Table Of Content

A NON-ARCHIMEDEAN OHSAWA–TAKEGOSHI EXTENSION THEOREM 7 MATTHEWSTEVENSON 1 0 2 n a Abstract. We prove an Ohsawa–Takegoshi-type extension theorem on the Berkovich closed unit disc over a J complete non-Archimedean field. As an application, we establish a non-Archimedean analogue of Demailly’s 7 regularizationtheoremforquasisubharmonicfunctionsontheBerkovichunitdisc. 1 1. Introduction ] G The Ohsawa–Takegoshitheorem is one of the fundamental extensions theorems in complex geometry. Orig- A inating in the foundational paper [OT87] of Ohsawa–Takegoshi, many generalizations and improvements have . sincebeenshown;see[Man93,Ber96,Siu96,Dem00,MV07]. ItsmanyapplicationsincludeSiu’smuch-celebrated h proof [Siu98] of the deformation invariance of plurigenera. In its simplest form, the classicalOhsawa–Takegoshi t a theorem asserts the following: given a plurisubharmonic function ϕ on the complex unit disc D, a point m z ∈D\{ϕ=−∞}, and a value a∈C, there is a holomorphic function f on D such that f(z)=a and [ |f(x)|2e−2ϕ(x)dλ≤π|f(z)|2e−2ϕ(z). 1 D v Z 9 The constant π is optimal, as shown in [Bl o13, Theorem 1]. There is also an adjoint formulation of the result, 3 which concerns the extension of a holomorphic 1-form rather than of a holomorphic function. 8 In [Ber90, Ber93], Berkovichintroducedhis theory ofanalytic spaces overa complete non-Archimedeanfield. 4 Inrecentyears,analoguesofmanytheoremsfromcomplexgeometryhavebeendevelopedinthenon-Archimedean 0 . setting, notably the solution in [BFJ15] of a non-Archimedean Calabi–Yau-type problem. It is natural to ask 1 for an analogue of the Ohsawa–Takegoshiextension theorem in the non-Archimedean setting, as well. 0 Let k be a field, complete with respect to a non-Archimedean absolute value |·|. Let k{T} denote the Tate 7 1 algebra in one variable over k, X =Ek(1) denote the Berkovich closed unit disc over k, and AX: X →[0,+∞] : denote Temkin’s canonical metric on the canonical bundle ω (more concretely, A is −log of the radius v X/k X function). For any quasisubharmonic function ϕ on X and analytic function f ∈k{T}, consider the norm i X kfk := sup |f|e−ϕ−AX r ϕ a X\Z(ϕ) whereZ(ϕ):={ϕ=−∞}. Thefunctionϕ+A canbethoughtofasametriconω andf asaglobalsection X X/k ofω , sothe normkfk measuresthe lengthofthe sectionf in the metric ϕ+A . See §2 for further details. X/k ϕ X Our first main result is a non-Archimedean version of the Ohsawa–Takegoshi extension theorem on the Berkovichclosed unit disc X =E (1). k Theorem A. Let ϕ be a quasisubharmonic function on X = E (1). For any z ∈ X, there exists a nonzero k polynomial f ∈k[T] such that lim kfk ≤|f(z)|e−ϕ(z). (1+ǫ)ϕ ǫ→0+ If ϕ(z) = −∞, then we may find f such that lim kfk < +∞. Moreover, if k is algebraically closed ǫ→0+ (1+ǫ)ϕ and z is a rigid point of X, then for any value a∈H(z)∗ =k∗, we may find f such that f(z)=a. Date:January19,2017. 1 2 MATTHEWSTEVENSON ToclarifytherelationshipbetweenTheoremAandtheadjointformulationoftheOhsawa–Takegoshiextension theorem, we must discuss the non-Archimedean analogues of (pluri)subharmonic functions and volume forms, which arise in the statement of the latter. We are interested in the class of quasisubharmonic functions on the Berkovich closed unit disc, which are the non-Archimedean analogue of (pluri)subharmonic functions on the complex unit disc. These are briefly discussed in §2.2; see [BR10, Jon15] for a comprehensive treatment. On complex manifolds, volume forms naturally correspond to smooth metrics on the canonical bundle. The non-ArchimedeananalogueofametriconthecanonicalbundlehasbeenexaminedbyTemkinin[Tem16]. Temkin produces a canonical lower-semicontinuous metric A on the canonical bundle ω of a k-analytic space X, X X/k extending the weight function of Mustat˘a–Nicaise [MN15]. Temkin’s metric is discussed in §2.3. This is the , natural candidate to replace the volume form that appears in the statement of the classical Ohsawa–Takegoshi theorem. In Theorem A, a section is measured using the sequence of norms k·k as ǫ → 0+ instead of with the (1+ǫ)ϕ single norm k·k , which is what one might expect from the classical Ohsawa–Takegoshitheorem. The