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Le déterminant de la cohomologie. (The determinant of the cohomology). (French) Zbl 0629.14008
Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math. 67, 93-117 (1987).
[For the entire collection see Zbl 0615.00004.]
This paper originates with a letter, dated 20 June 1985, from the author to D. Quillen. The letter exploits the philosophy of S. Arakelov [Math. USSR, Izv. 8(1974), 1167-1180 (1976); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1179-1192 (1974; Zbl 0355.14002) and Proc. int. Congr. Math., Vancouver, 1974, Vol. 1, 405-408 (1975; Zbl 0351.14003)] to calculate the analytic torsion of a vector bundle, with a metric, on a Riemann surface.
The author develops these ideas in the context of the results on intersection pairings and Riemann-Roch theorems which appear in the papers by G. Faltings [in Ann. Math., II. Ser. 119, 387-424 (1984; Zbl 0559.14005) and by H. Gillet and C. Soulé [in C. R. Acad. Sci., Paris, Ser. I 299, 563-566 (1984; Zbl 0607.14003)]. The paper under review pursues the principle of the coherent introduction of a role for a given metric in such results as Riemann-Roch. For example, one of the main results (§ 11.4) shows that if E/X is a vector bundle, with a metric, over a smooth, projective curve (over \({\mathbb{C}})\) then the Riemann-Roch formula, involving the analytic torsion, can be strengthened and reformulated as an isometric isomorphism.
Reviewer: V.P.Snaith

MSC:
14C40 Riemann-Roch theorems
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry