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Spherical tropical curves are balanced

Desmond Coles

TL;DR

The paper proves a balancing condition for spherical tropicalizations of curves in a spherical homogeneous space $G/H$, extending the classical toric balancing to the spherical setting. It develops the necessary framework via the Luna–Vust theory of colored fans, spherical tropicalization, and toroidal embeddings, and then defines weights from intersection theory on tropical compactifications. The main result states that if $C\subset G/H$ is a curve, then $\sum_{\sigma} m_{\sigma} v_{\sigma} = -\sum_{j} m_j v_j$, where $m_{\sigma}$ are intersection numbers with toroidal boundary divisors and $m_j$ with colors, and $v_{\sigma}, v_j$ are the corresponding valuation directions. The paper also provides explicit examples verifying the balancing condition and discusses connections to Gross’s approach and potential generalizations, highlighting a pathway toward a broader structure theory for spherical tropical varieties with applications to intersection theory and tropical compactifications.

Abstract

One of the characterizing features of tropicalizations of curves in an algebraic torus is that they are balanced. Tevelev and Vogiannou introduced a spherical tropicalization map for spherical homogeneous spaces $G/H$, where $G$ is a reductive group. This map generalizes the tropicalization map for algebraic tori. We prove a balancing condition for spherical tropicalizations of curves in $G/H$ that generalizes the balancing condition for tropicalizations of curves contained in an algebraic torus. We give examples and describe the relationship with Andreas Gross's balancing condition for tropicalizations of subvarieties of toroidal embeddings.

Spherical tropical curves are balanced

TL;DR

The paper proves a balancing condition for spherical tropicalizations of curves in a spherical homogeneous space , extending the classical toric balancing to the spherical setting. It develops the necessary framework via the Luna–Vust theory of colored fans, spherical tropicalization, and toroidal embeddings, and then defines weights from intersection theory on tropical compactifications. The main result states that if is a curve, then , where are intersection numbers with toroidal boundary divisors and with colors, and are the corresponding valuation directions. The paper also provides explicit examples verifying the balancing condition and discusses connections to Gross’s approach and potential generalizations, highlighting a pathway toward a broader structure theory for spherical tropical varieties with applications to intersection theory and tropical compactifications.

Abstract

One of the characterizing features of tropicalizations of curves in an algebraic torus is that they are balanced. Tevelev and Vogiannou introduced a spherical tropicalization map for spherical homogeneous spaces , where is a reductive group. This map generalizes the tropicalization map for algebraic tori. We prove a balancing condition for spherical tropicalizations of curves in that generalizes the balancing condition for tropicalizations of curves contained in an algebraic torus. We give examples and describe the relationship with Andreas Gross's balancing condition for tropicalizations of subvarieties of toroidal embeddings.
Paper Structure (7 sections, 15 theorems, 31 equations, 3 figures)

This paper contains 7 sections, 15 theorems, 31 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Luna-Vust theory of spherical varieties
  3. Spherical tropicalization
  4. Tropicalization of toroidal spherical embeddings
  5. Balancing for spherical tropical curves
  6. Examples
  7. Closing remarks on a general structure theory

Key Result

Theorem 1.1

Let $G/H$ be a spherical homogeneous space and let $C$ be a curve in $G/H$ (defined over $k$). Then $\textrm{Trop}_G(C)$ satisfies the following balancing condition:

Figures (3)

  • Figure 1: Caption: A depiction of $N_{{\mathbb R}}(G/H)$, ${\mathcal{V}}$, and the palette, for $\textrm{SL}_2/U$.
  • Figure 2: On the left The vector space $N_{\mathbb{R}}(G/H)$, valuation cone ${\mathcal{V}}$, the color $\varrho(\textrm{val}_{E_1})$, for $n=2$. On the right, the fan of $X$.
  • Figure 3: The spherical tropicalization of $C$ with its weights.

Theorems & Definitions (33)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Proposition 2.7
  • Theorem 2.8
  • Proposition 2.9
  • ...and 23 more