Finite morphisms between projective varieties and Skeleta

Abstract

Abstract. In this paper we study finite surjective morphisms between irreducible projective varieties in terms of the morphisms they induce between the respective analytifications. The background for the principal result is as follows. Let V ′ and V be irreducible, projective varieties over an algebraically closed, non-Archimedean real valued field k with V normal and φ : V ′ → V be a finite surjective morphism . Let x ∈ V an (L) (Remark 1.1), where L/k is an algebraically closed complete non-Archimedean real valued field extension. We associate canonically to x an L-point of the space (V × k L) an which lies on the fibre over x and denote this point x L . The embedding of V into some n-dimensional projective space defines in a natural way a family of open neighbourhoods of x L contained in (V × k L) an . Each element of this family is parametrized by an (n + 1) 2 -tuple which quantifies its size. Of particular interest to us will be those neighbourhoods O whose preimage for the morphism (φ × id L ) an decomposes into the disjoint union of homeomorphic copies of O via (φ × id L ) an . Let Gx L denote the sub collection of elements of this form. Theorem 1.7 shows that there exists a deformation retraction of the space V an onto a finite simplicial complex such that along the fibres of the retraction the size of the largest element belonging to Gx L is constant.