Toric Pluripotential Theory

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Toric Pluripotential Theory

Vincent Guedj, Ahmed Zeriahi, Dan Coman, Sibel Sahin

To cite this version:

Vincent Guedj, Ahmed Zeriahi, Dan Coman, Sibel Sahin. Toric Pluripotential Theory. 2018. �hal- 01761959�

DAN COMAN, VINCENT GUEDJ, SIBEL SAHIN, AHMED ZERIAHI

A tribute to Professor J´ozef SICIAK

Abstract. We study finite energy classes of quasiplurisubharmonic (qpsh) functions in the setting of toric compact K¨ahler manifolds. We characterize toric qpsh functions and give necessary and sufficient con- ditions for them to have finite (weighted) energy, both in terms of the associated convex function in Rⁿ, and through the integrability prop- erties of its Legendre transform. We characterize Log-Lipschitz con- vex functions on the Delzant polytope, showing that they correspond to toric qpsh functions which satisfy a certain exponential integrability condition. In the particular case of dimension one, those Log-Lipschitz convex functions of the polytope correspond to H¨older continuous toric quasisubharmonic functions.

Contents

Introduction 2

1. Finite energy classes 4

1.1. Bedford-Taylor theory 4

1.2. The class E(X, ω) 4

1.3. Weighted energy classes 5

2. Facts on convex functions 5

2.1. Subgradients and Monge-Amp`ere measures 5

2.2. Growth properties 8

3. Toric energy classes 10

3.1. Toric qpsh functions 10

3.2. The class Etor(X, ω) 12

3.3. The classes Eχ,tor(X, ω) 15

3.4. Finite moments 17

4. Higher regularity 19

4.1. Continuous toric functions 19

4.2. Log-Lipschitz Legendre transforms 19

References 24

Date: April 9, 2018.

Dan Coman is partially supported by the NSF Grant DMS-17000111.

Vincent Guedj and Ahmed Zeriahi are partially supported by the ANR project GRACK.

Sibel Sahin is supported by the TUBITAK 2219 postdoctoral grant.

1

Introduction

Atoriccompact K¨ahler manifold (X, ω, T) is an equivariant compactifica- tion of the torus T = (C^∗)ⁿ equipped with a (S¹)ⁿ-invariant K¨ahler metric ω. Then ω can be written as

ω=dd^cF0◦Lin (C^∗)ⁿ,

where F₀ :Rⁿ→Ris a smooth strictly convex function and (1) L: (C^⋆)ⁿ→Rⁿ, L(z1, . . . , zn) = (log|z1|, . . . ,log|zn|).

The celebrated Atiyah-Guillemin-Sternberg theorem asserts that the mo- ment map ∇F₀ :Rⁿ→ Rⁿ sends Rⁿ onto the interior of a compact convex polytope

P ={ℓi(s)≥0, 1≤i≤d} ⊂ Rⁿ,

where d≥n+ 1 is the number of (n−1)-dimensional faces ofP and ℓ_i(s) =hs, u_ii −λ_i,

with λi ∈R andui a primitive element of Zⁿ.

Delzant observed in [Del88] that in this case P is “Delzant”, i.e. there are exactly n faces of dimension (n−1) meeting at each vertex, and the corresponding u_i’s form a Z-basis of Zⁿ. He conversely showed that there is exactly one (up to symplectomorphism) toric compact K¨ahler manifold (XP,{ωP}, T) associated to a Delzant polytopeP ⊂Rⁿ. Here{ωP}denotes the cohomology class of theT-invariant K¨ahler form ω_P.

Let

G₀(s) := sup

x∈Rⁿ

(hx, si −F₀(x))

denote the Legendre transform of F₀. One has that G₀ = +∞ in Rⁿ\P and, for s∈intP =∇F₀(Rⁿ),

G₀(s) =hx, si −F₀(x)⇐⇒s∈ ∇F₀(x)⇐⇒x∈ ∇G₀(s).

Guillemin observed in [Gui94] that a “natural” representative of the co- homology class {ω_P}is given by

G(s) = 1 2

Xd

i=1

ℓ_i(s) logℓ_i(s).

We refer the reader to [CDG02] for a neat proof of this beautiful formula of Guillemin. Observe thatGis only Log-Lipschitz regular onP, although the original K¨ahler potential is smooth.

The purpose of this note is to undertake a systematic study of toric pluripotential analysis. There are three ways to understand a toric quasi- plurisubharmonic (qpsh) function and its Monge-Amp`ere measure:

• by working directly on X and imposing toric symmetries,

• by looking at the corresponding object (convex function, real Monge- Amp`ere measure) in Rⁿ after a logarithmic transformation, and un- derstanding the asymptotic properties at infinity,

• by understanding the behavior near the boundary of the polytope of the Legendre transform of the corresponding convex function.

We refer to Section 3 for the definition of toric ω-plurisubharmonic (ω- psh) functions on X and the corresponding energy classes. If ϕ is ω-psh, we denote by F_ϕ the corresponding convex function on Rⁿ and by G_ϕ its Legendre transform (see Sections 2 and 3).

Our main results are as follows. We first describe the class of toric ω-psh functions (see Proposition 3.2):

Proposition A. Let F_P(x) = max_s∈Phx, si denote the support function of the polytope P. The following are equivalent:

(i) ϕ∈P SH_tor(X, ω);

(ii) F_ϕ ≤F_P +C for some constant C;

(iii) G_ϕ = +∞ on Rⁿ\P;

(iv) ∇F_ϕ(Rⁿ)⊂P.

We then characterize finite energy toricω-psh functions and their weighted versions, showing in particular the following (see Theorem 3.6):

Theorem B. Let ϕ∈P SH_tor(X, ω). The following are equivalent:

(i) ϕ∈ Etor(X, ω);

(ii) G_ϕ is finite on intP;

(iii) Fϕ has full Monge-Amp`ere mass;

(iv) the Lelong numbers ν(ϕ, p) = 0 for all p∈X.

In Theorem 4.4 we study more regular toric ω-psh functions, characteriz- ing the maximal Log-Lipschitz regularity of Legendrian potentials:

Theorem C. Let ϕ∈ Etor(X, ω). The following properties are equivalent:

(i) There exists ε >0 such that exp(−εP SH_tor(X, ω))⊂L¹(M A(ϕ));

(ii) The function G_ϕ is Log-Lipschitz on P.

