JULIUS ROSS AND DAVID WITT NYSTROM¨ arXiv:1210.2220v2 ... · arXiv:1210.2220v2 [math.CV] 15 Jul...

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ENVELOPES OF POSITIVE METRICS WITH PRESCRIBEDSINGULARITIES

JULIUS ROSS AND DAVID WITT NYSTROM

ABSTRACT. We investigate envelopes of positive metrics with a prescribed singularitytype. First we generalise work of Berman to this setting, proving C1,1 regularity of suchenvelopes, showing their Monge-Ampere measure is supported on a certain “equilibriumset” and connecting with the asymptotics of the partial Bergman functions coming frommultiplier ideals. We investigate how these envelopes behave on certain products, and howthey relate to the Legendre transform of a test curve of singularity types in the contextof geodesic rays in the space of Kahler potentials. Finallywe consider the associatedexhaustion function of these equilibrium sets, connectingit both to the Legendre transformand to the geometry of the Okounkov body.

1. INTRODUCTION

In this paper we study a number of features of envelopes of positive metrics with pre-scribed singularities. The setting we shall consider consists of a compact complex manifoldX with a locally bounded metricφ on a line bundleL and a positive singular metricψ onan auxiliary line bundleF . Themaximal envelopeof this data is defined to be

φ[ψ] = sup{γ ∈ PSH(L) : γ ≤ φ andγ ≤ ψ +O(1)}∗

where the notation means the upper semicontinuous regularisation of the supremum of allpositive metricsγ onL that are bounded byφ and have the same singularity type asψ. Thismaximal envelope is itself a positive metric onL which, as the notation suggests, dependsonφ and the singularity type ofψ.

Before turning to precise statements, we begin with an overview of the contents of thispaper which starts with some general statements about theseenvelopes, and then moveson to a number of applications and special cases. The main technical result is Theorem1.1 in which we prove, under suitable hypothesis, thatφ[ψ] is C1,1 away from an obvioussingular locus. The reader may choose on first reading to takethis statement as given,at which point the applications to the Monge-Ampere measures (Theorem 1.2) and to thepartial Bergman function (Theorems 1.3 and 1.4) follow rather easily. The proof of thesestatements form the largest part of the paper, and take up Sections 2 through to Section 4.

Following this, in Section 5 we consider the case of maximal envelopes on products,which we show, at least in the algebraic case, is related to the Mustata summation formulafor multiplier ideals. This is independent of the above technical results, and relies only onthe statement in Theorem 1.3 concerning the partial Bergmanfunction.

In Section 6 we show how a previous construction of the authors [29] of geodesics inthe space of Kahler potentials has an interpretation as a maximal envelope (on a product).This too is independent of the above technical results, but the real interest is that theygive, as a corollary, certain regularity of these geodesic rays (Theorem 1.8). This topic iscontinued in Section 7 in which this regularity allows us to interpret the time derivative of

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http://arxiv.org/abs/1210.2220v2

2 JULIUS ROSS AND DAVID WITT NYSTROM

this associated geodesic as a certain “exhaustion function” that appears naturally from thedefinition of maximal envelopes. Finally we consider in Section 8 a very special case ofthis construction, essentially given by a singularity typealong a divisor, and show how thisexhaustion map gives a natural candidate for the “first cooordinate” of a kind of moment-map from a polarised manifold to its Okounkov body.

1.1. Regularity, Monge Ampere measures and Partial Bergman Kernels.Our first setof results generalise work of Berman [3] to this setting, andfor convenience we collect theprecise statements here. In the followingψ ∈ PSH(F ) will assumed to be exponentiallyHolder continuous (which holds, for instance, ifψ has analytic singularities).

Theorem 1.1. If φ is Lipschitz (resp. in the classC1,1) overX then the same is true ofφ[ψ] overX − B+(L − F ) ∪ Sing(ψ).

