Berkovich spaces embed in Euclidean spaces
aa r X i v : . [ m a t h . AG ] O c t BERKOVICH SPACES EMBED IN EUCLIDEAN SPACES
EHUD HRUSHOVSKI, FRANÇOIS LOESER, AND BJORN POONEN
Abstract.
Let K be a field that is complete with respect to a nonarchimedean absolutevalue such that K has a countable dense subset. We prove that the Berkovich analytification V an of any d -dimensional quasi-projective scheme V over K embeds in R d +1 . If, moreover,the value group of K is dense in R > and V is a curve, then we describe the homeomorphismtype of V an by using the theory of local dendrites. Introduction
In this article, valued field will mean a field K equipped with a nonarchimedean absolutevalue | | (or equivalently with a valuation taking values in an additive subgroup of R ). Let K be a complete valued field. Let V be a quasi-projective K -scheme. The associated Berkovichspace V an [Ber90, §3.4] is a topological space that serves as a nonarchimedean analogue ofthe complex analytic space associated to a complex variety. (Actually, V an carries morestructure, but it is only the underlying topological space that concerns us here.) Althoughthe set V ( K ) in its natural topology is totally disconnected, V an is arcwise connected if andonly if V is connected [Ber90, Proposition 3.4.8(iii)]. Also, V an is locally contractible: see[Ber99, Ber04] for the smooth case, and [HL12, Theorem 13.4.1] for the general case.Our goal is to study the topology of V an under a mild countability hypothesis on K withits absolute value topology. For instance, we prove the following: Theorem 1.1.
Let K be a complete valued field having a countable dense subset. Let V bea quasi-projective K -scheme of dimension d . Then V an is homeomorphic to a subspace of R d +1 .Remark . The hypothesis that K has a countable dense subset is necessary as well assufficient. Namely, K embeds in ( A K ) an , so if the latter embeds in a separable metric spacesuch as R n , then K must have a countable dense subset. Remark . The hypothesis is satisfied when K is the completion of an algebraic closure ofa completion of a global field k , i.e., when K is C p := b Q p or its characteristic p analogue \F p (( t )) , because the algebraic closure of k in K is countable and dense. It follows that thehypothesis is satisfied also for any complete subfield of these two fields. Date : October 23, 2012.2010
Mathematics Subject Classification.
Primary 14G22; Secondary 54F50.
Key words and phrases.
Berkovich space, analytification, dendrite, local dendrite, Euclidean embedding.E.H. was supported by the European Research Council under the European Union’s Seventh FrameworkProgramme (FP7/2007-2013) / ERC Grant agreement no. 291111/ MODAG.F.L. was supported by the European Research Council under the European Union’s Seventh FrameworkProgramme (FP7/2007-2013) / ERC Grant agreement no. 246903/NMNAG.B.P. was partially supported by the Guggenheim Foundation and National Science Foundation grantDMS-1069236. ecall that a valued field is called spherically complete if every descending sequence of ballshas nonempty intersection. Say that K has dense value group if | | : K × → R > has denseimage, or equivalently if the value group is not isomorphic to { } or Z . Remark . The separability hypothesis fails for any spherically complete field K with densevalue group. Proof: Let ( t i ) be a sequence of elements of K such that the sequence | t i | isstrictly decreasing with positive limit. For each sequence ǫ = ( ǫ i ) with ǫ i ∈ { , } , define U ǫ := { x ∈ K : | x − P ni =1 ǫ i t i | < | t n | for all n } . The U ǫ are uncountably many disjoint open subsets of K , and each is nonempty by definitionof spherically complete.Let us sketch the proof of Theorem 1.1. We may assume that V is projective. The key is aresult that presents V an as a filtered limit of finite simplicial complexes. Variants of this limitdescription have appeared in several places in the literature (see the end of [Pay09, Section 1]for a summary); for convenience, we use [HL12, Theorem 13.2.4], a version that does notassume that K is algebraically closed (and that proves more than we need, namely that themaps in the inverse limit can be taken to be strong deformation retractions). Our hypothesison K is used to show that the index set for the limit has a countable cofinal subset. Tocomplete the proof, we use a well-known result from topology, Proposition 3.1, that an inverselimit of a sequence of finite simplicial complexes of dimension at most d can be embeddedin R d +1 .Our article is organized as follows. Sections 2 and 3 give a quick proof of Proposition 3.1.Sections 4 and 5 prove results that are needed to replace K by a countable subfield, in orderto obtain a countable index set for the inverse limit. Section 6 combines all of the above toprove Theorem 1.1. The final sections of the paper study the topology of Berkovich curves:after reviewing and developing the theory of dendrites and local dendrites in Sections 7 and 8,respectively, we show in Section 9 how to obtain the homeomorphism type of any Berkovichcurve over K as above, under the additional hypothesis that the value group is dense in R > .For example, as a special case of Corollary 9.2, we show that ( P C p ) an is homeomorphic to atopological space first constructed in 1923, the Ważewski universal dendrite [Waż23]2. Approximating maps of finite simplicial complexes by embeddings If X is a topological space, a map f : X → R n is called an embedding if f is a homeo-morphism onto its image. For compact X , it is equivalent to require that f be a continuousinjection. When we speak of a finite simplicial complex, we always mean its geometric real-ization, a compact subset of some R n . A set of points in R n is said to be in general position if for each m ≤ n − , no m + 2 of the points lie in an m -dimensional affine subspace. Lemma 2.1.
