Pluripotential theory and convex bodies: large deviation principle

c 2019 by Institut Mittag-Leffler. All rights reserved

Pluripotential theory and convex bodies: large deviation principle

Turgay Bayraktar, Thomas Bloom, Norman Levenberg and Chinh H. Lu

Abstract. We continue the study in [2] in the setting of weighted pluripotential theory arising from polynomials associated to a convex bodyP in (R⁺)^d. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms ofP−pluripotential- theoretic notions. As an important preliminary step, we first give an existence proof for the solution of a Monge-Ampère equation in an appropriate finite energy class. This is achieved using a variational approach.

1. Introduction

As in [2], we fix a convex body P⊂(R⁺)^d and we define the logarithmic indi- cator function

(1.1) H_P(z) := sup

J∈P

log|z^J|:= sup

(j1,...,jd)∈Plog[|z₁|^j1...|z_d|^jd].

We assume throughout that

(1.2) Σ⊂kP for somek∈Z⁺

where

Σ :={(x1, ..., xd)∈R^d: 0≤xi≤1, d j=1

xi≤1}.

Then

HP(z)≥1 k max

j=1,...,dlog⁺|zj|

N. Levenberg is supported by Simons Foundation grant No. 354549.

Key words and phrases:convex body,P−extremal function, large deviation principle.

2010 Mathematics Subject Classification:32U15, 32U20, 31C15.

where log⁺|zj|=max[0,log|zj|]. We define

LP=LP(C^d) :={u∈P SH(C^d) :u(z)−HP(z) =O(1), |z| −→∞}, and

LP,+=LP,+(C^d) ={u∈LP(C^d) :u(z)≥HP(z)+Cu}.

These are generalizations of the classical Lelong classes whenP=Σ. We define the finite-dimensional polynomial spaces

P oly(nP) :={p(z) =

J∈nP∩(Z⁺)^d

c_Jz^J:c_J∈C}

forn=1,2,... wherez^J=z^j₁¹...z_d^jdforJ=(j₁, ..., j_d). Forp∈P oly(nP), n≥1 we have

1

nlog|p|∈LP; also eachu∈LP,+(C^d) is locally bounded in C^d. ForP=Σ, we write P oly(nP)=Pn.

Given a compact set K⊂C^d, one can define various pluripotential-theoretic notions associated toK related toL_P and the polynomial spacesP oly(nP). Our goal in this paper is to prove some probabilistic properties of random point pro- cesses on K utilizing these notions and their weighted counterparts. We require an existence proof for the solution of a Monge-Ampère equation in an appropri- ate finite energy class; this is done in Theorem 2.8 using a variational approach and is of interest on its own. The third section recalls appropriate definitions and properties in P−pluripotential theory, mostly following [2]. As in [2], our spaces P oly(nP) do not necessarily arise as holomorphic sections of tensor powers of a line bundle. Subsection 3.3 includes a standard elementary probabilistic result on almost sure convergence of probability measures associated to random arrays onK to aP−pluripotential-theoretic equilibrium measure. Section 4 sets up the machin- ery for the more subtle large deviation principle (LDP), Theorem5.1, for which we provide two proofs (analogous to those in [9]). As in [9], the first proof was inspired by [6] and the second proof was utilized by Berman in [5]. The reader will find far-reaching applications and interpretations of LDP’s in the appropriate settings of holomorphic line bundles over a compact, complex manifold in [5]. In particular, the case whereP is a convex integral polytope (vertices inZ^d) which is the moment polytope for a toric manifold (P is Delzant) is covered in [5].

2. Monge-Ampère andP−pluripotential theory 2.1. Monge-Ampère equations with prescribed singularity

In this section, (X, ω) is a compact Kähler manifold of dimensiond.

2.1.1. Quasi-plurisubharmonic functions

A functionu:X→R∪{−∞}is called quasi-plurisubharmonic (quasi-psh) if lo- callyu=ρ+ϕ, whereϕis plurisubharmonic andρis smooth.

We letP SH(X, ω) denote the set ofω-psh functions, i.e. quasi-psh functions usuch thatωu:=ω+dd^cu≥0 in the sense of currents onX.

Given u, v∈P SH(X, ω) we say that u is more singular than v (and we write u≺v) ifu≤v+ConX, for some constantC. We say thatuhas the same singularity asv (and we writeu v) ifu≺v andv≺u.

Given φ∈P SH(X, ω), we let P SH(X, ω, φ) denote the set ofω-psh functions uwhich are more singular thanφ.

