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The complement of tropical curves in moderate position on tropical surfaces

Yuki Tsutsui

Abstract

López de Medrano, Rincón and Shaw defined the Chern classes on tropical manifolds as an extension of their theory of the Chern-Schwartz-MacPherson cycles on matroids. This makes it possible to define the Riemann-Roch number of tropical Cartier divisors on compact tropical manifolds. In this paper, we introduce the notion of a moderate position, and discuss a conjecture that the Riemann-Roch number $\operatorname{RR}(X;D)$ of a tropical submanifold $D$ of codimension $1$ in moderate position on a compact tropical manifold $X$ is equal to the topological Euler characteristic of the complement $X\setminus D$. In particular, we prove it and its generalization when $\dim X=2$ and $X$ admits a Delzant face structure.

The complement of tropical curves in moderate position on tropical surfaces

Abstract

López de Medrano, Rincón and Shaw defined the Chern classes on tropical manifolds as an extension of their theory of the Chern-Schwartz-MacPherson cycles on matroids. This makes it possible to define the Riemann-Roch number of tropical Cartier divisors on compact tropical manifolds. In this paper, we introduce the notion of a moderate position, and discuss a conjecture that the Riemann-Roch number of a tropical submanifold of codimension in moderate position on a compact tropical manifold is equal to the topological Euler characteristic of the complement . In particular, we prove it and its generalization when and admits a Delzant face structure.
Paper Structure (14 sections, 14 theorems, 102 equations)

This paper contains 14 sections, 14 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Outline of this paper
  5. Tropical submanifolds in moderate position
  6. Tropical manifolds
  7. Moderate position
  8. Relatively uniform tropical submanifolds
  9. For tropical surfaces
  10. The Riemann--Roch number of pairs of tropical submanifolds
  11. Tropical homology
  12. Tropical Cartier divisors
  13. Chern classes and Todd classes of tropical manifolds
  14. Compatibility of intersections of tropical cycles

Key Result

Theorem 1.5

Let $D$ be a tropical submanifold of codimension $1$ in moderate position on a compact tropical surface $X$. Then,

Theorems & Definitions (56)

  • Conjecture 1.1
  • Example 1.2
  • Conjecture 1.3
  • Example 1.4
  • Theorem 1.5: Main theorem
  • Corollary 1.6
  • Conjecture 1.7
  • Definition 1.8
  • Remark 1.9
  • Definition 2.2
  • ...and 46 more