It is tempting to think that these conditions are all equivalent to the fact that ϕis H¨older continuous. This is easily seen to be the case when n= 1.

We refer the interested reader to [DDGHKZ14] for more information, geo- metric motivations, and related questions connecting the H¨older continuity of Monge-Amp`ere potentials to the integrability properties of the associated complex Monge-Amp`ere measure.

The paper is organized as follows. In Section 1 we recall some basic facts about ω-psh functions on any compact K¨ahler manifold (X, ω), together with the definition and main properties of various energy classes following [GZ07]. Section 2 deals with the relevant properties of convex functions and their Legendre transforms. In Section 3 we study energy classes of toric ω-psh functions on a toric compact K¨ahler manifold (X, ω), and in Section 4 we conclude by looking at questions about the higher regularity of such functions.

Acknowledgement. This article has been written during the postdoctoral research period of the third named author at l’Institut de Math´ematiques de Toulouse. She is grateful to her co-authors for their endless support and hospitality during the stay.

1. Finite energy classes

In this section we let (X, ω) be a compact K¨ahler manifold of dimensionn, and we recall the definition of finite energy classes of quasiplurisubharmonic (qpsh) functions following [GZ07].

1.1. Bedford-Taylor theory. A function on X is qpsh if it is locally the sum of a psh function and a smooth one. In particular qpsh functions are upper semicontinuous and integrable.

Definition 1.1. A function ϕ : X → R∪ {−∞} is ω-plurisubharmonic (ω-psh) if it is qpsh and if the currentω+dd^cϕis positive on X.

Let P SH(X, ω) denote the set of all ω-psh functions on X. This is a closed subset ofL¹(X, ωⁿ).

Bedford and Taylor showed in [BT82] that one can define the complex Monge-Amp`ere operator

M A(ϕ) := (ω+dd^cϕ)ⁿ= (ω+dd^cϕ)∧. . .∧(ω+dd^cϕ)

for all bounded ω-psh functions. They showed that whenever (ϕj) is a se- quence of boundedω-psh functions decreasing locally toϕ, the sequence of measuresM A(ϕ_j) converges weakly towards the measureM A(ϕ). Note also

that Z

X

M A(ϕ) = Z

X

ωⁿ=:Vω.

At the heart of Bedford-Taylor’s theory lies the following maximum prin- ciple: if u, v are bounded ω-psh functions, then

(M P) 1_{v<u}M A(max(u, v)) = 1_{v<u}M A(u).

The maximum principle (M P) implies the so calledcomparison principle:

if u, v are boundedω-psh functions then Z

{v<u}

M A(u)≤ Z

{v<u}

M A(v).

1.2. The class E(X, ω). Ifϕ∈P SH(X, ω), we let

ϕ_j := max(ϕ,−j)∈P SH(X, ω)∩L^∞(X).

It follows from the Bedford-Taylor theory that the measures M A(ϕ_j) are well defined measures of total massV_ω. The following monotonicity property holds:

µj :=1_{ϕ>−j}M A(ϕj) is an increasing sequence of Borel measures.

The proof is an elementary consequence of (M P) (see [GZ07, p.445]). Since µ_j have total mass bounded above by V_ω, we can define

µ_ϕ := lim

j→+∞µ_j,

which is a positive Borel measure on X of total mass ≤V_ω. Definition 1.2. We let

E(X, ω) :={ϕ∈P SH(X, ω) : µ_ϕ(X) =V_ω}. For ϕ∈ E(X, ω), we setM A(ϕ) :=µ_ϕ.

The definition is justified by the following important fact proved in [GZ07]:

the complex Monge-Amp`ere operator ϕ7→M A(ϕ)is well defined on the class E(X, ω), in the sense that ifϕ∈ E(X, ω) then for every decreasing sequence of bounded ω-psh functions ϕ_j ցϕ, the measures M A(ϕ_j) converge weakly on X towards µ_ϕ.

Every bounded ω-psh function clearly belongs to E(X, ω). The class E(X, ω) also contains many ω-psh functions which are unbounded. When X is a compact Riemann surface,E(X, ω) is the set ofω-sh functions whose Laplacian does not charge polar sets.

Remark 1.3. If ϕ ∈ P SH(X, ω) is normalized so that ϕ ≤ −1, then

−(−ϕ)^ε belongs to E(X, ω) whenever 0 ≤ ε < 1 (see e.g. [CGZ08]). The functions which belong to the class E(X, ω), although usually unbounded, have relatively mild singularities. In particular they have zero Lelong num- ber at every point.

It is shown in [GZ07] that the maximum principle (M P) and the com- parison principle continue to hold in the class E(X, ω). The latter can be characterized as the largest class for which the complex Monge-Amp`ere op- erator is well defined and the maximum principle holds.

1.3. Weighted energy classes. Let W denote the set of all functions χ: R⁻→R⁻ such that χis increasing and χ(−∞) =−∞.

Definition 1.4. We let Eχ(X, ω) be the set of ω-psh functions with finite χ-energy,

Eχ(X, ω) :=

ϕ∈ E(X, ω) : χ(−|ϕ|)∈L¹(X, M A(ϕ)) . When χ(t) =−(−t)^p,p >0, we set E^p(X, ω) =Eχ(X, ω).

We list here a few important properties of these classes and refer the reader to [GZ07, BEGZ10] for the proofs:

• E(X, ω) =S

χ∈W Eχ(X, ω);

• P SH(X, ω)∩L^∞(X) =T

χ∈WE^χ(X, ω);

• the classes E^p(X, ω) are convex;

• ϕ ∈ E^p(X, ω) if and only if for any (resp. for one) sequence of boundedω-psh functionsϕ_j ցϕ, sup_jR

X|ϕ_j|^pM A(ϕ_j)<+∞.

• if ϕ_j is a sequence of ω-psh functions decreasing to ϕ ∈ E^p(X, ω), then the measures |ϕ_j|^pM A(ϕ_j) converge weakly to|ϕ|^pM A(ϕ).

