The Collected Papers of Wei-Liang Chow

The following sections are included:

• PREFACE

• Biography

• CONTENTS

The following sections are included:

• The associated form of a manifold M

• Algebraic systems of manifolds

The following sections are included:

• Einleitung

• Algebraische Mannigfaltigkeiten. Dimension

• Der Satz von Bezout

• Lineare Scharen und rationale Abbildungen

• Die Auflösung der Singularitäten

• Punktgruppen. Äquivalenz. Vollscharen

• Der Riemann-Rochsche Satz

• Der Beweis des Existenzsatzes

• Literaturverzeichnis

Deuten wir die komplexen Zahlen x bzw. y als Punkte auf den Kugeloberflächen K_x bzw. K_y, so definiert jede Gleichung y = xⁿ + a₁x^{n - 1} + … + a_n eine (singuläre) 2-Kette Γ auf K_y, deren Urbild K_x ist. Γ ist also ein Zyklus. Gäbe es einen Punkt auf K_y, der nicht von Γ überdeckt wäre, dann würde Γ ~ 0, wie man leicht einsieht. Andererseits läßt Γ sich durch eine Deformation, die etwa durch

Wir werden hier einen einfachen rein-algebraischen Beweis für den folgenden Satz bringen:

n Hyperflächen F_i im n-dimensionalen projektiven Raume schneiden sich in einem Punkt ξ, der für die Hyperflächen F_i bzw. s_i-fach ist, im allgemeinen Π s_i-fach; die Schnittmultiplizität wird dann und nur dann größer, wenn die Hyperflächen F_i in ξ eine gemeinsame Tangente besitzen…

C. Carathéodory hat bei seiner Begründung des zweiten Hauptsatzes der Thermodynamik¹) den folgenden Satz über eine Pfaffsche Gleichung bewiesen: Wenn eine Pfaffsche Gleichung in jedem Punkte die Eigenschaft hat, daβ es in jeder Umgebung von ihm Punkte gibt, die sich nicht durch eine Integralkurve der Gleichung mit ihm verbinden lassen, dann ist die Gleichung vollständig integrierbar…

Consider an electric network N with n conductors F₁, F₂, …, F_n, of resistances R₁, R₂, … , R_n respectively, and d + 1 junctions P₁, P₂, … , P_d+1. Let a direction be assigned to each conductor in a quite arbitrary way. To each junction P_i we shall associate a row n-vector, that is a (1, n)-matrix (P_i1, P_i2, … , P_in), where p_ij is equal to +1, -1, or 0 according as F_i is directed toward P_i, directed away from P_i, or not connected with P_i. The d + 1 vectors thus defined shall be denoted again by P₁, P₂, … , P_d+1…

Artin¹ has recently given a new elegant and completely rigorous treatment of his theory of braids. In the same connection Bohnenblust² has derived some of the main results, in particular the completeness of the relations of the braid group, by an algebraical analysis of the abstract group defined by the relations (1) (2) below, which he calls the algebraical braid group. In the present paper we shall analyze the algebraical braid group by a different method, which we believe is simpler and more direct. Moreover, we shall be able not only to derive the same results as contained in Bohnenblust's paper, but also determine the center of the braid group…

The following sections are included:

• Introduction

• Analytic varieties

• Intersection of analytic varieties

• Proof of the main theorem

• Meromorphic transformations

• REFERENCES

The following sections are included:

• Introduction
  • Four Classes of Homogeneous Spaces
  • Two Problems
  • Remarks

• Projective Characterization of the Basic Group. The Fundamental Theorems
  • The Grassmann Space
  • The Space of an Involutoric Correlation
  • The Invariant Elements
  • The Space of a Null System
  • The Fundamental Theorem in the General Case
  • Applications. The Space of a Polar System and the Hyperquadric
  • The Irreducible Space of a Polar System

• Birational Characterization of the Basic Group
  • Representation as Algebraic Varieties
  • Two Fundamental Lemmas
  • Birational Characterization. Extension of Ground Field
  • The Main Theorem of Birational Characterization
  • Field of Complex Numbers. Analytic Characterization

• REFERENCES

In einer vor kurzem erschienenen Arbeit¹) hat Herr W. Habicht den folgenden interessanten Satz bewiesen:

K sei ein Körper; f₁, … , f_n seien Formen aus dem Ringe K [x₁, …, x_n]; zwischen ihnen bestehe die Relation ; es seien nicht gleichzeitig n gerade und die Grade sämtlicher f_i gleich 1. Dann besitzen die Formen f_i eine gemeinsame (nichttriviale) Nullstelle (in der algebraisch-abge-schlossenen Hülle von K)…

Professor C. Chevalley has recently pointed out to me the desirability of having a simple algebraic proof of the following theorem: The genus of any curve of an algebraic system of algebraic curves is not greater than the genus of the generic curve of the system. The theorem seems to be well known, but I have sought in vain for a proof of it in the literature(¹). We shall give here a purely algebraic proof of this theorem, based on the idea of the associated form of an algebraic variety, which was introduced by van der Waerden and myself some years ago(²)…

In the system of algebraic geometry as developed by A. Weil in his recent book, Foundations of algebraic geometry, a variety U in the n-space is defined as the set of all equivalent couples (k, P), each consisting of a field k and a point P in the n-space such that the field k(P) is a regular extension of k. Two such couples (k', P') and (k″, P″) are called equivalent if every finite specialization of P' over k' is also one of P″ over k″ and conversely. Any field k which enters into such a couple is called a field of definition of the variety U. It has been shown by Weil in his book that among all the fields of definition of a variety U there is a smallest one which is contained in all of them, which we shall call the defining field of the variety U. A d-cycle G in a variety U of dimension r is a finite set of simple subvarieties of dimension d in U, to each of which is assigned an integer called its multiplicity; a cycle is called positive if the multiplicity of each of its component varieties is positive. Let K be a field of definition of U. Then the G is said to be rational over K if it satisfies the following conditions: (1) each component variety of G is algebraic over K; (2) if a variety is a component of G, then all the conjugate varieties over K are also components of G with the same multiplicity; (3) the multiplicity of each component of G is a multiple of its order of inseparability. The question arises whether there is a smallest one among all the fields over which the cycle G is rational. If such a smallest field exists, we shall call it the defining field of the cycle G . One observes that since by definition every field over which the cycle G is rational must be a field of definition of the variety U, it follows that the defining field of G, if it exists, must contain the defining field of U…

The following sections are included:

• Introduction

• Statement of the problem and result

• Two lemmas

• Proof of the theorem

• APPENDIX. On the Extensions of Local Domains

• REFERENCES

Let A and B be two Abelian varieties and let F be a homomorphism of A onto B; then the kernel of the homomorphism F is a subgroup in A which consists of a bunch of subvarieties in A. The question naturally arises whether conversely for any given such subgroup X in A there exist an Abelian variety B and a homomorphism of A onto B with X as its kernel. In other words, the question is whether the quotient group A/X of A over X can also be represented by an Abelian variety A(X), such that the canonical homomorphism of the group A onto A/X is represented by a homomorphism of the Abelian variety A onto A(X). It is the purpose of this note to show that such a quotient variety A(X) always exists…

The following sections are included:

• Introduction

• Analytic mapping of cycles

• Dual pairing of complex tori

• The Jacobian varieties

• The Picard varieties

• Representative system of divisors

Let V be a compact complex analytic variety of complex dimension r, defined in abstracto without any embedding space.¹ According to a recent result of Chow,² the field 𝔉(V) of all meromorphic functions on V is a finite algebraic function field whose degree of transcendency s is at most equal to r. There exists therefore an algebraic variety V* of dimension s in a projective space, uniquely determined up to a birational transformation, and a meromorphic transformation Φ of V onto V*, such that every meromorphic function on V is the image of a meromorphic (and hence rational) function on V* under the inverse transformation Φ^-1. Thus the meromorphic transformation Φ induces an isomorphism between the two fields 𝔉(V) and 𝔉(V*), so that we can identify the two and set 𝔉(V) = 𝔉(*) = 𝔉. The variety V* will be called the algebraic equivalent of V. Now, we shall say that the analytic variety is algebraic (in an extended sense), if its algebraic equivalent V* can be so chosen that the transformation Φ is bi-regular between V and V*. Since from the abstract point of view two bi-regularly equivalent analytic varieties can be considered as not essentially distinct, it follows that in this case we can identify V with V*, so that our definition of an algebraic variety is a natural extension of the usual one for a variety in a projective space. It is obvious that an algebraic variety V of dimension r possesses the maximum possible number r of algebraically independent meromorphic functions. The question naturally arises whether the converse is also true: If a compact analytic variety has as many algebraically independent meromorphic functions as its dimension, is it then necessarily algebraic? So far as we know, this question has not yet been decided except for some special cases. It is therefore probably not without some interest to present here an answer to this question for the case of a compact analytic manifold of complex dimension 2 with a Kählerian metric, which we shall call a Kählerian surface….

