Then for any smooth positive measure µ on M with R Mµ =R Mc1(L)∧dimM, there is a pos- itive metric k k on L, unique up to a constant multiple, such that c1(L,k k)∧dimM =µ

89(2011) 87-110

A NON-ARCHIMEDEAN ANALOGUE OF THE CALABI-YAU THEOREM FOR TOTALLY

DEGENERATE ABELIAN VARIETIES Yifeng Liu

Abstract

We show an example of a non-archimedean version of the exis- tence part of the Calabi-Yau theorem in complex geometry. Pre- cisely, we study totally degenerate abelian varieties and certain probability measures on their associated analytic spaces in the sense of Berkovich.

Introduction

The theorem of Calabi-Yau is one of the most important results in complex geometry and has many applications (e.g., Yau’s famous paper [16]). In one version, it claims the following. Let M be a compact complex manifold with an ample line bundle L. Then for any smooth positive measure µ on M with R

Mµ =R

Mc₁(L)^∧dimM, there is a pos- itive metric k k on L, unique up to a constant multiple, such that c1(L,k k)^∧dimM =µ.

One would like to ask a similar question for a non-archimedean field, for example, if we replace C by C_p and complex manifolds by non- archimedean analytic spaces, or Berkovich spaces in [1]. Hence letXbe the analytic space associated to a projective variety over C_p equipped with (the analytification of) an ample line bundle L. Given any in- tegrable metric k k (cf. [21]) on L, although we do not have a nice analogue of (1,1)-form for c₁(L,k k) in the non-archimedean situation so far, we can still talk about its top wedge product, i.e.,c1(L,k k)^∧dimM which is defined by Chambert-Loir in [5]. The top wedge product is a measure on the underlying compact (metrizable) topological space ofX.

If k k is semipositive in the sense of [20, 21], then c₁(L,k k)^∧dimM is a positive measure in the following sense: R

Xf c₁(L,k k)^∧dimM ≥0 for any non-negative continuous real functionf onX. Then analogously to the complex case, given a positive measure µon X withR

Xµ= deg_L(X), we can ask the following two questions:

Received 11/30/2010.

87

(E): Does there exist a semipositive metric k kon L such that c1(L,k k)^∧dimM =µ?

(U): If it does, is it unique up to a constant multiple?

The question (U) has been answered positively by Xinyi Yuan and Shou-Wu Zhang in [18]. Moreover, they have already used this unique- ness result to prove several exciting theorems in algebraic dynamic sys- tems. The answer to the question (E) is known to be positive for curves and measures compactly supported on type II or III points, or the lim- its of such measures, which is implied by the theory in [19]. In any dimension, there is a necessary condition for the equality in (E) to hold, due to a result of Chambert-Loir and Thuillier (cf. [6, corollaire 4.2]), that µ cannot be supported on a proper algebraic subset of X, which is well-known in the archimedean case. But even under this restriction, the answer to (E) is stillnegativein general, due to several reasons. One conceptual reason is that the notions of being positive or smooth in the metric side and the measure side are not quite compatible as in the complex case. We are still not clear about the nature of this question, especially for its correct formulation. Nevertheless, we would like to give one example where the answer to (E) is positive (of course, under certain restrictions).

Let us consider a totally degenerate abelian variety A, say over C_p, i.e., the special fibre of its N´eron model is a torus, which is also equiv- alent to saying that the associated analytic space A^an is isomorphic to a quotient G^d_man

/M for a complete lattice M ⊂G^d_m(C_p) (cf. Section 1). We have an evaluation map τ_A :A^an = G^d_man

/M → R^d/Λ with a complete lattice Λ ⊂ R^d, which is continuous and surjective. This map has a continuous section iA : R^d/Λ ֒→ A^an. In fact, iA identifies R^d/Λ as a strong deformation retract ofA^an for which i_A◦τ_A= Φ(·,1) for a strong retraction map Φ :A^an×[0,1]→A^an. Then we prove the following theorem, which is a certain non-archimedean analogue of the Calabi-Yau theorem.

Theorem (4.2). Let A be ad-dimensional totally degenerate abelian variety over k (e.g. C_p) and L ∈ Pic(A)⊗ZQ an ample class on A.

For any measure µ = fdx of R^d/Λ with f a strictly positive smooth function, dx the Lebesgue measure on R^d/Λ, and R

R^d/Λµ = deg_L(A), there is a semipositive metrick konL, unique up to a constant multiple, such that c₁(L)^d= (i_A)_∗µwhere L= (L,k k).

The main ingredient of the proof is a limit formula for the measure as- sociated to certain integrable metrics, which is Theorem 3.3. According to this formula, the existence part ends up with a (real) Monge-Amp`ere equation on a real torus, quite similar to the complex case, although we are doing non-archimedean geometry. In fact, let R^d/Λ be the real

torus as above with usual coordinate x₁, . . . , x_d and Lebesgue measure dx= dx₁· · ·dx_d. Let (g_ij)i,j=1,...,d be a positive definite (real symmet- ric) matrix. Then for any smooth real function f on R^d/Λ such that R

R^d/Λe^fdx= covol(Λ), we will show that there exists a unique smooth (real) function φon R^d/Λ such that:

• the matrix

g_ij +_∂x^∂2φ

i∂xj

is positive definite;

• R

R^d/Λφdx= 0;

• it satisfies the following real Monge-Amp`ere equation:

det

g_ij + ∂²φ

∂x_i∂x_j

= det (g_ij)·e^f.

The above type of real Monge-Amp`ere equation has been solved by Cheng-Yau in [7, §2] without invoking the complex one. There is a second solution (see [7, §3]) which applies to more general types of real Monge-Amp`ere equations on compact K¨ahler affine flat manifolds, using a tube domain construction and reducing to the complex case which has already been solved by Yau [17] when attacking the (classical) Calabi conjecture. Hence the same PDE problem applies to this non- archimedean Calabi-Yau theorem as well. We will resolve the equation along the second line for our special case here, not only in order of this PDE problem, but also to suggest relations between archimedean (complex) and non-archimedean pictures.

At last, we would like to make some remarks about notations. We use| |for non-archimedean norms; k kfor metrics on line bundles. But due to the conventions, we will also use | |for the usual absolute value of real numbers and total measures; k k for the Euclidean norm of R^d when d >1.

For simplicity, we will assume thatkis the completion of the algebraic closure of ap-adic local field, i.e., isomorphic toC_porF\_p((t)), the comple- tion of the algebraic closure of F_p((t)). Such restriction onk is unneces- sary. In fact, all results and argument remain valid for anyalgebraically closed non-archimedean Banach field whose valuation is non-trivial: for example, the completion of the algebraic closure of C((t)). Then one only needs to use the generalized definition of Chambert-Loir’s measure given by Gubler in [11,§3].

Acknowledgements.This paper is motivated by works of Xinyi Yuan and Shou-Wu Zhang [18] on the uniqueness part of the non-archimedean Calabi-Yau theorem, and Gubler [11] on the tropical geometry of totally degenerate abelian varieties. The author would like to thank Xander Faber, Shing-Tung Yau, Xinyi Yuan, and Shou-Wu Zhang for stimulat- ing discussions. This work was presented in the Number Theory Semi- nar at Harvard University and AMS Sectional Meeting #1065, Special Session on Applications of Non-Archimedean Geometry. The author

thanks the organizers of both. Finally, the author would like to thank referees for their helpful comments.

1. Mumford’s Construction

In this section, we briefly recall Mumford’s construction of (formal) models of totally degenerate abelian varieties in [15, §6]; see also [11,

§4, §6].

1.1. Valuation map.Letk be the completion of the algebraic closure of ap-adic local field. Let| |be the norm onkand its extended valuation fields,k^◦ the subring ofkconsisting of elements x∈kwith|x| ≤1,k^◦◦

the maximal ideal of k^◦ consisting of elements x ∈k with|x|<1, and ek = k^◦/k^◦◦ the residue field which is algebraically closed. We fix a logarithm log such that log|x| ∈Qfor all x∈k^×.

Fix a split torus T = G^d_m,k of rank d ≥ 1 over k. We have the following valuation map:

τ :T^an=

G^d_m,kan

−→R^d, t7→(−log|T₁(t)|, . . . ,−log|T_d(t)|) where

G^d_m,kan

is the associated k-analytic space (cf. [1,§3.4]) and T_i are coordinate functions. It is surjective and continuous with respect to the underlying topology of the analytic space and the usual topology of R^d. Moreover, τ has a continuous section

i:R^d−→T^an, x7→ξ_x

where ξ_x is (the unique point of) the Shilov boundary of the affinoid domainτ⁻¹(x) (cf. [11, corollary 4.5]). Precisely, thek-affinoid algebra associated to τ⁻¹(x) is

khτ⁻¹(x)i=



 X

m∈Z^d

a_mu^m₁¹· · ·u^m_d^d lim

|m|→∞−log|a_m|+m·x=∞



 and the point ξ_x corresponds to the norm

X

m∈Z^d

a_mu^m

= max

m∈Z^d|a_m|e^−m·x.

Now consider a totally degenerate abelian variety A over k,i.e., the special fibre of its N´eron model is a torus. Then A^an ∼= T^an/M for a complete lattice M ⊂ T(k). By a complete lattice, we mean that M is a (discrete free) subgroup of T(k) and maps bijectively to a rational complete lattice Λ ⊂ R^d under τ. Here, we say a complete lattice is rational if it has a basis whose coordinates are in Q. Hence we have the induced map τ_A :A^an → R^d/Λ and i_A : R^d/Λ ֒→ A^an. In fact, i_A identifies R^d/Λ with a strong deformation retract or a skeleton of A^an as in [1,§6.5].

1.2. Rational polytopes.A compact subset ∆ ofR^dis called a poly- topeif it is an intersection of finitely many half-spaces{x∈R^d|m_i·x≥ c_i}. We say ∆ isrationalif we can choose allm_i∈Z^dand c_i ∈Q. The dimension dim(∆) of ∆ is its usual topological dimension and we denote by int(∆) the topological interior of ∆ inR^d. Aclosed faceof ∆ is either

∆ itself or B∩∆ where B is the boundary of a half-space containing

∆. It is obvious that a closed face of a (rational) polytope is again a (rational) polytope. An open face is a closed face with all its properly contained closed faces removed.

A (rational) polytopal complex C in R^d is a locally finite set of (ra- tional) polytopes such that (1) if ∆∈ C, then all its closed faces are in C and (2) if ∆,∆^′ ∈ C, then ∆∩∆^′ is either empty or a closed face of both ∆ and ∆^′. The polytopes of dimension 0 are called vertices. We sayC is a(rational) polytopal decomposition ofS ⊂R^dif S is the union of all polytopes in C. In particular, if S =R^d, we say C is a (rational) polytopal decomposition ofR^d.

For a (rational) complete lattice Λ ⊂ R^d, we say C is Λ-periodic if ∆ ∈ C implies ∆ +λλλ ∈ C for all λλλ ∈ Λ. A (rational) polytopal decomposition CΛ of R^d/Λ for a (rational) complete lattice Λ is a Λ- periodic (rational) polytopal decomposition C of R^d such that ∆ maps bijectively to its image under the projection R^d→R^d/Λ for all ∆∈ C. A polytope, a closed face, or an open face of CΛ is a Λ-translation equivalence class of the corresponding object of C.

A continuous real function f on R^d is called (rational) polytopal if there is a (rational) polytopal decomposition C of R^d such that f re- stricted to all ∆∈ C is affine (and takes rational values on all vertices).

We denote by C_poly(R^d) (C_rpoly(R^d)) the space of (rational) polytopal continuous functions onR^d. We have the following simple lemma. More- over,f is calledstrongly polytopal convexwith respect toCif it is convex and the maximal connected subsets on which f is affine are ∆∈ C with dim(∆) =d.

Lemma 1.1. Let Λ be a (rational) complete lattice of R^d and f in C_poly(R^d) (C_rpoly(R^d))satisfying

1) there exist affine functions z_λλλ for allλλλ∈Λ satisfying f(x+λλλ) = f(x) +z_λλλ(x) for allλλλand x∈R^d;

2) if ∆ is any maximal connected subset on which f is affine, then

∆ is a bounded convex subset ofR^d.

Then such ∆ is a (rational) polytope and if C is the polytopal complex generated by all such ∆ and their closed faces, then C is a Λ-periodic (rational) polytopal decomposition of R^d.

Proof. Since ∆ is closed, it is compact by (2). Since f is (ratio- nal) polytopal, there is a finite (rational) polytopal decomposition of

∆; hence ∆ itself is a (rational) polytope since it is convex. Let C

be the (rational) polytopal complex generated by all such ∆ and their closed faces, which is obviously a polytopal decomposition of R^d. The

Λ-periodicity is implied by (1). q.e.d.

1.3. Formal models.LetXbe a projective scheme overk; ak^◦-model X ofX is a scheme projective and flat over Speck^◦ whose generic fibre X_η ∼= X. A formal k^◦-model X of X is an admissible formal scheme over Spfk^◦ (cf. [11, 2.6]) whose generic fibreXη ∼=X^an in the sense of analytic spaces (cf. [2,§1]). We denote byXf(resp. Xe) the special fibre of X (resp. X) which is a proper scheme over Specek. The following result is due to Mumford in the case that A is the base change of an abelian variety over a p-adic local field (cf. [15, corollary 6.6]) and extended by Gubler in the general case (cf. [11, proposition 6.3]).

Proposition 1.2. Given a rational polytopal decomposition CΛ of R^d/Λ, we may associate a formal k^◦-model A of A whose special fibre Aeis reduced and the irreducible components Y of Aeare toric varieties and one-to-one correspond to the verticesv ofCΛ by v=τ_A(ξ_Y), where ξ_Y ∈A^an is the point corresponding to Y. The formal scheme A has a covering by formal open affine sets U∆ for ∆∈ CΛ. Moreover, if A is the base change of an abelian variety over a p-adic local field, then A can be constructed as a k^◦-model.

2. Toric Metrized Line Bundles

In this section, we briefly recall the theory of metrized line bundles and their associated measure for general varieties. We introduce line bundles and toric metrized line bundles on A. Then we prove the main result, identifying certain toric integrable metrics.

2.1. Metrized line bundles and measure.The general theory of metrized line bundles is developed in [20], [21]; see also [5] and [10].

Let X be a projective scheme over k, and L a line bundle over X (or more generally, a class in Pic(X)⊗ZQ). A metric k k on L assigns to every open subset U of X^an and a section s∈ Γ(U, L^an), a continuous functionkskU :U →R_≥0 such that

1) kf skU =|f| · kskU for all f ∈Γ(U,OU);

2) kskU(x) = 0 if and only if s(x) = 0 for x∈U;

3) for any other s^′ ∈Γ(U^′, L^an) such that s^′|U∩U^′ =s|U∩U^′, we have ks^′kU^′|U∩U^′ =kskU|U∩U^′.

We say a metric isalgebraicif it is defined by a model (X,L) where X is ak^◦-model ofX andL is a line bundle onX such thatL_η ∼=L^e for some integer e≥ 1. A metric is formal if we replace the k^◦-model X by a formal k^◦-model X, and L by a formal line bundle L on X such that Lη ∼= (L^e)^an. In fact, all formal metrics are algebraic. An algebraic (resp. formal) metric is calledsemipositive if L (resp. L) has

non-negative degree on every closed curve on the special fibreXf(resp.

Xe) over k^◦. In general, a metric on L is called semipositive if it is the uniform limit of algebraic semipositive metrics. A metrized line bundle is called integrable if it is isomorphic to a quotient of two semipositive metrized line bundles.

Next we recall the construction of measures by Chambert-Loir in [5,

§2]. For simplicity, we only recall the algebraic case (which we only need for calculation later), and for the general case, one needs to pass to the limit, for which we refer to [5, §2] for details. Let X be as above of dimensiond≥1, and considerdclassesL_i ∈Pic(X)⊗ZQ(i= 1, . . . , d) of line bundles. We endow L_i with an algebraic measure k ki induced by (X,L_i) with (L_i)η ∼=L^e_iⁱ on a common model X which is assumed to be normal. Let Y_j be the reduced irreducible components ofXfand ξ_j the unique point in the inverse image of the generic point ofY_j under the reduction map π:X^an →Xf. Then we define

c1(L1)∧ · · · ∧c1(L_d) = 1 e₁· · ·e_d

X

j

mj(c1(L₁)· · ·c1(L_d)|Yj)δ_ξj (2.1)

where Li = (Li,k ki), mj is the multiplicity of Yj inXfand δ_ξj is the normalized Dirac measure supported at ξ_j. In general, the measure c₁(L₁)∧ · · · ∧c₁(L_d) is symmetric and Q-multilinear, and we have

Z

X^an

c₁(L₁)∧ · · · ∧c₁(L_d) =c₁(L₁)· · ·c₁(L_d)|X.

(2.2)

2.2. Line bundles on A.The theory of line bundles on totally degen- erate abelian varieties is very similar to that over the complex field. We refer to [3,§2] and [8, chapter 6] for more details.