2. Facts on convex functions

We collect here a few properties of convex functions which will be used later. Some of these are well known and proofs are included for the conve- nience of the reader (see also [BeBe13, Section 2]).

2.1. Subgradients and Monge-Amp`ere measures. LetF :Rⁿ→Rbe a convex function. The subgradient of F at x is the set

∇F(x) ={s∈Rⁿ: F(y)≥F(x) +hy−x, si, ∀y∈Rⁿ}. We let

∇F(Rⁿ) := [

x∈Rⁿ

∇F(x).

The Legendre transformGofF is the lower semicontinuous convex func- tion defined by

G:Rⁿ→(−∞,+∞], G(s) = sup

x∈Rⁿ

(hx, si −F(x)). Then F is the Legendre transform of G,

F(x) = sup

s∈Rⁿ

(hx, si −G(s)), and one has

G(s) =hx, si −F(x)⇐⇒s∈ ∇F(x)⇐⇒x∈ ∇G(s). Lemma 2.1. Let F :Rⁿ→R be a convex function.

(i) If F is smooth and strictly convex then ∇F : Rⁿ → Rⁿ is injective, and hence an open map.

(ii) If s0 ∈ ∇F(Rⁿ) then G(s0)<+∞. Conversely, if G(s)<+∞for all s in an open ball B(s₀, r) then s₀∈ ∇F(Rⁿ).

(iii) LetFj :Rⁿ→R,j≥1, be convex functions. ThenFj ց F pointwise on Rⁿ if and only if the Legendre transforms G_j րG pointwise on Rⁿ. Proof. (i) If p 6=q and f(t) := F((1−t)p+tq) then f^′′(t) >0, sof^′(0) = h∇F(p), q−pi< f^′(1) =h∇F(q), q−pi. Hence∇F(p)6=∇F(q).

(ii) By the definition of the subgradient, if s₀ ∈ ∇F(x) then hy, s₀i − F(y) ≤ hx, s₀i −F(x) for all y ∈ Rⁿ, so G(s₀) = hx, s₀i −F(x) < +∞. Conversely, by shrinkingrwe may assume thatG < M onB(s₀, r) for some constant M, hence hx, si −F(x) ≤ M for all x ∈ Rⁿ and s ∈ B(s₀, r).

Let Fe(x) = F(x)− hx, s₀i+M. It follows that Fe(x) ≥ hx, s−s₀i for all s∈ B(s₀, r), hence Fe(x) ≥rkxk. ThereforeFe assumes a global minimum, i.e. there exists x₀ ∈ Rⁿ such that Fe(x) ≥ Fe(x₀). Thus 0 ∈ ∇Fe(x₀) =

∇F(x₀)−s₀. (Note that ifF(x) =e^x,x∈R, then G(0) = 0 but 06∈F^′(R), so the hypothesis that G(s)<+∞in a neighborhood of s₀ is needed.)

(iii) Assume thatFj ց F. ThenGj րG, wheree Ge is lower semicontin- uous, convex and Ge ≤G. IfFe is the Legendre transform ofGe we have that F_j ≥Fe≥F. We conclude thatFe=F and so Ge=G. The converse follows

by a similar argument.

Lemma 2.2. Let F : Rⁿ → R be a convex function. If χ is a continuous function with compact support on Rⁿ then

Z

(C^⋆)ⁿ

(χ◦L) (dd^cF◦L)ⁿ= Z

Rⁿ

χ M AR(F),

where L is defined in (1), d=∂+∂, d^c = _2πi¹ (∂−∂), and M AR(F) is the real Monge-Amp`ere measure of F.

Proof. Approximating F by a decreasing sequence of smooth convex func- tions it suffices to assume thatF is smooth. Recall that in this caseM AR(F) is the measure defined by

M AR(F) =n! det

∂²F

∂x_i∂x_j

dV ,

where V denotes the Lebesgue measure on the corresponding Euclidean space. Note that the function F◦L is psh on (C^⋆)ⁿ and

∂²(F◦L)

∂z_i∂z_j = 1 4z_iz_j

∂²F

∂x_i∂x_j ◦L

, hence

det

∂²(F◦L)

∂z_i∂z_j

= 1

4ⁿQ

j|z_j|²

det

∂²F

∂x_i∂x_j

◦L

. It follows that

(dd^cF ◦L)ⁿ= i

π n

∂∂F ◦Ln

=n!

i π

n

det

∂²(F◦L)

∂z_i∂z_j

dz₁∧dz₁∧. . .∧dz_n∧dz_n

=n!

2 π

n

det

∂²(F◦L)

∂z_i∂z_j

dV(z)

=n!

2 π

n

1 4ⁿQ

j|z_j|²

det

∂²F

∂x_i∂x_j

◦L

dV(z)

= n!

(2π)ⁿ det ∂²F

∂xi∂xj

(logr₁, . . . ,logr_n)

dr₁. . . dr_n r1. . . rn

dθ₁. . . dθ_n, where we used polar coordinatesz_j =r_je^iθj. Changing variablesx_j := logr_j we obtain

Z

(C^⋆)ⁿ

(χ◦L) (dd^cF ◦L)ⁿ=

=n!R

(0,+∞)ⁿ χdeth

∂²F

∂xi∂xj

i (logr₁, . . . ,logr_n)^dr_r^1...drn

1...rn

=n!R

Rⁿχ(x) deth

∂²F

∂xi∂xj(x)i

dV(x) =R

Rⁿχ M AR(F).

For a non-smooth convex functionF the positive measureM AR(F) is the real Monge-Amp`ere measure ofF (in the sense of Alexandrov [Gut01]).

The following lemma is proved using an idea of Al Taylor [T82].

Lemma 2.3. If F₁, F₂ : Rⁿ → R are convex functions such that F₂(x) → +∞ as kxk →+∞ andF₁(x)≤F₂(x) for all x∈Rⁿ, then

Z

Rⁿ

M AR(F₁)≤ Z

Rⁿ

M AR(F₂).

