DEGENERATIONS OF AMOEBAE AND BERKOVICH SPACES

MATTIAS JONSSON

Abstract. We prove a continuity result for the fibers of the Berkovich an-

alytification of a complex algebraic variety with respect to the maximum of

the Archimedean norm and the trivial norm. As a consequence, we obtaingeneralizations of a result of Mikhalkin and Rullgard about degenerations of

amoebae onto tropical varieties.

Introduction

LetX ⊂ (C∗)n be an algebraic subvariety of the n-dimensional complex algebraictorus. The amoeba AX ⊂ Rn of X is the image of X under the map

L : (C∗)n → Rn

defined by1 L = (− log |z1|, . . . ,− log |zn|), where (z1, . . . , zn) are coordinates on(C∗)n. See Figure 1 for a picture of the amoeba of X = {z1 + z2 + 1 = 0}.

More generally, let (K, | · |) be any complete valued field and let X ⊂ K∗n be analgebraic variety. For any valued field extension L/K, let XL ⊂ L∗n be the basechange. Define the tropicalization of X to be the subset Xtrop ⊂ Rn defined by

Xtrop =⋃L

L(XL),

where L ranges over all valued field extensions of K and L : (L∗)n → Rn is definedusing the same formula as above.

For example, suppose K = C. If | · |∞ is the usual Archimedean norm on C, then(C, | · |∞) does not admit any nontrivial valued field extensions, so Xtrop = AX inthis case. On the other hand, we can also equip C with the trivial norm | · |0, forwhich |a|0 = 1 for all a ∈ C∗. Then the tropicalization of X is equal to the cone overthe logarithmic limit set of X introduced in [Berg71]. The case X = {z1+z2+1 = 0}is depicted to the right in Figure 1. We see that the tropicalization looks like thelarge scale limit of the amoeba. This is a general fact:

Theorem A. The large scale limit of the amoeba AX equals the tropicalizationof X:

limρ→0+

ρ ·AX = Xtrop,

where the tropicalization is computed using the trivial norm on C.

Date: April 8, 2015.2010 Mathematics Subject Classification. Primary: 14T05, Secondary: 32A60, 32P05, 14M25.

Partially supported by NSF grants DMS-1001740 and DMS-1266207.1We use negative signs to match the standard convention for valuations. All logarithms are

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2 MATTIAS JONSSON

Figure 1. The amoeba and the tropicalization of the curve z1 +z2 + 1 = 0 in C∗2.

Here ρ · AX := {ρ · v | v ∈ AX} for ρ ∈ R∗+ and the limit can be understood,for example, in the sense of Kuratowski convergence. When X is a hypersurface,Theorem A is a special case of a result by Rullgard and Mikhalkin; see below. Thegeneral case of Theorem A is proved in [Berg71] in a slightly different language andconditional on a conjecture that was later establied in [BG84].

As a more global version, consider a (complex) toric variety Y . There is anatural topological space Y trop canonically associated to Y , see §3. If Y = C∗n,then Y trop = Rn; in general Y trop contains Rn as an open dense subset and comeswith a multiplicative action by R∗+ extending the usual action on Rn.

The two absolute values on C above define two different tropicalization mapsof Y onto Y trop. If X is an algebraic subvariety of Y , let AX and Xtrop denotethe images of X in Y trop under these two maps. When Y is projective, AX ishomeomorphic to the compactified amoeba defined in [GKZ].

Theorem A′. We have limρ→0+ ρ ·AX = Xtrop.

Note that the notation is somewhat abusive since both AX and Xtrop dependon the embedding of X in a toric variety Y . Theorem A is the special case ofTheorem A′ when Y is the algebraic torus.

New we consider one-parameter families of subvarieties. Let X ⊂ C∗ × Y be aclosed algebraic subvariety such that the projection of X onto the first factor C∗

is surjective. Write Xa ⊂ Y for the fiber of X above a ∈ C∗ and AXa⊂ Y trop

for the amoeba as in Theorem A′. Define X trop as the tropicalization of the basechange X ×Gm

Spec C((t)), where Gm = Spec C[t±1] ' C∗ and the field C((t)) offormal Laurent series is equipped with the usual non-Archimedean absolute valuefor which |t| = e−1.

Theorem B. We have lima→0(log |a|−1)−1 ·AXa = X trop.

See Figure 2 for an illustration of Theorem B in the case Y = C∗2 and X ={z1 + z2 + t = 0} ⊂ C∗ × Y , and Figure 3 for the same situation with Y = P2.When X = C∗ ×X is a product, Theorem B reduces to Theorem A′.

Theorem B is due to Rullgard [Rul01, Thm. 9] and Mikhalkin [Mik04a, Cor 6.4]in the case when Y = C∗n and X ⊂ C∗ × C∗n is a hypersurface; see also [Tei08,Thm. 7.1]. The proofs in loc. cit. use the characterization of Xtrop as the locuswhere the tropicalization of the Laurent polynomial defining X fails to be affine.The approach in [Mik04a] also emphasizes the analogy with the patchworking con-struction of Viro [Vir06]. As these proofs show, the scaled amoebae in fact convergeto the tropicalization in the Hausdorff metric; see also [AKNR13].

DEGENERATIONS OF AMOEBAE AND BERKOVICH SPACES 3

Figure 2. These pictures illustrating Theorem B show two scaledamoebae and the tropicalization of the curve V (ft) in the torusY = C∗2, where ft = (z1 + z2 + 1)(tz1 + t−1z2 + 1). The first twopictures show the amoeba of the complex curve V (fa), scaled by afactor (log |a|−1)−1, for a = 0.5 and a = 0.2, respectively. The lastpicture shows the tropicalization of the curve V (ft) over C((t)).