Let X be a finite simplicial complex of dimension at most d . Let ǫ ∈ R > .For any continuous map f : X → R d +1 , there is an embedding g : X → R d +1 such that | g ( x ) − f ( x ) | ≤ ǫ for all x ∈ X .Proof. The simplicial approximation theorem implies that f can be approximated within ǫ/ by a piecewise linear map g . For each vertex x i in the corresponding subdivision of X ,in turn, choose y i ∈ R d +1 within ǫ/ of g ( x i ) so that the y i are in general position. Let g : X → R d +1 be the piecewise linear map, for the same subdivision, such that g ( x i ) = y i .Then g is injective, and g is within ǫ/ of g , so g is within ǫ of f . (cid:3) . Inverse limits of finite simplicial complexes
Proposition 3.1.
Let ( X n ) n ≥ be an inverse system of finite simplicial complexes of dimen-sion at most d with respect to continuous maps p n : X n +1 → X n . Then the inverse limit X := lim ←− X n embeds in R d +1 .Proof. For m ≥ , let ∆ m ⊆ X m × X m be the diagonal, and write ( X m × X m ) − ∆ m = S ∞ n = m C mn with C mn compact. For ≤ m ≤ n , let D mn be the inverse image of C mn in X n × X n . Let K n = S nm =1 D mn . Since K n is closed in X n × X n , it is compact.For n ≥ , we inductively construct an embedding f n : X n → R d +1 and numbers α n , ǫ n ∈ R > such that the following hold for all n ≥ :(i) If ( x, x ′ ) ∈ K n , then | f n ( x ) − f n ( x ′ ) | ≥ α n .(ii) ǫ n < α n / .(iii) ǫ n < ǫ n − / (if n ≥ ).(iv) If x ∈ X n +1 , then | f n +1 ( x ) − f n ( p n ( x )) | ≤ ǫ n .Let f : X → R d +1 be any embedding (apply Lemma 2.1 to a constant map, for instance).Now suppose that n ≥ and that f n has been constructed. Since f n is injective and K n is compact, we may choose α n ∈ R > satisfying (i). Choose any ǫ n ∈ R > satisfying (ii)and (iii). Apply Lemma 2.1 to p n ◦ f n to find f n +1 satisfying (iv). This completes theinductive construction.Now P ∞ i = n ǫ i < ǫ n < α n / by (iii) and (ii). Let b f n be the composition X → X n f n → R d +1 .For x ∈ X , (iv) implies | b f n +1 ( x ) − b f n ( x ) | ≤ ǫ n , so the maps b f n converge uniformly to acontinuous map f : X → R d +1 satisfying | f ( x ) − f n ( x n ) | < α n / .We claim that f is injective. Suppose that x = ( x n ) and x ′ = ( x ′ n ) are distinct pointsof X . Fix m such that x m = x ′ m . Fix n ≥ m such that ( x m , x ′ m ) ∈ C mn . Then ( x n , x ′ n ) ∈ D mn ⊆ K n . By (i), | f n ( x n ) − f n ( x ′ n ) | ≥ α n . On the other hand, | f ( x ) − f n ( x n ) | < α n / and | f ( x ′ ) − f n ( x ′ n ) | < α n / , so f ( x ) = f ( x ′ ) . (cid:3) Remark . Proposition 3.1 was proved in the 1930s. Namely, following a 1928 sketchby K. Menger, in 1931 it was proved independently in by S. Lefschetz [Lef31], G. Nöbel-ing [Nöb31], and L. Pontryagin and G. Tolstowa [PT31] that any compact metrizable spaceof dimension at most d embeds in R d +1 . The proofs proceed by using P. Alexandroff’s ideaof approximating compact spaces by finite simplicial complexes (nerves of finite covers), soeven if it not obvious that the 1931 result applies directly to an inverse limit of finite simpli-cial complexes of dimension at most d (i.e., whether such an inverse limit is of dimension atmost d ), the proofs still apply. And in any case, in 1937 H. Freudenthal [Fre37] proved thata compact metrizable space is of dimension at most d if and only if it is an inverse limit offinite simplicial complexes of dimension at most d . See Sections 1.11 and 1.13 of [Eng78] formore about the history, including later improvements.4. Fiber coproducts of valued fields
We work in the category whose objects are valued fields and whose morphisms are fieldhomomorphisms respecting the absolute values. For example, if K is a valued field, wehave a natural morphism from K to its completion b K . Given morphisms i : K → L and i : K → L of valued fields, an amalgam of L and L over K is a triple ( M, j , j ) where M is a valued field and j : L → M and j : L → M are morphisms such that ◦ i = j ◦ i and such that M is generated by j ( L ) and j ( L ) . An isomorphism ofamalgams ( M, j , j ) → ( M ′ , j ′ , j ′ ) is an isomorphism φ : M → M ′ such that φ ◦ j = j ′ and φ ◦ j = j ′ . Proposition 4.1.
