Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry

Let L be an ample line bundle on a (geometrically reduced) projective variety X over any complete valued field. Our main result describes the leading asymptotics of determinant cohomology large powers L, with respect to supnorm continuous metric Berkovich analytification L. As consequence, we establish in this setting existence transfinite diameters and equidistribution Fekete points, following strategy going back Berman, Witt Nyström first author for complex manifolds. In non-Archimedean case, our approach relies version Knudsen–Mumford expansion models (possibly non-Noetherian) valuation ring, as replacement asymptotic Bergman kernels reduced fiber theorem, Bernstein–Markov inequalities. Along way, systematic study spaces norms associated Fubini–Study type metrics is undertaken.