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Let me recall the Standard Conjecture B (see [1,2] below):

The $\Lambda$-operation of Hodge theory is algebraic.

It more or less says that the partial inverse to “cupping with the class of a hyperplane” comes from an algebraic cycle. This is true, for example, for abelian varieties.

Question

On the fifth page of his article on the Standard Conjectures [2] (page 197 of the journal) in the second paragraph from the top Grothendieck writes:

I have an idea of a possible approach to Conjecture $B$, which relies in turn on certain unsolved geometric questions, and which should be settled in any case.

Q1: Did Grothendieck write/speak anywhere else about this idea? Did he work it out?

Q2: What are the “unsolved geometric questions” that he mentions? What is the status of these questions?

References

  1. https://en.wikipedia.org/wiki/Standard_conjectures_on_algebraic_cycles
  2. http://www.math.jussieu.fr/~leila/grothendieckcircle/StandardConjs.pdf
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    $\begingroup$ Kleiman's 1994 overview (cited in the Wikipedia entry) is quite explicit that Grothendieck published only this single 1968 paper on the standard conjectures. $\endgroup$
    – Carlo Beenakker
    Commented Feb 3, 2015 at 11:29

2 Answers 2

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My guess is that he was thinking about crystalline cohomology.

It fits rather nicely in Grothendieck research at that time. He had obviously in mind the success of Dwork's p-adic approach to the Weil conjectures, and the limitations of the other cohomologies avaible at the time (see sections 1.5 to 1.8 of "Crystals and the de Ram cohomology of schemes").

The standard conjectures were worked out in 1965 (according to Grothendieck's 1968 paper), so he had to have them in mind while working on his p-adic cohomology, that he presented to Bourbaki on december 1966. He then gave it to his student Pierre Berthelot to develop. The intro of the Bourbaki notes reads:

The content of the notes are by no means intended to be a complete theory. Rather, they outline the start of a program of work which has still not been carried out (*).

(*) For a more detailed exposition and progress in this direction, we refer to the work of P. Berthelot, to be developped presumably in SGA 8.

Berthelot's complete exposition was not presented as SGA 8, but as in independient work in 1974. So even if the cohomology was ready long before that, Grothendieck had to regard it as unsolved in 1968, when he wrote about the standard conjectures. It is also reasonable to imagine that he had hopes at those early stages for crystalline cohomology to be an important tool in the yoga of motives.

Again, this is just a guess. I'm not sure that he would refer to this as "unsolved geometric questions" (perhaps in the sense of its aplication to Hodge/Betti coefficients?). Maybe someone who knows more about all of this can add some details.

Some relevant references:

"On the de Rham cohomology of algebraic varieties" (Grothendieck, written in 1963)

"Crystals and the de Rham cohomology of schemes" pp. 254-306 (Grothendieck, written in 1966)

"Letter to Tate" (Grothendieck, written in 1966)

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    $\begingroup$ This may be true. At least, in some papers written around 1970 (not by Grothendieck:)) I have met the hope that crystalline cohomology (or, maybe, some other $p$-adic cohomology theory) will yield the proof of the last remaining Weil conjecture. $\endgroup$
    – Mikhail Bondarko
    Commented Mar 9, 2015 at 22:15
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    $\begingroup$ Thanks for your answer. It clears up some history, thought what Grothendieck really thought will probably remain a mistery. $\endgroup$
    – jmc
    Commented Mar 10, 2015 at 1:08
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In the paper Smirnov, Oleg N., Graded associative algebras and Grothendieck standard conjectures// Invent. Math. 128 (1997), no. 1, 201–206 it is proved that Standard Conjecture D (numerical equivalence vs. homological equivalence) implies Lefschetz type Standard Conjecture (Conjecture B).

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  • $\begingroup$ hmm, how does this answer Q1 or Q2? $\endgroup$
    – Carlo Beenakker
    Commented Feb 4, 2015 at 14:37
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    $\begingroup$ Well, Standard Conjecture D is an open problem.:) As you have noted, there are no publications of Grothendieck that answer the question; yet he could have suspected that the implication proved by Smirnov is valid. $\endgroup$
    – Mikhail Bondarko
    Commented Feb 4, 2015 at 14:45
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    $\begingroup$ That conjecture D implies conjecture B was known to Grothendieck, and can be found in the various articles of Kleiman on the subject. What Smirnov proved is that conjecture D for one variety $X$ implies conjecture B for $X$. I doubt very much that Grothendieck had such a detail in mind. $\endgroup$
    – abx
    Commented Mar 2, 2015 at 14:14
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    $\begingroup$ Well, this was just a guess; you can write down your own one. $\endgroup$
    – Mikhail Bondarko
    Commented Mar 2, 2015 at 14:25
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    $\begingroup$ Smirnov only showed (using Jannsen) that $D(X\times X)$ implies $B(X)$, but Grothendieck knew that $D(X\times X)\implies A(X\times X)\implies B(X)$ --- see Kleiman 1994. $\endgroup$
    – anon
    Commented Sep 16, 2020 at 13:33

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