Global Analysis: Papers in Honor of K. Kodaira (PMS-29)

D. C. Spencer
S. Iyanaga
Pages: 424
https://www.jstor.org/stable/j.ctt13x10qw

1. Front Matter
(pp. [i]-[iv])
2. Foreword
(pp. [v]-[vi])
S. IYANAGA

Kunihiko Kodaira, to whom this volume is dedicated, lived in the United States from 1949, when he was thirty four years old, until 1967. Thus he spent there eighteen years of his scholarly life.

During the summer of 1967, an informal meeting of some twenty mathematicians took place at Stanford University. At this meeting it was agreed to bring out a collection of papers in honor of Kodaira. This volume is the outcome of that agreement. Most of the authors who have contributed papers to this volume were present at Stanford during at least part of the summer, 1967.

Kodaira...

(pp. [vii]-[viii])
4. The second Lefschetz theorem on hyperplane sections
(pp. 1-20)
Aldo Andreotti and Theodore Frankel

The Lefschetz theorems on hyperplane sections describe how the topology of a projective algebraic manifoldXis related to the topology of a (generic) hyperplane sectionX0. We have (letting η be the complex dimension of X)

the${H_i}({X_o};z) \to {H_i}(X;Z)$

is bijective if$i < n - 1$

surjective if$i = n - 1$.

We shall call this the first Lefschetz theorem. The second Lefschetz theorem describes in detail

ker${H_{n - 1}}({X_0};Z) \to {H_{n - 1}}(X;Z)$

in terms of a pencil of hyperplanes having X0 as generic member.

In an unpublished lecture given at Princeton in 1957, Rene Thom described the geometric situation of the Lefschetz theorems in terms of Morse’s theory of...

5. Algebraization of formal moduli: I
(pp. 21-72)
M. Artin

Since the fundamental work of Kodaira and Spencer [7] on deformation theory appeared, their methods have become standard tools for the study of complex analytic spaces. Grothendieck (FGA, 195) and others have treated the algebro-geometric analogues, whose formal aspects are now well understood. In particular, readily applicable criteria for the existence of a formal universal deformation of a given structure are available. The criterion of Schlessinger [14] is the simplest of these. However an algebraic analogue of a convergence theorem was lacking, and it is the purpose of this paper to derive such a theorem from the approximation theory of...

6. The signature of fibre-bundles
(pp. 73-84)
M. F. Atiyah

For a compact oriented differentiable manifoldXof dimension 4kthe signature (or index) ofXis defined as the signature of the quadratic form inH2k(X; R) given by the cup product. Thus

Sign (X) = ρ – q

where ρ is the number of + signs in a diagonalization of the quadratic form and q is the number of — signs. If dimXis not divisible by 4 one defines Sign (X) to be zero. Then one has the multiplicative formula

Sign (ΧxF) = Sign (X)-Sign (F) . In [5] it was proved that this multiplicative formula...

7. On Hensel’s lemma and exponential sums
(pp. 85-100)
Walter L. Baily Jr.

It is our purpose to present here a result generalizing Hensel’s lemma [1, 2, 5], together with some related ideas which may be applied to the evaluation of certain exponential sums, connected with the Fourier coefficients of Eisenstein series. It is the latter application which motivated our original investigations. However, we think that the generalization of Hensel’s lemma may be considered of some interest in itself.

Part of what we do here is based on ideas suggested by P. Cartier and we wish to acknowledge that fact here. When we indicated to him an interest in certain ideas of Siegel...

8. An intrinsic characterization of harmonic one-forms
(pp. 101-118)
Eugenio Calabi

The author recalls hearing the question since about 1955, do the zeros of a harmonic 1-form in a compact riemannian manifold possess certain combinatorial properties that depend only on the differential topological structure of the manifold, and not on the particular riemannian metric. In the case of surfaces the Riemann-Roch theorem (or alternately the Hopf-Stiefel theorem) on the indices of the zeros of any harmonic 1-form on a closed surface provided a positive answer to the question, rendering an analogous conjecture at least plausible for perhaps some higher dimensional manifolds. The fact that such a conjecture fails completely is the...

