New Approach To Arakelov Geometry

New Approach to Arakelov Geometry arXiv:0704.2030v1 [math.AG] 16 Apr 2007 Nikolai Durov April 17, 2007 Introduction The principal aim of this work is to provide an alternative algebraic framework for Arakelov geometry, and to demonstrate its usefulness by presenting several simple applications. This framework, called theory of generalized rings and schemes, appears to be useful beyond the scope of Arakelov geometry, providing a uniform description of classical scheme-theoretical algebraic geometry (“schemes over Spec Z”), Arakelov geometry (“schemes over \Z”), tropical geometry (“schemes over Spec T and Spec N”) Spec Z∞ and Spec and the geometry over the so-called field with one element (“schemes over Spec F1 ”). Therefore, we develop this theory a bit further than it is strictly necessary for Arakelov geometry. The approach to Arakelov geometry developed in this work is completely algebraic, in the sense that it doesn’t require the combination of schemetheoretical algebraic geometry and complex differential geometry, traditionally used in Arakelov geometry since the works of Arakelov himself. \Z of algebraic varieties X/Q However, we show that our models X¯ /Spec define both a model X / Spec Z in the usual sense and a (possibly singular) Banach (co)metric on (the smooth locus of) the complex analytic variety X(C). This metric cannot be chosen arbitrarily; however, some classical metrics like the Fubini–Study metric on Pn do arise in this way. It is interesting to note that “good” models from the algebraic point of view (e.g. finitely presented) usually give rise to not very nice metrics on X(C), and conversely, nice smooth metrics like Fubini–Study correspond to m