Celebratio Mathematica

Michael F. 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.


[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

[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

[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