9. Intrinsic norms on a complex manifold
(pp. 119-140)
S. S. Chern, Harold I. Levine and Louis Nirenberg

We propose to define in this paper certain norms (or more precisely, semi-norms) on the homology groups of a complex manifold. They will be invariants of the complex structure and do not increase under holomorphic mappings. Their definitions depend on the bounded plurisubharmonic functions on the manifold and are modelled after the notion of harmonic length introduced by H. Landau and R. Osserman [8] for Rieriiann surfaces (= onedimensional complex manifolds). It is possible to extend the definition to certain families of chains. In particular we get in this way an intrinsic pseudo-metric on the manifold which is closely related...

10. On the area of complex manifolds
(pp. 141-148)
G. de Rham

This is part of a Seminar given in 1957 at the Institute for Advanced Study, Princeton, where I had the privilege of many fruitful discussions with K. Kodaira. It contains a proof of a theorem of Lelong, according to which an anlytic set in a complex manifold defines a closed current. Another proof has been given by Federer [8].

Let${z_\alpha } = {x_\alpha } + i{y_\alpha }(\alpha = 1,2,...,n)$be coordinates inCnThe hermitian parabolic metric defined by

$d{s^2} = \sum\nolimits_a^n {} = 1d{z_\alpha }d\overline {{z_a}} = \sum\nolimits_1^n {dx_\alpha ^2}$

is nothing but the euclidean metric. Taken with this metric, Cnis the hermitian parabolic space. The parabolic motions ofCnare the linear (in general non-homogeneous) transformations...

11. On the deformation of sheaves of rings
(pp. 149-158)
Murray Gerstenhaber

Among the current “deformation theories”, two exhibit remarkable formal similarities. The first, initiated in its modern form by Froelicher-Nijenhuis [1], and brilliantly elaborated by Kodaira-Spencer [8, a, b], (cf. also [10]), concerns itself with the variation of the complex analytic structure of a manifoldX. Denoting by$\Theta$the sheaf of germs of holomorphic tangent vectors, the group of “infinitesimal deformations” is shown to be${H^1}(X,\Theta ).$. There is a natural quadratic map${H^1}(X,\Theta ) \to {H^2}(X,\Theta )$which assigns to every infinitesimal deformation the obstruction to prolonging it one step. If such prolongation is possible, then one meets another obstructions, which like all subsequent...

12. Die Azyklizität der affinoiden Überdeckungen
(pp. 159-184)
Von Lothar Gerritzen and Hans Grauert

In der komplexen Analysis hat die Methode der Steinschen Uberdeckungen zu wichtigen Resultaten geführt. Man denke etwa an das 1953 von H. Cartan und J. P. Serre bewiesene “théorème de finitude”, das aussagt, daB auf geschlossenen komplexen Mannigfaltigkeiten die Kohomologiegruppen mit Koeffizienten in einer koharenten analytischen Garbe endlichdimensionale Vektorraume bilden. Beim Beweis dieses Satzes ist es wichtig zu wissen, daB auf einem Steinschen Raum alle diese Kohomologiegruppen für Dimensionen grosser als 0 verschwinden.

In der nichtarchimedischen Funktionentheorie treten an Stelle der Uberdeckungen mit Steinsehen Raumen gewisse Uberdeckungen mitaffinoiden Raumen. Aueh in diesem Falle muB gezeigt werden, daB entsprechend a...

13. Hermitian differential geometry, Chern classes, and positive vector bundles
(pp. 185-252)
Phillip A. Griffiths

(a)Statement of results. Our purpose is to discuss the notion of positivity for holomorphic vector bundles. This paper is partly expository, and in so doing we hope to clarify, simplify, and unify, some of the existing material on the subject. There are several new results, mostly relating the analytic notion of positivity to the topological properties of the bundle. We also correct two errors in our previous paper [10].

We now give the main results to be proved in this paper.

LetVbe a compact, complex manifold and$E \to V$a holomorphic vector bundle; we denote by$\Gamma (E)$the...

14. The signature of ramified coverings
(pp. 253-266)
F. Hirzebruch

LetXbe a compact oriented differentiable manifold of dimensionmwithout boundary on which the cyclic group of ordernacts by orientation preserving diffeomorphisms. The setYof fixed points of this action is a differentiable submanifold ofXnot necessarily connected. The various connectedness components ofYcan have different dimensions, they are not necessarily orientable.

