Tropical Hopf manifolds and contracting germs

Abstract

Classical Hopf manifolds are compact complex manifolds whose universal covering is $$\mathbb {C}^d \backslash \{0\}$$ . We investigate the tropical analogues of Hopf manifolds, and relate their geometry to tropical contracting germs. To do this we develop a procedure called monomialization which transforms non-degenerate tropical germs into morphisms, up to tropical modification. A link is provided between tropical Hopf manifolds and the analytification of Hopf manifolds over a non-archimedean field. We conclude by computing the tropical Picard group and (p, q)-homology groups.