Applications of Canonical Metrics on Berkovich Spaces

Abstract

This thesis examines the nature of Temkin’s canonical metrics on the sheaves of differentials of Berkovich spaces, and discusses 3 applications thereof. First, we show a comparison theorem between Temkin’s metric on the beth-analytification of a smooth variety over a trivially-valued field of characteristic zero, and a weight metric defined in terms of log discrepancies. This result is the trivially-valued counterpart to a comparison theorem of Temkin between his metric and the weight metric of Mustata–Nicaise in the discretely-valued setting.

These weight metrics are used to define an essential skeleton of a pair over a trivially-valued field; this is done following the approach of Brown–Mazzon in the discretely-valued case, and we show a compatibility result between the essential skeletons of pairs in the two settings. Furthermore, a careful study of the closures of these skeletons enables us to realize the toric skeleton of a toric variety as an essential skeleton.

On the Berkovich unit disc, Temkin’s metric acts a substitute for the Lebesgue measure. Adopting this philosophy, we show a non-Archimedean version of the Ohsawa–Takegoshi extension theorem. As a corollary, we deduce a non-Archimedean analogue of Demailly’s regularization theorem for quasisubharmonic functions on the Berkovich disc.

Finally, we employ Temkin’s metric and essential skeletons to compute the dual boundary complexes of two classes of character varieties that arise in non-abelian Hodge theory. These two results provide the first non-trivial evidence for the geometric P = W conjecture of Katzarkov–Noll–Pandit–Simpson in the compact case. For each result, we give two proofs: one using non-Archimedean geometry over a trivially-valued field, and another in the discretely-valued setting. The latter produces degenerations of compact hyper-Kahler manifolds, which are of independent interest.