Figure 3. This picture illustrates Theorem B for the closure inP2 of the curve V (ft) in Figure 2. The triangle is the momentpolytope of Y = P2 with its canonical polarization.

The higher codimension case of Theorem B for Y = C∗n is stated without proofin [IMS09, Thm. 1.4]. I have not been able to locate a proof nor the generalversion of Theorem B in the literature. At least in the case Y = C∗n, one mayin principle reduce Theorem B to the hypersurface case, using, on the one hand,Artin’s Approximation Theorem together with the approach in [Tei08, p.112] and,on the other hand, the fact that the tropicalization of a subvariety is the intersectionof the tropicalizations of finitely many hypersurfaces containing the subvariety.However, the latter fact is quite nontrivial, with incomplete proofs appearing in theliterature: a correct argument can be found by combining [BJSST07] and [CP12],or in [MS15].

Our proof of Theorem B is quite different and does not rely on reduction tothe hypersurface case. Indeed, the purpose of this paper is to show that theseresults on degenerations of amoebae are rather direct consequences of a continuityproperty of the fibers of certain Berkovich spaces that were introduced in [Berk09]and contain both Archimedean and non-Archimedean information. Our results givefurther evidence to the suggestion on p.51 of loc. cit. that such spaces are “worthstudying”.

Let us explain all this in the context of Theorem A′, leaving the setting ofTheorem B to §4. Consider the field C equipped with the norm

‖ · ‖ := max{| · |∞, | · |0}, (�)

4 MATTIAS JONSSON

0 1 ρ

Figure 4. The analytification P1,An of the complex projective linewith respect to the norm ‖ · ‖ on C, together with the canonicalmap λ : P1,An → [0, 1]. The fiber λ−1(0) is the analytification ofP1 with respect to the trivial norm, and is homeomorphic to a coneover P1(C). All the other fibers are homeomorphic to a sphere.The points on top form a continuous section of λ. The smallercircle in the fiber λ−1(ρ) is of radius e−1/ρ; these circles convergeas ρ→ 0 to a unique point in the fiber λ−1(0).

Note that ‖ · ‖ is only submultiplicative, but (C, ‖ · ‖) is nevertheless a Banach ring.Given a complex algebraic variety X, Berkovich introduced in [Berk09] a naturalanalytification XAn of X with respect to the norm ‖ · ‖ on C. See §2 for moredetails on this and on what follows. The space XAn is a locally compact Hausdorffspace and comes with a natural continuous and surjective map

λ : XAn → [0, 1].

The fiber λ−1(1) is the usual complex analytic space Xh associated to X.2 For0 < ρ ≤ 1, λ−1(ρ) is homeomorphic to Xh. Finally, the fiber λ−1(0) is the Berkovichanalytification of X with respect to | · |0. See Figure 4 for an illustration of (P1)An.

Theorem C. The map λ : XAn → [0, 1] is open.

This result essentially says that the Archimedean fibers λ−1(ρ) converge to thenon-Archimedean fiber Xan = λ−1(0) as ρ→ 0+. The latter convergence propertyimplies Theorem A′ since there is a natural continuous, proper and surjective trop-icalization map Y An → Y trop that takes the fiber λ−1(0) to Xtrop and takes anyother fiber λ−1(ρ) to the scaled amoeba ρ · AX .

We prove Theorem C using the fact that the points of Xan of maximal rationalrank are dense. Using resolution of singularities, such points can be realized ona blowup as monomial valuations with rationally independent weights, and thenthe proof is concluded by a direct computation. See §2.4 for details. A statementrelated to Theorem C appears as Corollary 6.8 in [Poi13a].

Let us make some bibliographical comments. Amoebae (with the opposite signconvention of ours) were introduced in [GKZ] and have been intensively studied.

When X = V (f) ⊂ C∗n is a hypersurface, the complement of the amoeba AX inRn is convex and its connected components correspond to Laurent series expansionsof 1/f at the origin [FPT00, PR02, Rul00, Rul01, TdW13]. Hypersurface amoebaecan also be effectively studied using Ronkin functions [MR01, PR04, Rul01, PT05].Their boundaries are studied in [MN13a, Mik00, Mik04a, SdW13]

2The superscript “h” stands for “holomorphic”.

DEGENERATIONS OF AMOEBAE AND BERKOVICH SPACES 5

In dimension n = 2, there is an inequality between the area of the amoeba Afdefined by a polynomial f and the area of the Newton polygon of f [PR04]: thecase of equality was characterized in [MR01] as arising from Harnack curves inreal algebraic geometry. Further interesting relations between real algebraic curvesand amoebae are studied in [Mik00]. The degeneration of amoebae in dimensiontwo onto tropical varieties is used in a striking way in [Mik05] for enumerativeproblems. Planar amoebae also arise in certain considerations in statistical ther-modynamics [KOS07, PPT13].

In higher codimension, amoebae may or may not have finite volume [MN13b]but their complements retain certain weaker convexity properties [Hen04, Ras09].Computational aspects are studied in [Pur08, The02, TdW11, deW13]. For moreinformation and further references, see the surveys [Ite04, Mik04b].

