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On the Work of Phillip A. Griffiths 1 1 Eduard Looijenga On the work of Phillip A. Griffiths, 2008 Brouwer Laureate NAW 5/9 nr. 3 september 2008 171 Eduard Looijenga Universiteit Utrecht Budapestlaan 6 3584 CD Utrecht [email protected] 2008 Brouwer Laureate On the work of Phillip A. Griffiths This text is the laudatio delivered by Eduard about 2600 pages in total. Let me emphasize ople professionally involved in science poli- Looijenga on the occasion of the awarding of that this was a selection indeed, as it certain- tics. This has happened in his capacity of Pro- the Brouwer Medal to Phillip A. Griffiths. ly does not contain all the laureate’s publis- vost of Duke University and even more so as hed mathematical papers. And given the ra- Scientific Director of the prestigious Institute When the Executive Council of the Dutch Ma- te at which he continues to publish research for Advanced Study and as secretary of the In- thematical Society instructed the selection at a high level, it would not surprise me if ternational Mathematical Union. In speeches committee to nominate a Brouwer lecturer eventually another volume will be added. In and essays he formulated his vision and ex- in the area of geometry, it was not so clear addition he authored or co-authored about a pressed his opinions on the role that mathe- what kind it had in mind: algebraic, complex- dozen books, some of which have become matics and science plays or should play in analytic or differential geometry. Given the si- classics. One of these, Principles of Algebraic society. ze of the area covered by that simple word, Geometry, coauthored with Harris and publis- it was not to be expected that the committee hed in 1978, is still generally regarded as the The Dutch connection would find someone who could represent that standard text from which to learn complex al- I confess that I find the task of giving you an field in its full glory, so that a further restric- gebraic geometry. And another, Geometry of idea of the laureate’s work rather daunting tion of the domain seemed in order. Fortuna- algebraic curves, Volume 1, dating from 1985 and so I hope you will allow me to begin this tely this turned out to be unnecessary, for the and written with Arbarello, Cornalba and Har- by recounting a personal experience. Almost committee, whose members were Hans Duis- ris is the only modern comprehensive text of forty years ago, in the spring of 1970, he was termaat, Jozef Steenbrink, Dirk Siersma and this venerable and beautiful subject. The title invited by Frans Oort, then at the Mathema- the chairman of the Royal Dutch Mathematical indicates that authors had boldly, if not so- tics Department of the University of Amster- Society, Henk Broer, readily and unanimously mewhat recklessly, committed themselves to dam, to give a series of lectures. At the time I agreed on a candidate who has been a towe- at least a Volume 2. I am happy to report that, was what we would now call a Master student ring figure in all these areas, namely tonight’s with such a volume now being virtually finis- of Nico Kuiper and my supervisor recommen- lecturer Phillip A. Griffiths. hed, they will redeem on this promise. Not ded that I attend these lectures. Since I knew A laudatio for a Brouwer lecturer must gi- quite in the form it had been conceived a quar- very little about algebraic geometry, I did not ve a general mathematical audience an idea ter of a century ago, I suppose, for the field has expect to get a lot out of these. I was howe- of the breadth and impact of the work of the spectacularly developed since. ver in for a pleasant surprise: I enjoyed these medallist and should also explain why this As I have hinted, the impact of this opus lectures immensely and found that I could ba- person was singled out for this award. These has been enormous. But the laureate has sically follow them to the very end. One rea- are clearly not independent duties, but in the not only exerted his influence through his son was surely that the exposition drew hea- present case, I confess that for me, the latter writings: when Griffiths took students, he of- vily on the kind of topological and geometric is easier to do than the former. ten attracted the best and as a result many reasoning that was not completely foreign to of the prominent algebraic geometers in the me, but it was most of all due to the quali- Mathematics, science and society US and abroad of a certain age group were ty of the lectures. These were not only very Let me then, before I begin to say anything formed by him. clear, making it easy to make notes (so that about the substance of his work, give you first A very