Proof. Fix a compact K ⊂ (C^⋆)ⁿ, a number ε > 0, and consider the psh function on (C^⋆)ⁿ,

u:= max{F₁◦L,(1 +ε)F₂◦L−C},

where the constantC >0 is chosen such thatu=F1◦Lin a neighborhood of K. SinceF₂(x)→+∞askxk →+∞it follows thatF₁≤F₂ ≤(1+ε)F₂−C on Rⁿ\ Kfor some compactK ⊂Rⁿ. ThenL⁻¹(K)⊂(C^⋆)ⁿis compact and u= (1 +ε)F₂◦L−C on (C^⋆)ⁿ\L⁻¹(K). We infer that

(1 +ε)ⁿ Z

(C^⋆)ⁿ

(dd^cF₂◦L)ⁿ= Z

(C^⋆)ⁿ

(dd^cu)ⁿ≥ Z

K

(dd^cF₁◦L)ⁿ.

The lemma follows by using Lemma 2.2 and by letting K ր (C^⋆)ⁿ and

εց0.

2.2. Growth properties. Let P be a (compact) convex body in Rⁿ. Its support function, which is also known as theindicator function, is the convex function

F_P(x) := max

s∈Phx, si. Its Legendre transform is the convex function

G_P(x) =

0, if x∈P, +∞, if x6∈P.

IfP_ϑ=θ+P is the image ofP under the translation byϑ, andP_λ =λP is the image of P under the dilation byλ >0, then

F_Pϑ(x) =F_P(x)+< ϑ, x > , G_Pϑ(s) =G_P(s−ϑ), F_Pλ(x) =F_P(λx) =λF_P(x), G_Pλ(s) =G_P s

λ .

Lemma 2.4. Let F :Rⁿ→Rbe a convex function with Legendre transform G. The following are equivalent:

(i) F ≤F_P +C for some constant C;

(ii) G= +∞ on Rⁿ\P;

(iii) ∇F(Rⁿ)⊂P.

Proof. To show that (i) ⇒ (ii), if F ≤ F_P +C then G ≥ G_P −C, so G=G_P = +∞onRⁿ\P. For (ii)⇒(iii), ifs∈ ∇F(Rⁿ) thenG(s)<+∞, hence s∈P by (ii).

To prove that (iii) ⇒ (i), let x ∈ Rⁿ and note that if s ∈ ∇F(Rⁿ) then s ∈ P, so hs, xi ≤ F_P(x). Since F is locally Lipschitz along the line t∈R→txwe have

F(x)−F(0) = Z 1

0

d

dtF(tx)dt= Z 1

0 h∇F(tx), xidt≤ Z 1

0

FP(x)dt =FP(x).

Lemma 2.5. Let F₀ : Rⁿ → R be a smooth strictly convex function such that F_P −C ≤F₀ ≤F_P +C for some constant C. Then∇F₀ :Rⁿ→intP is bijective and ∇G0 : intP → Rⁿ is its inverse, where G0 is the Legendre transform of F₀. Moreover, if χ is a continuous function with compact support on Rⁿ then

Z

Rⁿ

χ M AR(F₀) =n!

Z

intP

χ◦ ∇G₀ dV , and Z

Rⁿ

M AR(F₀) =n! vol(P). Proof. By Lemma 2.4 and Lemma 2.1 (i), ∇F₀ : Rⁿ → P is injective.

As F_P −C ≤ F₀ we have that G₀ ≤ G_P +C, so G₀ ≤ C on P. Thus intP ⊂ ∇F₀(Rⁿ) by Lemma 2.1 (ii), and hence∇F₀(Rⁿ) = intP since∇F₀ is open. If x, x^′ ∈ ∇G0(s) then s = ∇F0(x) = ∇F0(x^′), so x = x^′. Hence G₀ is differentiable on intP and∇G₀= (∇F₀)⁻¹. The remaining assertions of the lemma follow by the change of variables x=∇G₀(s),s=∇F₀(x), so

dV(s) = det

∂²F0

∂x_i∂x_j

dV(x) = 1

n!M AR(F0)(x).

Lemma 2.6. If 0∈intP then there exist constants a, b >0 such that bkxk ≤F_P(x)≤akxk, ∀x∈Rⁿ.

Proof. Ifa, b >0 are such that the closed ballsB(0, b)⊂P ⊂B(0, a), then bkxk=F_B(0,b) ≤F_P(x)≤F_B(0,a) =akxk.

Lemma 2.7. Assume that 0 ∈ intP and let F : Rⁿ → R be a convex function with Legendre transformG, such thatF ≤F_P+Cfor some constant C. The following are equivalent:

(i) G(s)<+∞ for alls∈intP;

(ii) for every ε∈(0,1) there is M_ε>0 s.t. F ≥(1−ε)F_P −M_ε on Rⁿ. Moreover, these conditions imply that R

Rⁿ M AR(F) =n! vol(P).

Proof. Note that

F_(1−ε)P(x) = sup

s∈Phx,(1−ε)si= (1−ε)F_P(x).

Assume that G(s) < +∞ for all s ∈intP. Since 0 ∈ intP, (1−ε)P ⊂ intP for ε∈(0,1), so there exists M_ε >0 such that G≤M_ε on (1−ε)P. It follows that

F(x)≥ sup

s∈(1−ε)P hx, si −G(s)

≥F_(1−ε)P(x)−Mε = (1−ε)FP(x)−Mε. Conversely, ifF ≥(1−ε)F_P−M_ε=F_(1−ε)P−M_ε, then G≤G_(1−ε)P+M_ε, so G(s)≤M_ε fors∈(1−ε)P. As εց0 this implies thatG(s) <+∞ for all s∈intP.

By Lemma 2.6 we have that F_P(x) → +∞ as kxk → +∞. Since F ≤ F_P +C, Lemmas 2.3 and 2.5 imply that

Z

Rⁿ

M AR(F)≤ Z

Rⁿ

M AR(F_P) = Z

Rⁿ

M AR(F₀) =n! vol(P), where F₀ is a function as in Lemma 2.5. Note that by (ii), F(x)→+∞ as kxk →+∞, hence Lemma 2.3 again shows that

Z

Rⁿ

M AR(F)≥(1−ε)ⁿ Z

Rⁿ

M AR(F_P), ∀ε∈(0,1).

Letting ε→0 finishes the proof.

We conclude this section with the following lemma:

Lemma 2.8. Let P be a compact convex body inRⁿ with nonempty interior and G:P →R∪ {+∞}be a lower semicontinuous convex function. Then

inf

P G

≤ 1 2^1/(n+1)−1

vol(P) Z

P |G|dV . Proof. If G ≥ 0 on P then R

PG dV ≥ inf_PG·vol(P) and we are done.

Otherwise, consider the convex set S = {G <0} ⊂ P. It suffices to show that if p∈intS then

−G(p)≤ 1 2^1/(n+1)−1

vol(P) Z

P|G|dV .

We assume without loss of generality that p = 0 and use spherical coor- dinates. For θ ∈ Sⁿ⁻¹ let 0 < a(θ) ≤ b(θ) be defined by a(θ)θ ∈ ∂S, b(θ)θ∈∂P. Ifσ is the area measure onSⁿ⁻¹ we have that

vol(P) = Z

Sⁿ⁻¹

b(θ)ⁿ

n dσ(θ). By convexity it follows that

G(tθ)≤ −G(0)

a(θ) (t−a(θ))≤0, if 0≤t≤a(θ), G(tθ)≥ −G(0)

a(θ) (t−a(θ))≥0, if a(θ)< t≤b(θ). ThusZ

P|G|dV ≥ Z

Sⁿ⁻¹

Z _a(θ)

0

−G(0)

a(θ) (a(θ)−t)tⁿ⁻¹dt dσ(θ) + Z

Sⁿ⁻¹

Z _b(θ)

a(θ)

−G(0)

a(θ) (t−a(θ))tⁿ⁻¹dt dσ(θ)

= −G(0) n(n+ 1)

Z

Sⁿ⁻¹

nb(θ)ⁿ⁺¹

a(θ) −(n+ 1)b(θ)ⁿ+ 2a(θ)ⁿ

dσ(θ). Note that

f(a) := nbⁿ⁺¹

a −(n+ 1)bⁿ+ 2aⁿ≥f b2⁻ⁿ⁺¹¹

= (n+ 1) 2ⁿ⁺¹¹ −1 bⁿ, for 0< a≤b.Therefore

Z

P|G|dV ≥ −G(0)

n 2ⁿ⁺¹¹ −1 Z

Sⁿ⁻¹

b(θ)ⁿdσ(θ)

=−G(0) 2^1/(n+1)−1

vol(P),

and we are done.

3. Toric energy classes

Let (X, ω) be a toric compact K¨ahler manifold of dimension n. Then X is a compactification of the complex torus (C^⋆)ⁿ such that the canonical action by multiplication of (C^⋆)ⁿ on itself extends to a holomorphic action of (C^⋆)ⁿ on X. Moreover, there exists a smooth strictly convex function F0 :Rⁿ →Rsuch that ω |(C^⋆)ⁿ=dd^cF0◦L, where L is defined in (1). If P is the compact convex polytope determined by X then∇F₀:Rⁿ→intP is bijective and we may assume that 0 ∈ intP. Let G₀ denote the Legendre transform of F₀.

3.1. Toric qpsh functions. A toricω-psh function on Xis anω-psh func- tion ϕthat is invariant under the (S¹)ⁿ action induced by the (C^⋆)ⁿ action on X. We denote by P SH_tor(X, ω) the class of such functions. It follows that there exists a convex function F_ϕ:Rⁿ→R such that

F_ϕ◦L=F₀◦L+ϕ on (C^⋆)ⁿ⊂X .

We denote byGϕthe Legendre transform onFϕ. Note thatFϕ is continuous on Rⁿ, henceϕis continuous on (C^⋆)ⁿ.

We define the energy classes of toric ω-psh functions by Etor(X, ω) =P SH_tor(X, ω)∩ E(X, ω), Etor^p (X, ω) =P SH_tor(X, ω)∩ E^p(X, ω), Eχ,tor(X, ω) =P SH_tor(X, ω)∩ Eχ(X, ω), where p >0 andχ∈ W (see sections 1.2, 1.3).

We begin with the following simple lemma:

Lemma 3.1. There exists a constant C >0 such that

−C≤F₀(x)−F_P(x)≤C , ∀x∈Rⁿ.

Proof. Since ∇F₀(Rⁿ) ⊂ P we have by Lemma 2.4 that F₀ ≤ F_P +C₁ for some constant C₁. Let ω^′ ∈ {ω} be a K¨ahler form with associated convex functionF such that its Legendre transformGis given by Guillemin’s formula. ThenF◦L=F₀◦L+θfor some smooth ω-psh functionθ. Hence F ≤F0+C2 and G≥G0−C2, for some constant C2. Since Gis bounded above on P it follows that G₀ ≤G_P +C₃, and so F₀ ≥F_P −C₃, for some

constant C₃.

Our next result gives a characterization of toric ω-psh functions:

Proposition 3.2. The following are equivalent:

(i) ϕ∈P SH_tor(X, ω);

(ii) F_ϕ ≤F_P +C for some constant C;

(iii) G_ϕ = +∞ on Rⁿ\P;

(iv) ∇Fϕ(Rⁿ)⊂P.

Proof. If ϕ ∈ P SH_tor(X, ω) then ϕ is bounded above on X, hence F_ϕ ≤ F₀+C^′ for some constant C^′, and (ii) follows by Lemma 3.1.

Conversely, if (ii) holds then by Lemma 3.1, F_ϕ ≤ F₀ +C^′ for some constant C^′, henceϕ≤C^′ on (C^⋆)ⁿ⊂X. SinceX\(C^⋆)ⁿis an analytic set invariant under the (S¹)ⁿ action, we conclude that ϕ extends to an ω-psh function on X which is (S¹)ⁿ invariant.

The remaining equivalences (ii) ⇔ (iii) ⇔ (iv) follow from Lemma 2.4.

Proposition 3.3. If ϕ∈P SH_tor(X, ω) then

sup

X

ϕ≤C_P + 1

2^1/(n+1)−1 vol(P)

Z

P|G_ϕ|dV , where CP = sup_P G0 = supRⁿ(FP −F0).

Proof. Note that, for a constant C, one hasF_ϕ−F₀ ≤C on Rⁿif and only if G0−Gϕ ≤C on P. It follows that

sup

X

ϕ= sup

Rⁿ

(F_ϕ−F₀) = sup

P

(G₀−G_ϕ)≤C_P −inf

P G_ϕ,

and the proposition follows from Lemma 2.8.

Example 3.4. Let X = F₁ be the blow up of P² at a toric point p. It is a geometrically ruled surface. We let F denote a generic fiber, E be the exceptional divisor, and H =E+F the total transform of a line through p. The cohomology classes ofF andH are both semi-positive and generate

H^1,1(X,R). Any K¨ahler class{ω}is cohomologous toaH+bF, witha, b >0.

In coordinates z∈(C^∗)² it can be represented by ω =aω₁+bω₂, whereω₁ = 1

2dd^clog(1 +kzk²), ω₂=dd^clogkzk. The convex function associated to ω is

F0(x) = a

2 log 1 +e^2x1 +e^2x2 +b

2 log e^2x1+e^2x2 , and P =∇F₀(R²) is the polytope

P ={s₁ ≥0, s₂≥0, b≤s₁+s₂ ≤a+b}.

Thus d = 4, ℓ₁(s) = s₁, ℓ₂(s) = s₂, ℓ₃(s) = a+b−s₁−s₂, and ℓ₄(s) = s1+s2−b. Fors∈P, the Legendre transform ofF0 is given by

G₀(s) = 1 2

s₁logs₁+s₂logs₂+ (a+b−s₁−s₂) log(a+b−s₁−s₂)+

(s₁+s₂−b) log(s₁+s₂−b)−(s₁+s₂) log(s₁+s₂)−aloga . 3.2. The class Etor(X, ω).

Definition 3.5. LetF :Rⁿ→Rbe a convex function such thatF ≤F_P+C.

We say that F hasfull Monge-Amp`ere mass if Z

Rⁿ

M AR(F) = Z

Rⁿ

M AR(F₀) =n! vol(P).

Recall that atoric pointofXis a point fixed by the action of the complex torus (C^⋆)ⁿ on X.

Theorem 3.6. Let ϕ∈P SH_tor(X, ω). The following are equivalent:

(i) ϕ∈ Etor(X, ω);

(ii) G_ϕ is finite on intP;

(iii) F_ϕ has full Monge-Amp`ere mass;

(iv) for every ε >0 there exists a compact setK_ε ⊂(C^⋆)ⁿ such that ϕ(z)≥ −εmaxlog|z₁|

, . . . ,

log|z_n|

on (C^⋆)ⁿ\K_ε. (v) the Lelong numbers ν(ϕ, p) = 0 for all p∈X.

(vi) the Lelong numbers ν(ϕ, p) = 0 at all toric points p∈X.

Proof. To prove that (i) ⇒ (ii), if ϕ∈ Etor(X, ω) then ϕ∈ Eχ,tor(X, ω) for some functionχ∈ W. By Proposition 3.9 following this proof,G_ϕ ∈L_χ(P), soG_ϕ <+∞ a.e. onP. Since G_ϕ is convex, this implies thatG_ϕ(s)<+∞ for all s∈intP. The implication (ii)⇒(iii) follows from Lemma 2.7.

We next prove that (iii) ⇒ (i). Consider the measure hωⁿ_ϕi defined as the non-pluripolar product of the positive closed currents ω_ϕ := ω +dd^cϕ [BEGZ10, Definition 1.1]. As ϕ is locally bounded on (C∗)ⁿ, the Bedford- Taylor product ω_ϕⁿ =ω_ϕ∧. . .∧ω_ϕ is well defined on (C^∗)ⁿ [BT76, BT82].

Since (C∗)ⁿ = X\A, where A is an analytic subset of X, it follows from [BEGZ10, p. 204, Proposition 1.6] that hω_ϕⁿi is the trivial extension ofω_ϕⁿto X. Then

Z

Xhωⁿ_ϕi= Z

(C^∗)ⁿ

ω_ϕⁿ = Z

(C^∗)ⁿ

(dd^cFϕ◦L)ⁿ

= Z

Rⁿ

M AR(F_ϕ) = Z

Rⁿ

M AR(F₀) = Z

X

ωⁿ.

Therefore hωⁿ_ϕi has full mass, so ϕ ∈ Etor(X, ω) and M A(ϕ) = hω_ϕⁿi [BEGZ10, Sect. 2].

To show that (ii)⇒(iv), let ε >0. Using Lemmas 2.7 and 3.1 we get ϕ= (F_ϕ−F₀)◦L≥ −εF_P ◦L−C−M_ε,

with some constantsC, Mε>0. By Lemma 2.6 there exists a constanta >0 such that F_P(x)≤amax{|x₁|, . . . ,|x_n|}. These imply that

ϕ(z)≥ −2aεmaxlog|z₁| , . . . ,

log|z_n| forz∈(C^⋆)ⁿ\K_ε, whereK_ε={εF_P ◦L≤C+M_ε}.

Conversely, to prove that (iv) ⇒ (ii), we let ε∈ (0,1) and by applying Lemma 2.7 we need to show that there exists Mε > 0 such that Fϕ ≥ (1−ε)F_P −M_ε. By Lemma 2.6 we haveF_P(x) ≥bmax{|x₁|, . . . ,|x_n|} for some constant b >0. Using (iv) and Lemma 3.1 we obtain that

F_ϕ(L(z)) =F₀(L(z)) +ϕ(z)

≥F_P(L(z))−C−bεmaxlog|z₁|, . . . ,log|z_n|

≥(1−ε)F_P(L(z))−C ,

for z ∈ (C^⋆)ⁿ\K_ε, where K_ε ⊂ (C^⋆)ⁿ is a compact set. Since F_ϕ, F_P are continuous this implies thatF_ϕ(L(z))≥(1−ε)F_P(L(z))−M_ε on (C^⋆)ⁿ, for some constant M_ε > C.

Recall that functions in the class E(X, ω) have zero Lelong number at each point. To complete the proof we assume that ϕ ∈ P SHtor(X, ω) has zero Lelong number at all the toric points ofX and show that (iv) holds.

Letε >0 and letp1, . . . , pN be the toric points ofX. We denote as before by z = (z₁, . . . , z_n) the coordinates on the complex torus (C^⋆)ⁿ ⊂X. For 1 ≤ j ≤ N, there exists an open set p_j ∈ V_j ⊂ X and a biholomorphic map Φ_j :V_j → Cⁿ such that Φ_j(p_j) = 0, (C^⋆)ⁿ ⊂ V_j, Φ_j((C^⋆)ⁿ) = (C^⋆)ⁿ, and X = V₁ ∪. . .∪V_N (see e.g. [ALZ16, Proposition 4.4] and its proof).

Moreover, if Φ_j(z) =ζ = (ζ₁, . . . , ζ_n) then

(log|ζ₁|, . . . ,log|ζ_n|) =A_j(log|z₁|, . . . ,log|z_n|), where A_j ∈GL_n(Z). We denote bykA_jk∞ the operator norm ofA_j with respect to the sup norm (i.e. kAjxk∞≤ kAjk∞kxk∞) and letγ := max{kA1k∞, . . . ,kANk∞}. Since X is compact we can findR >0 such that X=SN

j=1Φ⁻_j¹ ∆ⁿ(0, R)

, where

∆ⁿ(0, R)⊂Cⁿ is the open polydisc of radius R centered at 0.

We have that Φ⁻_j¹⋆

ω |(C^⋆)ⁿ

= dd^cF₀^j ◦L, where L(ζ₁, . . . , ζ_n) = (log|ζ₁|, . . . ,log|ζ_n|) and F₀^j :Rⁿ→R is a smooth strictly convex function such that F₀^j ◦L extends to a smooth psh function on Cⁿ. It follows that

∇F₀^j(Rⁿ)⊂(0,+∞)ⁿ. Moreover, there exists a convex functionFϕ^j :Rⁿ→ R such that∇Fϕ^j(Rⁿ)⊂[0,+∞)ⁿ and Fϕ^j◦L=F₀^j◦L+ϕ◦Φ⁻_j¹ on (C^⋆)ⁿ. The functionF_ϕ^j◦Lextends to a psh function onCⁿand has Lelong number ν(F_ϕ^j ◦L,0) = 0, since Lelong numbers are invariant under biholomorphic maps. Hence there exists r_j = r_j(ε) > 0 such that Fϕ^j(logr, . . . ,logr) ≥

ε

2 logr for 0< r < r_j. SinceF_ϕ^j is increasing in each variables this implies F_ϕ^j(log|ζ₁|, . . . ,log|ζ_n|)≥ ε

2 min{log|ζ₁|, . . . ,log|ζ_n|},

if min{|ζ₁|, . . . ,|ζ_n|}< r_j.So there exists a constantM_ε>0 such that ϕ◦Φ⁻_j¹(ζ) =F_ϕ^j◦L(ζ)−F₀^j◦L(ζ)≥ ε

2 min{log|ζ₁|, . . . ,log|ζ_n|} −M_ε, for ζ∈∆ⁿ(0, R) with min{|ζ₁|, . . . ,|ζ_n|}< r_j. By shrinkingr_j we obtain ϕ◦Φ⁻_j¹(ζ)≥εmin{log|ζ₁|, . . . ,log|ζ_n|}=−εmaxlog|ζ₁|

, . . . ,

log|ζ_n| , for ζ∈∆ⁿ(0, R) with min{|ζ₁|, . . . ,|ζ_n|}< r_j. It follows that

ϕ(z)≥ −γεmaxlog|z1| , . . . ,

log|zn| ,

ifz∈U_j := (C^⋆)ⁿ∩Φ⁻_j¹ ∆ⁿ(0, R)∩{ζ ∈Cⁿ: min{|ζ₁|, . . . ,|ζ_n|}< r_j} . We note that U_ε:=SN

j=1U_j ⊂(C^⋆)ⁿ is open and K_ε := (C^⋆)ⁿ\U_ε is compact.

Moreover the above lower estimate onϕholds on (C^⋆)ⁿ\K_ε. This concludes

the proof.

Corollary 3.7. If ϕ∈ Etor(X, ω) and χ is a nonnegative continuous func- tion on Rⁿ then

Z

(C^⋆)ⁿ

(χ◦L) (dd^cFϕ◦L)ⁿ= Z

Rⁿ

χ M AR(Fϕ) =n!

Z

intP

χ(∇Gϕ(s))dV(s).

Proof. The first equality follows from Lemma 2.2. The second one follows from [BeBe13, Lemma 2.7], since F_ϕ has full Monge-Amp`ere mass by The-

orem 3.6.

Examples 3.8. If X = Pⁿ is the complex projective space and ω is the Fubini-Study K¨ahler form, then F₀(x) = ¹₂ log 1 +Pn

i=1e^2xi

and P =

∇F₀(Rⁿ) is the simplex P =

(

s_i ≥0,1≤i≤n, Xn

i=1

s_i≤1 )

.

Thus d= n+ 1, ℓ_i(s) = s_i for 1≤i≤n, and ℓ_n+1(s) = 1−P_n

i=1s_i. The Legendre transform of F₀ is

G₀(s) = 1 2



 Xn

i=1

s_ilogs_i+



1− Xn

j=1

s_j



log



1− Xn

j=1

s_j







. This coincides with the function given by Guillemin’s formula.

1) Let [z] = [z0 :z1 :. . .:zn] denote the homogeneous coordinates onPⁿ. The function

ϕ₁[z] = log|z₁| −logkzk

is ω-psh and toric. It does not belong to the class Etor(X, ω) since it has positive Lelong numbers along the toric hyperplane (z₁ = 0). The associated convex function F1 : Rⁿ → R is given by F1(x) = x1 and its Legendre transform is

G₁(s) = 0 ifs= (1,0, . . . ,0), and G₁(s) = +∞ otherwise.

2) The function

ϕ2[z] = max

1≤i≤nlog|zi| −logkzk

is ω-psh and toric. It does not belong to the class Etor(X, ω) since it has one positive Lelong number at the point [1 : 0 :· · ·: 0]. The corresponding convex function is F₂(x) = max_1≤i≤nx_i and its Legendre transform is

G₂(s) = 0 if s∈ (

s_i ≥0,1≤i≤n, Xn

i=1

s_i= 1 )

, and G₂(s) = +∞ otherwise.