Tropical varieties have appeared in many different contexts. We have definedthem here as images under the tropicalization map, but they can also be charac-terized in terms of so-called initial degenerations [EKL06, SS04, Dra08, Pay09].They have a polyhedral structure [BG84, EKL06] that satisfies a balancing condi-tion [Spe05, ST08, Gub13]. Tropical geometry, especially for curves, can also, tosome extent, be developed intrinsically, see [BN07, BPR11, IMS09, Mik06]. It hasseen striking applications to algebraic geometry [CDPR12, JP14, Mik00, Mik05].

The relation between Berkovich spaces (over a valued field) and tropical geom-etry appears implicitly already in the work of Bieri-Groves [BG84] which predatesthe general theory developed by Berkovich himself. Since then, it has been sys-tematically studied by many authors. For finer properties of the tropicalizationmap, see e.g. [ABBR13, ABBR14, BPR11, Duc12, Gub07, GRW14, OP13, Rab12].In [Pay09, FGP13] it is shown that the Berkovich analytification of an algebraicvariety over a non-Archimedean field is the limit of its tropicalizations over allembeddings into toric varieties.

The general idea of using non-Archimedean techniques to study various kindsof limiting behavior of complex analytic objects is also not new. Morgan andShalen [MS84] used valuations to compactify complex affine varieties. Favre re-cently used the spaceXAn to recast and generalize their construction using Berkovichspaces; a statement close to Theorem C (and even closer to Theorem C′ in §4)appears in [Fav12]. Other examples of how Berkovich spaces, especially analytifica-tions of complex algebraic varieties with respect to the trivial norm, can be used tostudy complex analytic phenomena can be found in [Berk09, BFJ08, FJ05a, FJ05b,FJ07, FJ11, Kiw06, MS84]. Also related—at least in spirit—is the procedure knownas Maslov dequantization: see [Lit05, IM12] and the references therein.

A version of Theorem A for a non-Archimedean absolute value was proved byGubler, see [Gub13, §8, Cor. 11.13]. The techniques in this paper could likely beadapted to give a new proof of this result, at least in residue characteristic zero,but we leave this for future work. It would also be interesting to study the adelicamoebae associated to varieties defined over a number field, see [EKL06, Pay08].The results in this paper have recently been used to study the topology of thecomplements of certain tropical vareities [NS].

The organization of the paper is as follows. In §1 we recall the notion of con-tinuously varying families of spaces in the sense of Kuratowski. In §2 we discussvarious analytification procedures and prove Theorem C. Then, in §3 we study the

6 MATTIAS JONSSON

tropicalization map from Y An to Y trop for a toric variety Y . In particular, we proveTheorem A′. Finally, in §4 we study one-parameter families of varieties and proveTheorem B, as well as the required fact, Theorem C′, about Berkovich spaces.

Acknowledgment. I thank V. Berkovich for the proof of Lemma 2.2, and M. Baker,S. Boucksom, A. Ducros, W. Gubler, S. Payne and A. Werner, for comments on apreliminary version of this manuscript. I have also benefitted from discussions withE. Brugalle, C. Favre, G. Mikhalkin and B. Teissier.

1. Continuous families of subspaces

Consider a surjective continuous map π : X → B between topological spaces.Write Xb := π−1(b) for b ∈ B. We’d like to study the continuity properties ofb 7→ Xb. To this end, suppose that X embeds as a subset of B × Y , for sometopological space Y , and that π is the restriction of the projection of B × Y ontothe first factor. We can then view Xb as a subset of Y for all b ∈ B.

Definition 1.1. We say that b→ Xb is upper semicontinuous (usc) if given b0 ∈ Band y ∈ Y \ Xb0 , there exist neighborhoods U of y in Y and B0 of b0 in B suchthat Xb ∩ U = ∅ for all b ∈ B0.

Definition 1.2. We say that b→ Xb is lower semicontinuous (lsc) if given b0 ∈ Band y ∈ Xb0 and given any neighborhoods U of y in Y and B0 of b0 in B, we haveXb ∩ U 6= ∅ for all b ∈ B0.

Naturally, b → Xb is continuous if it is both usc and lsc. These continuityproperties are in the sense of Kuratowski [Kur]. The proof of the following resultis left to the reader.

Lemma 1.3. The map b → Xb is usc iff X is closed in B × Y . It is lsc iffπ : X → B is open.

Now suppose Y is a metric space. We can then consider continuity of b → Xb

in the Hausdorff topology, which means the following: for every b0 ∈ B and everyε > 0 there exists a neighborhood B0 of b in B such that, whenever b ∈ B0, anypoint in Xb0 (resp. Xb) is at distance at most ε from some point in Xb (resp. Xb0).

Suppose that Xb is a closed subset of Y for all b. Then continuity of b→ Xb inthe Hausdorff topology implies continuity in the sense of Kuratowski. The converseis true when Y is compact.

2. Analytification

We recall a special case of the construction in [Berk09, §2]; see also [Berk, §1.5]and [Poi10, Poi13a]. Consider a Banach ring (k, ‖ · ‖). This means that k is acommutative ring with unit and that ‖ · ‖ : k → R+ satisfies ‖a‖ = 0 iff a = 0;‖a− b‖ ≤ ‖a‖+ ‖b‖ and ‖ab‖ ≤ ‖a‖ · ‖b‖ for all a, b ∈ k; and k is complete in themetric induced by ‖ · ‖. In fact, we will only consider the case when k is a field.

Let X be a separated scheme of finite type over k. The construction in [Berk09]associates an analytification XAn of X with respect to the norm ‖ · ‖ on k. It isdefined as follows.3 For any affine open subset U = SpecA ofX, where A is a finitelygenerated k-algebra, let UAn be the (nonempty) set of multiplicative seminorms on

3While we shall only consider the analytification as a topological space, one can also equip itwith a structure sheaf.

