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.

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 |