Celebratio Mathematica — Atiyah

Early papers (1952–58)

Commentary by M. Atiyah

My first paper [1] was written when I was a second-year undergraduate and arose from a course on higher-dimensional projective geometry given by J. A. Todd. At that time I was fascinated by classical projective geometry, a subject well represented at the time in Cambridge by Todd, Babbage, and White, with the ghost of H. F. Baker still prominent in the background. Todd was my supervisor for several terms and he arranged for my little note to be published, something which gave me disproportionate pleasure and encouragement. When it came to decide on my graduate work I oscillated between Todd and Professor W. V. D. Hodge who represented a more modern approach based on differential geometry. Hodge’s greater international standing swung the balance and, in 1952, I became his research student.

In my first year Newton Hawley from America was visiting Cambridge and through him I became interested in analytic fibre bundles. At the same time great things were happening in France and I was an avid reader of the Comptes Rendus, following the developments in sheaf theory. Peter Hilton also showed me a letter of Serre (addressed to André Weil), that was circulating at the time, which gave the sheaf-theory treatment of Riemann–Roch for an algebraic curve. Given my interest in classical geometry I naturally looked at the old results on ruled surfaces from the new point of view an this led to [3], for which I received the Smith’s Prize in 1954. This came at a crucial time when I was unsure whether I should continue with mathematical research. In fact I toyed quite seriously with subjects like architecture and archeology, but the Smith’s Prize decided my fate.

The role of fibre bundles in algebraic geometry had been hinted at in André Weil’s 1938 paper and was developed by him in his Chicago lectures in the early fifties. He spent a term in Cambridge in 1953 lecturing on the topic but I received no encouragement from this quarter. In subsequent years this whole subject has seen extensive development in different directions. The particular problems studied in [3] concerning the classification of fibre bundles over curves were taken up more systematically by the Tata Institute School (Narasimhan, Seshadri, Ramanan) and also by my student Newstead.

My supervisor Hodge took a keen interest in my work and he also was following the new developments in algebraic geometry. One day he outlined to me the way in which Lefschetz’s theorems on integrals of the second kind should fit into the sheaf-theory framework. I developed this idea in great detail, resulting eventually in our joint paper [4] (summarized in [2]) which Hodge reported on when he attended a major conference in Princeton. This attracted the attention of Kodaira and Spencer and was instrumental in my going to the Institute for Advanced Study in 1955.

Princeton in 1955–56 was enormously stimulating. In particular, I got to know Serre, Hirzebruch, Bott, and Singer all at this time, with long-term implications for my subsequent work. Serre ran a seminar on vector bundles which I attended and my next three papers [5], [7], [6] were all influenced by him in one way or another.

Back in England I met Milnor, then in Oxford, and had a discussion with him about Kummer surfaces. Attempting to understand the effect of double points on the topology of algebraic surfaces then led to [8]. This paper was later the starting point for Brieskorn’s beautiful work on rational double points.

Around this time (1957) the first of Hirzebruch’s Arbeitstagungen began my long series of visits to Bonn. [9] was an expository lecture given there based on work of Calabi which I had learnt about in Princeton.

Works

[1]M. F. Atiyah: “A note on the tangents of a twisted cubic,” Proc. Cambridge Philos. Soc. 48 (1952), pp. 204–205. MR 0048079 Zbl 0046.14604 article

[2]W. Hodge and M. Atiyah: “Formes de seconde espèce sur une variété algébrique” [Forms of the second kind on an algebraic variety], C. R. Acad. Sci. Paris 239 (1954), pp. 1333–1335. MR 0068869 article

[3]M. F. Atiyah: “Complex fibre bundles and ruled surfaces,” Proc. London Math. Soc. (3) 5 (1955), pp. 407–434. MR 0076409 Zbl 0174.52804 article

[4]W. V. D. Hodge and M. F. Atiyah: “Integrals of the second kind on an algebraic variety,” Ann. Math. (2) 62 (1955), pp. 56–91. MR 0074082 Zbl 0068.34401 article

[5]M. Atiyah: “On the Krull–Schmidt theorem with application to sheaves,” Bull. Soc. Math. France 84 (1956), pp. 307–317. MR 0086358 Zbl 0072.18101 article

It is well-known that many standard algebraic results in the theory of groups, rings, modules, etc., can be proved more generally for suitable categories, in the sense of Eilenberg–Maclane [1945]. This has the usual advantages of abstraction. It singles out those features of a given algebraic structure which are essential to the results in question, and by so doing it extends the validity of these results to other domains. In this Note we shall be concerned with the Krull–Schmidt theorem for modules, which asserts under suitable conditions the existence and essential uniqueness of a direct decomposition into indecomposable factors. It is clear that if such a theorem is to have a meaning in some general category, then such notions as kernel, image and direct sum must be defined in the category, and must possess the usual properties. Such a category, called an exact category, has been considered by Buchsbaum [1955]. Basing ourselves on his paper we then have at our disposal all the necessary notions with the usual properties. Our purpose will be to investigate conditions under which the Krull–Schmidt theorem holds in an exact category. This categorical formulation will then enable us to obtain a Krull–Schmidt theorem for suitable categories of sheaves. This is of special interest in algebraic geometry, and it was this case of the theorem which provided our motivation.

[6]M. F. Atiyah: “Vector bundles over an elliptic curve,” Proc. London Math. Soc. (3) 7 : 1 (1957), pp. 414–452. MR 0131423 Zbl 0084.17305 article

[7]M. F. Atiyah: “Complex analytic connections in fibre bundles,” Trans. Amer. Math. Soc. 85 : 1 (1957), pp. 181–207. See also Symposium internacional de topología algebraica (1958). MR 0086359 Zbl 0078.16002 article

In the theory of differentiable fibre bundles, with a Lie group as structure group, the notion of a connection plays an important role. In this paper we shall consider complex analytic connections in complex analytic fibre bundles. The situation is then radically different from that in the differentiable case. In the differentiable case connections always exist, but may not be integrable; in the complex analytic case connections may not exist at all. In both cases we are led therefore to certain obstructions, an obstruction to the integrability of a connection in the differentiable case, an obstruction to the existence of a connection in the complex analytic case. It is a basic theorem that, if the structure group is compact, the obstruction in the differentiable case (the curvature) generates the characteristic cohomology ring of the bundle (with real coefficients). What we shall show is that, in a large class of important cases, the obstruction in the complex analytic case also generates the characteristic cohomology ring. Using this fact we can then give a purely cohomological definition of the characteristic ring. This has a number of advantages over the differentiable approach: in the first place the definition is a canonical one, not depending on an arbitrary choice of connection; secondly we remain throughout in the complex analytic domain, our characteristic classes being expressed as elements of cohomology groups with coefficients in certain analytic sheaves; finally the procedure can be carried through without change for algebraic fibre bundles.

[8]M. F. Atiyah: “On analytic surfaces with double points,” Proc. Roy. Soc. London. Ser. A 247 : 1249 (1958), pp. 237–244. MR 0095974 Zbl 0135.21301 article

[9]M. F. Atiyah: Some examples of complex manifolds. Bonner Mathematische Schriften 6. Universität Bonn, 1958. MR 0105718 Zbl 0080.37502 book

[10]M. F. Atiyah: “Complex analytic connections in fibre bundles,” pp. 77–82 in Symposium internacional de topología algebraica (Universidad Nacional Autónoma de México). Universidad Nacional Autónoma de México and UNESCO (Mexico City and Paris), 1958. See also Trans. Amer. Math. Soc. 85:1 (1957). MR 0098194 Zbl 0123.16503 incollection

[11]M. Atiyah: Collected works, vol. 1: Early papers; general papers. Oxford Science Publications. The Clarendon Press and Oxford University Press (Oxford and New York), 1988. MR 951892 Zbl 0935.01034 book