It is well known that any 1-cycle in an algebraic surface can be deformed into a 1-cycle lying in a generic plane section of the surface.¹ The usual proof of this theorem, which can be easily generalized from a surface to any non-singular algebraic variety, is topological and consists of a simple construction of the deformation chain. In the transcendental theory there is a generalization of this theorem, at least in its homology aspect, which can be stated as follows: There exist exactly 2p independent 1-cycles in an algebraic surface which are not homologous to 1-cycles belonging to a generic curve of an irrational pencil of genus p . In this paper we shall show that this theorem is a special case of a more general theorem about the fundamental group of an algebraic variety under a rational transformation. Our method of proof will be purely topological; the essential idea is that although a rational transformation is not in general a fibre mapping, the covering homotopy theorem is nevertheless true, in a somewhat modified form, for the mapping of a 1-simplex…

The following sections are included:

• Preliminary remarks

• The problem

• Construction of the variety 𝔅

• The derived normal model 𝔚

• Proof that the variety 𝔅 is non-singular

• The canonical homomorphism

• REFERENCES

We should like to make two remarks on this paper, which was published in vol. 76 (1954), pp. 453-76, of this Journal…

In a recent paper¹ on Abelian varieties over function fields, we have shown that to every algebraic system of Abelian varieties, defined over a field K, can be assigned two invariants, called the K-image and the K-trace. Let K(u) be a primary extension of K, and let A* be an Abelian variety defined over K(u); then the K-image A of A* over K(u) is an Abelian variety over K characterized by the existence of a rational homomorphism F (called the canonical homomorphism) of A* onto A, denned over K(u), such that, if H is a rational homomorphism of A* into an Abelian variety B, and if B is defined over a separably generated extension K₁ of K, independent with respect to K(u) over K, and H is defined over K₁(u), then H is the product of F and a rational homomorphism of A into B, defined over K₁. The K-trace A' of A* over K(u) is defined in case K(u) is a regular extension of K, and is an Abelian variety over K characterized by the existence of a rational isomorphism F (called the canonical isomorphism) of A' into A*, defined over K(u), such that if H is a rational homomorphism of an Abelian variety B into A*, and if B is defined over an extension K₁ of K, independent with respect to K(u) over K, and H is defined over K₁(u), then H is the product of F and a rational homomorphism of B into A', defined over K₁. We observe that, while the K-trace is defined only for the case of a regular extension K(u) of K, it possesses as a compensation the stronger property that the field K₁ involved is any extension of K which is independent with respect to K(u) over K, not necessarily a separably generated one…

The following sections are included:

• Introduction

• Algebraic subgroups in an Abelian variety

• An existence theorem

• The K-image

• The K-trace

• References

It is well-known that there is a close analogy between the intersection theories in the abstract algebraic geometry and in topology; in fact, this similarity is more than an analogy, for, in the algebraic geometry over the complex field, the algebraic cycles are also topological cycles and their algebraic intersections coincide with their topological intersections. However, as has been observed by Weil in the preface to his Foundations of Algebraic Geometry [6]¹, there is one important difference; that is, in topology the theory of intersection can be extended to an algebra of homology classes of cycles, while in the abstract algebraic geometry, because of the lack of a suitably developed concept of equivalence, one has to restrict oneself to operate with the cycles themselves, with the resulting inconvenience that the intersection-product is not always denned. The purpose of this paper is to show that this restriction can be removed, and that also in abstract algebraic geometry we can operate with classes of cycles (under a suitably chosen concept of equivalence) instead of the cycles themselves, and develop an algebra of equivalence classes closely analogous to the algebra of homology classes in topology…

