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An arithmetic Lefschetz-Riemann-Roch theorem. With an appendix by Xiaonan Ma. (English) Zbl 1474.14015

Summary: In this article, we consider regular projective arithmetic schemes in the context of Arakelov geometry, any of which is endowed with an action of the diagonalizable group scheme associated to a finite cyclic group and with an equivariant very ample invertible sheaf. For any equivariant morphism between such arithmetic schemes, which is smooth over the generic fiber, we define a direct image map between corresponding higher equivariant arithmetic \(K\)-groups and we discuss its transitivity property. Then we use the localization sequence of higher arithmetic \(K\)-groups and the higher arithmetic concentration theorem developed in our work [Math. Z. 290, No. 1–2, 307–346 (2018; Zbl 1430.14061)] to prove an arithmetic Lefschetz-Riemann-Roch theorem. This theorem can be viewed as a generalization, to the higher equivariant arithmetic \(K\)-theory, of the fixed-point formula of Lefschetz type proved by K. Köhler and D. Roessler [Invent. Math. 145, No. 2, 333–396 (2001; Zbl 0999.14002)].

MSC:

14C40 Riemann-Roch theorems
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14L30 Group actions on varieties or schemes (quotients)
19E08 \(K\)-theory of schemes
58J52 Determinants and determinant bundles, analytic torsion
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