by Phillip Griffiths
John and I go back to the period 1968–72 in Princeton when I was back and forth between the Institute for Advanced Study and Princeton University where John was a postdoc. Toward the later part of that period Dennis Sullivan was developing his approach to rational homotopy theory using differential forms. I particularly recall his recommending to the two of us the book Geometric Integration Theory by Hassler Whitney [e1], in which the use of forms on singular spaces was treated. Dennis’s theory was extraordinarily intriguing, e.g., to see Massey triple products computed using differential forms. My recollection is that during that time John and I hatched the idea of having some sort of seminar to work through Dennis’s emerging theory.
Before moving to Princeton I had spent my post PhD years 1962–67 in Berkeley, a period which overlapped with an extended visit by the mathematician Francesco Gherardelli from Florence, Italy. Mathematics in Florence was centered in the Istituto Ulisse Dini. The main tradition there was in analysis, in the classic Italian style. Gherardelli’s father had been a mathematician, in fact an algebraic geometer, as was Francesco, and the latter was interested in building up geometry at the Istituto. This led to an extended period — most of the summer of 1970 — that my wife, Taffy, and I spent in Florence. Although there was some tourism, this singular gem of a city was still recovering from the historic flood of 1966. Back in the United States after our period in Princeton, John and I were both in Cambridge, and together in discussion with a number of our mathematical colleagues, including Jim Carlson, Mark Green and Eric Friedlander, the idea arose of putting together a summer school at the Istituto in Florence, the goal being a systematic treatment of Dennis’s work. Francesco was delighted with the idea and this led to the “summer in Florence”.
Related to Francesco’s interest, one motivation was Dennis’s strong belief that because of the miraculous properties of differential forms on Kähler manifolds, his theory should have significant application to the topology of these manifolds, and therefore to complex projective algebraic varieties. In part this was because of the very special case of a compact symmetric Riemannian manifold where the harmonic forms constitute a ring. Thus the product of harmonic forms is a harmonic form, and then from Dennis’s theory there are no implications on the rational homotopy type beyond the cohomology ring. Such spaces are said to be “formal.” On a compact Kähler manifold the Kähler form is harmonic, as miraculously is its product with any other harmonic form. This is the central fact behind Hodge’s proof (the first complete one) of the hard Lefschetz theorem. This suggested that something very interesting should be going on for the homotopy type of compact Kähler manifolds. In preparing this account, Mark and I discussed the summer in Florence and some of his reminiscences were: All of us, including Susan Friedlander, Nicole Guillemet, and Taffy stayed in a former convent which was attached to a villa where Galileo had lived. Occasionally lights went on and off, and we wondered if Galileo’s ghost was responsible. (Perhaps he wanted to join in the discussion.) During the lectures, there were students at the back translating them into Italian. I think John’s Texas accent threw them a bit.
My memory of the festive meals was: first Tuscan wine would take one to an exalted plane, then espresso would bring a much-needed dose of reality, and finally grappa would be served, bringing everyone to a lovely state of equilibrium.
There was a pizzeria on the way back to the convent from Florence. This was the first time any of us had calzone, which of course was delicious.
Pinball was one of the favorite entertainments of the students — I think they called it “flipper.” Learning how to tilt without setting off the machine’s alarm was part of a PhD education.
And finally, frisbee had just been invented in a high school in New Jersey. Back in the US, Taffy was charged with the important task to find one, and when she came to join us, to bring it for use in the courtyard of the convent. This led to the development of the new game of fris-hockey. A player got points for a throw into the opposite center arch, fewer points for the adjacent arches, and ended the session if the frisbee went down the well in the center of the courtyard. John was by far the best player.
Notes for the summer course we were giving at the Istituto were taken by Moishe Breiner. They were edited and written out in detail in polished mathematical form and in Moishe’s beautiful script. Subsequently with further editing they were copied and circulated back in the US. Being a summer course, the first part of the notes was on basic algebraic topology; homology, homotopy, etc. This part was supplemented by an extensive set of exercises, many of them involving examples, amplifications and extensions of the general theory. The material was John’s approach to the subject; intuitive yet rigorous and directly treating the subject, getting at the essential points. I wish it had been available when I was first trying to learn topology.
The second part of the notes began with de Rham homotopy theory, including de Rham’s theorem over the rationals for simplicial complexes. Some of this harkened back to Geometric Integration Theory, mentioned above. With all the background now in place the culmination of the “Florence notes” was the construction of the minimal model, à la Sullivan, from which one is able to determine the rational homotopy type. Over time significant demand for the notes led to a book published by Springer [3] which is now in its second edition.
Not long after we were back in Cambridge, Deligne visited Harvard. John and I discussed Dennis’s theory with him, in particular how it applied to the calculation of Massey triple products. My recollection is that more or less out of the blue Deligne remarked, “By a weight argument these should all vanish.” This was before his proof of the Weil conjectures, but assuming them the result would follow. By then we knew that something special had to happen for the rational homotopy of compact Kähler manifolds, and Deligne’s observation let the cat out of the bag. The upshot of all this was the four-author paper on formality for these spaces [1]. For later reference I want to mention that a corollary of all of the above was to show that the higher homotopy groups of a compact Kähler manifold have a functorial mixed Hodge structure.
During the period after this work, when I had moved from Cambridge and had become primarily involved in administrative matters, John and I still occasionally crossed paths, directly and indirectly. One such time was a joint interview about “administration,” as well as mathematics, during the period when he was commuting to Stony Brook as the Director of the Simons Center for Geometry and Physics (SCGP). An instance of an indirect one was John’s collaboration with Bob Friedman in their work on algebraic surfaces and the topology of 4-manifolds.
A more direct interaction occurred very recently. Dennis’s de Rham homotopy theory applies to the unipotent completions of the fundamental group as well as to the higher homotopy groups of compact manifolds. Subsequent to our work on the four-author paper and the book that grew out of the Florence notes, John worked on extending the theory to smooth quasiprojective algebraic varieties. His wonderful paper in Publ. Math. IHES [2] essentially worked out the story. It is quite a bit more subtle than the complete case as formality no longer holds, but as John proved, the derivation from that property can be precisely described. This included the case of the unipotent completion of the fundamental groups of general smooth quasiprojective varieties.
In the 1990s Ludmil Katzarkov had the idea to apply this theory in the compact case as the central tool in establishing the Shafarevich conjecture for a smooth projective variety \( X \) with a nilpotent fundamental group.1 A central issue was to show that a closed cycle of rational curves on \( X \) did not open out into a connected infinite chain of such curves on the universal cover of \( X \), and this was established by a weight argument using the mixed Hodge structure on the unipotent completion of the fundamental group. The question arose of extending this result to the quasiprojective case. Mark and I were able to do this, the essential point being John’s quite subtle description in this case of the weights in the mixed Hodge structure in the unipotent completion of the fundamental group.
One of the great treasures in life is long-standing friendships, both personal and professional. My friendship with John is such a treasure, an especially meaningful one.
Phillip Griffiths has been a member of the mathematics department faculty at a number of institutions, including Princeton University, where he first met John and began a friendship and professional collaborations.
