Definable equivariant retractions onto skeleta in non-archimedean geometry

By Martin Hils (Münster)
Abstract: For a quasi-projective variety V over a non-archimedean valued field, Hrushovski and Loeser recently introduced a pro-definable space Vb, the stable completion of V , which is a model-theoretic analogue of the Berkovich analytification of V . They showed that Vb admits a pro-definable strong deformation retraction onto a skeleton, i.e., onto a space which is internal to the value group and thus piecewise linear. If the underlying variety is an algebraic group, the group naturally acts on its stable completion by translation. In the talk, we will sketch various ways to construct an S-equivariant pro-definable strong deformation retraction of Sb onto a skeleton, in case S is a semiabelian variety. This is joint work with Ehud Hrushovski and Pierre Simon.