A fixed point formula of Lefschetz type in Arakelov geometry. I: Statement and proof (Q5944961)

The authors prove a Bott residue formula in the context of Arakelov geometry. More precisely, for any arithmetic variety \(X\) endowed with the action of a diagonalisable torus \(T\), they obtain a formula which computes arithmetic Chern numbers of equivariant Hermitian vector bundles on \(X\) in terms of arithmetic Chern numbers of bundles on the fixed point scheme \(X^T\) and an anomaly term derived from the equivariant and non-equivariant analytic torsion on \(X({{\mathbb C}})\). The formula and the method of proof are similar to those in the papers by \textit{M. F. Atiyah} and \textit{I. M. Singer} [Ann. Math. (2) 87, 484-530, 531-545, 546-604 (1968; Zbl 0164.24001, Zbl 0164.24201, Zbl 0164.24301)]. [A more recent approach to the classical formula can be found in a paper by \textit{D. Edidin} and \textit{W. Graham}, Am. J. Math. 120, No. 3, 619-636 (1998; Zbl 0980.14004).]NEWLINENEWLINENEWLINEThe fundamental step of the proof is a passage to the limit on both sides of the authors' arithmetic Lefschetz fixed point formula [\textit{K. Köhler} and \textit{D. Roessler}, Invent. Math. 145, No. 2, 333-396 (2001; Zbl 0999.14002)] where the limit is taken over finite group schemes of increasing order inside \(T\). The determination of the anomaly term relies on results of \textit{J.-G. Bismut} and \textit{S. Goette} [Geom. Funct. Anal. 10, No. 6, 1289-1422 (2000; Zbl 0974.58033)].NEWLINENEWLINENEWLINEThe appendix contains a concise conjectural relative fixed point formula in Arakelov geometry. In a further paper [see \textit{C. Kaiser} and \textit{K. Köhler}, Invent. Math. 147, No. 3, 633-669 (2002; Zbl 1023.14008)], the residue formula proved in this paper is applied to compute the height of some flag varieties.