Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry
Journal article, 2021

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 the determinant of cohomology of large powers of L, with respect to the supnorm of a continuous metric on the Berkovich analytification of L. As a consequence, we establish in this setting the existence of transfinite diameters and equidistribution of Fekete points, following a strategy going back to Berman, Witt Nystrom and the first author for complex manifolds. In the non-Archimedean case, our approach relies on a version of the Knudsen-Mumford expansion for the determinant of cohomology on models over the (possibly non-No etherian) valuation ring, as a replacement for the asymptotic expansion of Bergman kernels in the complex case, and on the reduced fiber theorem, as a replacement for the Bernstein-Markov inequalities. Along the way, a systematic study of spaces of norms and the associated Fubini-Study type metrics is undertaken.

Transfinite diameter

Berkovich spaces

Determinant of the cohomology

Fubini-Study metrics

Fekete points


Sebastien Boucksom

École polytechnique

Dennis Eriksson

University of Gothenburg

Chalmers, Mathematical Sciences, Algebra and geometry

Advances in Mathematics

0001-8708 (ISSN) 1090-2082 (eISSN)

Vol. 378 107501

Subject Categories

Algebra and Logic


Mathematical Analysis



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3/5/2021 3