much larger group has felt his be- we could prepare for the next installment), an idea of its magnitude in published form: neficial influence also in other ways. I mean but the material was also great: it was ba- five years ago a four volume set of ‘Selec- here not a group of mathematicians, but the sed on his two part paper called The periods ta’ of the laureate was published, comprising scientific community as a whole as well as pe- of rational integrals that had just appeared 1 1 2 2 172 NAW 5/9 nr. 3 september 2008 On the work of Phillip A. Griffiths, 2008 Brouwer Laureate Eduard Looijenga tributed under these headings in more than one way, I find it practical to give you an idea of his accomplishments by picking an item from each of these. From the first category, Analytic Geometry, I mention his work in what we may call com- plex integral geometry, which you might think of as a kind of fusion of differential and al- gebraic geometry. In order to have some idea what this is about, let me mention Crofton’s formula as the archetypical example. This for- mula says that there is on the space of lines in a Euclidean plane an easily described mea- sure with the property that for any decent cur- ve in the plane, its length is equal to the in- tegral of the (integer valued) function which associates to a line in that plane its number of intersections with the curve. That little gem of integral geometry is generalized by Griffiths in a beautiful manner to a setting that is high- er dimensional and also complex-analytic: si- milarly defined integrals turn out to reproduce interesting complex analytic, metric and topo- logical invariants involving for example Chern and Milnor numbers. As for the second category, Algebraic Geo- metry, it is hard and admittedly, also unfair, to choose a single paper from his many contribu- tions in this area. My favourite is the one with his erstwhile student Herb Clemens about the Intermediate Jacobian of the cubic threefold. Photograph: Jan Schipper, CWI Amsterdam Griffiths receives the Brouwer Medal from the chairman of the Royal Dutch Mathematical Society. I hope you bear with me when I take a minu- te to be moderately technical, so that I can describe the essence of this paper. in the Annals of Mathematics. This is not the rent way. As a consequence, the laureate thus Some of you may recall from a course place to describe the mathematical content of very much influenced the development of al- on Riemann surfaces that to a compact Rie- this work, but in case you know a little about gebraic geometry in the Netherlands. mann surface, or what is the same, a com- the classical theory of meromorphic differen- I told you this story also because it reflects plex projective nonsingular curve, is associ- tials on Riemann surfaces and the cohomo- a remarkable quality of the laureate to which ated a complex torus, its so-called Jacobian. logy classes they can represent: it is essenti- I alluded earlier, namely his gift of attrac- The standard and easiest way to introduce ally a beautiful generalization of that theory ting and inspiring students. Until 1983, the this Jacobian is in terms of the periods of the to the case of smooth complex hypersurfaces year Griffiths left Harvard to become Provost holomorphic differentials of the curve. The fa- in a fixed complex projective space. Griffiths at Duke University, he had had about twenty mous Abel-Jacobi theorem asserts that this mentions in a commentary in the Selecta that students, most of whom have become well- torus can also be understood differently, na- a referee of that paper had berated him for known, if not famous, experts in algebraic ge- mely as parameterizing the curve’s degree ze- developing this theory not in the most gene- ometry, differential geometry or Lie groups. ro divisor classes. The Jacobian comes with a ral setting that his methods allowed, but that That diversity also illustrates that Griffiths is a little bit of extra linear structure, called a po- this criticism was ignored by the Annals’ edi- geometer in the broadest sense of the word: in larization and another famous theorem, the tors. Now these lectures were also attended each of these areas he has made very funda- Torelli theorem, asserts that no information is by two fellow students, who, I am happy to mental contributions making him a true heir lost if we pass from the hard-to-grasp curve say, are here with us tonight as well, namely of the legacy of some of the great geometers (a curved object after all) to the linear data Jozef Steenbrink (who provided the musical of a century ago, such as Henri Poincaré and that are embodied by its polarized Jacobian.
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