Introduction to $PSL_2$ phase tropicalization

TL;DR

The paper addresses the limitation of classical tropicalization, which loses phase data, by introducing a PSL_2(\mathbb{C}) phase tropicalization built on a degeneration via Hahn series and the matrix valuation $\operatorname{VAL}$. It provides a cone-model framework, a compactification through $\mathbb{C}P^3$, a circle-bundle description over the quadric $Q$, and a complete description of the phase tropicalization for constant families, illustrated by explicit examples. By preserving phase information with the $\operatorname{VAL}$ construction and the spherical coamoeba map, the work generalizes tropical limits from commutative tori to the non-commutative group $PSL_2(\mathbb{C})$ and lays groundwork for broader group-based tropicalizations and potential applications in topological invariants and curve counting.

Abstract

The usual approach to tropical geometry is via degeneration of amoebas of algebraic subvarieties of an algebraic torus $(\mathbb{C}^*)^n$. An amoeba is logarithmic projection of the variety forgetting the angular part of coordinates, called the phase. Similar degeneration can be performed without ignoring the phase. The limit then is called phase tropical variety, and it is a powerful tool in numerous areas. In the article is described a non-commutative version of phase tropicalization in the simplest case of the matrix group $PSL_2(\mathbb{C})$, replacing here $(\mathbb{C}^*)^n$ in the classical approach.

Figures (2)

• Figure 1: This picture represents the image under $\operatorname{VAL}$ of a line tangent to the quadric $Q(\mathbb{K})$.
• Figure 2: The image of a quadric surface under $\operatorname{VAL}.$ Note that a curve at the critical height 1 is missing.

Theorems & Definitions (13)

• Definition 1
• Theorem 1
• Definition 2
• Definition 3
• Remark 1
• Proposition 1
• proof
• Theorem 2
• Remark 2
• Theorem 3