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A fixed point formula of Lefschetz type in Arakelov geometry. I: Statement and proof. (English) Zbl 0999.14002

The authors develop and prove a fixed point formula of Lefschetz type in Arakelov geometry which may be viewed as an analog of the regular case of the theorem proved by P. Baum, W. Fulton and G. Quart for varieties over an algebraically closed field [see Acta Math. 143, 193-211 (1979; Zbl 0454.14009)]. It implies an equivariant refinement of the arithmetic Riemann-Roch theorem, in a form conjectured by E. Bismut [Compos. Math. 93, 291-354 (1994; Zbl 0817.32014)], and by C. Soulé, D. Abramovich, J.-F. Burnol and J. Kramer [“Lectures on Arakelov geometry”, Camb. Stud. Adv. Math. 33 (1992; Zbl 0812.14015)]. The following explanations aim to give a rough picture of the main result.
Let \(f: Y \rightarrow \text{Spec}({{\mathbb Z}})\) be a projective regular scheme equipped with an action of the diagonalisable group scheme \(\mu_n\) of \(n\)-th roots of unity. Choosing the action of \(\mu_n\) (rather than the action of an automorphism of order \(n\)) ensures that the fixed point scheme \(h: Z \rightarrow \text{Spec}({\mathbb Z})\) is regular again. Let \(\overline{E} = (E,h)\) be a \(\mu_n\)-equivariant hermitian bundle on \(Y\), i.e., a vector bundle \(E\) on \(Y\) together with an action of \(\mu_n\) (compatible with the given action on \(X\)) and a hermitian metric \(h\) on the associated bundle \(E_{{\mathbb C}}\) on \(Y({\mathbb C})\) which is invariant under complex conjugation and under the action of \(\mu_n\). The authors define equivariant arithmetic Grothendieck groups \(\widehat{K}_0^{\mu_n}(Y)\), \(\widehat{K}_0^{\mu_n}(Z)\), and they construct push-forward homomorphism \(f_*:\widehat{K}_0^{\mu_n}(Y) \rightarrow \widehat{K}_0^{\mu_n}({\mathbb Z})\), \(h_*: \widehat{K}_0^{\mu_n}(Z) \rightarrow \widehat{K}_0^{\mu_n}({\mathbb Z})\) (dependent on a chosen \(\mu_n\)-invariant Kähler metric \(\omega_Y\) on \(Y\)) and a restriction homomorphism \(\rho: \widehat{K}_0^{\mu_n}(Y) \rightarrow \widehat{K}_0^{\mu_n}(Z)\) in a natural way. Furthermore, a primitive \(n\)-th root of unity \(\zeta_n\) is fixed; it induces a ring homomorphism from the representation ring \(R(\mu_n)\) to \({\mathbb C}\) and a holomorphic automorphism \(g\) of \(Y({\mathbb C})\). The main result of the paper is the following formula in \(\widehat{K}_0^{\mu_n}({\mathbb Z}) \otimes_{R(\mu_n)} {\mathbb C}\): \[ f_*(\overline{E}) = h_*\left(\left(\lambda_{-1}(\overline{N}_{Y/Z}^\vee)\right)^{-1} \rho(\overline{E})\right) - \int_{Z({\mathbb C})} \text{Td}_g(TY_{{\mathbb C}}) R_g(N_{Y_{{\mathbb C}}/Z_{{\mathbb C}}}) \text{ch}_g(E_{{\mathbb C}}). \] As in the classical case, \(\lambda_{-1}(\overline{N}_{Y/Z}^\vee)\) denotes the alternating sum of the exterior powers of the normal bundle \(\overline{N}_{Y/Z}^\vee\) (which is equipped with the metric it inherits from \(\omega_Y\)), and the authors show that this sum is invertible in \(\widehat{K}_0^{\mu_n}(Z) \otimes_{R(\mu_n)} {\mathbb C}\). The expressions \(\text{Td}_g(TY_{{\mathbb C}})\), \(R_g(N_{Y_{{\mathbb C}}/Z_{{\mathbb C}}})\) and \(\text{ch}_g(E_{{\mathbb C}})\) represent complex characteristic classes depending on \(g\). In order to gain a better understanding of the push-forward homomorphism \(f_*\), the following expression for the equivariant degree of \(f_*(\overline{E})\) (which holds in the case that \(f\) is flat and \(R^if_*(E) = 0\) for \(i \geq 1\)) may be helpful: \[ \widehat{\text{deg}}_{\mu_n}(f_*(\overline{E})) = \sum_{k \in {\mathbb Z}/n {\mathbb Z}} \zeta_n^k \widehat{\text{deg}}\left((R^0f_*(E))_k\right) - T_g(Y({\mathbb C}),\overline{E}_{{\mathbb C}}) \] where the subscript \(k\) denotes the \(k\)-th term in the natural grading on the direct image \(R^0f_*(E)\) associated with the obvious action of \(\mu_n\), and \(T_g(Y({\mathbb C}), \overline{E}_{{\mathbb C}})\) is the equivariant analytic torsion of \(E_{{\mathbb C}}\), a purely analytic term which depends on \(\omega_Y\) and the metric on \(\overline{E}_{{\mathbb C}}\). Thus the main theorem sketched above computes the arithmetic Lefschetz trace of \(\overline{E}\) as a contribution of the fixed point scheme \(Z\) and an anomaly term, the equivariant analytic torsion.
The proof of the main theorem is globally parallel to the proof of the arithmetic Adams-Riemann-Roch theorem by D. Roessler [Duke Math. J. 96, No. 1, 61-126 (1999; Zbl 0961.14006)]. Its algebro-geometric part is based on the deformation to the normal cone, while its differential-geometric part relies on the results of J.-M. Bismut [J. Differ. Geom. 41, No. 1, 53-157 (1995; Zbl 0826.32024)]. On the analytic side, the main original ingredient is a double complex formula which generalizes a result of J.-M. Bismut, H. Gillet and C. Soulé [in: The Grothendieck Festschrift. Vol. I, Prog. Math. 86, 249-331 (1990; Zbl 0744.14015)]. Several applications of the main result to heights on arithmetic varieties are discussed in separate papers, for example see K. Köhler and D. Roessler, Ann. Inst. Fourier 52, 81-103 (2002; Zbl 1001.14006).

MSC:

14C40 Riemann-Roch theorems
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14L30 Group actions on varieties or schemes (quotients)
58J20 Index theory and related fixed-point theorems on manifolds
58J52 Determinants and determinant bundles, analytic torsion
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