Concentration theorem and relative fixed point formula of Lefschetz type in Arakelov geometry (Q2883847)

In this paper, the author proves a concentration theorem for arithmetic \(K_0\)-theory, which can be viewed as an arithmetic analogue of a result of \textit{R. W. Thomason} [Duke Math. J. 68, No. 3, 447--462 (1992; Zbl 0813.19002)]. This result is then applied to give a simpler proof of a relative fixed point theorem of Lefschetz type in Arakelov geometry, originally due to \textit{K. Köhler} and \textit{D. Roessler} [Invent. Math. 145, No. 2, 333--396 (2001; Zbl 0999.14002)], and to \textit{J. M. Bismut} and \textit{X. Ma} [J. Reine Angew. Math. 575, 189--235 (2004; Zbl 1063.58019)].NEWLINENEWLINELet \(D\) be an arithmetic ring (e.g., \(\mathbb Z\)), and let \(\mu_n\) be the group scheme over \(D\) of \(n\)-th roots of unity. Let \(X\) be a \(\mu_n\)-equivariant arithmetic variety (for details of these definitions see Section 3 of the paper). Choose a Kähler metric on \(X(\mathbb C)\), and let it induce a Kähler metric on \(X_{\mu_n}(\mathbb C)\) and (by a quotient metric) a metric on the normal bundle \(N_{X/X_{\mu_n}}\).NEWLINENEWLINEThe concentration theorem asserts that there is a well-defined embedding morphism \(i_{*}:\widehat K_0(X_{\mu_n},\mu_n)\to \widehat K_0(X,\mu_n)\), and that this is a group isomorphism with inverse \(\lambda_{-1}^{-1}(\overline N^\vee_{X/X_{\mu_n}})\cdot i^{*}\).NEWLINENEWLINEThe fixed point theorem can be described as follows. Let \(f: X\to Y\) be an equivariant morphism of \(\mu_n\)-equivariant arithmetic varieties. Assume also that \(f\) is flat and smooth over \(\mathbb C\), and that the fiber product \(f^{-1}(Y_{\mu_n})\) is regular. (The latter condition was not required by Köhler and Roessler, but does not affect the applications considered by the author.)NEWLINENEWLINEThe chosen Kähler metric on \(X(\mathbb C)\) gives a short exact sequence NEWLINE\[CARRIAGE_RETURNNEWLINE\overline{\mathcal N} : 0 \longrightarrow \overline N_{f^{-1}(Y_{\mu_n})/X_{\mu_n}} \longrightarrow \overline N_{X/X_{\mu_n}} \longrightarrow \overline N_{Y/Y_{\mu_n}} \longrightarrow 0CARRIAGE_RETURNNEWLINE\]NEWLINE of metrized line bundles (whose metrics are defined as quotient metrics). Define NEWLINE\[CARRIAGE_RETURNNEWLINE\begin{split} M(f) &= (\lambda_{-1}^{-1}(\overline N^\vee_{X/X_{\mu_n}}) \lambda_{-1}(f^{*}\overline N^\vee_{Y/Y_{\mu_n}}) + \widetilde{\text{Td}}_g(\overline{\mathcal N}) \text{Td}_g^{-1}(f^{*}\overline N_{Y/Y_{\mu_n}})) \\ &\qquad \cdot (1 - R_g(N_{X/X_{\mu_n}}) + R_g(f^{*}N_{Y/Y_{\mu_n}}))\;. \end{split}CARRIAGE_RETURNNEWLINE\]NEWLINENEWLINENEWLINEFinally, let \(p\) be the unique prime ideal in \(R(\mu_n):=K_0(D)[\mathbb Z/n\mathbb Z]\cong K_0(D)[T]/(1-T^n)\) whose intersection with \(\mathbb Z[T]/(1-T^n)\) is the kernel of the canonical morphism \(\mathbb Z[T]/(1-T^n)\to \mathbb Z[T]/(\Phi_n)\), where \(\Phi_n\) is the \(n\)-th cyclotomic polynomial.NEWLINENEWLINEThen the relative fixed point formula asserts that the diagram NEWLINE\[CARRIAGE_RETURNNEWLINE\begin{tikzcd} \widehat K_0(X,\mu_n) \rar["M(f)\cdot\tau"]\dar["f_\ast"'] & \widehat K_0(X_{\mu_n},\mu_n)_p \dar["f_{\mu_n\ast}"] \\ \widehat K_0(Y,\mu_n) \rar["\tau" '] & \widehat K_0(Y_{\mu_n},\mu_n)_p \end{tikzcd}CARRIAGE_RETURNNEWLINE\]NEWLINE commutes, where \(\tau\) stands for the restriction map.