2.1.2. Nonpluripolar Monge-Ampère measure

For bounded ω-psh functions u₁, ..., u_d, the Monge-Ampère product (ω+ dd^cu1)∧...∧(ω+dd^cud) is well-defined as a positive Radon measure onX (see [14], [3]). For generalω-psh functionsu1, ..., ud, the sequence of positive measures

1_∩{uj>−k}(ω+dd^cmax(u1,−k))∧...∧(ω+dd^cmax(ud,−k))

is non-decreasing ink and the limiting measure, which is called the nonpluripolar product ofω_u1, ..., ω_ud, is denoted by

ω_u1∧...∧ωud.

When u1=...=ud=uwe write ω_u^d:=ωu∧...∧ωu. Note that by definition

Xωu1∧ ...∧ωud≤

Xω^d.

It was proved in [20, Theorem 1.2] and [11, Theorem 1.1] that the total mass of nonpluripolar Monge-Ampère products is decreasing with respect to singularity type. More precisely,

Theorem 2.1. Let ω₁, ..., ω_d be K¨ahler forms onX. If u_j≺v_j,j=1, ..., d, are ω_j-psh functions then

X

(ω1+dd^cu1)∧...∧(ωd+dd^cud)≤

X

(ω1+dd^cv1)∧...∧(ωd+dd^cvd).

As noted above, for a general ω-psh function uwe have the estimate

Xω_u^d≤

Xω^d. Following [15] we let E(X, ω) denote the set of all ω-psh functions with maximal total mass, i.e.

E(X, ω) :=

u∈P SH(X, ω) :

X

ω_u^d=

X

ω^d

.

Givenφ∈P SH(X, ω), we define E(X, ω, φ) :=

u∈P SH(X, ω, φ) :

X

ω_u^d=

X

ω_φ^d

.

Proposition 2.2. Letφ∈P SH(X, ω). The following are equivalent:

(1)E(X, ω, φ)∩E(X, ω)=∅;

(2)φ∈E(X, ω);

(3)E(X, ω, φ)⊂E(X, ω).

Proof. We first prove (1)=⇒(2). Ifu∈E(X, ω, φ)∩E(X, ω) then

Xω_u^d=

Xω^d. On the other hand, sinceuis more singular thanφ, Theorem2.1ensures that

X

ω^d=

X

ω^d_u≤

X

ω^d_φ≤

X

ω^d, hence equality holds, proving thatφ∈E(X, ω).

Now we prove (2)=⇒(3). Ifφ∈E(X, ω) andu∈E(X, ω, φ) then

X

ω^d_u=

X

ω^d_φ=

X

ω^d, henceu∈E(X, ω).

Finally (3)=⇒(1) is obvious.

Proposition 2.3. Assume that φ_j∈P SH(X, ωj), j=1, ..., d with

X(ω_j+ dd^cφ_j)^d>0. If u_j∈E(X, ω_j, φ_j), j=1, ..., d,then

X

(ω1+dd^cu1)∧...∧(ωd+dd^cud) =

X

(ω1+dd^cφ1)∧...∧(ωd+dd^cφd).

Proof. Theorem 2.1 gives one inequality. The other one follows from [11, Proposition 3.1 and Theorem 3.14].

2.1.3. Model potentials

For a functionf:X→R∪{−∞}, we letf^∗denote its uppersemicontinuous (usc) regularization, i.e.

f^∗(x) := lim sup

Xy→x

f(y).

Givenφ∈P SH(X, ω), following J. Ross and D. Witt Nyström [18], we define P_ω[φ] :=

t→lim+∞P_ω(min(φ+t,0)) _∗

.

Here, for a functionf,Pω(f) is defined as

Pω(f) := (x−→sup{u(x) :u∈P SH(X, ω), u≤f})^∗.

It was shown in [11, Theorem 3.8] that the nonpluripolar Monge-Ampère measure ofPω[φ] is dominated by Lebesgue measure:

(2.1) (ω+dd^cP_ω[φ])^d≤1_{Pω[φ]=0}ω^d≤ω^d.

This fact plays a crucial role in solving the complex Monge-Ampère equation. For the reader’s convenience, we note that in the notation of [11] (on the left)

P_[ω,φ](0) =P_ω[φ].

Definition 2.4. A functionφ∈P SH(X, ω) is called a model potential if

Xω_φ^d>

0 andPω[φ]=φ. A functionu∈P SH(X, ω) has model type singularity if uhas the same singularity asPω[u]; i.e.,u−Pω[u] is bounded onX.

There are plenty of model potentials. If ϕ∈P SH(X, ω) with

Xω_ϕ^d>0 then, by [11, Theorem 3.12], Pω[ϕ] is a model potential. In particular, if

Xω_ϕ^d=

Xω^d (i.e. ϕ∈E(X, ω)) thenPω[ϕ]=0.

We will use the following property of model potentials proved in [11, Theorem 3.12]: ifφis a model potential then

(2.2) u∈P SH(X, ω, φ) ==⇒u−sup

X

u≤φ.

In the sequel we always assume that φ has model type singularity and small unbounded locus; i.e.,φis locally bounded outside a closed complete pluripolar set, allowing us to use the variational approach of [7] as explained in [11].

2.1.4. The variational approach

We call a measure which puts no mass on pluripolar sets anonpluripolar mea- sure. For a positive nonpluripolar measureμ onX we letLμ denote the following linear functional onP SH(X, ω, φ):

Lμ(u) :=

X

(u−φ)dμ.

Foru∈P SH(X, ω) withu φ, we define the Monge-Ampère energy

(2.3) Eφ(u) := 1

(d+1) d k=0

X

(u−φ)ω_u^k∧ω_φ^d−k.

It was shown in [11, Theorem 4.10] (by adapting the arguments of [7]) thatEφ is non-decreasing and concave along affine curves, giving rise to its trivial extension toP SH(X, ω, φ).

We define

(2.4) E¹(X, ω, φ) :={u∈P SH(X, ω, φ) :Eφ(u)>−∞}.

The following criterion was proved in [11, Theorem 4.13]:

Proposition 2.5. Letu∈P SH(X, ω, φ). Thenu∈E¹(X, ω, φ)iffu∈E(X, ω, φ) and

X(u−φ)ω_u^d>−∞.

Lemma 2.6. If E is pluripolar then there exists u∈E¹(X, ω, φ)such that E⊂ {u=−∞}.

Proof. Without loss of generality we can assume that φis a model potential.

Then (2.1) gives

X|φ|ω_φ^d=0. It follows from [7, Corollary 2.11] that there exists v∈E¹(X, ω,0),v≤0, such thatE⊂{v=−∞}. Setu:=P_ω(min(v, φ)). ThenE⊂{u=

−∞}and we claim thatu∈E¹(X, ω, φ). For each j∈Nwe setvj:=max(v,−j) and uj:=Pω(min(vj, φ)). Then uj decreases to uand uj φ. Using [11, Theorem 4.10 and Lemma 4.15] it suffices to check that{

X|u_j−φ|ω_u^d_j}is uniformly bounded. It follows from [11, Lemma 3.7] that

X

|uj−φ|ωu^dj≤

X

|uj|ω^duj≤

X

|vj|ωv^dj+

X

|φ|ωφ^d

=

X

|vj|ωv^dj. The fact that

X|vj|ω_v^d_j is uniformly bounded follows from [15, Corollary 2.4] since v∈E¹(X, ω,0). This concludes the proof.

Lemma 2.7. Assume thatE¹(X, ω, φ)⊂L¹(X, μ). Then,for eachC >0,L_μ is bounded on

E_C:={u∈P SH(X, ω, φ) : sup

X

u≤0 andE_φ(u)≥ −C}.

Proof. By concavity of E_φ the set E_C is convex. We now show that E_C is compact in theL¹(X, ω^d) topology. Let {u_j}be a sequence inE_C. We claim that {sup_Xuj}is bounded. Indeed, by [11, Theorem 4.10]

E_φ(u_j)≤

X

(u_j−φ)ω_φ^d

≤(sup

X

u_j)

X

ω^d_φ+

X

(u_j−sup

X

u_j−φ)ω^d_φ.

It follows from (2.2) thatuj−sup_Xuj≤Pω[φ]≤φ+C0, whereC0is a constant. The boundedness of{sup_Xu_j} then follows from that of{E_φ(u_j)} and the above esti- mate. This proves the claim.

A subsequence of {uj}, still denoted by{uj}, converges in L¹(X, ω^d) to u∈

P SH(X, ω) with sup_Xu≤0. Since uj−sup_Xuj≤φ+C0, we have u−sup_Xu≤φ+

C₀. This proves that u∈P SH(X, ω, φ). The upper semicontinuity of E_φ (see [11, Proposition 4.19]) ensures thatEφ(u)≥−C, hence u∈EC. This proves thatEC is compact in theL¹(X, ω^d) topology.

The result then follows from [7, Proposition 3.4].

The goal of this section is to prove the following result:

Theorem 2.8. Assume thatμis a nonpluripolar positive measure onX such thatμ(X)=

Xω^d_φ. The following are equivalent

(1)μhas finite energy, i.e., Lμ is finite onE¹(X, ω, φ);

(2)there existsu∈E¹(X, ω, φ)such thatω^d_u=μ;

(3)there exists a uniqueu∈E¹(X, ω, φ)such that Fμ(u) = max

v∈E¹(X,ω,φ)Fμ(v)<+∞

whereFμ=Eφ−Lμ.

Remark 2.9. It was shown in [11, Theorem 4.28] that a unique (normalized) solutionuinE(X, ω, φ) always exists (without the finite energy assumption on μ).

But that proof does not give a solution inE¹(X, ω, φ). Below, we will follow the proof of [11, Theorem 4.28] and use the finite energy condition,E¹(X, ω, φ)⊂L¹(X, μ), to prove thatubelongs toE¹(X, ω, φ).

Lemma 2.10. Assume that E¹(X, ω, φ)⊂L¹(X, μ). Then there exists a posi- tive constantC such that, for allu∈E¹(X, ω, φ)withsup_Xu=0,

(2.5) Lμ(u)≥ −C(1+|Eφ(u)|^1/2).

The proof below uses ideas in [7], [15].

Proof. Since φhas model type singularity, it follows from [11, Theorem 4.10]

thatE_φ−E_Pω[φ]is bounded. Without loss of generality we can assume in this proof that φ=Pω[φ]. Fix u∈E¹(X, ω, φ) such that sup_Xu=0 and |Eφ(u)|>1. Then, by [11, Theorem 3.12], u≤φ. Set a=|Eφ(u)|^−1/2∈(0,1), and v:=au+(1−a)φ∈

E¹(X, ω, φ). We estimate Eφ(v) as follows

(d+1)Eφ(v) =a d k=0

X

(u−φ)ω_v^k∧ω_φ^d−k

=a d k=0

X

(u−φ)(aωu+(1−a)ωφ)^k∧ω^d_φ^−k

≥C(d)a

X

(u−φ)ω_φ^d+C(d)a² d k=0

X

(u−φ)ω_u^k∧ω^d_φ,

whereC(d) is a positive constant which only depends ond. It follows fromφ=Pω[φ]

and [11, Theorem 3.8] thatω^d_φ≤ω^d(recall (2.1)). This together with [14, Proposition

2.7] give

X

(u−φ)ω^d_φ≥ −C1, for a uniform constantC₁. Therefore,

(d+1)Eφ(v)≥ −C1C(d)a+C2a²Eφ(u)≥ −C3.

It thus follows from Lemma 2.7 that Lμ(v)≥−C4 for a uniform constant C4>0.

Thus

X

(u−φ)dμ≥ −C₄/a, which gives (2.5).

We are now ready to prove Theorem2.8.

Proof of Theorem2.8. Without loss of generality we can assume that φ is a model potential. We first prove (1)=⇒(2). We write μ=f ν, where ν is a non- pluripolar positive measure satisfying, for all Borel subsetsB⊂X,

ν(B)≤ACap_φ(B),

for some positive constant A, and 0≤f∈L¹(X, ν) (cf., [11, Lemma 4.26]). Here Cap_φ is defined as

Cap_φ(B) := sup

B

ω^d_u:u∈P SH(X, ω), φ−1≤u≤φ

. Set, fork∈N,μk:=ckmin(f, k)ν where ck>0 is chosen so thatμk(X)=

Xω_φ^d; this is needed in order to solve the Monge-Ampère equation in the class E¹(X, ω, φ).

For k large enough, 1≤ck≤2 and ck→1 as k→+∞. It follows from [11, Theo- rem 4.25] that there existsu_j∈E¹(X, ω, φ), sup_Xu_j=0, such thatω^d_uj=μ_j; by [11, Theorem 3.12], uj≤φ. A subsequence of {uj} which, by abuse of notation, will be denoted by{uj}, converges in L¹(X, ω^d) to u∈P SH(X, ω) with u≤φ. Define vk:=(sup_j≥kuj)^∗. Thenvkuand sup_Xvk=0. It follows from (2.5) and [11, The- orem 4.10] that

|Eφ(uj)| ≤

X

|uj−φ|ω_u^d_j≤2

X

|uj−φ|dμ

≤2C(1+|Eφ(uj)|^1/2).

Therefore{|Eφ(u_j)|}is bounded, hence so is{|Eφ(v_j)|}sinceE_φ is non-decreasing.

It then follows from [11, Lemma 4.15] thatu∈E¹(X, ω, φ).

Now, repeating the arguments of [11, Theorem 4.28] we can show thatω_u^d=μ, finishing the proof of (1)=⇒(2).

We next prove (2)=⇒(3). Assume thatμ=ω_u^d for someu∈E¹(X, ω, φ). For all v∈E¹(X, ω, φ), by [11, Theorem 4.10] and Proposition2.5we have

L_μ(v) =

X

(v−φ)ω^d_u

=

X

(v−u)ω_u^d+

X

(u−φ)ω^d_u

≥E_φ(v)−Eφ(u)+

X

(u−φ)ω_u^d>−∞.

HenceLμ is finite onE¹(X, ω, φ). Now, for allv∈E¹(X, ω, φ), by [11, Theorem 4.10]

we have

Fμ(v)−Fμ(u) =Eφ(v)−Eφ(u)−

X

(v−u)ω_u^d≤0.

This gives (3). Finally, (3)=⇒(1) is obvious.

2.2. Monge-Ampère equations onC^d with prescribed growth

As in the introduction we let P be a convex body contained in (R⁺)^d and fix r>0 such that P⊂rΣ. We assume (1.2); i.e., Σ⊂kP for some k∈Z⁺. This ensures that HP in (1.1) is locally bounded on C^d (and of course HP∈L⁺_P(C^d)).

Letu∈LP(C^d) and define

(2.6) u(z) :=˜ u(z)−r

2log(1+|z|²), z∈C^d.

Consider the projective spaceP^dequipped with the Kähler metricω:=rωF S, where ω_{F S}=dd^c1

2log(1+|z|²)

onC^d. Then ˜uis bounded from above on C^d. It thus can be extended toP^d as a function inP SH(P^d, ω).

For a plurisubharmonic functionuonC^d, we let (dd^cu)^ddenote its nonpluripo- lar Monge-Ampère measure; i.e., (dd^cu)^d is the increasing limit of the sequence of measures1_{u>−k}(dd^cmax(u,−k))^d. Then

ω^d_u˜= (ω+dd^cu)˜ ^d= (dd^cu)^d onC^d. Ifu∈LP(C^d) then

C^d(dd^cu)^d≤

C^d(dd^cHP)^d=d!V ol(P) =:γd=γd(P) (cf., equation (2.4) in [2]). We define

EP(C^d) :=

u∈L_P(C^d) :

C^d(dd^cu)^d=γ_d

.

By the construction in (2.6) we have thatHP∈P SH(P^d, ω). We define ΦP:=Pω[HP].

The key point here, which follows from [12, Theorem 7.2], is thatHP has model type singularity (recall Definition2.4) and hence the same singularity as ΦP. Defining ΦP onC^d using (2.6); i.e., forz∈C^d,

ΦP(z) =ΦP(z)+r

2log(1+|z|²),

we thus have ΦP∈LP,+(C^d). The advantage of using ΦP is that, by (2.1), (dd^cΦP)^d≤ω^d on C^d. Note that LP,+(C^d)⊂EP(C^d). For u, v∈L⁺_P(C^d) we define

(2.7) E_v(u) := 1

(d+1) d j=0

C^d

(u−v)(dd^cu)^j∧(dd^cv)^d−j. The corresponding global energy (see (2.3)) is defined as

E_v˜(˜u) := 1 (d+1)

d j=0

P^d(˜u−v)(ω+dd˜ ^cu)˜ ^j∧(ω+dd^cv)˜ ^d−j.

ThenEv is non-decreasing and concave along affine curves inLP,+(C^d). We extend E_vtoL_P(C^d) in an obvious way. Note thatE_v may take the value−∞. We define

E_P¹(C^d) :={u∈LP(C^d) :EHP(u)>−∞}.

We observe that in the above definition we can replace EHP by EΦP, since for u∈LP,+(C^d), by the cocycle property (cf. Proposition 3.3 [2]),

EHP(u)−EHP(ΦP) =EΦP(u).

We thus have the following important identification (see (2.4)):

(2.8) u∈ EP¹(C^d) ⇐==⇒ u˜∈ E¹(P^d, ω,Φ_P).

We then have the following local version of Proposition2.5:

Proposition 2.11. Let u∈L_P(C^d). Then u∈E_P¹(C^d) iff u∈EP(C^d) and

C^d(u−HP)(dd^cu)^d>−∞. In particular, if supp(dd^cu)^d is compact, u∈E_P¹(C^d) iff

C^d(dd^cu)^d=γd and

C^du(dd^cu)^d>−∞.

Proof. SinceH_P Φ_P,

P^d

(˜u−H_P)ω^d_u˜>−∞iff

P^d

(˜u−Φ_P)ω_u^d_˜>−∞

where ˜u∈P SH(P^d, ω) anduare related by (2.6). Moreover, ΦP∈LP,+(C^d) implies u≤ΦP+Cso that ˜u∈P SH(P^d, ω,ΦP). But

P^d(˜u−HP)ω_u^d_˜=

C^d(u−HP)(dd^cu)^d

and the result follows from (2.8) by applying Proposition 2.5 to ˜u. For the last statement, note that for generalu∈LP(C^d) we may have

C^dHP(dd^cu)^d=+∞, but if (dd^cu)^d has compact support then

C^dH_P(dd^cu)^d is finite.

Note that Theorem2.1and Proposition2.3 give the following result:

Theorem 2.12. Let u1, ..., ud be functions inEP(C^d). Then

C^ddd^cu1∧...∧dd^cud=γd.

Foru₁, ..., u_n∈LP,+(C^d) Theorem2.12was proved in [1, Proposition 2.7].

Having the correspondence (2.8) we can state a local version of Theorem2.8;

this will be used in the sequel. Let MP(C^d) denote the set of all positive Borel measuresμonC^d withμ(C^d)=d!V ol(P)=γd.

Theorem 2.13. Assume that μ∈MP(C^d) is a positive nonpluripolar Borel measure. The following are equivalent

(1)E_P¹(C^d)⊂L¹(C^d, μ);

(2)there existsu∈E_P¹(C^d)such that (dd^cu)^d=μ;

(3)there existsu∈E_P¹(C^d)such that Fμ(u) = max

v∈EP¹(C^d)Fμ(v)<+∞. A priori the functionalFμ is defined foru∈E_P¹(C^d) by

Fμ,ΦP(u) :=EΦP(u)−

C^d(u−ΦP)dμ.

However, using this notation, since

Fμ,ΦP(u)−Fμ,HP(u) =Fμ,ΦP(HP),

in statement (3) of Theorem2.13we can take either of the two definitionsFμ,ΦP or Fμ,HP forFμ.

Remark 2.14. If μhas compact support inC^d then

C^dΦ_Pdμ and

C^dH_Pdμ are finite. Therefore, the functionalFμ can be replaced by

u−→EHP(u)−

C^du dμ.

Using the remark, forμ∈MP(C^d) with compact support, it is natural to define the Legendre-type transform ofE_HP:

(2.9) E^∗(μ) := sup

u∈EP¹(C^d)

[EHP(u)−

C^du dμ].

This functional, which will appear in the rate function for our LDP, will be given a more concrete interpretation usingP−pluripotential theory in section 4; cf., equa- tion (4.18).

Finally, for future use, we record the following consequence of Lemma2.6and the correspondence (2.8).

Lemma 2.15. If E⊂C^d is pluripolar then there exists u∈E_P¹(C^d) such that E⊂{u=−∞}.

3. P−pluripotential theory notions Given E⊂C^d, theP−extremal function ofE is

V_P,E^∗ (z) := lim sup

ζ→z

V_P,E(ζ) where

VP,E(z) := sup{u(z) :u∈LP(C^d), u≤0 onE}.

For K⊂C^d compact, w:K→R⁺ is an admissible weight function on K if w≥0 is an uppersemicontinuous function with {z∈K:w(z)>0} nonpluripolar. Setting Q:=−logw, we writeQ∈A(K) and define theweightedP−extremal function

V_P,K,Q^∗ (z) := lim sup

ζ→z VP,K,Q(ζ) where

VP,K,Q(z) := sup{u(z) :u∈LP(C^d), u≤QonK}.

IfQ=0 we writeV_P,K,Q=V_P,K, consistent with the previous notation. For P=Σ, VΣ,K,Q(z) =VK,Q(z) := sup{u(z) :u∈L(C^d), u≤QonK}

is the usual weighed extremal function as in Appendix B of [19].

We write (omitting the dependence onP)

μ_K,Q:= (dd^cV_P,K,Q^∗ )^d andμ_K:= (dd^cV_P,K^∗ )^d

for the Monge-Ampère measures ofV_P,K,Q^∗ andV_P,K^∗ (the latter ifKis not pluripo- lar). Proposition 2.5 of [2] states that

supp(μ_K,Q)⊂ {z∈K:V_P,K,Q^∗ (z)≥Q(z)} andV_P,K,Q^∗ =Qq.e. onsupp(μ_K,Q), i.e., off of a pluripolar set.

3.1. Energy

We recall some results and definitions from [2]. Foru, v∈LP,+(C^d), we define themutual energy

E(u, v) :=

C^d(u−v) d j=0

(dd^cu)^j∧(dd^cv)^d−j.

For simplicity, when v=HP, we denote the associated (normalized) energy func- tional byE:

E(u) :=EHP(u) = 1 d+1

d j=0

C^d(u−HP)dd^cu^j∧(dd^cHP)^d−j (recall (2.7)).

Foru, u, v∈LP,+(C^d), and for 0≤t≤1, we define f(t) :=E(u+t(u−u), v), From Proposition 3.1 in [2],f(t) exists for 0≤t≤1 and

f(t) = (d+1)

C^d(u−u)(dd^c(u+t(u−u)))^d Hence, takingv=H_P, we have, forF(t):=E(u+t(u−u)), that

F(t) =

C^d(u−u)(dd^c(u+t(u−u)))^d. ThusF(0)=

C^d(u−u)(dd^cu)^d and we write (3.1) < E(u), u−u >:=

(u−u)(dd^cu)^d.

We need some applications of a global domination principle. The following ver- sion, sufficient for our purposes, follows from [11], Corollary 3.10 (see also Corollary A.2 of [8]).

Proposition 3.1. Letu∈L_P(C^d)andv∈EP(C^d)withu≤va.e. (dd^cv)^d. Then u≤v inC^d.

This will be used to prove an approximation result, Proposition 3.3, which itself will be essential in the sequel. First we need a lemma.

Lemma 3.2. Assume thatϕ≤u, v≤H_P are functions inE_P¹(C^d). Then for all t>0,

{u≤HP−2t}(HP−u)(dd^cv)^d≤2^d+1

{ϕ≤HP−t}(HP−ϕ)(dd^cϕ)^d. In particular, the left hand side converges to0 ast→+∞uniformly inu, v.

Proof. Fors>0, we have the following inclusions of sets:

(u≤HP−2s)⊂

ϕ≤v+HP

2 −s

⊂(ϕ≤HP−s).

We first note that the left hand side in the lemma is equal to

{u≤HP−2t}(HP−u)(dd^cv)^d

= 2t

{u≤HP−2t}(dd^cv)^d+ _∞

2t

{u≤HP−s}(dd^cv)^d

ds.

(3.2)

We claim that, for alls>0, (3.3)

{u≤HP−2s}(dd^cv)^d≤2^d

{ϕ≤HP−s}(dd^cϕ)^d.

Indeed, the comparison principle ([11, Corollary 3.6]) and the inclusions of sets above give

{u≤HP−2s}(dd^cv)^d≤

{ϕ≤^v+2^HP−s}(dd^cv)^d≤2^d

{ϕ≤^v+2^HP−s}

dd^cv+HP

2 d

≤2^d

{ϕ≤^v+2^HP−s}

(dd^cϕ)^d≤2^d

{ϕ≤HP−s}

(dd^cϕ)^d. The claim is proved. Using (3.3) and (3.2) we obtain

{u≤HP−2t}

(H_P−u)(dd^cv)^d

≤2^d+1t

{ϕ≤HP−t}(dd^cϕ)^d+2^d+1 +∞

t

{ϕ≤HP−s}(dd^cϕ)^d

ds

= 2^d+1

{ϕ≤HP−t}(HP−ϕ)(dd^cϕ)^d.

Proposition 3.3. Let u∈E_P¹(C^d) with (dd^cu)^d=μ having support in a non- pluripolar compact setKso that

Ku dμ>−∞from Proposition2.11. Let{Qj}be a sequence of continuous functions onKdecreasing touonK. Thenuj:=V_P,K,Q^∗ _j↓u onC^d andμj:=(dd^cuj)^d is supported inK. In particular, μj→μ=(dd^cu)^d weak-*.

Moreover,

(3.4) lim

j→∞

K

Qjdμj= lim

j→∞

K

Qjdμ=

K

u dμ >−∞.

Proof. We can assume{Qj}are defined and decreasing touon the closure of a bounded open neighborhood Ω ofK. By adding a negative constant we can assume that Q1≤0 on Ω. Since {Qj} is decreasing, so is the sequence {uj}. Moreover, by [4, Proposition 5.1]uj≤Qj onK\Ej whereEj is pluripolar. Butuis a competitor in the definition ofVP,K,Qj so thatu≤ujonC^d. Thus ˜u:=limj→∞uj≥ueverywhere and ˜u≤uonK\E, whereE:=∪jE_j is a pluripolar set. Since (dd^cu)^d puts no mass on pluripolar sets,

{u<u}˜ (dd^cu)^d≤

E∪(C^d\K)(dd^cu)^d= 0.

It thus follows from Proposition3.1that ˜u≤u, hence ˜u=uonC^d.

The second equality in (3.4) follows from the monotone convergence theorem.

It remains to prove that

jlim→∞

K

(−Qj)dμj=

K

(−u)dμ.

For eachkfixed andj≥kwe have

K

(−Qj)dμj≥

K

(−Qk)dμj=

Ω

(−Qk)dμj, hence lim inf_j→∞

K(−Q_j)dμ_j≥

K(−Q_k)dμ since Ω is open and μ_j, μ are sup- ported onK. Letting k→+∞we arrive at

lim inf

j→∞

K

(−Q_j)dμ_j≥

K

(−u)dμ.

It remains to prove that lim sup

j→∞

K

(−Q_j)dμ_j≤

K

(−u)dμ.

The sequence {uj} is not necessarily uniformly bounded below on K. However, using the facts thatQj≥uand HP is continuous inC^d, it suffices to prove that

(3.5) lim sup

j→∞

K

(H_P−u)(dd^cu_j)^d≤

K

(H_P−u)(dd^cu)^d. To verify (3.5), we use Lemma3.2.

By adding a negative constant we can assume thatuj≤HP. For a functionv and fort>0 we definev^t:=max(v, H_P−t). Note that for eachtthe sequence{u^t_j} is locally uniformly bounded below. Define

a(t) := 2^d+1

{u≤HP−t/2}(H_P−u)(dd^cu)^d.

Sinceu∈E_P¹(C^d), from Proposition2.11we havea(t)→0 ast→+∞. By Lemma3.2 we have

(3.6) sup

j≥1

{u≤HP−t}(HP−u)(dd^cuj)^d≤a(t).

By the plurifine property of non-pluripolar Monge-Ampère measures [10, Proposi- tion 1.4] and (3.6) we have

K

(H_P−u)(dd^cu_j)^d≤

K∩{u>HP−t}(H_P−u)(dd^cu_j)^d+a(t)

=

K∩{u>HP−t}(HP−u^t)(dd^cu^t_j)^d+a(t)

≤

K

(H_P−u^t)(dd^cu^t_j)^d+a(t).

Since HP is bounded in Ω, it follows from [16, Theorem 4.26] that the sequence of positive Radon measures (H_P−u^t)(dd^cu^t_j)^d converges weakly on Ω to (H_P− u^t)(dd^cu^t)^d. SinceK is compact it then follows that

lim sup

j

K

(HP−u)(dd^cuj)^d≤

K

(HP−u^t)(dd^cu^t)^d+a(t).

We finally lett→+∞to conclude the proof in the following manner:

K

(HP−u^t)(dd^cu^t)^d≤

K∩{u>HP−t}(HP−u^t)(dd^cu^t)^d+a(t)

≤

K

(H_P−u)(dd^cu)^d+a(t),

where in the first estimate we have used{u≤HP−t}={u^t≤HP−t}and Lemma3.2 and in the last estimate we use again the plurifine property.

We now give an alternate description of the Legendre-type transformE^∗from (2.9) which will be related to the rate function in a large deviation principle. Given K⊂C^dcompact, we letMP(K) denote the space of positive measures onKof total massγ_d and we letC(K) denote the set of continuous, real-valued functions onK.

Proposition 3.4. Let K be a nonpluripolar compact set and μ∈MP(K).

Then

E^∗(μ) = sup

v∈C(K)

[E(V_P,K,v^∗ )−

K

v dμ].

Proof. We first treat the case whenE^∗(μ)=+∞. By Theorem2.13there exists u∈E_P¹(C^d) such that

Ku dμ=−∞. We take a decreasing sequenceQ_j∈C(K) such that Qj↓uonK and setuj:=V_P,K,Q^∗ _j. Then {uj} are decreasing; sinceu∈E_P¹(C^d) andE is non-decreasing,{E(uj)} is uniformly bounded and we obtain

E(V_P,K,Q^∗ _j)−

K

Qjdμ−→+∞, proving the proposition in this case.

Assume now thatE^∗(μ)<+∞. Theorem2.13ensures that

C^du dμ>−∞for all u∈E_P¹(C^d). By Lemma2.15,μputs no mass on pluripolar sets. From monotonicity ofE and the definition ofE^∗ in (2.9) we have

E^∗(μ)≥ sup

v∈C(K)[E(V_P,K,v^∗ )−

K

v dμ].

Here we have used that

V_P,K,v^∗ ≤v q.e. onK forv∈C(K).

For the reverse inequality, fix u∈E_P¹(C^d). Let{Qj} be a sequence of continuous functions on K decreasing to u on K and set u_j:=V_P,K,Q^∗ _j. Given ε>0, we can choosej sufficiently large so that, by monotone convergence,

K

Q_jdμ≤

K

u dμ+ε;

and, by monotonicity ofE,

E(V_P,K,Q^∗ _j)≥E(u).

Hence

E(V_P,K,Q^∗ _j)−

K

Q_jdμ≥E(u)−

K

u dμ−ε

so that

sup

v∈C(K)

[E(V_P,K,v^∗ )−

K

v dμ]≥E^∗(μ) and equality holds.