The geometry of the mixed Hodge structure on the fundamental group. (English) Zbl 0654.14006

Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 2, Proc. Symp. Pure Math. 46, 247-282 (1987).
[For the entire collection see Zbl 0626.00011.]
The paper under review is an exposition (with many proofs included) of the theory of mixed Hodge structures (MHS) on the (truncations of the group ring of the) fundamental group of a pointed algebraic variety (V,x). It starts with a self-contained account of Chen’s theory of iterated integrals [cf. K.-T. Chen, Bull. Am. Math. Soc. 83, 831- 879 (1977; Zbl 0389.58001)]; an elementary proof of Chen’s \(\pi_ 1\quad de Rham\) theorem is provided. Then the construction of the MHS on \(\pi_ 1(V,x)\) is explained; the Hodge and the weight filtrations are defined in terms of length and weights of iterated integrals representing elements of the truncated group ring \(\pi_ 1.\)
Various applications are discussed such as: Torelli for pointed varieties (especially pointed curves, cf. work of the author and M. Pulte), link with B. Harris’ harmonic volume [cf. B. Harris, Acta Math. 150, 91-123 (1983; Zbl 0527.30032)] and link with the Riemann-Hilbert problem [cf. the author, Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 609-27 (1986; Zbl 0616.14004)].
Reviewer: A.Buium


14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F35 Homotopy theory and fundamental groups in algebraic geometry
14E20 Coverings in algebraic geometry