local homeomorphisms of the space which is closed under composition, formation of inverses and restriction to open subsets (see Ehresmann [6]). For any pseudo-group of transformations, the corresponding concept of pseudo-group structures on manifolds can be defined by means of local coordinates (see Definition 1 below; for an intrinsic definition of pseudo-group structures in terms of sheaves, see Spencer [20]). A notable example is the concept of complex structures which corresponds to the pseudo-group of all biholomorphic transformations in the space of several...

23. 50 MULTIFOLIATE STRUCTURES
(pp. 1093-1142)
K. Kodaira and D. C. Spencer

Among the most important global structures on differentiable manifolds are those defined by the requirement that the jacobians of the coordinate transformations belong to a given linear Lie groupG.These structures are called (integrable)G-structures and they have been studied, from various points of view, by several writers—for example, by S.-S. Chern [2], C. Ehresmann [3], [4], A. HaefIiger [7], P. Libermann [14], [15], and others. The complexG-structures, defined by complex linear Lie groupsG, have the underlying structure of complex analytic manifolds, but may be regarded as realG-structures by imbedding the complex groupGas...

24. Back Matter
(pp. 1143-1143)