Let A be a totally degenerate abelian variety as above and ˇM = Hom_k(T,G_m,k) the character group of T. Let ˇT be the split torus with character groupM, i.e., ˇT = Hom_k(M,G_m,k). Then ˇA^an is canonically isomorphic to ˇT^an/Mˇ where ˇA is the dual abelian variety of A. Let L be a line bundle on A; the pull-back of L^an to T^an is trivial and is identified withT^an×G^an_a,k. HenceL^anis identified with a quotient (T^an× G^an_a,k)/M where the action ofM on T^an is given by an element µ7→Z_µ of H¹(M,O(T)^×). The functionZ_µhas the formZ_µ=d_µσ_µwhered_µ∈ k^×, µ7→ σ_µ is a group homomorphism σ :M → Mˇ and d_µνd⁻¹_µ d⁻¹_ν = σ_ν(µ). By the isomorphism τ : M → Λ, we get a unique symmetric bilinear form b on R^d such that b(τ(µ), τ(ν)) = −log|σ_ν(µ)|. Then b is positive definite if and only if L is ample. Since σ_µ is a character,

|Z_µ|factors through τ and hence uniquely determines a function z_λλλ on R^d, such that z_λλλ(τ(t)) = −log|Z_µ(t)| for all µ ∈ M and t ∈T, where

λλλ=τ(µ). The function z_λλλ is affine with

z_λλλ(x) =z_λλλ(0) +b(x, λλλ), λλλ∈Λ, x∈R^d. (2.3)

By linearity, we can also assign band z_λλλ to any class L∈Pic(A)⊗ZQ.

Before stating the next lemma, we introduce some notations. We fix a Z-basis (λλλ₁, . . . , λλλ_d) of Λ once for all and let F ={x= x₁λλλ₁+· · ·+ x_dλλλ_d |0≤x_i <1} be a fundamental domain of Λ. The volume of the closure F under the usual Lebesgue measure dx ofR^d only depends on Λ and will be denoted by covol(Λ). We define RF = max_x,x′∈Fkx−x^′k and rF to be the maximal radius of balls contained inF. We denote by S^d−1⊂R^dthe standard unit ball, SO_d the special orthogonal group of R^d, and S_dthe group of d-permutations.

Let (e₁, . . . ,e_d) be the standard basis of the Euclidean space R^d, C^l(R^d) (l ≥ 0 and C = C⁰) the space of real functions whose l-th partial derivatives exist and are continuous, C^∞(R^d) the space of real smooth functions, and C^k,α(R^d) (k ≥ 0, α ∈ [0,1)) the spaces of real functions whosek-th partial derivatives exist and are H¨older continuous with exponential α. We denote by C^l

≥0(R^d) the subspace of functions non-negative everywhere and C^l

>0(R^d) that of functions positive every- where; similarly, we have C^∞

≥0(R^d), C^∞

>0(R^d), C^k,α

≥0 (R^d), and C^k,α

>0 (R^d).

For a certain function f, we write fi = ∇eif, fij = ∇ei∇ejf, and f_ijk =∇ei∇ej∇ekf if the corresponding directional derivatives exist.

Let q(x) = ¹₂b(x,x) be the associated quadratic form and H_q=d! det (q_ij)i,j=1,...,d

which is a constant associated to q. We have the following lemma.

Lemma 2.1. If L∈Pic(A)⊗ZQis an ample class, thendeg_L(A) = covol(Λ)H_q.

Proof. Consider the morphismφL:A→Aˇassociated toL. Its lifting T →Tˇrestricts toµ:M →Mˇ onM. It is easy to see thatµis injective sinceL is ample and

deg(µ) = [ ˇM :µ(M)] = covol(Λ)⁻¹det (b(λλλ_i, λλλ_j))i,j=1,...,d

= covol(Λ) det (q_ij)i,j=1,...,d.

By [3, theorem 6.15], deg(φ_L) = deg(µ)², and by the Riemann-Roch the- orem [14,§16], (deg_L(A)/d!)² = deg(φ_L). SinceLis ample, deg_L(A)>

0 and hence equals covol(Λ)H_q. q.e.d.

2.3. Toric metrized line bundles.Recall that an element f inside C_rpoly(R^d) is a continuous real function on R^d such that there is a ra- tional polytopal decomposition C of R^d satisfying that f restricted to all ∆∈ C is affine and takes rational values on vertices. The following proposition is due to Gubler.

Proposition 2.2. LetL= (T×G_a,k)/M be a line bundle on Agiven by a cocycle(Z_µ)_µ∈M as above. LetA be a formalk^◦-model determined by a rational polytopal decomposition CΛ of R^d/Λ given by C.

1) There is a one-to-one correspondence between the following two sets: The first consists of all formal metrics of L, i.e., formal modelLofL^e(witheminimal) onA, with trivialization(U∆)_∆∈CΛ; and the second consists of functions g ∈ C_rpoly(R^d), affine on ∆ for any ∆∈ C, satisfying

g(x+λλλ) =g(x) +z_λλλ(x); λλλ∈Λ, x∈R^d. (2.4)

Moreover, if we denote by k k the corresponding formal metric on L, then we have

g◦τ =−log(p^∗k1k) (2.5)

on T^an, where p:T^an →A^an is the natural projection and p^∗k · k is the pull-back metric on the trivial line bundle on T^an.

2) The reduction Le is ample if and only if g is strongly polytopal convex with respect to C.

Proof. For (1), by (2.4) and the fact that there exists m_λλλ ∈ Z^d for all λλλ ∈ Λ such that b(x, λλλ) = m_λλλ ·x, we can find a smallest integer e ≥ 1 such that e·g has integer gradient everywhere. Then by [11, proposition 6.6], e·g determines a formal model L of L^e. Hence g determines a formal metric on L. The last identity (2.5) follows from the same proposition. For (2), it follows from [11, corollary 6.7]. q.e.d.