Given morphisms K → L and K → L of valued fields such that K isdense in L , the fiber coproduct of L and L over K exists and is the unique amalgam of L and L over K .Proof. Since K is dense in L , the composition K → L → b L extends uniquely to L → b L .Hence we may view K , L , and L as subfields of b L , and the morphisms K → L → b L and K → L → b L as inclusions. Let M := L L ⊆ b L . Now, given any M ′ in a commutativediagram M ′ L = = ⑤⑤⑤⑤⑤⑤⑤⑤ L a a ❇❇❇❇❇❇❇❇ K a a ❈❈❈❈❈❈❈❈ = = ④④④④④④④④ we may view one of the two upper morphisms, say L → M ′ , as an inclusion. Then all thefields become subfields of c M ′ . Now the other upper morphism L → M ′ is an inclusion toosince the composition L → M ′ ֒ → c M ′ restricts to the inclusion morphism on the densesubfield K . Thus M = L L ⊆ M ′ , and there is a unique morphism M → M ′ compatiblewith the morphisms from L and L , namely the inclusion. Thus M is a fiber coproduct.The existence of a fiber coproduct implies that at most one amalgam exists. Since M isgenerated by L and L , it is an amalgam. (cid:3) Remark . In [HL12], the value groups of valued fields are not necessarily contained in R .Proposition 4.1 remains true in the larger category, and the amalgams in the two categoriescoincide when they make sense, i.e., when the valued fields in question happen to have valuegroup contained in R .5. Berkovich spaces over noncomplete fields
Berkovich analytifications were originally defined only when the valued field K was com-plete [Ber90]. For a quasi-projective variety V over an arbitrary valued field K , [HL12, Sec-tion 13.1] defines a topological space in terms of types, and proves that it is homeomorphicto V an when K is complete.This definition uses types over K ∪ R . Using quantifier elimination for the theory ofalgebraically closed valued fields in the two-sorted language consisting of the valued fieldand the value group, such types can be identified with pairs ( L, c ) with L an R -valuedfield extension of K , where ( L, c ) is identified with ( L ′ , c ′ ) if there exists a K -isomorphism f : L → L ′ of R -valued fields with f ( c ) = c ′ . This description makes it clear that if v is thetype of ( K ′ , a ) , then an extension of v to L ≥ K corresponds precisely to an amalgam of K ′ and L over K .The restriction map r from types over L to types over K ′ (where K ≤ K ′ ≤ L ) takes ( L, a ) to ( K ′ ( a ) , a ) . If h : V → W is a morphism of varieties over K , the restriction map r is clearly compatible with the natural map from types on V to types on W induced by h . e take this space of types as a definition of the topological space V an for arbitraryvalued fields K . The following proposition shows that no new spaces arise: it would havebeen equivalent to define V an as ( V b K ) an (the subscript denotes base extension). Proposition 5.1.
Let K ≤ L be an extension of valued fields such that K is dense in L .Let V be a quasi-projective K -variety. Then ( V L ) an is naturally homeomorphic to V an .Proof. Restriction of types defines a continuous map r V : ( V L ) an → V an . A point v ∈ V an is represented by the type of some a ∈ V ( K ′ ) for some valued field extension K ′ = K ( a ) of K ; then r − V ( v ) is in bijection with the set of amalgams of K ′ and L over K , which byProposition 4.1 is a set of size . Thus r V is a bijection.If V is projective, then V an and ( V L ) an are compact Hausdorff spaces [HL12, Proposi-tion 13.1.2], so the continuous bijection r V is a homeomorphism. If V is an open subschemeof a projective variety P , then ( V L ) an and V an are open subspaces of ( P L ) an and P an , respec-tively, and r P restricts to r V , so the result for P implies the result for V . (cid:3) Embeddings of Berkovich spaces
Proposition 6.1.
Let K be a valued field having a countable dense subset. Let V be a projective K -scheme of dimension d . Then V an is homeomorphic to a inverse limit lim ←− X n where each X n is a finite simplicial complex of dimension at most d and each map X n +1 → X n is continuous.Proof. First suppose that K is countable. Since V is projective, V an is compact, so we mayapply [HL12, Theorem 13.2.4] to V an to obtain that V an is a filtered limit of finite simplicialcomplexes over an index set I . Since K is countable, the proof of [HL12, Theorem 13.2.4]shows that I may be taken to be countable, so our limit may be taken over a sequence, asdesired.Now assume only that K has a countable dense subset. Since V is of finite presentationover K , it is the base extension of a projective scheme V over a countable subfield K of K .By adjoining to K a countable dense subset of K , we may assume that K is dense in K .By Proposition 5.1, V an is homeomorphic to ( V ) an , which has already been shown to be aninverse limit of the desired form. (cid:3) Proposition 6.2.
