Lectures on Arakelov geometry.(English)Zbl 0812.14015

Cambridge Studies in Advanced Mathematics. 33. Cambridge: Cambridge University Press. 177 p. (1992).
Let $$X$$ be an arithmetic variety and $$\overline E$$ an hermitian vector bundle on $$X$$. We shall attach to $$\overline E$$ characteristic classes with values in arithmetic Chow groups. More specifically, an arithmetic cycle is a pair $$(Z,g)$$ consisting of an algebraic cycle on $$X$$, i.e. a finite sum $$\sum_ \alpha n_ \alpha Z_ \alpha$$, $$n_ \alpha \in \mathbb{Z}$$, where $$Z_ \alpha$$ is a closed irreducible subscheme of $$X$$, of fixed codimension $$p$$, say, and a Green current $$g$$ for $$Z$$. By this we mean that $$g$$ is a real current on $$X_ \infty$$ which satisfies $$F^*_ \infty (g) = (-1)^{p-1}g$$ and $$dd^ cg + \delta_ Z = \omega$$, where $$\omega$$ is (the current attached to) a smooth form on $$X_ \infty$$, and $$\delta_ Z$$ is the current given by integration on $$Z_ \infty$$: $$\delta_ Z(\eta) = \sum_ \alpha n_ \alpha \int_{Z_ \alpha (\mathbb{C})} \eta$$, for any smooth form $$\eta$$ of appropriate degree. The arithmetic Chow group $$\widehat {CH}^ p (X)$$ is the abelian group of arithmetic cycles, modulo the subgroup generated by pairs $$(0, \partial u + \overline \partial v)$$ and $$(\text{div} f,- \log | f |^ 2)$$, where $$u$$ and $$v$$ are arbitrary currents of the appropriate degree and div $$f$$ is the divisor of a nonzero rational function $$f$$ on some irreducible closed subscheme of codimension $$p-1$$ in $$X$$.
In chapter III we study the groups $$\widehat {CH}^ p(X)$$, showing that they have functoriality properties and a graded product structure, at least after tensoring them by $$\mathbb{Q}$$. To prove these facts is rather difficult, for two reasons. First, the intersection theory on a general regular scheme such as $$X$$ cannot be defined in the usual way, since no moving lemma is available. We remedy this in chapter I by using algebraic $$K$$-theory and Adams operations. – A second difficulty is that, given two arithmetic cycles $$(Z,g)$$ and $$(Z',g')$$, we need a Green current for their intersection. The formula $$g'' = \omega g' + g \delta_{Z'}$$ is formally satisfactory, but involves a product of currents $$g \delta_{Z'}$$. To make sense of it in general we need to show that we can take for $$g$$ a smooth form on $$X_ \infty - Z_ \infty$$, of logarithmic type along $$Z_ \infty$$. This is done in chapter II.
After having set up arithmetic intersection theory, we define in chapter IV characteristic classes for hermitian vector bundles $$\overline E$$ on $$X$$. – Our next construction is some direct image map for hermitian vector bundles. Let $$f:X \to Y$$ be a proper flat map between arithmetic varieties, smooth on the generic fiber $$X_ \mathbb{Q}$$. According to F. Knudsen and D. Mumford [Math. Scand. 39, 19-55 (1976; Zbl 0343.14008)], there is a canonical line bundle $$\lambda (E)$$ on $$Y$$ whose fiber at every point $$y \in Y$$ is the determinant of the cohomology of $$X_ y = f^{-1} (y)$$ with coefficients in $$E$$: $$\lambda (E)_ y = \bigotimes_{q \geq 0} \bigwedge^{\max} (H^ q (X_ y,E))^{(-1)^ q}$$. To get a metric on $$\lambda (E)$$ let us fix a Kähler metric on $$X_ \infty$$, hence on each fiber $$X_ y$$, $$y \in Y_ \infty$$. According to D. Quillen [Funct. Anal. Appl. 19, 31-34 (1985); translation from Funkts. Anal. Prilozh. 19, No. 1, 37-41 (1985; Zbl 0603.32016)] we may then get a smooth metric $$h_ Q$$ on $$\lambda (E)_ \infty$$ by multiplying its $$L^ 2$$-metric by the exponential of the Ray- Singer analytic torsion: $$h_ Q = h_{L^ 2} \cdot \exp (T(E))$$, with $$T(E) = \sum_{q \geq 0} (-1)^{q+1} q \zeta_ q'(0)$$. Here $$\zeta_ q'(0)$$ is the derivative at the origin of the zeta function $$\zeta_ q(s)$$, $$s \in \mathbb{C}$$. – In chapter V we study $$\zeta_ q (s)$$ and the Quillen metric. Following J. M. Bismut, H. Gillet and C. Soulé [Commun. Math. Phys. 115, No. 1, 49-78; 79-126; No. 2, 301-351 (1988; Zbl 0651.32017)] we show that $$h_ Q$$ is smooth and compute the curvature on $$Y_ \infty$$ of the hermitian line bundle $$\lambda (E)_ Q = (\lambda (E), h_ Q)$$. It is given by a Riemann-Roch-Grothendieck formula at the level of forms. – When combining the above results with the Riemann-Roch-Grothendieck theorem for algebraic Chow groups, we get in chapter VI a Riemann-Roch-Grothendieck theorem for arithmetic Chow groups.

MSC:

 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14C05 Parametrization (Chow and Hilbert schemes) 14C40 Riemann-Roch theorems