[Paper Review] A comparison of positivity in complex and tropical toric geometry

José Ignacio Burgos Gil, Walter Gubler+2 more|University of Regensburg Publication Server (University of Regensburg)|Mar 19, 2020

Nonlinear Waves and Solitons4 references4 citations

TL;DR

This paper establishes a correspondence between invariant closed positive currents on smooth complex toric varieties and closed positive Lagerberg currents on their tropicalizations, using real and complex positivity structures. The key result is a correspondence theorem that identifies the cone of invariant closed positive currents on the complex side with those on the tropical side, enabling a Bedford–Taylor theory for plurisubharmonic functions on tropical toric varieties.

ABSTRACT

Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the cone of invariant closed positive currents on the complex toric variety with closed positive currents on the tropicalization. In a subsequent paper, this correspondence will be used to develop a Bedford-Taylor theory of plurisubharmonic functions on the tropicalization.

Motivation & Objective

• To compare positivity structures in complex toric geometry and tropical toric geometry.
• To develop a theory of positive currents on tropicalizations of complex toric varieties.
• To establish a correspondence between invariant closed positive currents on complex toric varieties and closed positive Lagerberg currents on their tropicalizations.
• To lay the foundation for a Bedford–Taylor theory of plurisubharmonic functions on tropical toric varieties.

Proposed method

• Uses real and complex vector space positivity structures to compare forms and currents.
• Applies Lagerberg's bigraded algebra of smooth differential forms on real vector spaces with compatibility conditions across strata.
• Defines Lagerberg currents via compatibility conditions under projections between strata of the tropicalization.
• Employs decomposition of positive currents along strata and local mass analysis to characterize positivity.
• Applies a tropical analogue of the Skoda–El Mir Theorem to establish closedness of currents.
• Uses Radon and regular Borel measures to define and analyze total variation and image measures under proper maps.

Experimental results

Research questions

• RQ1 How do invariant closed positive currents on complex toric varieties relate to those on their tropicalizations?
• RQ2 Can a Bedford–Taylor theory for plurisubharmonic functions be developed on tropical toric varieties?
• RQ3 What is the precise correspondence between complex invariant currents and tropical Lagerberg currents?
• RQ4 How do compatibility conditions under strata projections affect the positivity of Lagerberg forms and currents?
• RQ5 What is the role of the tropicalization map in transferring positivity structures from complex to tropical geometry?

Key findings

• The cone of invariant closed positive currents on a smooth complex toric variety is isomorphic to the cone of closed positive Lagerberg currents on its tropicalization.
• The correspondence is established via a canonical identification of currents that respects stratification and projection structures.
• The tropicalization map preserves closedness and positivity of currents, enabling transfer of analytic properties.
• The paper proves a tropical analogue of the Skoda–El Mir Theorem, ensuring that currents extend across lower-dimensional strata under positivity conditions.
• Lagerberg currents are characterized by compatibility conditions under strata projections, ensuring constancy in boundary directions.
• The total variation of a current is well-defined and finite, with image measures preserved under proper tropicalization maps.