A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof

We consider arithmetic varieties endowed with an action of the group scheme of n-th roots of unity and we define equivariant arithmetic K ₀-theory for these varieties. We use the equivariant analytic torsion to define direct image maps in this context and we prove a Riemann-Roch theorem for the natural transformation of equivariant arithmetic K ₀-theory induced by the restriction to the fixed point scheme; this theorem can be viewed as an analog, in the context of Arakelov geometry, of the regular case of the theorem proved by P. Baum, W. Fulton and G. Quart in [BaFQ]. We show that it implies an equivariant refinement of the arithmetic Riemann-Roch theorem, in a form conjectured by J.-M. Bismut (cf. [B2, Par. (l), p. 353] and also Ch. Soulé’s question in [SABK, 1.5, p. 162]). Oblatum 22-I-1999 & 20-II-2001?Published online: 4 May 2001