Commentary by M. Atiyah
My first paper  was written when I was a second-year undergraduate and arose from a course on higher-dimensional projective geometry given by . 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 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. , 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.also showed me a letter of (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
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  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 .
My supervisor  (summarized in ) which Hodge reported on when he attended a major conference in Princeton. This attracted the attention of and and was instrumental in my going to the Institute for Advanced Study in 1955.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
Princeton in 1955–56 was enormously stimulating. In particular, I got to know 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 , ,  were all influenced by him in one way or another.,
Back in England I met . This paper was later the starting point for Brieskorn’s beautiful work on rational double points., 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
Around this time (1957) the first of Hirzebruch’s Arbeitstagungen began my long series of visits to Bonn.  was an expository lecture given there based on work of Calabi which I had learnt about in Princeton.