former ϕ provestogivethecorrectanalogywiththecomplexsetting,asthefollowingexampledemonstrates. Considerthe (pluri)subharmonicfunctionϕ=αlog|z|,withα>0,onthecomplexunitdiscD;then,itiselementarytocheck that e−2ϕdλ < +∞ if and only if α < 1. Similarly, consider the quasisubharmonic function ϕ = αlog|T|, D with α>0, on the Berkovich unit disc X =E (1); then, lim k1k <+∞ if and only if α<1. R k ǫ→0+ (1+ǫ)ϕ As an application of Theorem A, we prove a regularization theorem for quasisubharmonic functions on the Berkovich unit disc. Certain results in this direction already exist in the literature: in [Jon15, Theorem 2.10], it was shown that any quasisubharmonic function on X is the decreasing limit of bounded quasisubharmonic functions. Asimilarargumentappearsin[FRL06,§4.6]. However,theseconstructionsuseonlythetreestructure onthe Berkovichunitdisc(inparticular,they donotincorporatetheanalyticstructure). Itisthereforeunlikely that these proofs can be generalized to higher dimensions. Asinspiration,weusethemuch-celebratedregularizationtheoremofDemaillyforaplurisubharmonicfunction φ on a bounded pseudoconvex domain Ω ⊂ Cn. To a plurisubharmonic function φ on Ω and a positive integer m, we associate the Hilbert space H of holomorphic functions on Ω satisfying the integrability condition mφ |f|2e−2mφdλ <+∞. ZΩ The Demailly approximation associated to H is a plurisubharmonic function φ on Ω with analytic singu- mφ m larities, and the sequence (φ )∞ converges pointwise and in L1 to φ. See [Dem92] for further details. m m=1 loc We adopt the same philosophy in the non-Archimedean setting: to a quasisubharmonic function ϕ on X, we associatethe ideal H of the Tate algebrak{T}consisting of those analytic functions f satisfying the finiteness ϕ condition kfk+ := lim sup |f|e−(1+ǫ)ϕ−AX <+∞ ϕ ǫ→0+X\Z(ϕ) For each positive integer m, we define the non-Archimedean Demailly approximation ϕ by the formula m ∗ 1 |f| ϕ := sup log , m m f∈Hmϕ\{0} kfk+mϕ! where (−)∗ denotes the upper-semicontinuous regularization. We show that ϕ is a quasisubharmonic function m with analytic singularities on X. The ideal sheaf on X associatedto H behaves like a non-Archimedean multiplier ideal associatedto ϕ, and ϕ may be of independent interest. This idea is briefly explored in §4.2, where we show that H generates the ϕ stalks of a locally-defined multiplier ideal sheaf associated to ϕ. These multiplier ideals are used to show that the ideals H satisfy a subadditivity property. ϕ We prove the following non-Archimedean analogue of Demailly’s regularizationtheorem. A NON-ARCHIMEDEAN OHSAWA–TAKEGOSHI EXTENSION THEOREM 3 Theorem B. If ϕ is a quasisubharmonic function on X with ϕ≤0, then ϕ≤ϕ ≤ϕ+AX. In particular, the m m sequence (ϕ )∞ converges pointwise to ϕ on {A <+∞}⊆X. m m=1 X InTheoremB,thecrucialinequalityϕ ≥ϕisaconsequenceofTheoremA.Inprinciple,astatementsimilar m toTheoremBoughttobepossibleinhigherdimensions,butthiswouldlikelyrequireahigher-dimensionalversion of the Ohsawa–Takegoshitheorem on analytic spaces. Nonetheless, there has been much work done on the development of pluripotential theory on analytic spaces. Quite generally, Chambert-Loir and Ducros have introduced in [CLD12] the notion of continuous plurisubharmonic functions. In addition, semipositive metrics on line bundles were studied in detail by [Zha95, Gub98], among others. On analytic curves, potential theory is well-established, due to the work of Thuillier [Thu05]. A regularization theorem similar to Theorem B is proven in the higher-dimensional setting in [BFJ16, Theorem B]. A related discussion appears in [BFJ08, §5]. The techniques involved in the proof of Theorem A rely crucially on the tree structure of the Berkovich unit disc, and thus the proof does not, a priori, generalize to higher dimensions. More precisely, the proof proceeds by first constructing a finite subtree Γ of X that captures the worst of the singularities of ϕ, and ϕ reducing to proving Theorem A on the convex hull of Γ ∪{z}. For some end of the tree Γ , we construct a ϕ ϕ new quasisubharmonic function φ such that Γ (Γ and reduce to proving Theorem A for φ. This inductively φ ϕ reduces Theorem