The aim of this article is to state a conjectural Grothendieck-Riemann- Roch theorem for metrized bundles on arithmetic varieties, which would extend the known results of Arakelov, Faltings and Deligne in the case of arithmetic surfaces. Let \(X\) be an arithmetic variety (i.e. a regular scheme, quasi-projective and flat over \(\mathbb{Z}\)). In a previous paper [Publ. Math., Inst. Hautes Etud. Sci. 72, 93-174 (1990; Zbl 0741.14012)] we defined arithmetic Chow groups \(\widehat{CH}^ p(X)\) for every integer \(p\geq 0\), generated by pairs of cycles and “Green currents” (log. cit.). We showed that these groups have basically the same formal properties as the classical Chow groups. They are covariant for proper maps (with a degree shift). In Ann. Math., II. Ser. 131, No. 1, 163-203 (1990; Zbl 0715.14018) and No. 2, 205-238 (1990; Zbl 0715.14006)], we attached to any algebraic vector bundle \(E\) on \(X\), endowed with a hermitian metric \(h\) on the associated holomorphic vector bundle, characteristic classes \(\widehat{\phi}(E,h) \in \bigoplus_{p\geq 0}\widehat{CH}^ p(X) \otimes \mathbb{Q} = \widehat{CH}(X)_ \mathbb{Q}\), for every symmetric power series \(\phi(T_ 1,\dots,T_{rk(E)})\) with coefficients in \(\mathbb{Q}\). For instance we have Chern characters \(\widehat{ch}(E,h) \in \widehat{CH}(X)_ \mathbb{Q}\). We also introduced in the Ann. Math. paper (loc. cit.) a group \(\widehat{K}_ 0(X)\) of virtual hermitian vector bundles on \(X\) and extended \(\widehat{ch}\) to \(\widehat{K}_ 0(X)\).
To state a Grothendieck-Riemann-Roch theorem one still needs two notions. First, given a smooth projective morphism \(f: X\to Y\) between arithmetic varieties, one needs a direct image morphism \(f_ !: \widehat{K}_ 0(X) \to \widehat{K}_ 0(Y)\). Given \((E,h)\) on \(X\), to get the determinant of \(f_ !(E,h)\) amounts to defining a metric on the determinant of the cohomology of \(E\) (on the fibers of \(f\)). This question was solved by Quillen using the Ray-Singer analytic torsion. In §3 we define higher analogs of the Ray-Singer analytic torsion and get a reasonable definition of \(f_ !\) (this is a variant of ideas from our joint work with J.-M. Bismut).
The second question we have to ask is what will play the role of the Todd genus. For this we proceed in a way familiar to algebraic geometry, namely we compute both sides of the putative Riemann-Roch formula for the trivial line bundle on the projective spaces \(\mathbb{P}^ n\) over \(\mathbb{Z}\), \(n\geq 1\). This normalizes the arithmetic Todd genus uniquely.
The paper is organized as follows. In §1 we define Quillen’s metric on the determinant of cohomology, recall the definitions from our papers cited above, and define the arithmetic Todd genus. We then give a conjecture computing the Quillen metric (1.3). The holomorphic variation of this equality is known to be true (section 1.4). When specialized to the moduli space of curves of a given genus, the conjecture 1.3 gives the value of some unknown constants in string theory (1.5). – In §2 we prove conjecture 1.3 for the trivial line bundle on \(\mathbb{P}^ n\) (theorem 2.1.1) by reduction to an identity of Zagier. In §3 we define higher analytic torsion using results of our joint paper with J.-M. Bismut [Commun. Math. Phys. 115, No. 1, 79-126 (1988; Zbl 0651.32017)], compute its holomorphic variation (3.1) and define the map \(f_ !\) (3.2). We then conjecture a general arithmetic Grothendieck-Riemann-Roch identity (3.3) the holomorphic variation of which holds.