Let K be a complete valued field. If U is an open subscheme of V , thenthe induced map U an → V an is a homeomorphism onto an open subspace.Proof. See [Ber90, Proposition 3.4.6(8)]. (cid:3)
Theorem 1.1 follows immediately from Propositions 3.1, 6.1, and 6.2.7.
Dendrites
When V is a curve, more can be said about V an . But first we recall some definitions andfacts from topology.7.1. Definitions. A continuum is a compact connected metrizable space (the empty space isnot connected). A simple closed curve in a topological space is any subspace homeomorphicto a circle. A dendrite is a locally connected continuum containing no simple closed curve.Dendrites may be thought of as topological generalizations of trees in which branching mayoccur at a dense set of points. A point x in a dendrite X is called a branch point if X − { x } has three or more connected components. .2. Ważewski’s theorems.
The following three theorems were proved by T. Ważewski inhis thesis [Waż23]. Theorem 7.1.
Up to homeomorphism, there is a unique dendrite W such that its branchpoints are dense in W and there are ℵ branches at each branch point. The dendrite W in Theorem 7.1 is called the Ważewski universal dendrite . Theorem 7.2.
Every dendrite embeds in W . Theorem 7.3.
Every dendrite is homeomorphic to the image of some continuous map [0 , → R .Remark . The key to drawing W in the plane is to make sure that the branches comingout of each branch point have diameters tending to .7.3. Pointed dendrites. A pointed dendrite is a pair ( X, P ) where X is a dendrite and P ∈ X . An embedding of pointed dendrites is an embedding of topological spaces mappingthe point in the first to the point in the second. Let P be the category of pointed dendrites,in which morphisms are embeddings. By the universal pointed dendrite , we mean W equippedwith one of its branch points w . Theorem 7.5.
Every pointed dendrite ( X, P ) admits an embedding into the universal pointeddendrite ( W, w ) .Proof. Enlarge X by attaching a segment at P in order to assume that P is a branch pointof X . Theorem 7.2 yields an embedding i : X ֒ → W . Then i ( P ) is a branch point of W . By[Cha91, Proposition 4.7], there is a homeomorphism j : W → W mapping i ( P ) to w . Then j ◦ i is an embedding ( X, P ) → ( W, w ) . (cid:3) Proposition 7.6.
Any dendrite admits a strong deformation retraction onto any of itspoints.Proof.
In fact, a dendrite admits a strong deformation retraction onto any subcontinuum [Ill96]. (cid:3) Local dendrites
Definition and basic properties. A local dendrite is a continuum such that everypoint has a neighborhood that is a dendrite. Equivalently, a continuum is a local dendriteif and only if it is locally connected and contains at most a finite number of simple closedcurves [Kur68, §51, VII, Theorem 4(i)]. Local dendrites are generalizations of finite connectedgraphs, just as dendrites are generalizations of finite trees. Proposition 8.1. (a)
Every subcontinuum of a local dendrite is a local dendrite. Actually, Ważewski used a different, equivalent definition: for him, a dendrite was any image D of acontinuous map [0 , → R n such that D contains no simple closed curve. A dendrite in Ważewski’s senseis a dendrite in our sense by [Nad92, Corollary 8.17]. Conversely, a dendrite in our sense embeds in R by [Nad92, Section 10.37] (or, alternatively, is an inverse limit of finite trees by [Nad92, Theorem 10.27]and hence embeds in R by Proposition 3.1), and is a continuous image of [0 , by the Hahn–Mazurkiewicztheorem [Nad92, Theorem 8.14]. b) An open subset of a local dendrite is arcwise connected if and only if it is connected. (c)
A connected open subset U of a local dendrite is simply connected if and only if it containsno simple closed curve. (d) A dendrite is the same thing as a simply connected local dendrite.Proof. (a) This follows from the fact that every subcontinuum of a dendrite is a dendrite [Kur68,§51, VI, Theorem 4].(b) This follows from [Why71, II, (5.3)].(c) If U contains a simple closed curve γ , [BJ52, Theorem on p. 174] shows that γ cannotbe deformed to a point, so U is not simply connected. If U does not contain a simpleclosed curve, then the image of any simple closed curve in U is a dendrite, and hence byProposition 7.6 is contractible, so U is simply connected.(d) This follows from (c). (cid:3) Local dendrites and quasi-polyhedra.
We now relate the notion of quasi-polyhedronin [Ber90, §4.1] to the notion of local dendrite.