DEGENERATIONS OF AMOEBAE AND BERKOVICH SPACES 7

A whose restrictions to k are bounded by the norm ‖ · ‖. The topology on UAn isthe weakest one for which UAn 3 | · | → |f | is continuous for every f ∈ A.

It is customary to denote the points in Uan by a letter such as x and the cor-responding seminorm by | · |x. The latter induces a multiplicative norm on A/px,where px is the kernel of | · |x. Let H(x) be the completion of the fraction field ofA/px with respect to this norm.

By gluing together the spaces UAn we construct a topological space XAn. Thisspace is Hausdorff, locally compact and countable at infinity. The assignmentx 7→ px above globalizes to a continuous map

π : XAn → X,

where X is viewed as a scheme, equipped with the Zariski topology. The assignmentX → XAn is functorial. If X ↪→ Y is an open (resp. closed) embedding, then so isXAn ↪→ Y An. If X → Y is surjective, then so is XAn → Y An.

The analytification of the zero-dimensional affine space is equal to the BerkovichspectrumM(k, ‖ · ‖) defined in [Berk, §1.2]. The canonical map X → A0 = Spec kinduces a surjective, continuous map

λ : XAn →M(k, ‖ · ‖).We shall study this general analytification functor X 7→ XAn for three types of

Banach fields (k, ‖ · ‖).

2.1. Archimedean case. First assume that k = C is the field of complex numbersand that ‖ · ‖ = | · |∞ is the usual Archimedean norm. Denote by Xh the usualcomplex analytic variety associated to X. Recall that the points of Xh can beidentified with the closed points of X.

It turns out that Xh can be identified with the analytification XAn above insuch a way that π maps a point of Xh to the corresponding closed point of X. Tosee this, first note thatM(C, | · |∞) = {| · |∞} is a singleton. Now consider an openaffine subset U . To each point x ∈ Uh we can associate a seminorm | · |x ∈ UAn by|f |x := |f(x)|∞. This gives rise to a injective continuous map Uh → UAn which issurjective by the Gelfand-Mazur Theorem, and easily seen to be a homeomorphism.

2.2. Non-Archimedean case. Next suppose that k is a non-Archimedean field.This means that ‖·‖ = |·|, where |·| is a non-Archimedean, multiplicative norm on k,that is |ab| = |a| · |b| and |a−b| ≤ max{|a|, |b|} for any a, b ∈ k. The analytificationsXAn are then special cases of the Berkovich spaces studied in [Berk, Berk93].4. Toconform with the notation in loc. cit. we write Xan instead of XAn.

To any non-Archimedean field (k, | · |) is associated a value group |k∗| := {|a| |a ∈ k∗} as well as its divisible version

√|k∗| := {r1/n | r ∈ |k∗|, n ≥ 1}. Now

suppose x ∈ Xan. We can view√|k∗| and

√|H(x)∗| as Q-vector spaces. Define

the rational rank t(x) of x as the codimension of√|k∗| in

√|H(x)∗|. If X has

dimension n, then t(x) ≤ n for all x ∈ Xan, see [Berk93, Lemma 2.5.2]. In fact,t(x) is bounded by the transcendence degree over k of the residue field of π(x). Wesay that x has maximal rational rank if t(x) = n. In this case, π(x) is the genericpoint of an irreducible component of dimension n, and x defines a valuation of theresidue field at this point.

Our approach to the proof of Theorem C in the introduction is based on

4They are good k-analytic spaces without boundary.

8 MATTIAS JONSSON

Lemma 2.1. Assume that X has pure dimension n and that the divisible valuegroup

√|k∗| has infinite codimension in R∗+ as a Q-vector space. Then the set of

points in Xan with maximal rational rank, t(x) = n, is dense in Xan.

In fact, a more general statement is true. I am grateful to V. Berkovich for thefollowing statement and proof. (Closely related results appear as Lemma 10.1.2of [Duc11] and Corollary 5.7 of [Poi13b].) Here we freely use terminology andresults from [Berk] and [Berk93].

Lemma 2.2. Let k be as in Lemma 2.1. Consider a k-analytic space X of puredimension n and let X ′ be the set of points x ∈ X such that t(x) = n. Then X ′ isdense in X.

Proof. We may assume that X is k-affinoid. Given positive numbers r1, . . . , rmwhose images in R∗+/

√|k∗| are linearly independent, define a valued field extension

Kr/k as in [Berk, p.22]. We can pick r such that the base change Y = X⊗kKr isstrictly Kr-affinoid and of pure dimension n. The image of the analogous subsetY ′ of Y in X under the continuous canonical map Y → X lies in X ′. This reducesthe situation to the case when X is strictly k-affinoid and k is nontrivially valued.

The set X0 of points x ∈ X with [H(x) : k] < ∞ is dense in X, and any pointof X0 has a fundamental system of strictly affinoid neighborhood, see Proposi-tion 2.1.15 and its proof in [Berk]. Hence it suffices to show that every strictlyk-affinoid space of pure dimension n contains a point x with t(x) = n. By Noethernormalization, the situation is reduced to the case when X is a closed polydiscof radii one. By the assumption on k, we can find numbers 0 < r1, . . . , rn < 1whose images in R∗+/

√|k∗| are linearly independent. Then the maximal point of

the closed polydisc of radii (r1, . . . , rn) belongs to X ′. �

Now we specialize to the case when k = C is the field of complex numbersand | · | = | · |0 is the trivial norm. Berkovich spaces over this non-Archimedeanfield has seen a surprising number of applications, see for example [Berk09, BFJ08,Fav12, FJ05a, FJ07, FJ11, Jon12, Thu07]. Their topological structure is partiallydescribed in [BFJ08, FJ04, Jon12].