HereB+(L−F ) denotes the augmented base locus ofL−F , andSing(ψ) is the locus onwhichψ is not locally bounded. We may as well assume thatL− F is big since otherwiseB+(L−F ) = X and the statement becomes vacuous. This result is in some sense optimal,since even in the case without the singular metric there are examples of maximal envelopesthat are no more thanC1,1.

Assume now thatφ is in factC2, denote byU the set on whichφ[ψ] is locally boundedand setX(0) = {x : ddcφx > 0}. Theequilibrium measureis defined as

µ(φ, ψ) := 1UMA(φ[ψ]) =1

n!1Udd

c(φ[ψ])n

where1U is the characteristic function ofU , and theequilibrium setis

D = D(φ, ψ) = {x ∈ X : φ[ψ](x) = φ(x)}.

Theorem 1.2. AssumeL− F is big. Then there is an equality of measures

µ(φ, ψ) = 1X−B+(L−F )∪Sing(ψ)MA(φ[ψ]) = 1DMA(φ) = 1D∩X(0)MA(φ).

We remark that this theorem justifies the terminology, sinceMA(φ[ψ]) = 0 away fromD and thusφ[ψ] is (locally) maximal there.

Turning to algebraic data, denote the multiplier ideal sheaf of kψ by I(kψ), and definethepartial Bergman functionof φ andψ to be the smooth function onX given by

Bk(φ, ψ) =∑

α

|sα|2φ,

where{sα} is any basis forH0(I(kψ) ⊗ Lk) that is orthonormal with respect to theL2-norm induced byφ and some fixed smooth volume formdV .

Theorem 1.3. There is a limit

k−1 lnBk(φ, ψ) → φ[ψ] − φ

ask tends to infinity that holds uniformly on compact subsets ofX−B+(L−F )∪Sing(ψ).

More precisely, for each such compact setK there is aCK > 0 such that for allk

C−1K e−k(φ−φ[ψ]) ≤ Bk(φ, ψ) ≤ CKk

ne−k(φ−φ[ψ]).

overK.

ENVELOPES OF POSITIVE METRICS WITH PRESCRIBED SINGULARITIES 3

Theorem 1.4. SupposeL− F is big. Then there is a pointwise limit

limk→∞

k−nBk(φ, ψ)dV = 1D(φ,ψ)∩X(0)MA(φ)

almost everywhere onX(0). Moreover

limk→∞

k−nBk(φ, ψ)dV → µ(φ, ψ)

weakly in the sense of measures.

1.2. Maximal Envelopes on Products.Following these technical results we turn to max-imal envelopes on products. Suppose that we have two sets of data of the above kind, givenby (Xi, Li, Fi, φi, ψi) for i = 1, 2 whereLi is a line bundle on a compact complex mani-fold Xi, φi a smooth metric onLi andψi a positive singular metric onFi. For simplicityassumeLi − Fi is ample andψi has algebraic singularities fori = 1, 2.

Theorem 1.5. Consider the product metricφ = φ1 + φ2 onL1 ⊗ L2 (where we suppressthe pullback notation), and let

ψ = sup{ψ1, ψ2}.

Thenφ[ψ] = sup{(φ1)[λψ1] + (φ2)[(1−λ)ψ2] : λ ∈ (0, 1)}∗.

This result resembles known formulae for the Siciak extremal function [2, 8, 30] andfor the pluricomplex Green function [28] on products. The particular proof we give usesthe connection with partial Bergman functions and gives an interesting interplay betweenthis circle of ideas and the the Mustata summation formulafor multiplier ideals. We donot suggest that the previous Theorem is optimal, and discuss conjectural generalisationsin Section 5. However rather than pursing this we move on to consider other aspects ofmaximal envelopes that can be thought of as a special case in whichX2 is the unit disc inC andψ2 has a logarithmic singularity at the origin.