A to a simple case, which is solved directly. The paper is organized as follows. In §2, we briefly review the definition of the Berkovich unit disc, and its potential theory. In §2.3, we discuss Temkin’s metric on the canonical bundle. In §3, we prove Theorem A and a variant thereof. In §4, we construct the ideal H and the non-archimedean Demailly approximation ϕ as- ϕ m sociatedtoaquasisubharmonicfunctionϕandapositiveintegerm≥1. ThesearethenusedtoproveTheoremB. Acknowledgements. I would like to thank my advisor, Mattias Jonsson, for suggesting the problem and for his invaluable help, guidance, and support. I am also grateful to Takumi Murayama and Emanuel Reinecke for helpful comments on a previous draft. This work was partially supported by NSF grant DMS-1600011. 2. Preliminaries Throughoutthepaper,letk beafieldcompletewithrespecttoa(possiblytrivial)non-Archimedeanabsolute value |·|, and let K :=ka denote its completed algebraic closure. Let k◦ :={|·|≤1} denote the valuation ring of k, and K◦ the valuation ring of K. Toestablishconventiocns,webrieflyrecallin§2.1thedefinitionoftheBerkovichunitdiscovera(notnecessarily algebraicallyclosed)non-Archimedeanfieldk,anddescribeitspoints;in§2.2,wereviewthemetrictreestructure and potential theory on the Berkovichunit disc. Finally, in §2.3, we discuss the canonical metric of Temkin on the canonical bundle of a k-affinoid space. 2.1. The Berkovich unit disc. The Tate algebra k{T}in the variable T is the subalgebra of k[[T]]consisting of those power series f = ∞ a Ti, with a ∈ k, such that |a | → 0 as i → ∞. The Tate algebra is a Banach i=0 i i i k-algebra when equipped with the Gauss norm P |f(x )|:=max|a |. G i i≥0 See [BGR84, §5.1] for further details. The Berkovich unit disc is the set X := M(k{T}) of multiplicative seminorms on the Tate algebra k{T} which extend the absolute value on k and which are bounded above by the Gauss norm x . When equipped G with the topology of pointwise convergence, X is a compact Hausdorff path-connected space. Let H(x) denote the completed residue field of x∈X. Let Gal(ka/k) denote the group of automorphisms of ka fixing k (though ka/k is not Galois when k is not perfect), and let X :=M(K{T}) denote the ground field extension of X to K, which comes equipped with a K continuous surjective map π: X → X. The ground field extension X carries a Gal(ka/k)-action, extending K K 4 MATTHEWSTEVENSON the natural action on K by isometries, such that π induces a homeomorphism X /Gal(ka/k) −∼→ X, where K X /Gal(ka/k) has the quotient topology. See [Ber90, p.18] and [BR10, §1] for further details. K Given z ∈ K◦ and r ∈ [0,1], we may construct a point x ∈ X as the sup-norm over the closed disc z,r K D(z,r)⊂K◦ of radius r about z; that is, |f(x )|:= sup |f(z′)|, f ∈K{T}. z,r z′∈D(z,r) Thepointπ(x )∈X willagainbedenotedbyx . WhenK 6=k,itispossiblethattwopairs(z,r)and(z′,r′) z,r z,r may define the same point of X; for example, if z′ ∈Gal(ka/k)·{z} and r >0, then xz′,r =xz,r. When z ∈(ka)◦ and r =0, the associated point x is called a rigid point; the seminorm x coincides with z,0 z,0 the seminorminduced by the maximalidealofk{T}generatedby the minimal polynomialof z overk. LetXrig denote the subset of rigid points. The points of X can be classified into 4 types: (i) a type 1 point is of the form x ∈X for z ∈K◦; z,0 (ii) a type 2 point is of the form x ∈X for z ∈K◦ and r ∈(0,1]∩|K∗|; z,r (iii) a type 3 point is of the form x ∈X for z ∈K◦ and r ∈(0,1]\|K∗|; z,r (iv) a type 4 point is the pointwise limit of x such that the corresponding discs D(z ,r ) form a decreasing zi,ri i i sequence with empty intersection. See [Ber90, p.18] for further details; when K 6= k, see also [Ked11, Proposition 2.2.7]. When k is nontrivially- valued,the setofrigidpoints andthe setof type 2 points areboth dense inX. Pointsoftype 4exist onlywhen k is not spherically complete. For any x ∈ X, the multiplicity of x is m(x) := #π−1(x) ∈ Z ∪{∞}. The points of type 1 with finite >0 multiplicity are precisely the rigid points. All points of type 2 and type 3 have finite multiplicity. TheBerkovichunitdiscX comesequippedwithastructuresheafO ontheG-topologyonX. Thek-algebra