Proposition 8.2. (a)
A connected open subset of a local dendrite is a quasi-polyhedron. (b)
A compact metrizable quasi-polyhedron is the same thing as a local dendrite. (c)
A compact metrizable simply connected quasi-polyhedron is the same thing as a dendrite. (d)
A compact metrizable quasi-polyhedron is special in the sense of [Ber90, Definition 4.1.5] .Proof. (a) Suppose that V is a connected open subset of a local dendrite X . By [Kur68, §51,VII, Theorem 1], each point v of V has arbitrarily small open neighborhoods U withfinite boundary. We may assume that each U is contained in a dendrite. Since V islocally connected, we may replace each U by its connected component containing x :this can only shrink its boundary. Now each U , as a connected subset of a dendrite, isuniquely arcwise connected [Why71, p. 89, 1.3(ii)]. So these U satisfy [Ber90, Defini-tion 4.1.1(i)(a)].By Proposition 8.8(a) (whose proof does not use anything from here on!), X is home-omorphic to a compact subset of R , so every open subset of X is countable at infinity(i.e., a countable union of compact sets). Thus V is a quasi-polyhedron.(b) If X is a local dendrite, it is a quasi-polyhedron by (a) and compact and metrizable bydefinition.Conversely, suppose that X is a compact metrizable quasi-polyhedron. In particular, X is a continuum. Condition ( a ) in [Ber90, Definition 4.1.1] implies that X is locallyconnected and covered by open subsets containing no simple closed curve. By compact-ness, this implies that there is a positive lower bound ǫ on the diameter of simple closedcurves in X . By [Kur68, §51, VII, Lemma 3], this implies that X is a local dendrite.(c) Combine (b) and Proposition 8.1(d).(d) A dendrite is special since each partial ordering as in [Ber90, Definition 4.1.5] arisesfrom some x ∈ X , and we can take θ there to be a radial distance function as in[MO90, Section 4.6], which applies since dendrites are locally arcwise connected and niquely arcwise connected. A local dendrite is special since any simply connected sub-quasi-polyhedron is homeomorphic to a connected open subset of a dendrite. (cid:3) The core skeleton.
By [Ber90, Proposition 4.1.3(i)], any simply connected quasi-polyhedron Q has a unique compactification b Q that is a simply connected quasi-polyhedron.The points of b Q − Q are called the endpoints of Q . Given a quasi-polyhedron X , Berkovichdefines its skeleton ∆( X ) as the complement in X of the set of points having a simplyconnected quasi-polyhedral open neighborhood with a single endpoint [Ber90, p. 76]. In thecase of a local dendrite, we can characterize this subset in many ways: see Proposition 8.4. Lemma 8.3.
Let X be a local dendrite. Let G be a subcontinuum of X containing all thesimple closed curves. Let C be a connected component of X − G . Then C is open in X and is a simply connected quasi-polyhedron with one endpoint, and its closure C in X is adendrite intersecting G in a single point.Proof. Since X is locally connected, X − G is locally connected, so C is open. By Proposi-tion 8.2(a), C is a quasi-polyhedron. Since C contains no simple closed curve, it is simplyconnected by Proposition 8.1(c).The complement of C ∪ G is a union of connected components of X − G , so C ∪ G isclosed, so it contains C . Since X is connected, C = C , so C ∩ G ) ≥ .If C had more than one endpoint, there would be an arc α in b C connecting two ofthem, passing through some c ∈ C since b C − C is totally disconnected by [Ber90, Propo-sition 4.1.3(i)]; the image of α under the induced map b C → X together with an arc in G connecting the images of the two endpoints would contain a simple closed curve passingthrough c , contradicting the hypothesis on G . Also, each point in C ∩ G is the image of apoint in b C − C . Now ≤ C ∩ G ) ≤ b C − C ) ≤ , so equality holds everywhere. (cid:3) Proposition 8.4.
