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Let L be an ample line bundle on a (geometrically reduced) projective varietyX over any complete valued field. Our main result describes the leadingasymptotics of the determinant of cohomology of large powers of L, with respectto the supnorm of a continuous metric on the Berkovich analytification of L. Asa consequence, we establish in this setting the existence of transfinitediameters and equidistribution of Fekete points, following a strategy goingback Berman, Witt Nystr\"om and the first author for complex manifolds. In thenon-Archimedean case, our approach relies on a version of the Knudsen-Mumfordexpansion for the determinant of cohomology on models over the (possiblynon-Noetherian) valuation ring, as a replacement for the asymptotic expansionof Bergman kernels in the complex case, and on the reduced fiber theorem, as areplacement for the Bernstein-Markov inequalities. Along the way, a systematicstudy of spaces of norms and the associated Fubini-Study type metrics isundertaken.

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