Arakelov geometry of Cuntz-Pimsner algebras

We use $K$-theory of the $C^{\ast }$-algebras to study the Arakelov geometry, i.e. a compactification of the arithmetic schemes $V\to Spec\mathbf{Z}$. In particular, it is proved that the Picard group of $V$ is isomorphic to the $K_0$-group of a Cuntz-Pimsner algebra associated to $V$. We apply the result to the finiteness problem for the algebraic varieties over number fields.