An arithmetic Riemann-Roch theorem in higher degrees. (English) Zbl 1152.14023

Let \(S\) be the spectrum of a Dedekind domain and \(g:Y\rightarrow B\) be a flat and projective \(S\)-morphism of quasi-projective regular and flat \(S\)-schemes. The Grothendieck-Riemann-Roch theorem, which asserts that the communication of the Chern character with push-forward maps can be obtained after multiplication of the Chern character with the Todd class of the virtual relative tangent bundle of \(g\), is one of the fundamental results in intersection theory.
The arithmetic intersection theory can be traced back to S. Y. Arakelov’s paper [Math. USSR, Izv. 8(1974), 1167–1180 (1976), translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1179–1192 (1974; Zbl 0355.14002)]. This theory has been systematically developed by H. Gillet and C. Soulé [Publ. Math., Inst. Hautes Étud. Sci. 72, 93–174 (1990; Zbl 0741.14012)]. In the article under review, the authors have proved an analogue of the Grothendieck-Riemann-Roch theorem in the framework of Gillet-Soulé’s arithmetic intersection theory for the case where \(S=\mathrm{Spec}(\mathbb Z)\) and \(g_{\mathbb Q}\) is smooth. They have actually established the following equality: \[ \forall y\in\widehat{K}_0(Y),\;\widehat{\mathrm{ch}}(g_*(y)) =g_*\big(\widehat{\mathrm{Td}}(g)\cdot(1-R(Tg_{\mathbb C}))\cdot\widehat{\mathrm{ch}}(y)\big) \] in \(\widehat{\mathrm{CH}}(B)_{\mathbb Q}\), where the first \(g_*\) denotes the push-forward map \(\widehat{K}_0(Y)\rightarrow\widehat{K}_0(B)\) of arithmetic Grothendieck groups, and the second one denotes the push-forward map of the arithmetic Chow groups \(\widehat{\mathrm{CH}}(Y)_{\mathbb Q}\rightarrow\widehat{\mathrm{CH}}(B)_{\mathbb Q}\), \(\widehat{\mathrm{ch}}\) is the arithmetic Chern character, \(\widehat{\mathrm{Td}}(g)\) is the arithmetic Todd class and \(R\) is the \(R\)-genus.     The proof of the theorem consists of establishing the invariance of the “error term” under closed immersions in projective spaces and the vanishing of the error term for relative projective spaces. The general case comes from a combination of these two results. For the first point, the authors have applied the arithmetic Riemann-Roch for closed immersions and Bismut’s immersion formula. The proof of the second points relies on the degree \(1\) version of the arithmetic Grothendieck-Riemann-Roch theorem previously proved by H. Gillet and C. Soulé [Invent. Math. 110, No. 3, 473–543 (1992; Zbl 0777.14008)] and the independence of the “error term” established in the article under review by using Bismut-Köhler’s anomaly formulae for the analytic torsion form.
Reviewer: Huayi Chen (Paris)


14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14C40 Riemann-Roch theorems
58J52 Determinants and determinant bundles, analytic torsion
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