Consider a complex algebraic variety X of pure dimension n. Here is an exampleof a point x ∈ Xan of maximal rational rank, t(x) = n. Suppose ξ ∈ X is a closedpoint, X is smooth at ξ and there exist coordinates z1, . . . , zn at ξ and positivenumbers αi > 0, 1 ≤ i ≤ n. Then we can define a monomial valuation v on

OX,ξ ' C[[z1, . . . , zn]] by setting

v(∑m∈Zn

+

amzm) := min{m1α1 + · · ·+mnαn | am 6= 0}.

The valuation v defines a point x = e−v in Xan with π(x) = ξ, and we have t(x) = niff the numbers αi are linearly independent over Q. We call x a monomial point.

Lemma 2.3. Assume that X has pure dimension n and that x ∈ Xan has maximalrational rank t(x) = n. Then there exists a surjective birational morphism ϕ : Y →X, with Y smooth, and a monomial point y ∈ Y an with t(y) = n and ϕan(y) = x.

Proof. The point x defines a real rank one valuation on the function field of X andthe condition t(x) = n implies that this valuation is an Abhyankar valuation. Thestatement to be proved is then an example of local uniformization of Abhyankar

DEGENERATIONS OF AMOEBAE AND BERKOVICH SPACES 9

valuations, see [KK05]. A simple proof using Hironaka’s theorem on resolutions ofsingularities is given in [ELS03, Proposition 2.8]; see also [JM12, Proposition 3.7].

�

2.3. Hybrid case. Finally we consider the “hybrid” construction of [Berk09, §2]that combines Archimedean and non-Archimedean information. Equip C with thenorm ‖ · ‖ defined in (�), that is,

‖ · ‖ := max{| · |∞, | · |0}.

The Berkovich spectrum M(C, ‖ · ‖) is the set of multiplicative seminorms | · |on C bounded by ‖ · ‖. Such a seminorm has to be of the form | · |ρ∞ for someρ ∈ [0, 1], where the case ρ = 0 is interpreted as the trivial norm. Thus we canidentify M(C, ‖ · ‖) with the interval [0, 1], so we get a surjective, continuous map

λ : XAn → [0, 1].

Concretely, this map can be defined by λ(x) = log |e|x.The fiber λ−1(ρ) is equal to the analytification of X with respect to the multi-

plicative norm | · |ρ∞ on C (where ρ = 0 is interpreted as the trivial norm.)In view of §2.1, the fiber λ−1(1) is therefore homeomorphic to (and will be

identified with) Xh in such a way that π maps a point of Xh to the correspondingclosed point of X.

For 0 < ρ ≤ 1, the fiber λ−1(ρ) is also homeomorphic to Xh: each seminorm | · |in XAn ∩ λ−1(ρ) is of the form |f | = |f(x)|ρ∞ for some x ∈ Xh. In fact, λ−1(]0, 1])is homeomorphic to the product ]0, 1]×Xh, see [Berk09, Lemma 2.1].

Finally, the fiber λ−1(0) is the Berkovich analytification of X with respect tothe trivial norm on C, as in §2.2. Following [Berk09], we denote this space by Xan.

Any closed point η ∈ X gives rise to a continuous section sη of λ: if η ∈ U =SpecA, then sη(ρ) is the multiplicative seminorm on A defined by f 7→ |f(η)|ρ∞.

See Figure 4 for a picture of the space XAn when X = P1.

2.4. Proof of Theorem C. We must prove that λ : XAn → [0, 1] is open. Recall

that there exists a homeomorphism λ−1(]0, 1])∼→ ]0, 1] × Xh that commutes with

λ, so the restriction of λ to XAn \Xan is open. Therefore, it suffices to prove thatfor any x ∈ Xan, the pair (X,x) satisfies:

(?) for any neighborhood U of x in XAn, λ(U) is a neighborhood of 0 in [0, 1]

In fact, it suffices to prove (?) for x of maximal rational rank, since by Lemma 2.1such points are dense in Xan. Thus assume t(x) = n. By Lemma 2.3 we can finda surjective birational morphism φ : Y → X and a monomial point y ∈ Y an suchthat φAn(y) = x. Since φAn is continuous and surjective, it suffices to prove (?) forthe pair (Y, y).

Thus we may assume that X is smooth and that x is a monomial point. Byassumption, there exists a closed point ξ ∈ X such that v = − log |·|x is a monomialvaluation on OX,ξ in some local coordinates z1, . . . , zn at ξ, say with weights αi =v(zi) > 0, where α1, . . . , αn are linearly independent over Q. Upon replacing X byan open affine neighborhood, we assume that X = SpecA is affine and that zi ∈ Afor all i.

For 0 < ρ� 1, consider the following polycircle in the coordinates zi

Z ′ρ = {η ∈ Xh | |zi(η)| = e−αi/ρ for 1 ≤ i ≤ n}.

10 MATTIAS JONSSON

Also write Zρ for the image of Z ′ρ under the isomorphism λ−1(1)∼→ λ−1(ρ). We

claim that if U is any neighborhood of x in XAn, and 0 < ε� 1, then then Zρ ⊂ Ufor 0 < ρ ≤ ε2. This will show that λ(U) ⊃ [0, ε2] and hence complete the proof.

