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Tropical Kummer quartic surfaces

Shu Kawaguchi, Kazuhiko Yamaki

TL;DR

The paper develops tropical analogues of classical Kummer quartic geometry by constructing tropical Kummer surfaces as images of principally polarized tropical abelian surfaces under tropical theta functions of second order. It proves that these tropical Kummer surfaces embed faithfully into tropical projective space for irreducible cases and describes them explicitly as parallelepipeds with eight vertices, linking to the action of a Klein four group and tropical Cremona involution. A key portion connects these tropical objects to nonarchimedean geometry, showing faithful tropicalizations of canonical skeletons and establishing when these coincide with Kontsevich–Soibelman skeletons. The work introduces rational polyhedral orbifolds and provides a robust framework for faithful embeddings, nonarchimedean Appell–Humbert theory, and the tropicalization of nonarchimedean theta functions, thereby bridging tropical, nonarchimedean, and mirror-symmetry perspectives on Kummer surfaces.

Abstract

We introduce tropical Kummer quartic surfaces in tropical projective $3$-space as the images of certain principally polarized tropical abelian surfaces under tropical theta functions of second order. We study some of their properties, showing that they are included in the tropicalizations of Kummer quartic surfaces defined over nonarchimdean valued fields. In the course of this work, we introduce the notion of a rational polyhedral orbifold and we provide faithful embeddings of tropical Kummer surfaces as such. Further, we show faithful tropicalizations of the canonical skeletons of certain Kummer surfaces over nonarchimdean valued fields. Under a suitable assumption on the base field, the canonical skeletons coincide with the Kontsevich--Soibelman skeletons.

Tropical Kummer quartic surfaces

TL;DR

The paper develops tropical analogues of classical Kummer quartic geometry by constructing tropical Kummer surfaces as images of principally polarized tropical abelian surfaces under tropical theta functions of second order. It proves that these tropical Kummer surfaces embed faithfully into tropical projective space for irreducible cases and describes them explicitly as parallelepipeds with eight vertices, linking to the action of a Klein four group and tropical Cremona involution. A key portion connects these tropical objects to nonarchimedean geometry, showing faithful tropicalizations of canonical skeletons and establishing when these coincide with Kontsevich–Soibelman skeletons. The work introduces rational polyhedral orbifolds and provides a robust framework for faithful embeddings, nonarchimedean Appell–Humbert theory, and the tropicalization of nonarchimedean theta functions, thereby bridging tropical, nonarchimedean, and mirror-symmetry perspectives on Kummer surfaces.

Abstract

We introduce tropical Kummer quartic surfaces in tropical projective -space as the images of certain principally polarized tropical abelian surfaces under tropical theta functions of second order. We study some of their properties, showing that they are included in the tropicalizations of Kummer quartic surfaces defined over nonarchimdean valued fields. In the course of this work, we introduce the notion of a rational polyhedral orbifold and we provide faithful embeddings of tropical Kummer surfaces as such. Further, we show faithful tropicalizations of the canonical skeletons of certain Kummer surfaces over nonarchimdean valued fields. Under a suitable assumption on the base field, the canonical skeletons coincide with the Kontsevich--Soibelman skeletons.
Paper Structure (32 sections, 32 theorems, 90 equations, 1 table)

This paper contains 32 sections, 32 theorems, 90 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Tropical Kummer surfaces
  4. Nonarchimedean Kummer surfaces
  5. Organization of this paper
  6. Tropical abelian varieties and tropical theta functions
  7. Tropical abelian varieties
  8. Tropical theta functions
  9. Tropical descent data
  10. Rational polyhedral orbifolds and faithful embeddings
  11. Rational polyhedral spaces
  12. Rational polyhedral orbifolds
  13. Faithful embeddings of rational polyhedral orbifolds
  14. Faithful embeddings of tropical Kummer surfaces
  15. Faithful embedding of tropical Kummer surfaces
  16. ...and 17 more sections

Key Result

Theorem 1.1

Assume that $(X, Q)$ is not isomorphic to the product of principally polarized tropical elliptic curves. Then the map $\psi$ faithfully embeds $Y$ into ${\mathbb{T}}{\mathbb{P}}^3$, i.e., $\psi$ is a homeomorphism onto its image preserving the integral structures.

Theorems & Definitions (60)

  • Theorem 1.1: see Theorem \ref{['thm:faith:embeddings']}
  • Theorem 1.2: see Theorem \ref{['thm:faith:embeddings2']}
  • Theorem 1.3: see Theorem \ref{['thm:main1:FT']}
  • Theorem 1.4: see Theorem \ref{['thm:FTmain2']}
  • Definition 2.1: Tropical abelian variety, torpical elliptic curve, torpical abelian surface
  • Definition 2.2: (Principally) polarized tropical abelian variety
  • Definition 2.3: products
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • ...and 50 more