Let X be a local dendrite. Each of the following conditions defines thesame closed subset ∆ of X . (i) If X is a dendrite, ∆ = ∅ ; otherwise ∆ is the smallest subcontinuum of X containingall the simple closed curves. (ii) The set ∆ is the union of all arcs each endpoint of which belongs to a simple closedcurve. (iii) The set ∆ is the skeleton ∆( X ) defined in [Ber90, p. 76] .Proof. Let L be the union of the simple closed curves in X . If L = ∅ , then X is a dendriteand (i), (ii), (iii) all define the empty set. So suppose that L = ∅ .For each pair of distinct components of L , there is at most one arc α in X intersecting L intwo points, one from each component in the pair (otherwise there would be a simple closedcurve not contained in L ). Let D be the union of all these arcs α with L . Any arc β in X with endpoints in L must be contained in D , since a point of β outside D would be containedin some subarc β ′ intersecting L in just the endpoints of β ′ , which would then have to besome α . Thus D is the union of the arcs whose endpoints lie in L . By Proposition 8.1(b), X is arcwise connected, so D is arcwise connected. By definition, D is a finite union of compactsets, so D is a subcontinuum.By Proposition 8.1(b), any subcontinuum Y ⊆ X is arcwise connected, so if Y contains L , then for each α as above, Y contains an arc β with the same endpoints as α , and then = α (otherwise there would be subarcs of α and β whose union was a simple closed curvenot contained in L ); thus Y ⊇ D . Hence D is the smallest subcontinuum containing L .Let ∆ be the ∆( X ) of [Ber90, p. 76]. If x were a point in a simple closed curve γ in X with a neighborhood Q as in the definition of ∆ , then Q must contain γ , since otherwise Q ∩ γ would have a connected component homeomorphic to an open interval I , and the twopoints of b I − I would map to two distinct points of b Q − Q , contradicting the choice of Q .Thus ∆ ⊇ L . But D is the smallest subcontinuum containing L , so ∆ ⊇ D . On the otherhand, Lemma 8.3 shows that the points of X − D lie outside ∆ . Hence ∆ = D . (cid:3) We call ∆ the core skeleton of X , since in [HL12, Section 10] the term “skeleton” is usedmore generally for any finite simplicial complex onto which X admits a strong deformationretraction. If ∆ = ∅ , then ∆ is a finite connected graph with no vertices of degree less thanor equal to [Ber90, Proposition 4.1.4(ii)].8.4. G -dendrites.Proposition 8.5. For a subcontinuum G of X , the following are equivalent. (i) G contains the core skeleton of X . (ii) G is a deformation retract of X . (iii) G is a strong deformation retract of X . (iv) There is a retraction r : X → G such that there exists a homotopy h : [0 , × X → X between h (0 , x ) = x and h (1 , x ) = r ( x ) satisfying r ( h ( t, x )) = r ( x ) for all t and x (i.e.,“points are moved only along the fibers of r ”); moreover, r is unique, characterized bythe condition that it maps each connected component C of X − G to the singleton C ∩ G .Proof. First we show that a retraction r as in (iv) must be as characterized. Suppose that C is a connected component of X − G . Any c ∈ C is moved by the homotopy along a pathending on G , and if we shorten it to a path γ so that it ends as soon as it reaches G then γ stays within X − G until it reaches its final point g and hence stays within C until it reaches g ; Hence g ∈ C ∩ G , and r ( c ) = g . Thus r ( C ) ⊆ C ∩ G . By Lemma 8.3, C ∩ G ) = 1 , so r is as characterized.(i) ⇒ (iv): See [Ber90, Proposition 4.1.6] and its proof.(iv) ⇒ (iii): Trivial.(iii) ⇒ (ii): Trivial.(ii) ⇒ (i): The result of deforming the inclusion of a simple closed curve γ in X is a closedpath whose image contains γ [BJ52, Theorem on p. 174], so if G is a deformation retract of X , then G must contain each simple closed curve, so G contains the core skeleton. (cid:3) Given an embedding of local dendrites
G ֒ → X , call X equipped with the embedding a G -dendrite if the image of G satisfies the conditions of Proposition 8.5; we generally identify G with its image. Let D G be the category whose objects are G -dendrites and whose morphismsare embeddings extending the identity G : G → G . Given a G -dendrite X and g ∈ G , let X g be the fiber r − ( g ) with the point g distinguished; say that g is a sprouting point if X g is nota point. Theorem 8.6 below makes precise the statement that any G -dendrite is obtained byattaching dendrites to countably many points of G . Theorem 8.6.
There is a fully faithful functor F : D G → Q g ∈ G P sending a G -dendrite X to the tuple of fibers ( X g ) g ∈ G , and its essential image consists of tuples ( D g ) such that { g ∈ G : D g > } is countable. roof. Let X be a G -dendrite. For each g ∈ G , the homotopy restricts to a contractionof X g to g , so X g is a (pointed) dendrite. By [Kur68, §51, IV, Theorem 5 and §51, VII,Theorem 1], { g ∈ G : X g > } is countable.The characterization of the retraction in Proposition 8.5(iv) shows that a morphism of G -dendrites X → Y respects the retractions, so it restricts to a morphism X g → Y g in P for each g ∈ G . This defines F .Given ( D g ) g ∈ G ∈ Q g ∈ G P with { g ∈ G : D g > } countable, choose a metric d D g on D g such that the diameters of the D g with D g > tend to if there are infinitely many ofthem. Identify the distinguished point of D g with g . Let X be the set ` g ∈ G D g with themetric for which the distance between x ∈ D g and x ′ ∈ D g ′ is ( d D g ( x, x ′ ) , if g = g ′ , d D g ( x, g ) + d G ( g, g ′ ) + d D g ′ ( g ′ , x ′ ) , if g = g ′ .It is straightforward to check that X is compact and locally connected and that the map G → X is an embedding. By Proposition 7.6, there is a strong deformation retraction of D g onto { g } ; running these deformations in parallel yields a strong deformation retractionof X onto G . Thus X is a G -dendrite. Moreover, F sends X to ( D g ) g ∈ G . Thus the essentialimage is as claimed.Given X, Y ∈ D G , and given morphisms f g : X g → Y g in P for all g ∈ G , there existsa unique morphism f : X → Y in D G mapped by F to ( f g ) g ∈ G ; namely, one checks thatthe union f of the f g is a continuous injection, and hence an embedding. Thus F is fullyfaithful. (cid:3) The universal G -dendrite. Let G be a local dendrite. Given a countable subset G ⊆ G , Theorem 8.6 yields a G -dendrite W G,G whose fiber at g ∈ G is the universalpointed dendrite ( W, w ) if g ∈ G and a point if g / ∈ G . By Theorems 8.6 and 7.5, any G -dendrite with all sprouting points in G admits a morphism to W G,G .Now let G be a finite connected graph. Fix a countable dense subset G ⊆ G containing allvertices of G . Define W G := W G,G , and call it the universal G -dendrite . Its homeomorphismtype is independent of the choice of G , since the possibilities for G are permuted by the self-homeomorphisms of G fixing its vertices. Any G -dendrite has its sprouting points containedin some G as above (just take the union with a G from above), so every G -dendrite embedsas a topological space into W G . Theorem 8.7.