To prove the claim, we may assume that U is of the form

U+(f, t) := {y ∈ XAn | |f |y < t} or U−(f, t) := {y ∈ XAn | |f |y > t}

where f ∈ A and t > 0. Indeed, finite intersections of such sets form a basis ofneighborhoods of x in XAn. We consider only the case U = U+(f, t), leaving thecase U = U−(f, t) to the reader. Pick a real number s > 0 such that

|f |x < s < t

Expand f as a power series

f =∑m∈Zn

+

amzm

in OX,ξ ' C[[z1, . . . , zn]]. This series converges in some neighborhood of ξ in Xh,so there exists R ≥ 1 such that

|am|∞ ≤ R|m| (2.1)

for all m, where we write |m| = m1 + · · ·+mn.Since the αi are rationally independent, there exists m ∈ Zn+ such that am 6= 0

and 〈m, α〉 < 〈m,α〉 :=∑ni=1miαi for all m 6= m such that am 6= 0. Note that

e−〈m,α〉 = |f |x < s. We choose ε small enough so that if 0 < ρ ≤ ε2, then

Rρ|m| ≤√t

s, (2.2)

Rρ|m|e−〈m,α〉 ≤ Rρ|m|e−〈m,α〉−ε|m| when am 6= 0 and m 6= m, (2.3)

and ∑m∈Zm

+

e−|m|/ε

ρ

<

√t

s. (2.4)

We claim that Zρ ⊂ U for 0 < ρ ≤ ε2 for such ε. To see this, pick y ∈ Zρ. Weuse (2.1)-(2.4) to estimate the terms in the series expansion of f . First,

|amz(η)m|∞ = |am|∞ · |zm|1/ρy ≤ R|m|e−〈m,α〉/ρ.

Second, if m 6= m and am 6= 0, then

|amz(η)m|∞ = |am|∞ · |zm|1/ρy ≤ R|m|e−〈m,α〉/ρ

≤ R|m|e−〈m,α〉/ρe−ε|m|/ρ ≤ R|m|e−〈m,α〉/ρe−|m|/ε.

DEGENERATIONS OF AMOEBAE AND BERKOVICH SPACES 11

Since m 6= 0 when m 6= m and am 6= 0, this leads to

|f |y = |f(η)|ρ∞ ≤

(∑m

|amz(η)m|∞

)ρ

≤(R|m|e−〈m,α〉/ρ

)ρ1 +∑

m 6=m,am 6=0

e−|m|/ε

ρ

≤ R|m|ρe−〈m,α〉 ∑m∈Rn

+

e−|m|/ε

ρ

<

√t

s· s ·

√t

s= t,

and hence y ∈ U = U+(f, t), completing the proof.

3. Toric varieties and tropicalizationWe recall some basic definitions about toric varieties from [Ful]. Let N ' Zn

be a lattice, M = Hom(N,Z) the dual lattice, and Σ a fan in N . To each coneσ ∈ Σ is associated a finitely generated monoid Sσ := σ ∩M , a finitely generatedalgebra Z[Sσ] and an affine variety Uσ = Spec Z[Sσ]. By suitably gluing togetherthe different affine varieties Uσ over σ ∈ Σ, we obtain a toric variety Y = YΣ.

We can also associate a tropical object Y trop = Y tropΣ to Σ following [Ful, §4.1]

or [AMRT]; see also [Kaj08] or [Pay09].5 Namely, consider the additive monoidR := R ∪ {+∞} equipped with the natural topology. For each cone σ ∈ Σ, letU tropσ = Hom(Sσ,R) be the set of monoid homomorphisms, and equip U trop

σ with

the topology of pointwise convergence. For example, U trop0 = NR := N⊗ZR ' Rn.

The space Y trop is obtained by gluing together U tropσ for σ ∈ Σ and contains NR as

an open dense subset. It comes with the scaling action by R∗+ induced by the same

action on R. For a polarized projective toric variety Y , the moment map gives ahomeomorphism of Y trop onto the moment polytope in MR = M ⊗Z R.

3.1. Tropicalization. As in §2, let Y An be the analytification of Y ×Z C withrespect to the norm ‖ · ‖ on C. We have a continuous map

trop: Y An → Y trop

defined as follows. Let σ be a cone in Σ. A point in UAnσ is a multiplicative

seminorm | · | on C[Sσ] whose restriction to C is bounded by ‖ · ‖. In particular,− log | · | defines a monoid homomorphism from Sσ to R, and hence an element inU tropσ . It is easy to verify that the maps UAn

σ → U tropσ glue together to a globally

defined continuous map trop: Y An → Y trop.Let λ : Y An → [0, 1] be the canonical map, and set

Y Trop := [0, 1]× Y trop and Trop := λ× trop .

This leads to a commutative diagram

Y An

λ ##

Trop // Y Trop

��[0, 1]

5As with the case of the analytification, the tropicalization Y trop will only be considered as atopological space (together with an action by R∗

+) and not equipped with a structure sheaf.

12 MATTIAS JONSSON

where the map Y Trop → [0, 1] is the projection onto the first factor.

Proposition 3.1. For any toric variety Y , the map

Trop: Y An → Y Trop

is continuous, proper and surjective; hence it is also closed.

Proof. We basically argue as in Lemma 2.1 and §3 of [Pay09], but include somedetails as our setting is slightly different. The statements to be proved are local oneither the source or target, so it suffices to consider the case when Y = Uσ is affine.