X of global sections of O is the Tate algebra k{T}. See [Ber90, 2.3] for further details. X 2.2. Quasisubharmonic functions on the Berkovich unit disc. In this section, we discuss the metric tree structure and potential theory on the Berkovich unit disc. For a comprehensive treatment, see [BR10] and [Jon15]. Define a partial order ≤ on X by declaring that x ≤ y if and only if |f(x)| ≤ |f(y)| for all f ∈ k{T}; in this way, the pair (X,≤) becomes a rooted tree with root at the Gauss point x . For each x ∈ X, define an G equivalence relation ∼ on the set X\{x} by declaring y ∼z if the paths (x,y] and (x,z] intersect. The tangent space T at x is the set of equivalences classes of X\{x} modulo ∼. For each~v ∈T , let U(~v) be the set of X,x X,x points of X representing~v. The subsets of the formU(~v), for some tangentdirection~v, forma subbasis ofopen sets for the topology on X. See [Jon15, §2.3] for further details. TheclosedunitdiscX maybeequippedwithageneralizedmetric,inthesensethatthedistancebetweentwo pointsmaybe infinite; inparticular,X gainsthe structure ofa metric tree. This generalizedmetric is described as follows. Let r: X → [0,1] denote the radius function, which can be thought of as a Gal(ka/k)-equivariant function on X . Define a function α: X →[−∞,+∞] by specifying that α(x )=0 and K G x 1 α(x)−α(y)=− d(logr(z)), m(z) Zy for any two distinct points x,y ∈X, where the integral is taken over the unique path in X joining the points x and y. These constraints completely determine a function α: X →[−∞,+∞] whose restriction to any segment is monotone decreasing. Observe that, when k =ka, α=−logr. This, in turn, induces a generalized metric d on X by setting d(x,y):=|α(x)−α(x∨y)|+|α(y)−α(x∨y)| for x,y ∈ X; here, x∨y is the least upper bound of x and y. The rooted tree (X,≤) acquires the structure of a metric tree when equipped with the generalized metric d. It is important to note that the topology on X induced by d is strictly finer than the native topology. A NON-ARCHIMEDEAN OHSAWA–TAKEGOSHI EXTENSION THEOREM 5 One can discuss potential theory and the notion of quasisubharmonic functions on any metric tree, as devel- opedin[Jon15,§2.5]. WebrieflyrecallthistheoryinthespecialcaseoftheBerkovichdisc. Thisisalsodiscussed in [BR10, §5] (though our conventions differ slightly). Fix a finite atomic measure ρ supported at x , i.e. ρ is a positive real multiple of the Dirac mass δ at 0 G 0 xG the Gauss point x . A function ϕ: X →[−∞,∞) is called ρ -subharmonic if it satisfies: G 0 (1) for every finite subtree Y ⊂X\Xrig containing x , ϕ| is a continuous function on Y such that: G Y (a) ϕ| is convex on any segment in Y that does not contain x ; Y G (b) for any y ∈Y, ρ {y}+ d (ϕ| )≥0, 0 ~v Y ~v∈XTyY where d (ϕ| ) denotes the directional derivative of ϕ| in the direction~v. ~v Y Y (2) ϕ is the limit of its retractions to finite subtrees containing x ; more precisely, if {Y } denotes the G i i∈I net of finite subtrees of X\Xrig containing x and if r : X → Y is the (continuous) retraction map of G i i X onto Y , then ϕ=limr∗ϕ. i i i The condition (b) is equivalent to the subaverage property: for any y ∈Y, there exists r >0 such that 1 ϕ(y)≤ ϕ(z)−mass(ρ )r·1 (y), |B (y,r)| 0 {xG} Y z∈BXY(y,r) where B (y,r):={z ∈Y : d(y,z)=r} is the ball of radius r about y in Y, and 1 is the indicator function Y {xG} of the point x . This is reminiscient of the classical definition of subharmonic functions on Rn. G A function is said to be quasisubharmonic if it is ρ -subharmonic for some measure ρ as above. The class 0 0 of ρ -subharmonic functions on X is a convex set, which is closed under taking finite maxima and decreasing 0 (pointwise) limits. A quasisubharmonic function is upper-semicontinuous, but it may take the value −∞; this can only occur at those points x∈X such that α(x)=−∞, i.e. x∈Xrig. Given a topological space Z and a function φ: Z → [−∞,∞) that is locally bounded above, the upper- semicontinuous (usc) regularization φ∗ of φ is defined by the formula φ∗(z):=limsupφ(y), z ∈Z. y→z The usc regularizationφ∗ is the smallest usc function such that φ∗ ≥φ. Lemma 2.1. If {ϕ } is a net of ρ -subharmonic