3.3. The classes E^χ,tor(X, ω). If χ ∈ W we defineLχ(P) to be the set of lower semicontinuous functions G:P →R∪ {+∞}such that

Z

P−χ min

P G−G(s)

dV(s)<+∞.

Proposition 3.9. Let ϕ ∈P SH_tor(X, ω). If χ ∈ W and ϕ∈ Eχ,tor(X, ω) then G_ϕ ∈ L_χ(P). Conversely, if p ≥ 1 and G_ϕ ∈ L^p(P) then ϕ ∈ Etor^p (X, ω).

Proof. For the first claim, let ϕ∈P SH_tor(X, ω)∩L^∞(X) be such that F_ϕ is smooth and strictly convex, and F_ϕ ≤F_P ≤F₀. We prove the following a priori estimate:

Z

intP−χ(−G_ϕ(s))dV(s)≤ 1 n!

Z

X−χ(ϕ)M A(ϕ).

Note that ϕ≤0 on X and G_ϕ ≥G_P = 0 on P. Moreover, by Proposition 4.1 and Lemma 2.5, ∇F_ϕ : Rⁿ → intP is bijective. Since F₀ ≥ F_P and Fϕ(x) =hx, si −Gϕ(s) for x=∇Gϕ(s), we obtain

(F₀−F_ϕ)◦ ∇G_ϕ(s) ≥ (F_P −F_ϕ)◦ ∇G_ϕ(s)

= F_P(∇G_ϕ(s))− h∇G_ϕ(s), si+G_ϕ(s)≥G_ϕ(s)≥0, where s ∈ intP and the last estimate follows from the definition of FP. Applying Lemmas 2.5 and 2.2 we get

Z

intP−χ(−G_ϕ(s))dV(s)≤ Z

intP−χ((F_ϕ−F₀)◦ ∇G_ϕ(s))dV(s)

= 1 n!

Z

Rⁿ−χ(F_ϕ−F₀)M AR(F_ϕ)

= 1 n!

Z

(C^⋆)ⁿ−χ(ϕ)M A(ϕ).

Let now ϕ ∈ Eχ,tor(X, ω) be such that F_ϕ ≤ F_P −1. There exists a sequence ϕ_j ∈ P SH_tor(X, ω)∩L^∞(X) such that ϕ_j ց ϕ, the associated functions F_ϕj are smooth and strictly convex, andF_ϕj ≤F_P. Then

Z

X−χ(ϕ_j)M A(ϕ_j)→ Z

X−χ(ϕ)M A(ϕ)

as j→+∞. SinceG_ϕj ր G_ϕ it follows by the a priori estimate applied to ϕ_j and the monotone convergence theorem that

Z

P−χ(−G_ϕ(s))dV(s)≤ 1 n!

Z

X−χ(ϕ)M A(ϕ) <+∞, so G_ϕ ∈L_χ(P). This concludes the proof of the first claim.

Conversely, let p ≥ 1 and consider the space of K¨ahler potentials H = {ϕ∈ C^∞(X) : ω+dd^cϕ >0} endowed with the metric

d_p(ϕ₁, ϕ₂) = inf Z ₁

0

Z

X|ϕ˙_t|^pM A(ϕ_t) 1/p

dt , ϕ₁, ϕ₂ ∈ H,

where the infimum is taken over all smooth pathst∈[0,1]→ϕt∈ Hjoining ϕ₁ to ϕ₂. It is shown in [Dar15, Theorem 3] that if ϕ₁, ϕ₂ ∈ Hthen

(2)

Z

X|ϕ₁−ϕ₂|^pM A(ϕ₁)≤C_pd_p(ϕ₁, ϕ₂)^p,

for some constantC_p>1 depending onp. On the other hand ifϕ₁, ϕ₂ ∈ H∩

P SH_tor(X, ω) are determined by the convex functionsF₁, F₂ with Legendre transforms G₁, G₂, then, by [G14, Proposition 4.3],

(3) d_p(ϕ₁, ϕ₂)^p = Z

P |G₁−G₂|^pdV .

Ifϕ∈ H ∩P SH_tor(X, ω) we apply (2) and (3) withϕ₁ =ϕandϕ₂ = 0, and obtain that

Z

X|ϕ|^pM A(ϕ)≤C_p Z

P|G_ϕ−G₀|^pdV ≤2^p−1C_p

kG_ϕk^p_L^p_(P)+kG₀k^p_L^p_(P) . Let nowϕ∈P SH_tor(X, ω) be such thatG_ϕ ∈L^p(P), and take a sequence ϕj ∈ H ∩P SHtor(X, ω) such that ϕj ց ϕ. Then Gϕj ր Gϕ, so by the above estimate applied to ϕ_j and dominated convergence we conclude that sup_jR

X|ϕ_j|^pM A(ϕ_j)<+∞. Hence ϕ∈ Etor^p (X, ω).

Example 3.10. Let X =P¹ and ω be the Fubini-Study K¨ahler form. By Examples 3.8 the corresponding convex function is F₀(x) = ¹₂ log 1 +e^2x and P = F₀^′(R) = [0,1]. Let ϕ be the toric ω-sh function associated to the convex function F(x) := F_P(x) = max(x,0). Note that the Legendre transform of F is G = 0 on [0,1], and dd^cF(log|z|) is the (normalized) Lebesgue measure on the unit circle S¹ ⊂P¹. We consider the sequence of toric ω-sh {ϕ_j} defined by the convex functions

Fj(x) = (1−εj)F(x) +εjmax(x,−Cj),

where ε_j decreases to 0, while C_j increases to +∞. A straightforward com- putation yields that the corresponding Legendre transforms are

G_j(s) = max C_j(ε_j−s),0

, 0≤s≤1.

Note that

ϕj(z)−ϕ(z) =−εjlog⁺|z|+εjmax(log|z|,−Cj)

=







−ε_jC_j, |z|< e^−Cj, ε_jlog|z|, e^−Cj ≤ |z|<1, 0, |z| ≥1.

Thus we obtain the following:

• ϕ_j −→ϕinL¹ if and only ifε_j →0;

• ϕ_j −→ϕinL^∞ if and only if ε_jC_j →0;

• ϕ_j −→ϕ in W^1,2 (the natural topology on E¹(X, ω)) if and only if ε²_jC_j →0.