Antoine Chambert-Loir and Antoine Ducros have recently written the preprint “Formes différentielles réelles et courants sur les espaces de Berkovich” (see [CD12]). This opens the door for applying methods from differential geometry also at non-archimedean places. We may think of possible applications for Arakelov theory or for non-archimedean dynamics. In the Arakelov theory developed by Gillet and Soulé [GS90], contributions of the $p$-adic places are described in terms of algebraic intersection theory on regular models over the valuation ring. The existence of such models usually requires the existence of resolution of singularities which is not known in general. Another disadvantage is that canonical metrics of line bundles on abelian varieties with bad reduction can not be described in terms of models. In the case of curves, there is an analytic description of Arakelov theory also at finite places due to Chinburgh–Rumely [CR93], Thuillier [Th05] and Zhang [Zh93]. Now the paper of Chambert-Loir and Ducros provides us with an analytic formalism including $(p,q)$-forms, currents and differential operators $d^{\prime},d^{\prime\prime}$ such that the crucial Poincaré–Lelong equation holds. This makes hope that we get also an analytic description of the $p$-adic contributions in Arakelov theory. In Amaury Thuillier’s thesis [Th05], he has given a non-archimedean potential theory on curves. For the case of the projective line, we refer to the book of Baker and Rumely [BR10] with various applications to non-archimedean dynamics. Again, we may hope to use the paper of Chambert-Loir and Ducros to give generalizations to higher dimensions.

The purpose of the present paper is to summarize the preprint [CD12] and to compare it with tropical algebraic geometry. We will assume that $K$ is an algebraically closed field endowed with a (non-trivial) complete non-archimedean absolute value $|\phantom{a}|$. Let $v:=-\log|\phantom{a}|$ be the corresponding valuation and let $\Gamma:=v(K^{\times})$ be the value group. Note that the residue field $\tilde{K}$ is also algebraically closed. For the sake of simplicity, we will restrict mostly to the case of an algebraic variety $X$ over $K$. In this case, there is quite an easy description of the associated analytic space ${X^{\rm an}}$ and so we require less knowledge about the theory of Berkovich analytic spaces than in [CD12]. The main idea is quite simple: Suppose that $X$ is an $n$-dimensional closed subvariety of the split multiplicative torus $T={\mathbb{G}}_{m}^{r}$. Then there is a tropicalization map ${\rm trop}:{T^{\rm an}}\rightarrow{\mathbb{R}}^{r}$. Roughly speaking, the map is given by applying the valuation $v$ to the coordinates of the points. Tropical geometry says that the tropical variety ${\rm Trop}(X):={\rm trop}({X^{\rm an}})$ is a weighted polyhedral complex of pure dimension $n$ satisfying a certain balancing condition. The thesis of Lagerberg [La12] gives a formalism of $(p,q)$-superforms on ${\mathbb{R}}^{r}$ together with differential operators $d^{\prime},d^{\prime\prime}$ similar to ${\partial},{\bar{\partial}}$ in complex analytic geometry. Using the tropicalization map, we have a pull-back of these forms and differential operators to ${X^{\rm an}}$. In general, we may cover an arbitrary algebraic variety $X$ of pure dimension $n$ by very affine open charts $U$ which means that $U$ has a closed immersion to ${\mathbb{G}}_{m}^{r}$ and we may apply the above to define $(p,q)$-forms and currents on ${X^{\rm an}}$. Chambert-Loir and Ducros prove that there is an integration of compactly supported $(n,n)$-forms on ${X^{\rm an}}$ with the formula of Stokes and the Poincaré–Lelong formula. The main result of the paper [CD12] is that the non-archimedean Monge-Ampère measures, which were introduced by Chambert-Loir [Ch06] directly as Radon measures on ${X^{\rm an}}$, may be written as an $n$-fold wedge product of first Chern currents. We will focus in this paper on the basics and so we will omit a description of this important result here.

Terminology

In $A\subset B$, $A$ may be equal to $B$. The complement of $A$ in $B$ is denoted by $B\setminus A$ as we reserve $-$ for algebraic purposes. The zero is included in ${\mathbb{N}}$ and in ${\mathbb{R}}_{+}$.

All occurring rings and algebras are with $1$. If $A$ is such a ring, then the group of multiplicative units is denoted by $A^{\times}$. A variety over a field is an irreducible separated reduced scheme of finite type. We denote by $\overline{F}$ an algebraic closure of the field $F$.

The terminology from convex geometry is introduced in §2 and §3. Note that polytopes and polyhedra are assume to be convex.

The author thanks Klaus Künnemann, Julius Hertel, Hartwig Mayer, Jascha Smacka and Alejandro Soto for helpful comments.