Definition 2.3. Given a class L ∈ Pic(A)⊗^Z Q such that L^e is a line bundle, we call a metric on L whose e-th power is a metric on L^e determined by the above proposition a toric formal metric,and the corresponding function g (divided by e) the associated formal Green function. We denote byG_for(L) the set of all formal Green functions of L, and G₊(L) the set of all uniform limits of formal Green functions of L associated to semipositive toric formal metrics which we call semi- positive Green functions. It is easy to see that g ∈G₊(L) also satisfies (2.4), and the metric determined by (2.5) is semipositive in its original sense. Similarly, we denote by G_int(L) the set of difference of functions in G₊(L^′) and G₊(L^′′) with L = L^′⊗(L^′′)⁻¹ which we call integrable Green functions; hence the corresponding metric is integrable. More- over, the set of integrable Green functions for all classesL∈Pic(A)⊗^ZQ:

G_int(A) =SG_int(L) is a vector space overQ. All classes in Pic(A)⊗ZQ equipped with a metric corresponding to a Green function inG_int(A) are called toric metrized line bundles. At last we denote by G(L) the set

of continuous real functions satisfying (2.4), and we simply call them Green functions of L.

The following theorem provides a certain large class of semipositive Green functions for an ample line bundle.

Theorem 2.4. LetL∈Pic(A)⊗ZQbe an ample class andg∈G(L)∩ C²(R^d) such that the Hessian matrix (g_ij(x))i,j=1,...,d is semipositive definite everywhere; then g is inside G₊(L).

Remark 2.5. This theorem and its Corollary 2.11 are crucial for our main results since they provide a large class of functions which should be the Green functions of toric metrics on L. The proof is based on Lemmas 2.6 to 2.10, in which Lemma 2.9 will also be used in the proof of Theorem 3.3. But we will not single it out, since it appears naturally in a series of estimates, as we will see in the following proof.

Proof. The proof is divided into several steps.

Step 1. We reduce to the case where (g_ij(x))i,j=1,...,d is positive defi- nite for all x∈R^d.

First, there exists a function g_can ∈ G(L) ∩C²(R^d) such that the matrix ((gcan)ij(x))is positive definite. Indeed, by [8, lemma 6.5.2 (4)], there is a group homomorphismc: Λ→Qsuch thatz_λλλ(0) =q(λλλ)+c(λλλ).

We linearly extendc to R^d and defineg_can=q+c; then g_can∈G(L)∩ C²(R^d) and ((gcan)ij(x)) is a constant positive definite matrix. Next, for anyg∈G(L)∩C²(R^d),g_ij is Λ-periodic for any (i, j) andg−g_can is a Λ-periodic C²-function. For any gin the proposition, letf =g−g_can and gt = gcan+tf for t ∈ [0,1]; then gt → g1 = g when t → 1 and ((g_t)_ij(x)) are positive definite for all t <1. The claim follows.

Step 2. Now fix a function g as in the proposition, but with the condition that (g_ij(x))i,j=1,...,d is positive definite; we are going to con- struct a sequence of functions g_n∈G_for(L)∩G₊(L) approachingg. For any u ∈ S^d−1, the function ∇u∇ug is Λ-periodic and strictly positive;

hence there exist 0< hg < Hg such thathg <∇u∇ug(x)< Hg for any u∈S^d−1 andx∈R^d.

LetN be a sufficiently large integer; for j= (j₁, . . . , j_d)∈ _N¹Zd

, we let λλλ_j = j₁λλλ₁ +· · ·+j_dλλλ_d. For each j such that λλλ_j ∈ F, we choose a positive number ǫ_N(j) < _N¹2 such that g(λλλ_j)−ǫ_N(j) ∈ Q and a vector ǫǫǫ_N(j) such thatkǫǫǫ_N(j)k< _N¹,∇g(λλλj)−ǫǫǫ_N(j)∈Q^d and the graph of the function

g_λλλ^(N)

j (x) = (∇g(λλλj)−ǫǫǫ_N(j)) (x−λλλj) +g(λλλj)−ǫ_N(j)

is below the graph of g, which is possible sinceg is strictly convex. For generalj, we letj₀ be the unique element in _N¹Zd

such thatj−j₀ ∈Z^d andλλλj₀ ∈F. Then we define

g_λλ^(N)_λ

j (x) = (∇g(λλλ_j)−ǫǫǫ_N(j₀)) (x−λλλ_j) +g(λλλ_j)−ǫ_N(j₀).

By construction, g_λ^(N_λλ ⁾

j ∈C_rpoly(R). We need the following lemma.

Lemma 2.6. For each j∈ _N¹Zd

andλλλ∈Λ, we have g(x)−g_λλλ^(N)

j (x) =g(x+λλλ)−g_λλ^(N)_λ

j+λλλ(x+λλλ).

Proof. By definition, we have g_λλλ^(N)

j+λλλ(x+λλλ)−g_λλ^(N)_λ

j (x)

= (∇g(λλλj+λλλ)− ∇g(λλλj)) (x−λλλj) +g(λλλj+λλλ)−g(λλλj)

=∇z_λλλ(λλλ_j)(x−λλλ_j) +z_λλλ(λλλ_j)

=z_λλλ(x)

=g(x+λλλ)−g(x)

where the third equality is because z_λλλ is affine. q.e.d.

Step 3. We define a functiong^(N) by g^(N)(x) = sup

j

g_λλλ^(N)

j (x)

which is less than g(x). We have

Lemma 2.7. For any compact subset V ⊂ R^d, there exists a finite subset JV ⊂ _N¹Zd

such that

g^(N)(x) = max

j∈JV

g_λλ^(N)_λ

j (x)

for all x∈V.

Proof. We only need to prove that for given M ∈ R, there are only finitely many j such that g_λλλ^(N)

j (x) ≥ M for some x ∈ V. For a given j, we try to give a lower bound for the difference g(x)−g_λλλ^(N)

j (x). Let

u=x−λλλ_j; by definition, g(x)−g_λ^(N_λλ ⁾

j (x)

>

g(x)−g_λλ^(N)_λ

j (x)

−

g(λλλj)−g_λλλ^(N)

j (λλλj)

= Z 1

0

d dt

g(λλλ_j+tu)−g_λλλ^(N)

j (λλλ_j+tu) dt

= Z 1

0 ∇ug(λλλ_j+tu)dt− ∇ug(λλλ_j) +ǫǫǫ_N(j₀)·u

= Z 1

0

∇ug(λλλj) + Z t

0 ∇u∇ug(λλλj+su)ds

dt− ∇ug(λλλj) +ǫǫǫ_N(j₀)·u

= Z 1

0

Z t

0 ∇u∇ug(λλλj+su)dsdt+ǫǫǫ_N(j₀)·u

>hgkuk²

2 − kǫǫǫ_N(j₀)k · kuk

>h_gkuk²

2 −kuk N .

We see that there is N_M > 0 such that kλλλ_j −xk > N_M implies g_λ^(N_λλ ⁾

j (x)< M. Hence the lemma follows. q.e.d.

The above lemma implies that g^(N) ∈C_rpoly(R^d) and is convex. On the other hand, if g^(N)(x) = g_λλ^(N)_λ

j (x) for some j, then g^(N)(x+λλλ) = g_λ^(N_λλ ⁾

j+λλλ(x+λλλ) for allλλλ∈Λ since, by Lemma 2.6,g_λλ^(N)_λ

j′ (x+λλλ)> g_λλλ^(N)

j+λλλ(x+λλλ) will imply that g_λλ^(N)_λ

j′−λλλ(x)> g_λ^(N_λλ ⁾

j (x), which is a contradiction. Again by the same lemma, we conclude thatg^(N)satisfies (2.4) and hence is inside G(L).

Step 4. Before proving thatg^(N)is semipositive formal, we would like to bound the difference of it and g.

Lemma 2.8. For anyx∈R^d, we have

0≤g(x)−g^(N)(x)< R²_F·H_g+ 2RF+ 2

2N² .

Proof. The proof follows the same line as in Lemma 2.7. Hence we have

g(x)−g^(N)(x)< H_g

2 kλλλ_j−xk²+ 1

Nkλλλ_j−xk+ 1 N²

for any j. In fact, we can choosej such thatkλλλ_j−xk ≤ ^R_N^F. Hence the

lemma follows. q.e.d.

Conversely, we have the following lemma on the estimate of the gra- dient, which will be used also in the next section.

Lemma 2.9. Letf be any convex rational polytopal continuous func- tion on R^d. Suppose that |f(x)−g(x)| < ǫ for any x ∈ R^d; then for any d-dimensional polytope ∆on which f is affine, km_∆− ∇g(x₀)k ≤ 2p

ǫH_g for all x₀∈∆, where m_∆ is the gradient of f on ∆.

Proof. By continuity, we can assume that x₀ ∈ int(∆) and x₀ = 0.

We only need to prove the lemma for ˜f and ˜g where ˜f(x) = f(x)−

∇g(0)·(x)−g(0) and ˜g(x) =g(x)− ∇g(0)·(x)−g(0). In this case, we need to prove that km_∆k <2p

ǫH_g. For any u ∈S^d−1, we assume that m_∆·u≥0, since otherwise we can take−u. For t >0, consider

f˜(tu)−˜g(tu)

−

f˜(0)−˜g(0)

= Z t

0

∇uf˜(su)− ∇ug(su)˜ ds

≥m_∆·ut− Z t

0

Z s

0 ∇u∇ug(ru)drds˜

>m_∆·ut−H_gt² 2

where the first inequality is due to the assumption thatf is convex. On the other hand, it is less than 2ǫ; hence we have

Hgt²

2 −m_∆·ut+ 2ǫ >0 for all t >0. Hence m_∆·u<2p

ǫH_g and then km_∆k ≤2p

ǫH_g. q.e.d.

The above lemma immediately implies the following:

Lemma 2.10. Let f and ∆be as above; then for any x,x^′∈∆, the distance kx−x^′k ≤ _h⁴_gp

ǫH_g.

Proof. Let x^′−x= tu with u ∈S^d−1 and t= kx−x^′k ≥0. Since x,x^′ ∈∆ andf is affine on ∆, we have

h_gt≤

Z t

0 ∇u∇ug(x+su)ds

=∇ug(x^′)− ∇ug(x)≤4p ǫH_g by the above lemma. Hencet=kx−x^′k ≤ _h⁴_gp

ǫH_g. q.e.d.

In particular, if we apply the above lemma to g^(N), we see that ∆ is compact for any maximal connected subset ∆ on which g^(N) is affine.

Step 5. We would like to apply Lemma 1.1. Hence we need to show that ∆ is convex. By Lemma 2.7, g^(N) coincides with some g_λλλ^(N)

j re-

stricted to ∆. Suppose that there are x₀,x₁ ∈ ∆ and t ∈ (0,1) such thatxt=tx1+ (1−t)x0 6∈∆; then g^(N)(xt) =g_λλ^(N)_λ

j′ (xt) for somej^′ 6=j.

Again by construction, g_λλλ^(N)

j′ (x_t) > g_λλ^(N)_λ

j (x_t). Hence there is one point

x∈ {x₀,x₁}such thatg_λλλ^(N)

j′ (x)> g_λ^(N_λλ ⁾

j (x) which is a contradiction. Now by Lemma 1.1, g^(N) determines a Λ-periodic rational polytopal decom- positionC ofR^d.

Finally, we prove that for any ∆∈ C, ∆ maps bijectively to its image under the projection p :R^d → R^d/Λ when N is sufficiently large. We can assume dim(∆) = d. By Lemmas 2.8 and 2.10, we see that this holds if

4 h_g

s

R_F²·Hg+ 2RF+ 2

2N² ·H_g <2rF.

Now by Proposition 2.2, g^(N) is a semipositive formal Green function for large N. Hence the theorem follows by Lemma 2.8. q.e.d.

The above theorem has the following direct corollary.

Corollary 2.11. For any class L ∈ Pic(A) ⊗ZQ, if g ∈ G(L) ∩ C²(R^d), then g is integrable, i.e., g∈G_int(L).

3. A Limit Formula for the Measure

In this section, we prove a formula for the measure of metrics deter- mined by certain integrable Green functions.

3.1. Measures on torus and mixed Hessian.Recall that we have a closed manifold R^d/Λ. Similar to R^d as in 2.2, we define various spaces of real functions C^?(R^d/Λ), C^?

≥0(R^d/Λ), and C^?

>0(R^d/Λ) for

? = l; ∞; k, α. A measure on R^d/Λ is a continuous linear functional µ:C(R^d/Λ)→R; it issemipositive if µ(f)≥0 for all f ∈C_≥0(R^d/Λ);

positiveifµ(f)>0 for 06=f ∈C_≥0(R^d/Λ). The space of all (resp. semi- positive, positive) measures is denoted byM(R^d/Λ) (resp. M_≥0(R^d/Λ), M_>0(R^d/Λ)). It is endowed with the weak-∗ topology; i.e., a sequence µ_n → µ if and only if µ_n(f) → µ(f) for all f ∈C(R^d/Λ). Recall that we have the Lebesgue measure dx on R^d/Λ; hence the spaces of func- tionsC^?(R^d/Λ) can be identified as subspacesM^?(R^d/Λ) of the space of measures by integration against dxfor ? =l; ∞; k, α. Under this iden- tification, being semipositive (i.e.,non-negative) or positive (i.e., strictly positive) for a function coincides with that for a measure. Hence we also introduce the notation M^?

≥0(R^d/Λ) or M^?

>0(R^d/Λ) for ? = l; ∞; k, α for their obvious meaning. We write µ ≤ µ^′ if µ^′ −µ ∈ M_≥0(R^d/Λ).

Finally, we denote by |µ|the total mass ofµ, i.e., |µ|=µ(1). It is easy to see that if µis semipositive and |µ|= 0, thenµ= 0.

Definition 3.1. For functions g₁, . . . , g_d ∈ C²(R^d), we define the so-called mixed Hessianofg₁, . . . , g_d to be the function

Hessg1,...,gd(x) = X

µ,ν∈Sd

ǫ(µν) Yd i=1

(gi)_µ(i)ν(i)(x)

where ǫ(µ) = 1 (resp. −1) if µ is an even (resp. odd) permutation.

(The subscript i in g_i is not to be confused with the one for the i-th derivative of g.) In particular, when g₁ =· · ·=g_d=g,

Hess_g(x) := Hess_g,...,g(x) =d! det(g_ij(x))i,j=1,...,d

is d! times the determinant of the usual Hessian matrix of g.

The following lemma is an easy exercise in calculus.

Lemma 3.2. LetL∈Pic(A)⊗ZQbe an ample class andg₁, . . . , g_d∈ G(L)∩C³(R^d); then Hess_g1,...,gd is Λ-periodic and hence in C¹(R^d/Λ).

We have

Z

R^d/Λ

Hess_g1,...,gd(x)dx= deg_L(A).

Proof. First, we check the caseg₁ =· · ·=g_d=g_can. Then Z

R^d/Λ

Hess_gcan(x)dx= Z

R^d/Λ

H_qdx= covol(Λ)H_q

which equals deg_L(A) by Lemma 2.1. In general, since any two g, g^′ ∈ G(L) ∩C³(R^d) differ by a Λ-periodic function f ∈ C³(R^d), by mul- tilinearity and symmetry, we only need to prove that for such f and g2, . . . , g_d∈G(L)∩C³(R^d),

Z

R^d/Λ

Hess_f,g2,...,gd(x)dx= 0.

(3.1)

By a linear change of coordinates, one can assume that λλλ_i = e_i for i= 1, . . . , d. Hence (3.1) becomes

Z

[0,1]^d

Hess_f,g2,...,gd(x)dx₁· · ·dx_d= 0.

For each i= 1, . . . , d, let us denote by

F_i⁺={x= (x₁, . . . , x_d)∈[0,1]^d|x_i = 1} F_i⁻={x= (x1, . . . , x_d)∈[0,1]^d|xi = 0}.

ThenZ

[0,1]^d

Hess_f,g2,...,gd(x)dx1· · ·dx_d

= Z

[0,1]^d

X

µ,ν∈Sd

ǫ(µν)f_µ(1)ν(1) Yd i=2

(g_i)_µ(i)ν(i)dx₁· · ·dx_d

= X

µ,ν∈Sd

ǫ(µν) Z

F_µ(1)⁺

f_ν(1) Yd i=2

(g_i)_µ(i)ν(i)dy− Z

F_µ(1)⁻

f_ν(1) Yd i=2

(g_i)_µ(i)ν(i)dy−S_µ,ν

!

=− X

µ,ν∈Sd

ǫ(µν)Sµ,ν

sincef and (g_i)_µ(i)ν(i) are Λ-periodic, where S_µ,ν =

Z

[0,1]^d

f_ν(1) Xd j=2

(g_j)µ(1)µ(j)ν(j)

Yd i=2i6=j

(g_i)_µ(i)ν(i)dx₁· · ·dx_d. But then

X

µ,ν∈Sd

ǫ(µν)S_µ,ν

= Xd j=2

X

ν∈Sd

ǫ(ν) Z

[0,1]^d

f_ν(1)



X

µ∈S_d

ǫ(µ)(g_j)µ(1)µ(j)ν(j)

Yd i=2i6=j

(g_i)_µ(i)ν(i)



dx₁· · ·dx_d

=0

since in the inner summation, the terms withµandµ^′such thatµ^′(1) = µ(j),µ^′(j) =µ(1), andµ^′(i) =µ(i) fori6= 1, jcancel each other. Hence

the lemma follows. q.e.d.

3.2. Limit formula for the measure.Before we state the formula, we would like to introduce the notion of dual polytopes. The main reference for this is [13, §2, §3]. Let C be a (rational) polytopal de- composition of R^d and f be a (rational) polytopal function, strongly polytopal convex with respect to C. Then for any vertex v ∈ C, let star(v) be the set of all d-dimensional polytopes ∆ ∈ C containing v and for such ∆, let m_∆ be the gradient of f on ∆ (which is just the pegin [13]). Then thedual polytope ofvwith respect to f is defined to be the convex hull of points m_∆ for all ∆∈ star(v), which we denote

by bv^f and is denoted by ∆(v)^f in [11]. It is ad-dimensional (rational) polytope.

Now we are going to prove the following main theorem of this section.

Recall that we have the embedding i_A:R^d/Λ֒→A^an.

Theorem 3.3. Let L_i = (L_i,k ki) (i = 1, . . . , d) be d integrable metrized line bundles on A, where k ki are toric metrics determined by Green functions g_i ∈ G(L_i)∩C³(R^d) (cf. Corollary 2.11). Then we have the following equality of measures on A^an:

c1(L1)∧ · · · ∧c1(L_d) = (iA)∗Hessg1,...,gddx.

Proof. The reader may compare this formula with the one stated in [22, theorem 3.4.2], where the author considers the triple product of curves and measures with singularities.

The proof of the theorem is divided into several steps.

Step 1. First we reduce to the case L₁ = · · · = L_d = L = (L,k k) where L is an ample line bundle on A and k k is determined by a Green functiong∈G₊(L)∩C³(R^d) such that the matrix (gij)i,j=1,...,d is positive definite everywhere. Assuming the theorem holds for this case, consider the subsetG^′(A)⊂G_int(A) consisting of such Green functions.

Then g, g^′ ∈G^′(A) impliesag+bg^′ ∈G^′(A) for (a, b)∈Z²_≥0− {(0,0)}. For any continuous functionf onA^an, consider the functional

ℓ_f(g₁, . . . , g_d) = c₁(L₁)∧ · · · ∧c₁(L_d)−(i_A)_∗Hess_g1,...,gddx (f) which is symmetric andQ-multilinear ing₁, . . . , g_dand ℓ_f(g, . . . , g) = 0 for g ∈ G^′(A) by our assumption. Then for g₁, . . . , g_d ∈ G^′(A) and t₁, . . . , t_d∈Z^d_>0,

0 =ℓ_f Xd

i=1

tigi, . . . , Xd i=1

tigi

!

= X

k1,...,kd≥0 k1+···+kd=d

d!

k₁!· · ·k_d!ℓ_f(. . . , g_i, . . . , g_i, . . .)t^k₁¹· · ·t^k_d^d

whereg_iappearsd_itimes in the secondℓ_f. Henceℓ_f(. . . , g_i, . . . , g_i, . . .) = 0. In particular, ℓ_f(g₁, . . . , g_d) = 0 for allg_i ∈G^′(A). But on the other hand, G^′(A) generates the whole space G_int(A)∩C³(R^d) by definition and the existence and smoothness of g_can. Then ℓ_f(g₁, . . . , g_d) = 0 for all g_i∈G_int(A)∩C³(R^d). The theorem follows.

Step 2. Now fix g ∈ G^′(A) as above and assume g ∈ G₊(L) for an ample line bundle L. By Proposition 2.2, there are g_n ∈ G_for(L) such thatg_n→gand the corresponding formalk^◦-models (Xn,Ln) satisfying thatLfnis ample onXfn. Hence the models are in fact algebrizable. Now we viewXnas schemes projective and flat over Speck^◦and (Ln)_η ∼=L^en. We denote the corresponding metrized line bundle determined byg_n by

L_n = (L,k kn) and by g by L = (L,k k). By (2.1) and Proposition 1.2, the measure c₁(L_n)^∧d is supported on i_A(R^d/Λ) and hence also for their limit. If we let µ_n be the measure c₁(L_n)^∧d restricted on R^d/Λ and µ= lim_nµ_n, then we only need to prove that µ=µ_g as elements inM(R^d/Λ), whereµ_g is the measure Hess_gdx∈M³

>0(R^d/Λ).

We claim that for anyδ >0, we have µ≤(1 +δ)µ_g. (3.2)

Assuming this, then µ≤ µ_g. But by (2.2) and Lemma 3.2,|µ|=|µ_g|; i.e., |µ_g−µ|= 0. Hence µ=µ_g, which confirms the theorem.

Step 3. The last step is dedicated to proving the above claim (3.2).

We prove that for any δ >0 andf ∈C_≥0(R^d/Λ), µ(f)≤(1 +δ)µ_g(f).

Given ǫ₁ > 0, take L_n as above such that |g_n −g| < ǫ₁; then the modelXndetermines a rational polytopal decomposition (Cn)_ΛofR^d/Λ which comes from a Λ-periodic rational polytopal decomposition Cn of R^d. The functiong_nis rational polytopal and strongly polytopal convex with respect toCn. By Proposition 1.2, the set of irreducible components of Xfn is identified with the set of Λ-translation classes of vertices inCn. Hence if we denote Y_v the irreducible component corresponding to v, then Y_v =Y_v′ if and only if v =v^′ +λλλ for some λλλ ∈Λ. By (2.1), we have

µ_n= 1 e^d_n

X

v∈F

deg_L

n(Yv)δ_p(v) (3.3)

where we recall that F⊂R^d is the fixed fundamental domain of Λ and p:R^d→ R^d/Λ is the projection. Here, the multiplicities m_j appearing in (2.1) are all 1 since the special fibre is reduced by Proposition 1.2.

We recall a formula in [11, p. 366 (36)] which is deduced from [9, p.112 corollary], that

deg_Ln(Yv) =d!·vol(vb^engn)

wherevb^engn is the dual polytope defined before the theorem. Hence we have

(3.3) =d!X

v∈F

vol(vb^gn)δ_p(v). (3.4)

For any integer N > 0, we divide F into N^d blocks as follows. For (b1, . . . , b_d)∈ {0,1, . . . , N −1}^d, let

F^(N_b ⁾

1,...,bd =

x=x₁λλλ₁+· · ·+x_dλλλ_d| bi

N ≤x_i < bi+ 1 N

.