Differentiability of the $\chi$-volume function over an adelic curve

Abstract

In this article, we show a differentiability property for the $\chi$-volume function on the ample cone of adelic line bundles over an adelic curve. This result is deduced from a non-Archimedean counterpart of a diffrentiability result of Witt Nyström. As an application, we give a logarithmic equidistribution result over adelic curves.

43 pages, comments are welcome! Second version, the differentiability property is extended to the non relative case (Theorem 1.2) and allows to give the logarithmic equidistribution result (Theorem 1.3)