Let V be a variety,¹ and let {W_i} be a finite set of varieties contained in V (which need not be closed in V and hence are not, in general, subvarieties in V) such that every point in V is contained in one and only one of these varieties; we shall call such a set of varieties {W_i} a dissection of V, and the dissection is said to be rational if each variety W_i is equivalent to an affine space by an everywhere biregular birational transformation. A dissection {W_i} of a variety V is said to be defined over a field K if K is a field of definition for all the varieties W_i (and hence necessarily also for V); and in case {W_i} is a rational dissection, we add the further condition that each W_i is equivalent to an affine space by an everywhere biregular birational transformation which is defined over K. It is evident that if W is an element in a (rational) dissection {W_i} of V, then the subset consisting of all W_i contained in the closure or the boundary of W in V is also a (rational) dissection of or , respectively. We shall be concerned in this note only with rational dissections, and we shall call an element W in a rational dissection a cell, or an r-cell if r is the dimension of W. It is clear that a variety having a rational dissection must be a rational variety; in fact, any dissection of a variety V of dimension d can contain only one element W of dimension d, and it is easily seen that V must coincide with the closure of W in V. The converse of this statement is, however, not true in general; for this, one need only observe that if a variety V has a rational dissection, then (in case the universal domain is the complex field) all the odd-dimensional Betti numbers of V must vanish, and that there exist rational varieties with nonvanishing odd-dimensional Betti numbers…

In an article [5] elsewhere in this volume, Weil has shown that an idea of Lefschetz on the projective embedding of an Abelian variety over the complex field, which seemingly depends upon the use of theta functions ([1], pp. 368–9), can actually be extended to the case of an abstract Abelian variety over a field of arbitrary characteristic. In this note we shall show that this idea can be further extended to get a projective embedding not only of an arbitrary group variety, but also of any homogeneous variety. We shall say that a variety V can be embedded in a projective space or has a projective embedding, if there is an everywhere biregular birational transformation of V onto a (not necessarily complete) variety contained in a projective space…

The Principle of Degeneration in algebraic geometry asserts that every specialization (over any field) of a connected positive cycle in a projective space is connected, whereby a positive cycle is said to be connected if its support, i.e., the set of its component varieties, is connected in the sense of the (absolute) Zariski topology. This principle, although more or less "known" for a long time in algebraic geometry and which can be easily proved in the classical case by transcendental methods, has been only recently given a purely algebraic proof by Zariski [7], as a special case of a more general principle called the Principle of Connectedness. In view of the fact that Zariski's proof of the Principle of Connectedness involves the very difficult theory of holomorphic functions on an algebraic variety, it is probably not without some interest to give here a new and very much simpler proof of the Principle of Degeneration. Furthermore, since the concept of a specialization is subsumed under the more general concept of a homomorphism of an integral domain onto its residue ring modulo a prime idea, it is natural to ask for a generalization of the Principle of Degeneration which is valid in this general case. In the present paper we shall give a proof of such a generalization; this will include in particular the case of reduction modulo a prime number of a cycle defined over the field of rational numbers, a case which has of late become increasingly important in number-theoretic applications…

Let k be a field with a discrete valuation, let p be the maximal prime ideal in the valuation ring on k, and let be the residue field of k. Let Sⁿ be the projective space of dimension n in the algebraic geometry over a universal domain containing k, and denote by the projective space of dimension n over a universal domain containing ; if Z is any cycle in Sⁿ, rational over k, we shall denote by the cycle obtained from Z by reduction modulo p, in the sense of Shimura [4], and we shall call the specialization of Z (with respect to the given valuation of k). This definition applies also in case Z is a variety, which is then to be considered as a prime cycle; the the specialization is then a positive cycle, but in general not necessarily a variety. According to the generalization of the Principle of Degeneration recently proved by Chow, the specialization of a variety is always connected, but we shall not need this fact here…

