Degenerations of amoebae and Berkovich spaces

TL;DR

This paper establishes a continuity property for Berkovich analytifications of complex varieties, leading to new insights into how amoebae degenerate onto tropical varieties, generalizing previous results.

Contribution

It introduces a continuity result for fibers of Berkovich analytifications combining Archimedean and trivial norms, extending known degenerations of amoebae onto tropical varieties.

Findings

Proves a continuity result for Berkovich analytifications.

Generalizes Mikhalkin and Rullgard's degeneration results.

Connects amoebae degenerations with Berkovich space fibers.

Abstract

We prove a continuity result for the fibers of the Berkovich analytification of a complex algebraic variety with respect to the the maximum of the Archimedean norm and the trivial norm. As a consequence, we obtain generalizations of a result of Mikhalkin and Rullgard about degenerations of amoebae onto tropical varieties.