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Fibers of phase tropicalizations

Andrei Bengus-Lasnier, Mikhail Shkolnikov

TL;DR

This work extends Kapranov-type tropicalization to affine varieties over the field of reversed Hahn series by developing the phase-tropical framework and the leading-term map $In_\nu$, yielding a fibration $In_\nu(X)=\bigsqcup_{\alpha\in\mathbb{R}} \mathbb{V}(IN_\alpha(I))$ with finitely many critical levels. It provides an algebraic backbone for non-abelian tropicalizations, particularly $SL_2$-tropicalization, and establishes a lifting principle that connects phase data to actual algebraic sets via $IN_{\underline{\alpha}}(I)$. The paper then offers a detailed geometric picture for $SL_2$ phase tropicalization, including hyperbolic and double-hyperbolic tropicalizations, and proves a lifting theorem and a layered level-structure description that apply to curves and general surfaces, culminating in a comprehensive description of valuative tropicalizations in this non-abelian setting. The results open paths toward patchworking-type phenomena in $SL_2$-tactors and lay foundations for broader non-commutative tropical theories, while also clarifying how topology of initial varieties is recovered in phase tropical limits.

Abstract

The subject of the present paper is phase tropicalization, which was used crucially in the context of Mikhalkin's correspondence theorem for curve counting in the complex coefficient case. The subject can be traced back to Viro's patchworking for constructing topological types of real algebraic curves. These two instances correspond to complex and real phases. Both fall into the category of what can be called "abelian" or classical tropicalization, referring to degenerations of varieties within an algebraic torus (or its compactification). In contrast, in "non-abelian" tropicalizations the ambient torus is replaced by a non-commutative group such as the special linear group. This is the beginning of a general theory valid for a wide array of coefficient systems and dimensions. As an application, the paper settles the question of phase tropicalization for the special linear group $\mathrm{SL}_2$. It also gives an algebraic explanation and phase extension of the case of curves, previously studied in the purely geometric framework. To accomplish these tasks we introduce valuative tools that allow us to prove an affine version of Kapranov's theorem on tropical hypersurfaces and its generalization to arbitrary tropical varieties. Most notably, we show the functorial properties of the graded ring of a valuation and exhibit the polynomial structure of the graded ring of monomial valuations.

Fibers of phase tropicalizations

TL;DR

This work extends Kapranov-type tropicalization to affine varieties over the field of reversed Hahn series by developing the phase-tropical framework and the leading-term map , yielding a fibration with finitely many critical levels. It provides an algebraic backbone for non-abelian tropicalizations, particularly -tropicalization, and establishes a lifting principle that connects phase data to actual algebraic sets via . The paper then offers a detailed geometric picture for phase tropicalization, including hyperbolic and double-hyperbolic tropicalizations, and proves a lifting theorem and a layered level-structure description that apply to curves and general surfaces, culminating in a comprehensive description of valuative tropicalizations in this non-abelian setting. The results open paths toward patchworking-type phenomena in -tactors and lay foundations for broader non-commutative tropical theories, while also clarifying how topology of initial varieties is recovered in phase tropical limits.

Abstract

The subject of the present paper is phase tropicalization, which was used crucially in the context of Mikhalkin's correspondence theorem for curve counting in the complex coefficient case. The subject can be traced back to Viro's patchworking for constructing topological types of real algebraic curves. These two instances correspond to complex and real phases. Both fall into the category of what can be called "abelian" or classical tropicalization, referring to degenerations of varieties within an algebraic torus (or its compactification). In contrast, in "non-abelian" tropicalizations the ambient torus is replaced by a non-commutative group such as the special linear group. This is the beginning of a general theory valid for a wide array of coefficient systems and dimensions. As an application, the paper settles the question of phase tropicalization for the special linear group . It also gives an algebraic explanation and phase extension of the case of curves, previously studied in the purely geometric framework. To accomplish these tasks we introduce valuative tools that allow us to prove an affine version of Kapranov's theorem on tropical hypersurfaces and its generalization to arbitrary tropical varieties. Most notably, we show the functorial properties of the graded ring of a valuation and exhibit the polynomial structure of the graded ring of monomial valuations.
Paper Structure (14 sections, 18 theorems, 80 equations, 1 figure)

This paper contains 14 sections, 18 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fibers of SL2 phase tropicalization
  3. Preliminaries
  4. Composing initial forms
  5. Functoriality
  6. Monomial valuations
  7. Changing variables for initial forms
  8. Lifting phase tropicalizations
  9. Phases
  10. Splittings
  11. A lifting theorem
  12. Level structure of phase tropicalizations
  13. Another approach to lifting
  14. SL2 valuation and the diffeomorphism

Key Result

Theorem 1.1

Consider an algebraic variety $X\subset\mathbb{K}^n\setminus\{0\}$, given by an ideal $I\subset\mathbb{K}[x_1,\ldots,x_n]$.

Figures (1)

  • Figure 1: The different levels of tropicalization

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Example 2.2
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • ...and 41 more