Tropical Realization Spaces for Polyhedral Complexes

TL;DR

This paper constructs a moduli space for algebraic varieties with a fixed tropicalization polyhedral complex using rigid analytic geometry, linking tropical and algebraic families.

Contribution

It introduces a rigid space representing the tropical realization moduli functor for quasiprojective toric varieties, connecting tropicalization with algebraic families.

Findings

The moduli functor is represented by a rigid space for quasiprojective toric varieties.

A tropical complex realized as a tropicalization of a formal family can be algebraically realized.

The approach combines rigid analytic geometry with combinatorics of Chow complexes.

Abstract

Tropicalization is a procedure that assigns polyhedral complexes to algebraic subvarieties of a torus. If one fixes a weighted polyhedral complex, one may study the set of all subvarieties of a toric variety that have that complex as their tropicalization. This gives a "tropical realization" moduli functor. We use rigid analytic geometry and the combinatorics of Chow complexes as studied by Alex Fink to prove that when the ambient toric variety is quasiprojective, the moduli functor is represented by a rigid space. As an application, we show that if a polyhedral complex is the tropicalization of a formal family of varieties then it is the tropicalization of an algebraic family of varieties.