functions on X which is locally bounded above, then the usc i i∈I 0 regularization ψ∗ of ψ :=sup ϕ is ρ -subharmonic. Furthermore, ψ∗ =ψ on X\Xrig. i∈I i 0 The proof of Lemma 2.1 is a minor variation on [BR10, Proposition 8.23(E)]. TherestofthissectionisdevotedtoabriefdiscussionoftheLaplacianofaquasisubharmonicfunction. Given aρ -subharmonicfunctionϕandanyfinitesubtreeY ⊂X containingx ,let∆(ϕ| )betheuniquesignedBorel 0 G Y measure on Y determined by the rule n n ∆(ϕ| ) U(v~) = d (ϕ| ) Y i v~i Y ! i=1 i=1 \ X for distinct tangent directions ~v ,...,~v ∈ T at a point x ∈ Y. The Laplacian is the signed Borel measure 1 n Y,x ∆ϕ uniquely characterized by the following property: for any finite subtree Y ⊂X containing x , G (r ) (ρ +∆ϕ)=ρ +∆(ϕ| ), Y ∗ 0 0 Y where r : X →Y is the (continuous) retraction map of X onto Y. Y Quasisubharmonic functions and their Laplacians behave well with respect to the ground field extension morphism π: X → X, in the following sense: given a ρ -subharmonic function ϕ on X, π∗ϕ is a π∗ρ - K 0 0 subharmonic function on X and ∆ϕ=π ∆(π∗ϕ). K ∗ 6 MATTHEWSTEVENSON Example 2.2. For any irreducible f ∈ k{T} and any positive real number c > 0, the function ϕ := c·log|f| is ρ -subharmonic, where ρ = c · δ . It is easy to check that ∆ϕ = c(δ −δ ), where x ∈ Xrig is the 0 0 xG x xG rigid point of X corresponding to the maximal ideal (f) of k{T}. More generally, for any f ∈ k{T}, clog|f| is quasisubharmonic and its Laplacian can be identified, up to scaling and adding a multiple of δ , with the xG divisor of zeros of f via the Poincaré–Lelongformula. See [BR10, Example 5.20]. For a quasisubharmonicfunction ϕ on X, we may construct a metric on O : to a local sectionf of O over X X an analytic domain V ⊂ X, we assign the function x 7→ |f(x)|e−ϕ(x), for x ∈ V. This convention mirrors how (semipositive)metrics online bundles oncomplex manifolds are locally givenby plurisubharmonic functions, as in e.g. [Dem12, §3]. 2.3. Temkin’s metric. In [Tem16], Temkin introduces a canonical lower-semicontinuous metric A on the Z canonicalbundle ω of any k-analytic space Z. We will explainthe constructionof A when Z is a k-affinoid Z/k Z space. In the sequel, the metric A will play a crucial role when Z is the closed unit disc. Z To a contractive morphism A → B of seminormed rings, the module Ω of Kähler differentials may be B/A equipped with the maximalseminormk·k making the derivationd: B →Ω into a contractiveA-module B/A B/A morphism. Explicitly, ksk = inf max|b |·|b′|, B/A s=Pibid(b′i) i i i wheretheinfimumistakenoverallrepresentationss= b d(b′). LetΩ denotetheseparatedcompletionof i i i B/A Ω with respect to k·k ; this is a BanachB-module. We say that Ω is the completed module of Kähler B/A B/A P B/A b differentials and we say its norm k·k is the Kähler norm on Ω . See [Tem16, 4.1] for further details. B/A B/A b Forak-affinoidalgebraA,themoduleΩ hasanalternatedescription: ifJ isthekernelofthemultiplication A/k map A⊗ A→A, then Ω =J/J2. The derivation d: A→Ωb is induced from the bounded map A→J k A/k A/k b of BanachA-modules givenby a7→1⊗a−a⊗1. The image of the derivationd generatesΩ as an A-module. A/k This isbthe approachtakben in [Ber93, 3.3]. b Let Z = M(A) be a k-affinoid spabce. Thbe finite Banach A-module Ω determines abcoherent O -module A/k Z Ω :=O (Ω )onZ. TemkinprovidesacanonicalmetriconeachstalkofΩ ,aswellasonthestalksofits Z/k Z A/k Z/k exterior powers (and in particular, on the canonical bundle). This was obriginally introduced as a generalization of the weightbfunction of [MN15]. It has arisen in other contexts more recently, e.g. as a log discrepancy-type function in [BJ16, 5.6]. For any z ∈Z, there is a natural homomorphism of k-algebras ≃ ψ : Ω −→Ω ⊗ H(z)−→Ω , z Z/k,z Z/k,z OZ,z H(z)/k whichextendsthenaturalisomorphismfromthecompletedfibreofΩ atz tothecompletedmoduleofKähler Z/kb differentials of H(z)/k. Let π: Z →Z denote the ground field extension of Z. K Fix z ∈Z. The canonicalmetric k·k onΩ is definedby the followingrule: forany germs∈Ω ,set z Z/k,z Z/k,z kskz :=kψz′(π∗s)kH(z′)/k for any z′ ∈π−1(z). This is independent of the choice of z′. In particular,if k =ka, then