Let X be a local dendrite, and let G be its core skeleton. Suppose that G = ∅ ,that the branch points of X are dense in X , and that there are ℵ branches at each branchpoint. Then X is homeomorphic to W G .Proof. The vertices of G of degree or more are among the branch points of X . Afterapplying a homeomorphism of G (to shift degree vertices), we may assume that all thevertices of G are branch points of X . Since the branch points of X are dense in X , thesprouting points must be dense in G . For each sprouting point g ∈ G , the fiber X g satisfies thehypotheses of Theorem 7.1, so X g is the universal pointed dendrite. Thus X is homeomorphicto W G , by construction of the latter. (cid:3) Euclidean embeddings.Proposition 8.8. a) Every local dendrite embeds in R . (b) Let X be a local dendrite, and let G ⊆ X be a finite connected graph containing all thesimple closed curves. Then the following are equivalent: (i) X embeds into R . (ii) G embeds into R . (iii) G does not contain a subgraph isomorphic to a subdivision of the complete graph K or the complete bipartite graph K , .Proof. (a) A local dendrite is a regular continuum [Kur68, §51, VII, Theorem 1], and hence ofdimension , so it embeds in R by as discussed in Remark 3.2.(b) See [Kur30]. (cid:3) Berkovich curves
Finally, we build on [Ber90] (especially Section 4 therein) and the theory of local dendritesto describe the homeomorphism type of a Berkovich curve. See also the forthcoming bookby A. Ducros [Duc12], which will contain a systematic study of Berkovich curves.
Theorem 9.1.
Let K be a complete valued field having a countable dense subset. Let V bea projective K -scheme of pure dimension . (a) The topological space V an is a finite disjoint union of local dendrites. (b) Suppose that V is also smooth and connected, and that K has nontrivial value group. (i) If V an is simply connected, then V an is homeomorphic to the Ważewski universaldendrite W . (ii) If V an is not simply connected, let G be its core skeleton; then V an is homeomorphicto the universal G -dendrite W G .Proof. (a) We may assume that V is connected, so V an is connected by [Ber90, Theorem 3.4.8(iii)].Also, V an is compact by [Ber90, Theorem 3.4.8(ii)]. It is metrizable by Theorem 1.1. Itis a quasi-polyhedron by the proof of [Ber90, Corollary 4.3.3]. So V an is a local dendriteby Proposition 8.2.(b) Let k be the residue field of K . Since K has a countable dense subset, k is countable,so any k -curve has exactly ℵ closed points.First suppose that K is algebraically closed. In particular K has dense value group.Choose a semistable decomposition of V an (see [BPR12, Definition 5.15]). Each openball and open annulus in the decomposition is homeomorphic to an open subspace of ( P K ) an , in which the branch points (type (2) points in the terminology of [Ber90, 1.4.4])are dense by the assumption on the value group, so the branch points are dense in V an .At each branch point, the branches are in bijection with the closed points of a k -curveby [BPR12, Lemma 5.66(3)], so their number is ℵ .Now suppose that K is not necessarily algebraically closed. Let K ′ be the completionof an algebraic closure of K . Then [Ber90, Corollary 1.3.6] implies that V an is thequotient of ( V K ′ ) an by the absolute Galois group of K . It follows that the branch pointsof V an are the images of the branch points of ( V K ′ ) an , and that the branches at eachbranch point of V an are in bijection with the closed points of some curve over a finite xtension of k . Thus, as for ( V K ′ ) an , the branch points of V an are dense, and there are ℵ branches at each branch point.Finally, according to whether G is simply connected or not, Theorem 7.1 or Theo-rem 8.7 shows that V an has the stated homeomorphism type. (cid:3) Corollary 9.2.