1.3. The Legendre Transform as a Maximal Envelope.In previous work of the authorsmaximal envelopes were used to construct solutions to a Dirichlet problem for the complexHomogeneous Monge-Ampere Equation (HMAE). The general idea was to start with aconcave “test curve”ψλ for λ ∈ (0, c) of singular metrics and consider the Legendretransform

φt := supλ{φ[ψλ] + λt}∗ for t ∈ R.

Lettingw be the standard coordinate on the closed unit discB ⊂ C and changing variablest = − ln |w|2 we considerΦ(z, w) := φt(z) as anS1-invariant metric over the productX×B. In [29] it is proved, under some mild assumptions onψλ, thatφt is a weak geodesicin the space of positive metrics onL emanating fromφ. That is,Φ is a positive metric onπ∗L whereπ : X × B → X is the projection, that satisfiesMA(Φ) = 0 overX × B andΦ|∂B = φ.

Here we show how the Legendre transform can itself be considered as a maximal enve-lope overX ×B. Let

(1) ψ′ = supλ{ψλ + λt}∗.

Theorem 1.6. Setφ′ = φ+ ct. Then the Legendre transform ofψλ is given by

φt = φ′[ψ′]

overX ×B.


As is well known, an important aspect of the study of Dirichlet problems for the HMAEequation is finding solutions with good regularity properties (see, for example [16] foran introduction). We see from what has been said thus far thatsolutions coming fromthe Legendre transform construction have as much regularity as the associated maximalenvelope.

Definition 1.7. We say a test curve isexponentially Holder continuousif the singulairtyψ′ is exponentially Holder continuous onX ×B.

Theorem 1.8. Let ψλ be an exponentially Holder continuous. Then for each fixed finitet ∈ R the associated weak geodesicφt isC1,1 as a function onX , and moreover is locallyLipschitz in the variablet.

This gives a regularity result for a (reasonably large) class of weak geodesic rays. Weexpect this to be suboptimal, and that in factφt is alsoC1,1 it the variablet (see Remark6.7). This is very much in the spirit of the regularity resultof Phong-Sturm [25] concerningweak geodesics associated to test configurations that will be discussed again below (oneobserves that, when it applies, the above is neither weaker or stronger than what is provedthere). Certainly ifψλ is the test-curve coming from the degeneration to the normalconeof a divisor inX then it is exponentially Holder-continuous, and it seems likely that holdsfor any test curve coming from a test-configuration, but we have not attempted to provethis.

1.4. Exhaustion functions. Our final use for maximal envelopes is through the associatedexhaustion functions of the equilibrium sets. Fix a singular metricψ ∈ PSH(F ) andconsiderH : X → R given by

H = H(φ, ψ) = supλ{φ[λψ] = φ}.

It turns out that this exhaustion function is essentially the “time derivative” of the associatedLegendre function:

Theorem 1.9. Supposeφt is the Legendre transform associated to the test curveψλ = λψfor λ ∈ (0, 1). Then

H =dφtdt

∣∣∣∣t=0+

.

A particularly interesting case of the exhaustion functionarises whenψ = ln |sD|2

wheresD is the defining function of some divisorD ⊂ X . In this case there is a naturalexpression for the exhaustion function as a limit of algebraic objects. Fixλ ∈ Q+ andfor each largek with kλ ∈ N let {sα} be anL2 orthonormal basis forH0(Lk) that iscompatible with the filtration determined by the order of vanishingνα = ordD(sα) alongD. That is, for eachj the set{sα : να ≥ j} is a basis forH0(Lk ⊗ IjD).

Theorem 1.10.We have

H = lim supk→∞

∑α να|sα|

2φ∑

α k|sα|2φ

almost everywhere onX .

The previous two theorems can put into context through work of Phong-Sturm on weakgeodesics in the space of Kahler metrics. We continue the same notation as above, so{sα}is a basis forH0(Lk) that respects the filtration by order of vanishing alongD. The nextis a special case of a construction from [23].


Definition 1.11. Let

Φk(t) :=1

kln(

∑

i

etνα |sα|2)

ThePhong-Sturmray is the limit

(2) Φt := limk→∞

(supl≥k

Φl(t))∗.

In [29] it is shown that the Legendre transform of the test curveψλ = λ|sD|2 equals the

Phong-Sturm ray, namelyφt = Φ(t).

So formally differentiating with respect tot, and ignoring various exchanges of limits,

d

dt

∣∣∣∣t=0+

φt =d

dt

∣∣∣∣t=0+

Φt ≃d

dt

∣∣∣∣t=0+

limk→∞

Φk(t)

= limk→∞

d

dt

∣∣∣∣t=0+

1

kln(

∑

i

etνα |sα|2φ)

= limk→∞

∑να|sα|

2φ

k∑

|sα|2φ.

Thus Theorems 1.9 and 1.10 combine to give the same conclusion (almost everywhere),in that both sides are in fact equal toH .

We end with a remark concerning the connection between the exhaustion functionH(φ, ln |sD|

2) associated to a divisorD = {sD = 0} and the geometry of the Okounkovbody∆(X,L) taken with respect to a flag with divisorial partD.

Theorem 1.12.LetH = H(φ, ln |sD|2) and letp : ∆(X,L) → R be the projection to the

first coordinate. ThenH∗(MA(φ)) = p∗dσ

wheredσ denotes the Lebesgue measure onRn.

This theorem is really nothing more than an unwinding of the definitions and an appli-cation of the technical results above. It partly resembles the Duistermaat-Heckman push-forward property of the moment map in toric geometry, and forthis reason we think ofHas a kind of weak “Hamiltonian” arising fromφ andD.

Comparison with other works: In the time between this article first appearing in preprintform and its publication, there have been some developmentsdirectly related that the readermay like to be aware of. It turns out that the envelopes considered in this paper are inti-mately related to the classE(X,ω) of ω-plurisubharmonic functions with finite weightedMonge-Ampere energy introduced by Guedj-Zeriahi [18]. Infact, Darvas proves in [11,Theorem 3] that this class can be characterised using the envelopesP[ψ](φ) (see Remark3.9). Similar envelopes have been studied by Darvas-Rubinstein, with a similar regularityresult to the one proved here given in [12, Theorem 2.5].

Acknowledgments: We wish to thank Bo Berndtsson, Robert Berman, Julien Keller,Reza Seyyedali, Ivan Smith and Richard Thomas for discussions about this work. We alsowith to thank Alexander Rashkovskii for pointing out an error in an earlier version of thispreprint. During this project the first author has been supported by a Marie Curie Grantwithin the 7th European Community Framework Programme and by an EPSRC CareerAcceleration Fellowship.


2. PRELIMINARIES

2.1. Singular metrics. LetX be a Kahler manifold of complex dimensionn, andL be aline bundle onX. A hermitian metrich = e−φ onL is a choice of hermitian scalar producton the complex lineLp at each pointp on the manifold. Iff is a local holomorphic framefor L onUf , we write

|f |2h = hf = e−φf ,

whereφf is a function onUf . We say thatφ is continuous if this holds for eachφf (withanalogous definitions for smooth, Lipschitz,C1,1 etc.). It is standard abuse of notationto let φ denote the metrich = e−φ and to confuseφ with φf if a given frame is to beunderstood. Thus ifφ is a metric onL, kφ is a metric onkL := L⊗k.

The curvature of a smooth metric is given byddcφ which is the(1, 1)-form locally de-fined asddcφf , wheref is any local holomorphic frame anddc is the differential operator

i

2π(∂ − ∂),

so ddc = (i/π)∂∂. The curvature form of a smooth metricφ is a representative for thefirst Chern class ofL, denoted byc1(L). A smooth metricφ is said to bestrictly positiveif ddcφ is strictly positive as a(1, 1)-form, i.e. if for any local holomorphic framef, thefunctionφf is strictly plurisubharmonic.