ksk =kψ (s)k , z z H(z)/k i.e. k·k is the norm on Ω defined by pulling back the norm on Ω . This metric naturally extends to z Z/k,z H(z)/k the the exterior powers of Ω and, in particular, to the canonical bundle ω . For the various properties of Z/k Z/k the canonical metric, see [Tem16, §6]. b The canonical metric, as defined above, is called the “geometric Kähler norm” in [Tem16, 6.3.15]. When k 6=ka,ifonedefinesthe metriconthe stalkΩ bypullingbackthemetric onΩ (withoutfirstpassing Z/k,z H(z)/k to the completedalgebraicclosure),then the metric may be quite poorlybehaved(for example when k has wild extensions). See [Tem16, 6.2]. b A NON-ARCHIMEDEAN OHSAWA–TAKEGOSHI EXTENSION THEOREM 7 We prefer to write Temkin’s metric on the canonical bundle ω additively. That is, consider the lower- Z/k semicontinuous function A : Z → [0,+∞] defined implicitly by the following property: for any z ∈ Z and for Z any nonzero germ s∈ω , write Z/k,z ksk =|s(z)|e−AZ(z). z ThefunctionA isindependentofthe choiceofgerms usedinthe definition. We writeA=A whenthespace Z Z Z is implicit. For our purposes, the important special case is when X = M(k{T}) is the closed unit disc. The completed module Ω of Kähler differentials is the free Banach k{T}-module on the differential dT and, follow- k{T}/k ing[Tem16,6.2.1],thecanonicalmetricadmitsasimpledescription: foranyx∈X andanyglobalsectionf·dT of Ω b, we have k{T}/k kf ·dTk =|f(x)|r(x)=|f(x)|e−AX(x), x b where r: X →[0,1] is the radius function. Thus, A =−logr. This description holds regardless of whether or X not k is algebraically closed. Observe that A =+∞ on the type 1 points of X. X In §3 and §4, we will be concerned with “twists” of the canonical metric A on Ω by a metric on O . X X/k X Moreprecisely,foranyquasisubharmonicfunctionϕonX andanyx∈X,weconsiderthe metrick·k which, ϕ,x to a global section f ·dT of Ω , assigns the number X/k kf ·dTk :=|f(x)|e−ϕ(x)−AX(x). ϕ,x In fact, we will be interested in sup kf ·dTk , which we write as kfk in the sequel. x∈X ϕ,x ϕ 3. An Ohsawa–Takegoshi-type extension theorem Let us briefly recall the notation established in §2. Let k be a complete (possibly trivially-valued) non- Archimedean field, let X = M(k{T}) be the Berkovich closed unit disc over k with Gauss point x , let G r: X →[0,1] be the radius function, and let A:=−logr: X →[0,+∞] be Temkin’s metric. In this section, we prove the following variant of Theorem A and derive Theorem A as an easy consequence. Theorem 3.1. Let ϕ be a quasisubharmonic function on X with ϕ(x )=0, and let Z(ϕ)={ϕ=−∞} denote G the polar locus of ϕ. For any z ∈X, there exists a constant ǫ >0 and a nonzero polynomial f ∈k[T] such that 0 kfk := sup |f(x)|e−(1+ǫ)ϕ(x)−A(x) ≤|f(z)|e−ϕ(z) for all ǫ∈[0,ǫ ]. (1+ǫ)ϕ 0 x∈X\Z(ϕ) If ϕ(z)=−∞, then we may find f such that kfk < +∞ for all ǫ∈ [0,ǫ ]. Moreover, if k is algebraically (1+ǫ)ϕ 0 closed and z ∈Xrig, then for any value a∈H(z)∗ =k∗, we may find f such that f(z)=a. WewillproveTheorem3.1in§3.1. ThehypothesesofTheorem3.1maybeweakenedtoallowϕ(x )≥0,but G itisfalseifϕ(x )<0(e.g.takeϕtobeaconstantfunction). Nonetheless,TheoremA,whichhasnohypothesis G on the value of ϕ(x ), may be easily deduced from Theorem 3.1. G Proof of Theorem A. Given a quasisubharmonic function ϕ on X and a point z ∈X, set φ:=ϕ−ϕ(x ). The- G orem 3.1 asserts that there is a nonzero polynomial f ∈ k[T] such that kfk ≤ |f(z)|e−φ(z) for all ǫ > 0 (1+ǫ)φ sufficiently small. Thus, lim kfk =eϕ(xG) lim eǫϕ(xG)kfk ≤|f(z)|e−φ(z)+ϕ(xG) =|f(z)|e−ϕ(z). (1+ǫ)ϕ (1+ǫ)φ ǫ→0+ ǫ→0+ This completes the proof of Theorem A. (cid:3) Asdiscussedin§1,theoptimalconstantappearingintheclassicalOhsawa–Takegoshitheoremforthecomplex unit disc is π. In the non-Archimedean setting, however,the optimal constant is 1, in the following sense. 