We assume that all components ofYhave codimension 2 and Gnoperates freely on X – Y. Then X/Gnis an oriented manifold. The natural projection

$\pi :X \to X/{G_n}$

mapsYbijectively onto a submanifold Y’ of X/Gn. For any point a a...

15. Deformations of compact complex surfaces
(pp. 267-272)
Shigeru Iitaka

By a surface we shall mean a compact connected complex manifold of complex dimension 2. For notation and terminology we follow that of Kodaira [4], Thus we fix our notation as follows.

S :a surface,

0(D). :an invertible sheaf on S associated to a divisor D,

I(D) = dim H°(S, O(D)) ,

s(D) = dim H¹(S, O(D)) ,

i(D) = dim H2(S, O(D)) ,

C :a complex plane,

D :a complex upper half plane,

E :linear equivalence of divisors,

K :a canonical divisor of S,

Pm= j(mK) :them-genus of S,

pg= j(K) :the geometric genus of S,...

16. On the volume of polyhedra
(pp. 273-288)
S. Iyanaga

Letnbe a positive integer and R be the field of real numbers as usual. Rnis then then-dimensional euclidean space with the wellknown metric. Elements of Rnwill be denoted as$a = ({a_1},...,{a_n}),b = ({b_1},...,{b_n}),X = ({x_1},...,{x_n}) = ({y_1},...,{y_n})$, etc. The inner product of a and x will be denoted by$(a,x) = \sum\nolimits_{i = 1}^n {{a_i}.{s_i}.}$If$a \ne 0$and$b\varepsilon R$,

${\pi _{a,b}} = \{ x;(a,x) = b\}$

is ahyperplane,and

${H_{a,b}} = \{ x;(a,x) \ge b\}$

is ahalf-spaceRn. Ha,bis a closed, convex and unbounded subset of Rn, whose boundary is${\pi _{a,b}},i.e.,\delta {H_{a,b}} = {\pi _{a,b}}$

The subsets of form a boolean lattice with respect to the set operations i u S (union), (intersection) and (complement). Let be the subboolean-lattice...

17. On the resolution of certain holomorphic mappings
(pp. 289-294)
Arnold Kas

Let f:X— Sbe a flat morphism of reduced analytic spaces.

Definition. By a resolution off, we mean a commutative diagram of reduced analytic spaces and morphisms

with the following properties

(i) , f’:X’$$\to T$$flat;

(ii) φ is surjective; ψ is proper and surjective;

(iii) each fibreX;off’ is non-singular, and

$$\Psi |\mathop X\nolimits_t^' :\mathop X\nolimits_t^' \to {X_{\varphi (t)}}$$

is a resolution of singularities in the sense of Hironaka [2],

THEOREM. (Brieskorn [1]). Let$f:X \to S$be a fiat morphism as above. Assume

(i)X and S are both non-singular;

(ii) dim X = 3; dim S = 1;

(iii) each fibre Xshas only...

18. Harmonic integrals for differential complexes
(pp. 295-308)
J. J. Kohn

The operatorsd(and δ) on forms on compact riemannian (and hermitian) manifolds have been studied by means of the theory of harmonic integrals (see [3]). This theory is easily generalized to the case of “elliptic differential complexes”. Consider vector bundles E, F, and G over a compact manifold M, then a differentiate complex is a sequence

(1.1)$E\underrightarrow AF\underrightarrow BG$

Where E, F, and G denote the sheaves of germs of local C°° sections andAandBare first order differential operators for whichBA= 0. For every$x\varepsilon M$and every non-zero cotangent vectorsηatxthere is...

19. A note on families of complex structures
(pp. 309-314)
Masatake Kuranishi

This paper consists of two remarks on complex analytic families of complex structures. The first is about the definition of families on complex structures, and the second is about automorphisms of the locally complete families. Both of these results were apparently known to Douady. The first appears without proof in [1] in a somewhat different form. In any case, we could not find any proofs in the literature.

When the writer developed a theory of complex families in [3], we put in (for technical reasons) side conditions in the definition of complex analytic families. However, it turns out that we...

20. A survey of some results on complex Kähler manifolds
(pp. 315-324)
James A. Morrow

One of the first questions that one asks about a given complex manifold is “What are the other complex structures on the under lying differentiable manifold?” The whole theory of classification of (complex) surfaces is motivated by this question, and the Kodaira-Spencer theory of deformations can be thought of as a study of the complex structures close to a given one. Sometimes one has uniqueness (e.g.S²), and it is conjectured that Pnhas only one complex structure. Emery Thomas [9] has computed the total Chern class for all possible almost complex structures on Pn, 1$\le n \le 4$. Kodaira and Hirzebruch...

