Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry
By S. Boucksom, Dennis Eriksson. 2018
78 citations recorded by the source corpus.
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 Berman, Witt Nystr\"om 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-Noetherian) 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.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/cdf20f6bb1f97f1a6751825ca7b6725aa305eccb
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- A non-Archimedean approach to K-stability (2018, methodology): “By taking the two graded norms as supremum norms of continuous metrics on L , we recover a version of the main results of [BE18, BGJKM16] in the trivially valued case.”
- K-stability of Fano varieties via admissible flags (2020, unknown): “The starting point is the following elementary observation (compare to [9, Proposition 1.14]).”