Tropical Kummer Quartic Surfaces

• Tropical Kummer quartic surfaces are two-dimensional combinatorial objects in tropical projective 3-space, defined via principally polarized tropical abelian surfaces and second-order theta functions.
• They admit faithful unimodular embeddings with a structure decomposed into quadrilaterals and triangles through piecewise-linear min-plus equations, forming rational polyhedral orbifolds.
• Their construction bridges tropical geometry with non-Archimedean tropicalizations and Berkovich skeleta, offering insights into the degeneration of Kummer K3 surfaces.

Tropical Kummer quartic surfaces are two-dimensional combinatorial objects in tropical projective 3-space $\mathbb{P}^3_\mathrm{trop}$, constructed as the images of principally polarized tropical abelian surfaces under tropical theta functions of second order. These surfaces serve as the tropical counterparts to the classical Kummer quartic surfaces in algebraic geometry, linking tropicalization, Berkovich skeleta, and the theory of theta functions on abelian varieties. Tropical Kummer quartic surfaces arise as rational polyhedral orbifolds, admit faithful unimodular embeddings, and are included in the tropicalizations of algebraic Kummer quartics defined over non-Archimedean fields (Kawaguchi et al., 13 Jan 2026).

1. Principally Polarized Tropical Abelian Surfaces and Second-Order Theta Functions

A tropical abelian surface is a real two-dimensional torus with integral structure $X=N_\mathbb{R}/M'$ where $N\cong\mathbb{Z}^2$ and $M'\subset N_\mathbb{R}$ is a full-rank lattice. Such a surface is equipped with a positive-definite bilinear form $Q:N_\mathbb{R}\times N_\mathbb{R}\to\mathbb{R}$ satisfying the tropical Riemann condition $Q(M',N)\subset\mathbb{Z}$. The associated map $\lambda:M'\to M$ defined by $\lambda(u')(v)=Q(u',v)$ is required to be an isomorphism for $(X,Q)$ to be principally polarized.

Second-order tropical theta functions are defined by selecting four coset representatives $\mathcal{B}=\{b_0,b_1,b_2,b_3\}\subset M$ of $M/2\lambda(M')$. For $b\in\mathcal{B}$,

$$\theta_b(x) = \min_{u'\in M'} \left[2Q(x, u' + \lambda^{-1}(b)) + Q(u' + \lambda^{-1}(b), u' + \lambda^{-1}(b))\right], \quad x\in N_\mathbb{R}.$$

Each $\theta_b$ is a piecewise-$\mathbb{Z}$-affine, even function, quasi-periodic under translation by $u'\in M'$, descending to a well-defined function $\theta_b:X \to\mathbb{R}$. These theta functions generate the semimodule of all second-order tropical theta functions and fulfill the min-plus addition law.

The tropicalization of a convergent Laurent series $f(u)=\sum a_u\chi^u$ on the analytic torus $T$ is given by $$f_\mathrm{trop}(v) = \min_{u\in M}(\langle u, v \rangle -\log |a_u|)$$ for $v\in N_\mathbb{R}$, establishing that the $\theta_b$ are tropicalizations of explicit non-Archimedean theta functions.

2. Quotients under Involution and the Tropical Kummer Quartic in $\mathbb{P}^3_\mathrm{trop}$

The involution $x\mapsto -x$ on $X$ induces a quotient $Y = X/\langle -1 \rangle$, with the four second-order tropical theta functions being even, permitting the map

$$\phi = (\theta_{b_0} : \theta_{b_1} : \theta_{b_2} : \theta_{b_3}): X\to\mathbb{P}^3_\mathrm{trop}$$

to factor through $Y$, yielding $\psi: Y \to \mathbb{P}^3_\mathrm{trop}$. The quotient $Y$ is homeomorphic to $S^2$ with eight distinguished vertices corresponding to the action of $V_4\times\{\pm1\}\subset S_4\times \{\pm1\}$ on $$\tau=(\theta_{b_0}(0):\cdots:\theta_{b_3}(0))\in\mathbb{P}^3_\mathrm{trop}$$. These vertices span a parallelepiped in $\mathbb{P}^3_\mathrm{trop}$, identified as the tropical Kummer quartic.