In this case, the continuity of the map C[Sσ]An → UTropσ is clear from the

definition. To prove properness, pick generators m1, . . . ,mN of the monoid Sσ. Itsuffices to prove that if 0 ≤ ρ ≤ ρ′ ≤ 1 and −∞ < si ≤ ti ≤ +∞ for 1 ≤ i ≤ N ,then the set

W := Trop−1([ρ, ρ′]× {v ∈ U trop

σ | si ≤ v(mi) ≤ ti for 1 ≤ i ≤ N})

is compact in C[Sσ]An. Now, the characters zi := χmi , 1 ≤ i ≤ N generate C[Sσ]as a C-algebra; we have

C[Sσ] ' C[z1, . . . , zN ]/a

for some (monomial) ideal a ⊂ C[z1, . . . , zN ]. Under this identification, W becomesthe set of multiplicative seminorms | · | on C[z1, . . . , zN ] whose restrictions to C are

bounded by ‖ · ‖, and such that eρ ≤ |e| ≤ eρ′, e−ti ≤ |zi| ≤ e−si for 1 ≤ i ≤ N ,

and |f | = 0 for all f ∈ a. It is then clear that W is compact, as a consequence ofTychonoff’s Theorem.

Finally, surjectivity can be established as follows. Pick any (ρ, v) ∈ UTropσ and

let mi, 1 ≤ i ≤ N , be generators of Sσ as before. Set ti := v(mi) ∈ R.First suppose ρ = 0. Define a multiplicative seminorm | · | on C[z1, . . . , zN ] by

|∑β aβz

β | = max{e−〈t,β〉 | aβ 6= 0}, where 〈t, β〉 =∑Ni=1 tiβi. This seminorm

vanishes on the ideal a, and hence induces a multiplicative seminorm | · | on C[Sσ]whose restriction to C is the trivial norm. It is then clear that Trop(| · |) = (0, v).

Now suppose 0 < ρ ≤ 1. Let η ∈ Spec C[z1, . . . , zN ] be the closed point withcoordinates zi(η) = e−ti , 1 ≤ i ≤ N , and define a multiplicative seminorm | · | onC[z1, . . . , zN ] by |f | = |f(η)|ρ∞. As before, this induces a multiplicative seminormon C[Uσ] whose restriction to C is equal to | · |ρ∞, so Trop(| · |) = (ρ, v). �

3.2. Proof of Theorem A′. Let X be a complex algebraic subvariety of Y ×Z C.Then XAn is a closed subset of Y An. Let Xtrop ⊂ Y trop and XTrop ⊂ Y Trop be theimages of XAn under the mappings trop and Trop, respectively. By Proposition 3.1,XTrop is closed in Y Trop. We have a commutative diagram

XAn

λ ##

Trop // XTrop

π1

��[0, 1]

The map λ : XAn → [0, 1] is continuous and surjective, and by Theorem C it isalso open. The map Trop: XAn → XTrop is surjective by definition and continuousby Proposition 3.1. It follows from these two properties that π1 : XTrop → [0, 1] isopen and surjective.

DEGENERATIONS OF AMOEBAE AND BERKOVICH SPACES 13

Write π−11 (ρ) = {ρ} × Xtrop

ρ for 0 ≤ ρ ≤ 1, where Xtropρ ⊂ Y trop. Lemma 1.3

implies that ρ 7→ Xtropρ is continuous. Theorem A′ will thus follow immediately if

we can prove that Xtrop0 = Xtrop and Xtrop

ρ = ρ ·AX for 0 < ρ ≤ 1.

Now, the fiber λ−1(1) of XAn is the analytification of X with respect to the

Archimedean norm | · |∞ on C. Hence the fiber Xtrop1 of XTrop is equal to the

amoeba AX . Similarly, for 0 < ρ ≤ 1, λ−1(ρ) is the analytification of X withrespect to the norm | · |ρ∞ on C, and this implies that Xρ is the scaled amoeba

ρ ·AX for 0 < ρ ≤ 1. Finally, the fiber Xtrop0 is the image of Xan ⊂ Y an under the

tropicalization map Y an → Y trop, where the analytifications are defined using thetrivial norm on C. This image is equal to the tropicalization Xtrop of X as definedin [Gub13]. We should check that this image also agrees with the definition of Xtrop

in the introduction. On the one hand, the tropicalization does not change undernon-Archimedean field extensions, see [Gub13, Prop. 3.7]. On the other hand, Xan

may be viewed as the set of equivalence classes of L-valued points, over all valuedfield extensions (L, | · |) of (C, | · |0), see [Berk, 3.4.2]. This completes the proof.

4. One-parameter families

Consider a complex algebraic variety X that admits a surjective morphism

p : X → Gm

where Gm = Spec C[t±1] ' C∗. We can view X as a one-parameter family ofcomplex algebraic varieties, and we are interested in the behavior as t→ 0.

As in §2.3, let XAn be the Berkovich analytification with respect to the norm ‖·‖on C, and consider the closed subset X ] ⊂ XAn of seminorms for which |t| = e−1.The morphism p gives rise to a continuous surjective map pAn : XAn → GAn

m thatsends X ] to G]

m, and is equivariant with respect to the continuous maps λ : XAn →[0, 1] and λ : GAn

m → [0, 1]. Write X]ρ (resp. G]

m,ρ) for the fiber λ−1(ρ) inside X ](resp. G]

m).Note that G]

m,ρ consists of all multiplicative seminorms | · | on C[t±1] such that

|t| = e−1 and |a| = |a|ρ∞ for all a ∈ C∗. In particular, G]m,0 is a singleton, consisting

of the restriction to C[t±1] of the multiplicative non-Archimedean norm on C((t))such that |t| = e−1 and |a| = 1 for a ∈ C∗. Now let 0 < ρ ≤ 1. Any seminorm| · | in G]

m,ρ is then of the form |f | := |f(a)|ρ∞ for some a ∈ C∗, and the condition

|t| = e−1 means exactly that |a|∞ = e−1/ρ. Thus G]m,ρ is in bijection with the

circle of radius e−1/ρ in C, so G]m can and will be identified with the closed disc

∆e−1 of radius e−1 in C. Under this identification we have

λ(a) =(log |a|−1

)−1for a ∈ ∆e−1 .

