Metrics and Chern Forms

Abstract

In this first chapter we establish the language of metrics on invertible sheaves, which is equivalent with the language of Néron functions or Néron divisors. Classically, over the complex numbers, given a divisor on a complex non-singular variety, one looks for a real-valued function g which has a logarithmic singularity on the divisor, and satisfies some PDE condition on . Such conditions can to some extent be expressed algebraically over any complete, algebraically closed valued field, thus giving rise to Néron functions. We shall give a description of the analytic conditions precisely on Riemann surfaces, specializing Griffiths-Harris to the case of dimension 1, in the language of metrics.