Within this parallelepiped, $\psi(Y)$ is defined by six piecewise-linear min-plus equations associated with codimension-one faces in the Voronoi decomposition of $X$ with respect to $Q$. The surface decomposes into four quadrilaterals and two triangles, each confined to affine planes of the form $T_1\pm T_2\pm T_3=\mathrm{const}$ for inhomogeneous tropical coordinates $(T_1,T_2,T_3)$.

3. Rational Polyhedral Orbifolds and Unimodular Embeddings

A rational polyhedral space is locally modeled by finite unions of rational polytopes in $\mathbb{R}^n$, with transition maps being $\mathbb{Z}$-affine. Tropically acting finite groups yield orbifold charts via quotients, and their gluings produce rational polyhedral orbifolds.

The quotient $Y=X/\langle-1\rangle$ acquires a canonical 2-dimensional orbifold structure from $X$. An embedding $\psi:Y\to\mathbb{P}^3_\mathrm{trop}$ is termed faithful if it is both injective and unimodular, i.e., locally described by piecewise $\mathbb{Z}$-affine linear maps with determinant $\pm1$. The principal result is that for irreducible $(X,Q)$ (not decomposable into two tropical elliptic curves), $\psi$ is a homeomorphism onto its image and preserves the integral structure (Kawaguchi et al., 13 Jan 2026).

4. Non-Archimedean Kummer Quartic Surfaces and Their Tropicalizations

Let $k$ be a non-Archimedean field of residue characteristic not equal to 2. A totally degenerate abelian surface $A/k$ is uniformized by an analytic torus $T$ via

$$0 \to M' \to T^\mathrm{an} \to A^\mathrm{an} \to 0,$$

with the valuation map $\mathrm{val}:T^\mathrm{an}\to N_\mathbb{R}$ and a section $\sigma$ identifying the canonical skeleton $\Sigma_A \cong N_\mathbb{R}/M'$. A symmetric line bundle $\widetilde{L}_1$ defining a principal polarization descends to the Kummer quotient $\widetilde{Y} = A/\langle -1 \rangle$. The associated second-order theta functions produce a morphism $\widetilde{Y} \to \mathbb{P}^3_k$ whose tropicalization maps the skeleton $\Sigma_{\widetilde{Y}}$ homeomorphically and unimodularly onto the parallelepiped $\psi(Y) \subset \mathbb{P}^3_\mathrm{trop}$. A non-Archimedean Kummer quartic surface $Z\subset \mathbb{P}^3_k$ is thus constructed, whose tropicalization contains the tropical Kummer quartic $\psi(Y)$, situating the latter within the tropicalization of an algebraic Kummer quartic (Kawaguchi et al., 13 Jan 2026).

5. Canonical Skeletons and Kontsevich–Soibelman Skeleta

Given a complete discretely-valued field $k$—exemplified by $\mathbb{K} = \mathbb{F}((t))$—the Berkovich analytification $\widetilde{Y}^\mathrm{an}$ of the Kummer surface supports the Kontsevich–Soibelman skeleton $\operatorname{Sk}(\widetilde{Y})$, or essential skeleton, constructed by either weight functions (Mustaţă–Nicaise) or Kulikov degenerations.

Under the hypotheses that $A$ is split-torus uniformized and $\widetilde{Y}\to\operatorname{Spec}k$ has a strictly semistable Kulikov model, the canonical skeleton $\Sigma_{\widetilde{Y}}$ coincides with the Kontsevich–Soibelman skeleton. The associated faithful tropicalization then serves as a faithful tropicalization of the KS-skeleton of the algebraic Kummer surface $\widetilde{Y}_k$ over $\mathbb{K}$. This identification confirms the explicit $2$-sphere cut out by min-plus quartic equations in $\mathbb{P}^3_\mathrm{trop}$ as the essential skeleton controlling degeneration in Kummer K3 surfaces (Kawaguchi et al., 13 Jan 2026).

6. Connections across Tropical, Non-Archimedean, and Classical Geometries

Tropical Kummer quartic surfaces provide a bridge between tropical geometry, non-Archimedean analytic settings, and algebraic theta function theory. The explicit tropical construction, the realization as skeletons of non-Archimedean Kummer quartics, and the coincidence with essential (Kontsevich–Soibelman) skeleta establish deep combinatorial and analytic correspondences. This framework confirms that the tropical Kummer quartic is not only a combinatorial model in $\mathbb{P}^3_\mathrm{trop}$ but also the geometric skeleton governing degenerations of Kummer K3 surfaces (Kawaguchi et al., 13 Jan 2026).