Berkovich Analytification, Non-Archimedean Skeletons, and the Dual Complex¶

Job: develop the Berkovich analytification (multiplicative seminorms), the skeleton / dual complex of an sncd model (the retraction, the homotopy type), the essential skeleton / K-stability / SYZ connection, and the Berkovich vs tropicalization comparison — with the Berkovich of $\mathbb{G}_m$ / $\mathbb{P}^1$ / the skeleton as the interval as the worked example.

Core¶

Berkovich analytification and the analytification functor¶

For a variety $X$ over a non-Archimedean field $K$ (e.g. $\mathbb{C}\{\{t\}\}$ the Puiseux series, or $\mathbb{Q}_p$), the Berkovich analytification $X^{\mathrm{an}}$ (the Berkovich spectrum / the multiplicative seminorms extending the valuation on $K$). The $X^{\mathrm{an}}$ is a locally ringed space / a non-Archimedean analytic space, with the points the multiplicative seminorms $|\cdot|_x:K[X]\to\mathbb{R}_{\ge0}$ extending the base valuation.

Berkovich analytification	Description
$X^{\mathrm{an}}$	the Berkovich spectrum (the multiplicative seminorms extending the base valuation)
Points	the multiplicative seminorms $
Berkovich vs rigid	the Berkovich is the "thickening" / the compactification of the rigid (the generic points / the valuations over all extensions)
Gelfand spectrum	the max-spectrum / the rigid points are a subset; the Berkovich adds the generic / the higher-rank valuations

The skeleton and the dual complex¶

For a sncd (simple normal crossing divisor) model $\bar X\setminus D$ of $X$ (a compactification $\bar X$ with $D=\bar X\setminus X$ a simple normal crossing divisor), the skeleton $\mathrm{Sk}(X)=\mathrm{Sk}(\bar X,D)\subset X^{\mathrm{an}}$ is the dual complex $\Delta(D)$ of the divisor $D$ (the simplex complex encoding the intersections of the components of $D$, embedded in $X^{\mathrm{an}}$ via the monomial valuations).

Skeleton / dual complex	Description
Dual complex $\Delta(D)$	the combinatorial complex: vertices = irreducible components $D_i$, edges = intersections $D_i\cap D_j$, etc.
Skeleton $\mathrm{Sk}(X)$	$\Delta(D)\subset X^{\mathrm{an}}$ (the monomial valuations, the "combinatorial core")
Retraction $X^{\mathrm{an}}\to\mathrm{Sk}(X)$	the deformation retraction onto the skeleton (the Berkovich / the Thuillier retraction)
Homotopy type	$X^{\mathrm{an}}\simeq\mathrm{Sk}(X)$ for the sncd (the dual complex as the homotopy type of the analytification)

The Kontsevich–Soibelman / the essential skeleton and the K-stability¶

Essential skeleton / K-stability	Description
Essential skeleton	the Kontsevich–Soibelman / the minimal skeleton over all sncd models (the invariant skeleton)
Calabi–Yau theorem	for a CY variety, the essential skeleton is the dual complex of the canonical / the sncd (the SYZ base)
K-stability (Odaka–Song)	the dual complex and the K-moduli (the relation of the dual complex to the CY / the Kähler–Einstein metric)
Berkovich and SYZ	the skeleton as the SYZ base (the non-Archimedean / the tropical mirror symmetry)

The Berkovich and the tropicalization¶

Berkovich vs tropicalization	Description
Tropicalization $\mathrm{trop}(V(I))\subset\mathbb{R}^n$	the closure of the valuation of the $K^\times$-points (recalled from foundations)
Partial tropicalization	the skeleton maps to the tropicalization (the coordinate valuations $X^{\mathrm{an}}\to\mathbb{R}^n$)
Berkovich vs tropical	the Berkovich is the full / intrinsic non-Archimedean space, the tropicalization is the projection / the coordinate shadow
Kapranov (recalled)	the tropicalization of a hypersurface = the tropical hypersurface (the non-Archimedean amoeba), the Berkovich / the coordinate-tropicalization

See it¶

Berkovich analytification of $\mathbb{G}_m$ / $\mathbb{P}^1$ and the skeleton. For $X=\mathbb{G}_m=\mathrm{Spec}\,K[t,t^{-1}]$ over the Puiseux $K$, the Berkovich analytification $\mathbb{G}_m^{\mathrm{an}}$ is the "line with the generic points" (the multiplicative seminorms, the valuations on $K[t,t^{-1}]$ extending the base). The skeleton $\mathrm{Sk}(\mathbb{G}_m)=\mathbb{R}$ (the real line, the valuations $v_a(f)=\mathrm{val}(a)+\mathrm{ord}_0(f)$ the monomial valuations at $0$, $v_a(t)=a$ for $a\in\mathbb{R}$) — the skeleton is the real line, the tropical line. The tropicalization $\mathrm{trop}(\mathbb{G}_m)=\mathbb{R}$ (the valuation of the $K^\times$-points). The retraction $\mathbb{G}_m^{\mathrm{an}}\to\mathbb{R}$ (the Berkovich / the skeleton).