8 MATTHEWSTEVENSON Corollary 3.2. For any z ∈ X, let c(z) be the smallest positive number such that for any quasisubharmonic function ϕ on X, there exists a nonzero f ∈k{T} satisfying lim kfk ≤c(z)|f(z)|e−ϕ(z). (1+ǫ)ϕ ǫ→0+ Then, c(z)=1. Proof. If ϕ ≡ 0, then |f(x )| = lim kfk ≤ c(z)|f(z)| ≤ c(z)|f(x )|. As |f(x )| 6= 0, it follows that G ǫ→0+ (1+ǫ)ϕ G G c(z)≥1. Theorem A asserts that the lower bound c(z)=1 is achieved for any z. (cid:3) 3.1. Proof of Theorem 3.1. Fix z ∈ X. Suppose ϕ is ρ -subharmonic for some finite (atomic) measure ρ 0 0 supported at the Gauss point x , and ϕ(x ) = 0. To simplify the exposition, we assume ϕ(z) > −∞. When G G ϕ(z)=−∞, the proof is similar: one proves analogues of the sequence of lemmas below, each of which is made easier because one is only concerned with ensuring that a quantity is finite, as opposed to ensuring that the same quantity is less than some fixed value. The strategyofthe proofis touse stronginductionon⌊mass(ρ )⌋. For thisreason,the followingterminology 0 will be quite helpful. Definition 3.3. Let ǫ >0. A nonzero polynomial f ∈k[T] is a (ϕ,ǫ )-extension at z if the inequality 0 0 kfk ≤|f(z)|e−ϕ(z) (1+ǫ)ϕ holds for all ǫ∈[0,ǫ ]. Equivalently, f is a (ϕ,ǫ )-extension if for all x∈X\Z(ϕ) and all ǫ∈[0,ǫ ], we have 0 0 0 log|f(x)|−log|f(z)|≤A(x)+(1+ǫ)ϕ(x)−ϕ(z). The key player in the proof of Theorem 3.1 is the finite subtree Γ := x∈X :(ρ +∆ϕ){y ≤x}≥m(x) . ϕ 0 The subtree Γ is crucial in reducing Tnheorem 3.1 to a “finite” problem. Oobserve that any point of Γ must ϕ ϕ have a finite Gal(ka/k)-orbit because mass(ρ ) is finite and, moreover, Γ ∩Xrig ⊆Z(ϕ). Variants of this tree 0 ϕ appear in [Jon15, Prop 2.8] and [FJ05, Lem 7.7]. The following is the base case for our induction. Lemma 3.4. If mass(ρ ) < 1, there exists a constant ǫ > 0 such that any nonzero constant function is a 0 0 (ϕ,ǫ )-extension at z. 0 Proof. Set ǫ := 1−mass(ρ0). Since mass(ρ )<1, Γ =∅ and |d (1+ǫ)ϕ|≤(1+ǫ)mass(ρ )≤1 for all tangent 0 mass(ρ0) 0 ϕ ~v 0 directions~v in X, provided ǫ∈[0,ǫ ]. To check that any constant function is a (ϕ,ǫ )-extension at z, it suffices 0 0 to check that 0≤A+(1+ǫ)ϕ−ϕ(z) on X\Z(ϕ). This inequality is satisfied at the Gauss point: A(x )+(1+ǫ)ϕ(x )−ϕ(z)=−ϕ(z)≥0. G G Moreover,theconvexityofϕensuresthatitisenoughtocheckthat,inanytangentdirectionatx ,thefunction G (1+ǫ)ϕ+A only increases in that direction. However, in any such direction~v, d ((1+ǫ)ϕ+A)≥−(1+ǫ)mass(ρ )+1≥0. ~v 0 (cid:3) Now, if mass(ρ )≥1, then x ∈Γ because m(x )=1. In particular, Γ 6=∅ and the set of ends Ends(Γ ) 0 G ϕ G ϕ ϕ is nonempty. Let Γ′ denote the convex hull of Γϕ∪{z}. Let rΓ′: X → Γ′ denote the retraction of X onto Γ′. Observe that a (r∗ ϕ,ǫ )-extension f at z is also a (ϕ,ǫ )-extension at z, because the functions y 7→|f(y)| and Γ′ 0 0 y 7→e−(1+ǫ)ϕ(y)−A(y) areboth either locallyconstantor decreasingoff ofΓ′. Thus,by replacingϕ with r∗ ϕ, we Γ′ may assume that ϕ is locally constant off of Γ′. As mentioned before, the strategy of the proof of Theorem 3.1 is by strong induction on ⌊mass(ρ )⌋. In the 0 following sequence of lemmas, we explain how to reduce the problem of the existence of a (ϕ,ǫ )-extension at 0 A NON-ARCHIMEDEAN OHSAWA–TAKEGOSHI EXTENSION THEOREM 9 z to the existence of a (φ,ǫ )-extension at z, where φ is a ρ-subharmonic for some measure ρ with mass(ρ) ≤ 1 mass(ρ )−1and0<ǫ ≤ǫ . Asmass(ρ )isfinite,afterfinitely-manysuchreductionswemustfindourselvesin 0 1 0 0 the setting of Lemma 3.4. The hypotheses of eachlemma are concernedwith whichtypes ofpoints may arisein the finite set Ends(Γ ); in particular, if one assumes that k is spherically complete, one can ignore Lemma 3.7. ϕ (This greatly simplifies the proof.) Lemma 3.5. Suppose that mass(ρ )≥1 and that there exists x∈Ends(Γ ) of type 1. Then, there exists ǫ >0 0 ϕ 1 and a ρ-subharmonic function φ, with mass(ρ) ≤ mass(ρ )−1, such that a (ϕ,ǫ )-extension f at z may be 0 0 constructed from a (φ,ǫ )-extension fãt z. 1 Proof. As x∈Γ , it satisfies m(x)<+∞ and so we must have x∈Xrig, because x is of type 1. In addition, as ϕ Γ ∩Xrig ⊆Z(ϕ) and z 6∈Z(ϕ), we have x6=z. Let m denote the maximal ideal of k{T} corresponding to x, ϕ x and let g be a polynomial generator of m with |g(x )| = 1. As type-1 points are minimal with respect to the x G partial order ≤, we have 1≤m(x)≤(ρ +∆ϕ){y ∈X: y ≤x}=∆ϕ{x}. 