A positive singular metricis a metric that can be written asψ := φ + u, whereφ is asmooth metric andu is addcφ-psh function, i.e.u is upper semicontinuous andddcψ :=ddcφ + ddcu is a positive(1, 1)-current. For convenience we also allowu ≡ −∞. Thesingular locus ofψ will be denoted bySing(ψ) is the set on whichψ is not locally bounded.We letPSH(L) denote the space of positive singular metrics onL. If Sing(ψ) is emptywe sayψ is locally bounded(we will mostly consider the caseX is compact in which casethis is equivalent to beingglobally bounded).

We note thatPSH(L) is a convex set, since any convex combination of positive metricsyields a positive metric. Moreover ifψi ∈ PSH(L) for i ∈ I are uniformly boundedabove by some fixed positive metric, then the upper semicontinuous regularisation of thesupremum denoted by(sup{ψi : i ∈ I})∗ lies inPSH(L) as well. Ifψ ∈ PSH(L), thenthe translateψ + c wherec is a real constant is also inPSH(L).

A plurisubharmonic functionu on a setW is maximal if for every relative compactU ⊂ W and upper semicontinuous functionv on U with v ∈ PSH(U) the inequalityv ≤ u on∂U impliesv ≤ u on all ofU .

If ψ andφ are metrics onL and there exists a constantC such thatψ ≤ φ + C, we saythatψ is more singularthanφ. When specific mention of the constantC is unimportantwe shall write this asψ ≤ φ+O(1). Of courseψ ≤ φ+O(1) if and only ifψ ≤ φ+O(1)holds on some neighbourhood ofSing(φ) and we will use this in the sequel without furthercomment.

More generally, ifψ in a metric onL1 andφ a metric onL2 we will write ψ ≤ φ+O(1)to mean there is a locally bounded metricτ on L1 ⊗ L∗

2 such thatψ ≤ φ + τ . Theconditionψ ≤ φ + O(1) andφ ≤ ψ + O(1) is an equivalence relation, which we denotebyψ ∼ φ+O(1), and following [15] an equivalence class[ψ] is called asingularity type.

If ψi is a metric onFi for i = 1, 2 then by abuse of notation we will occasionally write

(3) sup{ψ1, ψ2}


to mean the metric onF1+F2 given bysup{ψ1+φF2 , φF1 +ψ2} whereφFi is a choice ofglobally bounded metric onFi. Thus the singularity type ofsup{ψ1, ψ2} is independentof choice ofφFi .

Given a coherent analytic ideal sheafI ⊂ OX and a constantc > 0 we say thatψ hasanalytic singularities modeled on(I, c) if X is covered by open setsU on which we canwrite

(4) ψ = c(ln∑

|fi|2) + u

wherefi are generators forI(U), andu is a smooth function. IfI is algebraic,c isrational and we can arrange this to hold in the Zariski topology then we sayψ hasalgebraicsingularitiesmodeled on(I, c).

We say that a singular metricψ isexponentially Holder continuouswith exponentc > 0,if it is smooth away fromSing(ψ) and over the singular locus satisfies

|eψ(x) − eψ(y)| ≤ C|x− y|c

for some constantC (here we are taking a local expression forψ thought of as a functionon some coordinate chartU and the norm on the right hand side is taken to be the Euclideannorm on the coordinates). A metric isexponentially Lipschitzif it is Holder continuous withexponentc = 1. Note that ifψ has analytic singularities as in (4) then it is exponentiallyHolder continuous with exponentc.

Given a metricψ themultiplier idealI(ψ) is the ideal generated locally by holomorphicfunctionsf such that|f |2e−ψ ∈ L1

loc. So ifψ ≤ φ+O(1) then clearlyI(ψ) ⊂ I(φ).

TheMonge-Ampere measureof a metricφ is defined as the positive measure

MA(φ) := (1/n!)(ddcφ)n.

Whenφ is smooth this is defined by taking the wedge product of the(1, 1) formsddcφin the usual sense. Through the fundamental work of Bedford-Taylor, the Monge-Amperemeasure can in fact be defined on the set on whichφ is locally bounded, and this mea-sure does not put any mass on pluripolar sets (i.e. sets that are locally contained in theunbounded locus of a local plurisubharmonic function).

