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Deligne's Weil I has been published under the title "La conjecture de Weil: I" in 1974, and Weil II in 1980. So did Deligne know in 1974 that there would be a Weil II, and can one explain the period between the two publications?

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  • $\begingroup$ I recall this question being answered on this site, but it may have just been in the comments to another question because I can't find it anymore. $\endgroup$
    – Matt
    Commented Oct 17, 2013 at 16:12
  • $\begingroup$ Certainly, he knew that there would be a Weil II, and that it would include a proof of hard Lefschetz, but he probably didn't know exactly what it would include. So I expect that part of the six years was taken up with research. $\endgroup$
    – abz
    Commented Oct 17, 2013 at 17:22
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    $\begingroup$ I guess you have an answer. Regarding the first question, look at Messing, "Short sketch of Deligne's proof of the hard Lefschetz theorem". This was published in 1975 from a talk given a year earlier. It contains a sketch and lists "La conjecture de Weil II (to appear)" as a reference. Weil II has a huge amount of material. I don't find the gap surprising. $\endgroup$
    – Donu Arapura
    Commented Oct 18, 2013 at 14:46

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To complete Carlo's answer, I think that one thing that can explain the long gap (in addition of the amount of difficult material in Weil II) is that Deligne felt the need to consolidate his result of Weil I before going further.

It should be reminded that Weil I was criticized from various directions for relying on results that were not yet formally published. Those results were essentially the theory of Grothendieck of application of Etale Cohomology to L-functions (SGA 5) and the theory of Lefschetz as generalized to schemes by Grothendieck (SGA 7). At the time of the publishing of Weil I, Grothendieck had left the IHES and was not working anymore on the SGA's (nor the EGA's), and his students and colleagues were left with notes of his talks in various states of redaction. Serre tells, for example, in a letter to Grothendieck (published by the SMF in the volume "correspondence Serre-Grothendieck"), that Illusie, who was in charge to prepare for publication some crucial parts of SGA 5, told him he was not able to check the commutativity of certain diagrams, commutativity considered as obvious in the notes.

So something that occupied Deligne for quite a while between 1974 and 1980 was this huge work of publishing/completing the work that Grothendieck left abruptly in 1970 (how much this work was just cleaning, proofreading and publishing, and how much it involved original mathematical work is the subject of polemics in Grothendieck's "Récoltes et Semailles"). For instance, Deligne (with others) published SGA 4+1/2 in 1977 with the stated intent to be a partial substitute for the then missing SGA 5. Deligne also worked with Katz and the second part of SGA 7, etc.

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This is what Pierre Deligne writes himself in the introduction to part I about the need for a part II:

Dans un article faisant suite à celui-ci, je donnerai divers raffinements de résultats intermédiaires, et des applications, parmi lesquelles le théorème de Lefschetz "difficile".

In a follow-up article, I will present several refinements of intermediate results, as well as applications, among which the "hard" Lefschetz theorem.

I would think it took 6 years to complete part II because the Hard Lefschetz Theorem is, well, hard.

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  • $\begingroup$ Carlo, I had the impression that in the proofs of Weil's conjecture, Deligne found a way around the hard Lefschetz theorem which is still open. Did Deligne prove the hard Lefschetz theorem (or a harf Lefschetz theorem) as part of his proof? $\endgroup$
    – Gil Kalai
    Commented Jul 20, 2022 at 15:40
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    $\begingroup$ experts (which I'm not) somehow do seem to be of that opinion, see for example page 54 of page.mi.fu-berlin.de/esnault/preprints/helene/… $\endgroup$
    – Carlo Beenakker
    Commented Jul 20, 2022 at 15:48
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    $\begingroup$ Dear Carlo, What I heard (which is consistent with the quote) was that Deligne first proved the Weil conjectures and then later used his solution to deduce the hard Lefschetz theorem. (But the Hodge-Riemann relations are still open). It would make a good MO question to clarify it. $\endgroup$
    – Gil Kalai
    Commented Jul 26, 2022 at 20:40

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