21. Enriques’ classification of surfaces in char p: I
(pp. 325-340)
David Mumford

The principal assertion in Enriques’ classification of surfaces is:

Theorem.Let F be a non-singular projective surface, with out exceptional curves of the1stkind. Let KFbe the canonical divisor class on F. Then

(i)if|12KF| =$\emptyset$,then F is ruled,

(ii)if|12KF|$\ne \emptyset$,then either12KF= 0,or else| nKF|is a, linear system without base points for some n.

As a Corollary, if we introduce the notations:

(a)$H = tr{d_k}\sum\nolimits_{n = 0}^\infty {{H^0}} \left( {F,{0_F}\left( {nK} \right)} \right)$— 1

(b)F ellipticif$\exists$a morphismf:F➔C, Ca curve, with almost all fibres singular rational curvesEwith...

22. On modular imbeddings of a symmetric domain of type (IV)
(pp. 341-354)
Ichiro Satake

The purpose of this note is to give some supplementary results concerning the holomorphic imbeddings of a symmetric domain of type (IV) into a Siegel space constructed in [8]. Namely, in § 1, we will give a simple geometric interpretation of such an imbedding along with an “analytic expression” of it (in the sense as explained in the Appendix); and, in § 2, some results concerning the relationship between the fields of automorphic functions on these domains with respect to suitable arithmetic groups. Similar results for other kinds of holomorphic imbeddings have been obtained by Klingen, Hammond, Freitag, and K....

23. The curvature of 4-dimensional Einstein spaces
(pp. 355-366)
I. M. Singer and J. A. Thorpe

It is well known that the diagonalization or normal form of a self adjoint linear transformationTon a real inner product space {V, <,>} is equivalent to the analysis of the critical point behavior of the function$v \to \langle Tv,v\rangle$on the unit sphere (actually projective space) ofV. It is also classical that the curvature tensor at a point of a riemannian manifold is completely determined by the sectional curvature function a on the Grassman manifold of 2- planes at the point.

It seems natural, then, to attack the problem of a “normal form” for the curvature tensor by analyzing...

24. On deformation of pseudogroup structures
(pp. 367-396)
D. C. Spencer

In his papers ([10(a), (b)]) the author developed a general mechanism for the local deformation of structures on manifolds defined by transitive continuous pseudogroups. Meanwhile additional papers on this subject have appeared, notably those by Griffiths [3] and a paper by Guillemin and Sternberg [5] in which an approach is given based on the use of the fundamental form of the structure, which incorporates in an elegant way many of the formal aspects of the author’s approach. Recently B. Malgrange found a proof for the existence of local coordinates compatible with an integrable structure defined by an elliptic transitive, continuous...

25. Sur les variétés d’ordre fini
(pp. 397-402)
Par René Thom

Les problemes classiqués de Géometrie Finie (au sens de Juel, 0. Haupt, de Marchaud, etc) relévent probablement de la théorie des singularités affines des plongements différentiables. Le présent article vise á préciser ce rapport.

Définition1. Un ensemble A de Rnseradit de k-degré fini m,si tout k-plan de Rnrencontre A en au plusmpoints.

Exemple.Un ensemble algebrique A de Rn, de codimensionk, est de k-degré fini (majoré par le degré algébrique de A pris en tant que variété affine), á moins qu’il ne contienne une variéte linéaire.

On se propose d’établir le théoréme:...

26. Obstructions to the existence of a space of moduli
(pp. 403-414)
John J. Wavrik

Kuranishi has shown [8], [9], [1] that, if X⁰ is a compact complex analytic manifold, a complete family of deformations of X₀ always exists. Let Θ⁰ denote the sheaf of germs of holomorphic vector fields on X₀. If H⁰ (X₀, Θ0) = 0, the Kuranishi space is a (local) space of moduli for X⁰ [1] If H⁰(X₀, Θ₀) ≠ 0, X₀ may still have a space of moduli (e.g. X₀ = a complex torus). Examples are known, however, of manifolds not having a space of moduli (e.g. X₀ = a Hirzebruch surface). In this paper we will show that the...