The well-known criterion for unit multiplicity in algebraic geometry states that if two cycles 𝔛^r and 𝔜^s in a variety Vⁿ (n = r + s) are transversal to each other at a simple point α in V, then 𝔛 and 𝔜 intersect properly at α with the multiplicity 1. This is the version of the criterion as formulated in Weil's Foundations of Algebraic Geometry,² whereby we have restricted ourselves for simplicity to the case r + s = n. It is possible to give this criterion another formulation which is somewhat more elementary in the sense that it involves the multiplicity of a specialization rather than that of an intersection. It can be stated as follows: If two cycles 𝔛^r and 𝔜^s are specializations of two subvarieties 𝔛' and 𝔜' respectively in a variety Vⁿ (n = r + s) over a field of definition k for V, if 𝔛' ∩ 𝔜' consists of a finite set of (distinct) points a⁽ⁱ⁾, and if 𝔛 and 𝔜 are transversal to each other at a simple point α in V, then the point α occurs exactly once in every specialization of the set a⁽ⁱ⁾ over the specialization (𝔛', 𝔜') → (𝔛, 𝔜) over k. This version of the criterion for unit multiplicity appeared in the earlier intersection theory of van der Waerden,³ and played an essential role there in the development of the concept of the intersection-multiplicity itself. However, if one examines very closely the treatment of van der Waerden, one observes that only a part of this criterion, namely the assertion that the specialization-multiplicity is at most one, was given a simple proof before the introduction of the concept of the intersection-multiplicity, while the other part, which asserts that this specialization-multiplicity is at least one, was proved only later after the concept of the intersection-multiplicity had already been introduced.⁴ In fact, this second part concerning the positivity of this particular specialization-multiplicity was proved by van der Waerden in a rather complicated and indirect way, at first only for cycles in a projective space and later carried over to the general case. As we have recently developed a new theory of intersection which preserves the basic ideas of the original theory of van der Waerden, although in an extensively generalized form, we become naturally interested in finding a proof of the criterion for unit multiplicity which is completely free from any use of the notion of the intersection-multiplicity. It turns out that we are able not only to find a simple proof of this nature, but also at the same time, by virtue of the abstract local form in which we have recast this criterion itself, to obtain a more precise result which contains as a special case a certain generalization of the Hensel's Lemma conjectured by Weil some years ago. Although the subject forms an integral part of our intersection theory, which will be published elsewhere at a later date, we feel that this connection with the Hensel's Lemma justifies its separate publication here as a short note. In keeping with the foundational nature of the subject, we shall make our treatment as elementary as posible, using only a few elementary properties in the theory of local rings. Finally, we shall show in the last section that this connection with Hensel's Lemma can be extended to a more general result in the intersection-theory, namely the invariance of intersection-multiplicity under specialization; it turns out that this invariance can be easily deduced from the Associativity Formula, as generalized in a recent paper⁵ of Nagata. Here we shall of course need the more sophisticated results in the theory of local rings…

The well-known Theorem of Bertini on reducible linear systems of divisors in an algebraic variety asserts that any such system, assumed to be free from fixed components, is composite with a pencil. A somewhat special form of this theorem, which is actually a special case of it, states that if a linear system of divisors is obtained from a rational transformation of the variety onto a projective space of dimension >1, then a generic element of the system, apart from a possible fixed component, is absolutely irreducible. In a recent discussion on local algebraic geometry, Professor S. Abhyankar raised the question whether this Theorem of Bertini, at least in the special form, also holds for local varieties, with suitable modifications. In this note we shall answer this question in the affirmative by formulating and proving such a theorem for an arbitrary complete local domain…

The following sections are included:

• Introduction

• Models and Cohomology

• A Künneth Relation

• Principle of Upper Semicontinuity

The following sections are included:

• Introduction

• Rational transforms

• Quadratic transforms and linear connectedness theorem

• General connectedness theorem and extended principle of Degeneration

• Proof of Theorem 2

• REFERENCES

Let V be a complex-analytic variety of complex dimension n which is the complexification of its real trace V⁰, so that V⁰ is a real-analytic variety of real dimension n, and let X be a complex-analytic subvariety of complex dimension r in V such that X⁰ = X ∩ V⁰ has the real dimension r (and hence ). If we denote by ϕ(X) and ϕ(V⁰) the homology classes mod 2 determined by X and V⁰ respectively as topological cycles mod 2 in V, then their intersection class ϕ (X) · ϕ (V⁰), with due regard for their supports, can be considered as an element in H_2r-n(V⁰, Z₂). On the other hand, X⁰ is also a topological cycle mod 2 in V⁰ and hence determines an element ψ(X⁰) in H_r(V°, Z₂). It is natural to ask about the relation between these two elements ϕ(X) · ϕ(V°) and ψ(X⁰) in H_*(V⁰, Z₂), since roughly speaking they represent respectively the algebraic intersection and the geometric intersection of X and V⁰. We shall give here an answer to this question for the case where V and X are algebraic…

B. Dwork and K. Ireland, in their recent work on the zeta function of an algebraic variety, have raised the question whether the unmixedness theorem holds for the doubly or multiply projective space, or rather for the (generalized) Segre representation of such a space as a projective variety. In this paper we shall give an affirmative answer to this question. However, in course of our investigation, it turns out that our method is susceptible of a bit of generalization; in fact, we shall now express our result as a theorem on the validity of the unmixedness theorem for the "Segre" product of two Noetherian graded rings, assuming that unmixedness theorem holds for the two rings themselves. This generalization is not nominal, for, apart from giving a clearer picture of the nature of our reasoning, it yields also some additional results of quite a bit of interest, as we shall see presently…

The following sections are included:

• Algebroid geometry

• A criterion for algebraicity

• The main theorem

• The projective spaces

• References

Many years ago we proved a theorem which says that a closed analytic subspace in the projective space is algebraic. Since then the theorem has received a number of other proofs. Of particular interest is the proof given by Serre in his well-known paper entitled "Géométrie algébrique et géométrie analytique" (often quoted as GAGA), in which this theorem appears as a corollary to a systematic treatment of the relations between analytic and algebraic sheaves. Under the circumstances, it would seem slightly absurd to offer here yet another proof of this old theorem. A few words of explanation are therefore in order…

There are several mistakes in the section III of our paper which make the proof of Lemma 4 given there almost unintelligible. The basic ideas of the proof are sound, but in the process of formulating them in abstract general terms we unfortunately made some mistakes. We take this opportunity not only to correct these mistakes but also to add a few remarks which we hope will improve the exposition…

Let X be a homogeneous algebraic variety on which a group G acts, and let Z be a subvariety of positive dimension in X. Assume that Z generates X in the sense that , where is the subgroup in G generated by the component containing the unit element in the set G_p,Z of all elements in G which carry a given point p in Z each to another point in Z. One asks whether a formal rational function on X along Z is the restriction along Z of an algebraic function (or even a rational function) on X. In a paper [1] some years ago, the author gave an affirmative answer to this question, under the assumption that the subvariety Z is complete, but only for the complex-analytic case with the formal function replaced by the usual analytic function defined in a neighborhood of Z. The question remains whether the result holds also for the formal functions in the abstract case over any ground field. We had then some thoughts on this question, but we did not pursue them any further as we did not see a way to reach the desired conclusion at the time. In a recent paper [3], Faltings raised this same question and gave a partial answer to it in a slightly different formulation. This result of Faltings led us to reconsider this question again, and this time we are more fortunate. In fact, we have been able not only to solve the problem, but also to do it by using essentially the same method we used in our original paper…

I met Shiing-Shen Chern for the first time in Hamburg sometime in the fall of 1934. We were at that time both students at the Hamburg University; Chern was at that time studying with Blaschke, while I was attending the lectures of Artin. I purposely use the rather vague expression "attending the lectures of Artin" in order to indicate the tentative nature of my sojourn in Hamburg at that time; more correctly, I should say that I was at that time strictly speaking a student at the University of Leipzig, but I decided to stay in Hamburg for personal reasons, as I shall explain later, and I took the opportunity to learn something by attending the lectures of Artin. In order to explain this rather strange lack of definiteness in my mathematical aspirations and also in order to understand the importance of Chern's influence at a critical juncture later in my life, let me say a few words about my rather unusual educational background…

The following sections are included:

• WEI-LIANG CHOW, 1911–1995

• COMMENTS ON CHOW'S WORK

• Wei-Liang Chow's Bibliography

• Permission