Tropical Hopf manifolds and contracting germs

TL;DR

This paper explores tropical analogues of Hopf manifolds, introduces a monomialization process for tropical germs, and connects tropical geometry with non-archimedean analytic Hopf manifolds, including computations of their tropical invariants.

Contribution

It develops a monomialization procedure for tropical germs and establishes a link between tropical Hopf manifolds and non-archimedean analytification, advancing tropical geometry understanding.

Findings

Monomialization transforms tropical germs into morphisms.

Tropical Hopf manifolds relate to non-archimedean analytification.

Computed tropical Picard and (p,q)-homology groups.

Abstract

Classical Hopf manifolds are compact complex manifolds whose universal covering is $\mathbb{C}^d \setminus \{0\}$. 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.