Write p] : X ] → ∆e−1 for the restriction of pAn to X ], and X]a for the fiber above

a ∈ ∆e−1 . The central fiber X]0 is isomorphic to the analytification of the base

change X ×GmC((t)), with respect to the non-Archimedean norm on C((t)). Any

other fiber X]a, 0 < |a| ≤ e−1, is homeomorphic to the fiber above t = a of the

complex analytic space X h.

Theorem C′. For 0 < δ � 1, the map p] : X ] → ∆e−1 is open above ∆δ.

Remark 4.1. One can check that when X = Gm × X is a product, Theorem C′

implies Theorem C in the introduction.

14 MATTIAS JONSSON

Proof. Using Hironaka’s theorem on resolution of singularities, we can find a properand surjective birational morphism Y → X , with Y smooth. Then p] : Y] → ∆e−1

factors through a continuous surjective map Y] → X ]. Hence, if Y] → ∆δ is openfor some δ ∈ (0, 1), then so is X ] → ∆δ. We may therefore assume that X issmooth.

There exists a finite subset A ⊂ Gm such that p : X → Gm is flat above Gm \A,see for example [Har, Ch. III, Prop. 9.7]. By [Dou68, Corollary, p. 73] this impliesthat ph : X h → C∗, the analytification of p with respect to | · |∞, is open aboveC∗ \A. Pick δ > 0 small enough so that |a| > δ for all a ∈ A. Then p] : X ] → ∆e−1

is open above ∆δ \ {0}. It remains to see that p] is open also at points on the

non-Archimedean fiber X]0.

By the Nagata compactification theorem (see [Con07]) there exists a proper com-plex algebraic variety X , and an open immersion X ↪→ X , with dense image, suchthat p extends to a proper morphism p : X → P1. Using resolution of singularities,we may assume that X is smooth. Again by [Har, Ch. III, Prop. 9.7], p : X → P1

is automatically flat above P1 \A.The general properties of the analytification functor imply that XAn is an open

subset of XAn. We need to show that if x ∈ X]

0 and U is a neighborhood of x in

X ], then p(U) is a neighborhood of 0 in ∆e−1 . Since the Q-vector space√|C((t))∗|

is of dimension one, Lemma 2.2 applies. We may therefore assume that x is a

point of maximal rational rank, t(x) = n, since such points are dense in X]0. The

divisible value group√|H(x)∗| of x is a Q-vector space of dimension n+ 1; hence

x defines an Abhyankar valuation of the function field of X of rational rank n+ 1.That advantage of having X proper and smooth is now that this valuation admitsa unique center on X , as a consequence of the valuative criterion of properness.The center is a point ξ ∈ X 0 such that the valuation is nonnegative on the localring OX ,ξ and strictly positive on the maximal ideal.

Using [JM12, Proposition 3.7] or [BFJ12, Remark 3.8] we may, after a suitableblowup of X above 0 ∈ P1, assume that there exist local coordinates z1, . . . , zn+1

at ξ, positive integers b1, . . . , bn+1 and rationally independent positive real numbers

α1, . . . , αn+1 such that t = u∏n+1i=1 z

bii , with u a unit inOX ,ξ, and the point x defines

a monomial valuation v on OX ,ξ in these coordinates, with values v(zi) = αi for

1 ≤ i ≤ n+ 1. In particular,∑n+1i=1 biαi = v(t) = 1.

For 0 < |a| � 1, set

Za := X]a ∩ {|zi| = e−αi for 1 ≤ i ≤ n+ 1}.

The same type of estimates as in the proof of Theorem C now show that any openneighborhood U of x in X ] will contain Za for 0 < |a| � 1. Indeed, we may assumeU = {|f | > t} or U = {|f | < t}, where t > 0 and f ∈ OX ,ξ. This proves that p](U)is an open neighborhood of 0 in ∆δ, as was to be shown. �

4.1. Proof of Theorem B. The product Gm × Y is a toric variety, and we have(Gm × Y )trop = R× Y trop. The image of (Gm × Y )] in (Gm × Y )trop is given by

trop((Gm × Y )]) = {1} × Y trop ' Y trop.

DEGENERATIONS OF AMOEBAE AND BERKOVICH SPACES 15

Via the identification G]m ' ∆ := ∆e−1 above, this induces a commutative diagram

(Gm × Y )]

p

''

p×trop // ∆× Y trop

��∆

Now suppose X is a closed subvariety of (Gm×Y )×Z C ' C∗× (Y ×Z C) suchthat the projection of X to C∗ is surjective. Let X † ⊂ ∆ × Y trop be the image ofX ] under p× trop. Its fiber over a ∈ ∆ is then equal to

X†a := trop(X]a),

where, again, X]a = p−1(a). If a 6= 0, then X†a = λ(a) · AXa

. On the other hand,

X†0 is equal to X trop, the image of X]0 in Y trop. This completes the proof.

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E-mail address: mattiasj@umich.edu