Letter to G. Faltings. (Brief an G. Faltings.) (German, English) Zbl 0901.14002

Grothendieck, Alexandre

Schneps, Leila (ed.) et al., Geometric Galois actions. 1. Around Grothendieck’s “Esquisse d’un programme”. Proceedings of the conference on geometry and arithmetic of moduli spaces, Luminy, France, August 1995. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 242, 49-58; English translation 285-293 (1997).

This letter of Alexandre Grothendieck to Gerd Faltings was written on June 27, 1983, that is, at about the time when G. Faltings had published his fundamental results on the arithmetic of algebraic surfaces over number fields, including his affirmative answer to Mordell’s conjecture. In a foregoing letter to Grothendieck, Faltings had apparently made some comments on the theory of motives, which had been initiated by Grothendieck in the early 1960’s, and the present letter begins with Grothendieck’s response to Faltings’s respective remarks. Grothendieck’s reflections on what he calls the “yoga of motives” quickly flow over into particularly mathematical-philosophical explanations of his ideas about his more general concept of “anabelian algebraic geometry” (or “absolute algebraic geometry”), which is based on his experience suggesting that the geometry of certain algebraic schemes is completely determined by its (pro-finite) fundamental group. Grothendieck’s discourses on anabelien algebraic geometry, as they are made from his memory in this letter are closely related to what he had written up, a few months later, in his just as reflectory “Sketch of a programme” (see the preceding review).
This famous, so far unpublished treatise has finally been made public through the present conference proceedings, in a very rewarding manner, and Grothendieck’s letter to Faltings has been added to it in regard of the tight correlation that the two written depositions exhibit.
For the entire collection see [Zbl 0868.00041].

Reviewer: W.Kleinert (Berlin)

MSC:

14-03	History of algebraic geometry
14A20	Generalizations (algebraic spaces, stacks)
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
01A65	Development of contemporary mathematics
14F35	Homotopy theory and fundamental groups in algebraic geometry

Keywords:

theory of motives; algebraic fundamental groups; moduli spaces; anabelian varieties; birational geometry; Galois group actions

Citations:

Zbl 0901.14001