For $X=\mathbb{P}^1\setminus\{0,\infty\}=\mathbb{G}_m$, the skeleton is the interval $[0,1]$ (the dual complex of $\{0,\infty\}$, the two endpoints), and the $\mathbb{P}^1$ Berkovich retracts to the interval. The key point: the Berkovich analytification is the "thickening" of the tropicalization, the skeleton is the dual complex (the combinatorial core), and the retraction gives the homotopy type — the non-Archimedean / Berkovich is the intrinsic lift of the tropical.

Check¶

Show that the skeleton of $\mathbb{G}_m$ over a Puiseux field is $\mathbb{R}$, and identify the retraction.

$\mathbb{G}_m=\mathrm{Spec}\,K[t,t^{-1}]$, the Berkovich points are the multiplicative seminorms $|\cdot|_x$ on $K[t,t^{-1}]$ extending the base valuation. The monomial valuations $v_a$ (parametrized by $a\in\mathbb{R}$) defined by $v_a(\sum c_i t^i)=\min_i(\mathrm{val}(c_i)+ai)$ (the Gauss valuations / the monomial valuations at $0$, the $v_a(t)=a$) form a copy of $\mathbb{R}\subset\mathbb{G}_m^{\mathrm{an}}$ — the skeleton. The retraction: every point $x\in\mathbb{G}_m^{\mathrm{an}}$ retracts to the skeleton point $v_a$ where $a=v_x(t)$ (the valuation of $t$ at $x$), the deformation retraction $\mathbb{G}_m^{\mathrm{an}}\to\mathbb{R}=\mathrm{Sk}(\mathbb{G}_m)$. The homotopy type: $\mathbb{G}_m^{\mathrm{an}}\simeq\mathbb{R}$ (the Berkovich is homotopy-equivalent to the skeleton, the real line).

Why is the essential skeleton (the Kontsevich–Soibelman) the invariant, and what is the role of the sncd model?

The skeleton $\mathrm{Sk}(\bar X,D)$ depends on the choice of the sncd model $(\bar X,D)$ — different compactifications give different skeletons. The essential skeleton (Kontsevich–Soibelman) is the minimal / the invariant skeleton: the union (or the inverse limit) over all sncd models, the skeleton that is independent of the choice. For a Calabi–Yau variety, the essential skeleton is the dual complex of the canonical / the sncd model (the canonical divisor), the SYZ base. The role of the sncd: the skeleton is defined for each sncd model (the dual complex $\Delta(D)$), and the essential skeleton is the invariant over all models — the sncd is the input, the essential the output / the invariant.

Pitfalls¶

The Berkovich analytification is the "thickening" / the intrinsic (the tropicalization is the projection, not the full space) — the Berkovich has the generic points / the higher-rank valuations, the tropicalization is the coordinate shadow.
The skeleton requires the sncd model (the essential skeleton is the invariant / the minimal, not the arbitrary model) — the choice of the model matters, the essential is the canonical.
The homotopy type $X^{\mathrm{an}}\simeq\mathrm{Sk}(X)$ holds for the sncd model (the retraction, the deformation), but the essential skeleton may be smaller (the invariant).
The Berkovich vs the rigid: the Berkovich is the compactification / the thickening of the rigid (the rigid points are a subset, the Berkovich adds the generic / the boundary points).
The K-stability / the Odaka–Song is the relation of the dual complex to the K-moduli (the Calabi–Yau / the Kähler–Einstein metric), the deep / the recent connection, not a formal consequence.

Refs¶

Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields (1990); Baker–Rumely, Potential Theory on the Berkovich Projective Line
Thuillier, "Géométrie toroïdale et géométrie analytique non archimédienne" (2007); Kontsevich–Soibelman, "Affine structures and non-archimedean analytic spaces" (2006)
Payne, "Analytification is the limit of all tropicalizations" (2009); Abramchenko–Caporaso–Payne–Melo, "The tropicalization of the moduli space of curves" (2013)
Odaka–Song, "The Calabi–Yau theorem and the dual complex" (2018); the essential skeleton / the K-stability

Next¶

TG index · Foundations · Matroids & Linear Spaces · Moduli & Enumerative · Scattering & Mirror · Mirror symmetry / SYZ · Algebraic geometry

Skeleton / dual complex	Description
Dual complex \(\Delta(D)\)	the combinatorial complex: vertices = irreducible components \(D_i\), edges = intersections \(D_i\cap D_j\), etc.
Skeleton \(\mathrm{Sk}(X)\)	\(\Delta(D)\subset X^{\mathrm{an}}\) (the monomial valuations, the "combinatorial core")
Retraction \(X^{\mathrm{an}}\to\mathrm{Sk}(X)\)	the deformation retraction onto the skeleton (the Berkovich / the Thuillier retraction)
Homotopy type	\(X^{\mathrm{an}}\simeq\mathrm{Sk}(X)\) for the sncd (the dual complex as the homotopy type of the analytification)