0 Let c:=∆ϕ{x}≥1, m:=⌊c⌋, ρ:=ρ −mδ , and φ:=ϕ−mlog|g|. Observe that φ(x )=0. Suppose there 0 xG G exists ǫ > 0 and a (φ,ǫ )-extension f˜∈ k[T] at z. Let f := gmfãnd ǫ := ǫ c−m. For any x ∈ X\Z(ϕ) and 1 1 0 1 c any ǫ∈[0,ǫ ], 0 log|f(x)|−log|f(z)|≤A(x)+(1+ǫ )φ(x)−φ(z)+mlog|g(x)|−mlog|g(z)| 1 =A(x)+ϕ(x)+ǫ φ(x)−ϕ(z) 1 ≤A(x)+(1+ǫ)ϕ(x)−ϕ(z), wherethelastinequalityfollowsbecauseǫ φ≤ǫϕforallx∈X is equivalenttoǫ≤ǫ inf φ,asϕ≤0. Finally, 1 1 X ϕ we claim that inf φ = c−m, from which it follows that f is a (ϕ,ǫ )-extension at z. X ϕ c 0 Write ϕ = ψ+clog|g| for some quasisubharmonic function ψ on X; it follows that φ = ψ+(c−m)log|g| and ∆ψ{x} = 0. Take any y ∈ (x,x ) and any tangent direction ~v ∈ T pointing off of [x,x ] and towards G X,y G the leaves of X, i.e. a direction~v such that U(~v)∩[x,y]=∅ and U(~v)∩Xrig 6=∅. It is easy to check that φ (d ψ)(mlog|g|) d = ~v ≥0. ~v ϕ (ψ+clog|g|)2 (cid:18) (cid:19) Thus, φ only increases in any direction not along [x,x ]. In fact, the same argument holds for any tangent ϕ G direction in which |g| is constant; in particular, inf φ may be computed on [x,x ]. Since ψ does not have a X ϕ G pole at x, it is then clear that inf φ = c−m, as required. (cid:3) [x,xG] ϕ c Lemma 3.6. Suppose that mass(ρ ) ≥ 1 and that there exists x ∈ Ends(Γ )\{x } of type 2 or of type 3 such 0 ϕ G that r (z)6=x. Then, there exists ǫ >0 and a ρ-subharmonic function φ, with mass(ρ)≤mass(ρ )−1, such Γϕ 1 0 that a (ϕ,ǫ )-extension f at z may be constructed from a (φ,ǫ )-extension fãt z. 0 1 Proof. Pick x′ ∈ Xrig such that r (x′) = x; such a point exists because Xrig is dense when k is nontrivially- Γϕ valued, and all type 1 points are rigid when k is trivially-valued (because K = ka). Let c = ∆ϕ{x} ≥ 1. We construct a new ρ -subharmonic function ϕ′ by extending ϕ linearly with slope c from the end x to the rigid 0 point x′; more precisely, ϕ′ is given by the formula |g(y)| ϕ′(y):=ϕ(y)+min 0,clog , |g(x)| (cid:26) (cid:27) whereg isanypolynomialgeneratorofmx′. Itisclearthatϕ′(xG)=0,ϕ′(z)=ϕ(z),andϕ′ ≤ϕ. Inparticular, any (ϕ′,ǫ0)-extension at z is also a (ϕ,ǫ0)-extension at z. As x′ ∈Ends(Γϕ′), we can now apply Lemma 3.5 to ϕ′ to produce the desired ǫ and φ. (cid:3) 1 10 MATTHEWSTEVENSON Lemma 3.7. Suppose that mass(ρ )≥1 and that there exists x∈Ends(Γ ) of type 4. Then, there exists ǫ >0 0 ϕ 1 and a ρ-subharmonic function φ, with mass(ρ) ≤ mass(ρ )−1, such that a (ϕ,ǫ )-extension f at z may be 0 0 constructed from a (φ,ǫ )-extension fãt z. 1 x • G z∨x • • x˜ • r (z) • Γϕ u • • x z • • • • u′ Figure 1. One possible configuration for Γ in the setting of Lemma 3.7. ϕ Proof. As trivially-valuedfields are spherically complete, the existence of a type 4 point in X guaranteesthat k is nontrivially valued. In particular, the type 2 points of X are dense. Recall that Γ′ denotes the convex hull of Γ ∪{z} (we allow the possibility that x=z, so it is possible that ϕ Γ′ = Γ ). Let x˜ ∈ (x,x ] be the minimal point with the property that {y ∈ Γ′: y ≤ x˜} =6 [x,x˜]. That is, x˜ is ϕ G theminimalpointon(x,x ]suchthat(x˜,x]doesnotintersectanybranchesofΓ′ otherthanthe onecontaining G x. An example is in Fig. 1. Let ~v be the unique tangent direction at x˜ with x ∈ U(~v), c := −d ϕ ≥ 1, and m := ⌊c⌋. After possibly ~v replacing ϕ with a smaller quasisubharmonic function, we may assume that ϕ is linear of slope −c on [x,x˜], as in Lemma 3.6. Consider the quasisubharmonic functions ϕ˜(y):=mlogr(y∨x) and φ:=ϕ−ϕ˜, and observe that φ(x )=0 G and Γ ∩[x,x˜]={x˜}. Suppose there exists an ǫ >0 and a (φ,ǫ )-extension fãt z. φ 1 1 Pick η ∈ 0,−ǫ1φ(x) . As eη >1 and the type 2 points are dense in X, there exists u∈(x,x˜) of type 2 such 2m that r(u)<(cid:16)eηr(x). By(cid:17)possibly taking u closer to x, we may assume that φ(u)≤ 1φ(x). If 2 mη+ ǫ1φ(x) ǫ1φ(x)− ǫ1φ(x) δ := 2 > 2 2 =0, 0 1φ(x)+mlogr(x) − 1φ(x)+mlogr(x) 2 2 then for any ǫ∈[0,δ0], we have (cid:0) (cid:1) r(u) me(ǫ1−ǫ)φ(u)r(x)−mǫ ≤emη+ǫ21φ(x)−mǫlogr(x) ≤1. r(x) (cid:18) (cid:19) Picku′ ∈Xrig suchthatrΓϕ(u′)=u,andletg beapolynomialgeneratorofmu′ with|g(xG)|=1. Setf :=f˜gm. We claim that there exists δ1 ∈ (0,ǫ1) such that r(y∨x)−ǫme(ǫ1−ǫ)φ(y) ≤ 1 for all y ∈ X and all ǫ ∈ [0,δ1]. After rearranging,this is equivalent to requiring that φ(y) ǫ inf ≥ǫ 1 y∈X φ(y)+mlogr(y∨x) As in the proof of Lemma 3.5, the above infimum is equal to c−m, hence we may take δ :=ǫ c−m. c 1 1 c