Families of Berkovich spaces

This book investigates, roughly speaking, the variation of the properties of the fibers of a map between analytic spaces in the sense of Berkovich. First of all, we study flatness in this setting ; the naive definition of this notion is not reasonable, we explain why and give another one. We then describe the loci of fiberwise validity of some usual properties (like being Cohen-Macaulay, Gorenstein, geometrically regular...) ; we show that these are (locally) Zariski-constructible subsets of the source space. For that purpose, we develop systematic methods for « spreading out » in Berkovich geometry, as one does in scheme theory, some properties from a « generic » fiber to a neighborhood of it.