Let K be a complete valued field having a countable dense subset and densevalue group. Then ( P K ) an is homeomorphic to W .Proof. It is simply connected by [Ber90, Theorem 4.2.1], so Theorem 9.1(b)(i) applies. (cid:3)
Remark . Any finite connected graph with no vertices of degree less than or equal to can arise as the core skeleton G in Theorem 9.1(b)(ii): see [Ber90, proof of Corollary 4.3.4].In particular, there exist smooth projective curves V such that V an cannot be embedded in R . Remark . Theorem 9.1 also lets us understand the topology of Berkovich spaces associatedto curves that are only quasi-projective . Let U be a quasi-projective curve. Write U = V − Z for some projective curve V and finite subscheme Z ⊆ V . Then Z an is a closed subset of V an with one point for each closed point of Z , and U an = V an − Z an . Remark . Even more generally, the arguments apply equally well to Berkovich curves thatdo not arise as analytification of algebraic curves.
Remark . The smoothness assumption in Theorem 9.1(b) can be weakened to the state-ment that the normalization morphism e V → V has no fibers with three or more schematicpoints. Remark . If in Theorem 9.1(b) we drop any of the hypotheses, then the result fails; wedescribe the situations that arise. • If V is the non-smooth curve consisting of three copies of P K attached at a K -pointof each, then V an consists of three copies of W attached in the same way; this isa dendrite, but it has a branch point of order , so it cannot be homeomorphic to W . More generally, if the normalization e V has three distinct schematic points abovesome point a of V , the same argument applies. • If V is disconnected, then so is V an , so it cannot be homeomorphic to W or W G . Inthis case, V an is the disjoint union of the analytifications of the connected componentsof V . • Suppose that V is smooth and connected, but K has trivial value group. Then V an is adendrite consisting of ℵ intervals emanating from one branch point; cf. [Ber93, p. 71].Equivalently, V an is the one-point compactification of | V | × [0 , ∞ ) , where | V | is theset of closed points of V with the discrete topology. Remark . As is well-known to experts [Thu05, BPR12], there is a metrized variant ofTheorem 9.1. We recall a few definitions; cf. [MNO92]. An R -tree is a uniquely arcwiseconnected metric space in which each arc is isometric to a subarc of R . Let A be a countablesubgroup of R , and let A ≥ (resp. A > ) be the set of nonnegative (resp. positive) numbersin A . An A -tree is an R -tree X equipped with a point x ∈ X such that the distance fromeach branch point to x lies in A .More generally, we may introduce variants that are not simply connected. Let us define an R -graph to be an arcwise connected metric space X such that each arc of X is isometric to subarc of R and X contains at most finitely many simple closed curves. Define an A -graph to be an R -graph X equipped with a point x ∈ X such that the length of every arc from x to a branch point or to itself is in A . Given an A -graph ( X, x ) , let B ( X ) be the set of points y ∈ B not of degree such that y is an endpoint of an arc of length in A ≥ emanating from x . Then let E ( X ) be the A -graph obtained by attaching ℵ isometric copies of [0 , ∞ ) andof [0 , a ] for each a ∈ A > to each y ∈ B ( X ) (i.e., identify each with y ). Let E n ( X ) := E ( E ( · · · ( E ( X )) · · · )) . The direct limit of the E n ( X ) is an A -graph W AX . If X is a point,define W A := W AX , which is a universal separable A -tree in the sense of [MNO92, Section 2],because it contains the space obtained by attaching only copies of [0 , ∞ ) at each stage; thelatter is the universal separable A -tree constructed in [MNO92, Theorem 2.6.1].Let K be a complete valued field having a countable dense subset. Let A be the valuegroup of K , expressed as an additive subgroup of R . Let A ≤ R be the Q -vector spacespanned by A . Let V be a projective K -scheme of pure dimension . Let V an − be thesubset of V an consisting of the complement of the type (1) points (the points correspondingto closed points of V ). Then V an − admits a canonical metric, whose existence is relatedto the fact that on the segments of the skeleta of V an , away from the endpoints, one hasan integral affine structure [KS06, Section 2]. If V an − is simply connected, then V an − isisometric to W A ; otherwise V an − is isometric to W AG , where G is the core skeleton of V an with the induced metric. Warning . The metric topology on V an − is strictly stronger than the subspace topologyon V an − induced from V an : see [FJ04, Chapter 5] and [BR10, Section B.6]. Nevertheless,when V is smooth and complete, the topological space V an can be recovered from the metricspace V an − . Acknowledgements
We thank Sam Payne for suggesting the argument and references for the facts that aBerkovich curve satisfying our hypotheses has a dense set of branch points and that each has ℵ branches. The third author thanks the Centre Interfacultaire Bernoulli for its hospitality. References [BPR12] Matthew Baker, Sam Payne, and Joseph Rabinoff,
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