2.2. Augmented Base Locus.LetL be a big line bundle. The base locus ofL is the set

Bs(L) =⋂

s∈H0(L)

{x : s(x) = 0}

and the stable base locus isB(L) =⋂k Bs(kL). We denote byB+(L) the augmented

base locus ofL which is given by

B+(L) = B(L − ǫA) for any small rationalǫ > 0

whereA is any fixed ample line bundle onX . It is a fact thatL is ample if and only ifB+(L) is empty, and is big if and only ifB+(L) 6= X [10, Example 1.7].

2.3. Partial Bergman Functions. Now suppose we fix a smooth volume formdV onX .Then any metricφ onL induces anL2-inner product onH0(Lk) for all k, whose norm isgiven by

(5) ‖s‖2φ,dV =

∫

X

|s|2kφdV for s ∈ H0(Lk).

We will omit thedV from the notation when the volume form is understood.


Definition 2.1. Thepartial Bergman functionassociated toφ andψ is the function

Bk(φ, ψ) =∑

α

|sα|2φ

where{sα} is anyL2-orthonormal basis forH0(I(kψ)Lk).

If ψ is locally bounded then the associated multiplier ideal sheaf is trivial, andBk(φ, ψ)becomes the usual Bergman function forφ which for simplicity we shall denote byBk(φ).Thusk−1 lnBk(φ, ψ) + φ is a metric onL with singularities modeled on(I(kψ), k−1).

Remark 2.2. The partial Bergman function depends on the choice of smoothvolume form,but it is easy to verify directly that the limitk−1 lnB(φ, ψ) ask tends to infinity does notsince the quotient of any two volume forms is globally bounded.

Example 2.3. LetY ⊂ X be a smooth subvariety of codimensionr which is given by theintersection of a finite number of sections of some line bundleF . Then we can define asingular metricψ = ln

∑i |si|

2. To calculate the multiplier ideal letπ : X → X be theblowup alongY with exceptional divisorE and canonical divisorKX = π∗KX + (r −1)E. Then, by smoothness ofY , π∗OX(−uE) = IuY for all u ≥ 0. Following [14, 5.9]one computesI(kψ) = π∗OX((r − 1− k)E) = Ik−r+1

Y . ThusBk(φ, ψ) is precisely thepartial Bergman kernel consisting of sections that vanish to a particular order alongY(which for toric manifolds is studied in[26, 27]).

2.4. The Ohsawa-Takegoshi extension theorem.We will need the following version ofthe Ohsawa-Takegoshi extension theorem. To state it cleanly we shall say that a metricφFon a vector bundleF has theextension property with constantC if for any x ∈ X andζ ∈ Fx there is ans ∈ H0(X,F ) with s(x) = ζ and

‖s‖φF ≤ C|s(x)|φF .

Theorem 2.4. (Ohsawa-Takegoshi) Supposec > 0 is given. Then there exists ak′ and aC′ such that the following holds: for allk ≥ k′ and allL′ andF ′ with singular metricsφL′ andφF ′ onL′, F ′ respectively such that

(6) ddcφL′ ≥ cω ddcφF ′ ≥ −cω

the metrickφL′ + φF ′ has the extension property with constantC′.

This well-known statement is a consequence of the more general result proved in [13,Proposition 12.4]

3. MAXIMAL ENVELOPES

Fix a complex manifoldX (which we shall assume is compact unless indicated other-wise) along with line bundlesL andF . Let φ be a (not necessarily positive) continuousmetric onL and pickψ ∈ PSH(F ). Fix also a smooth metricφF onF and define

ψ′ = φ− φF + ψ.

Definition 3.1. LetPψ′ be the envelope

(7) Pψ′φ := sup{γ ≤ min{φ, ψ′}, γ ∈ PSH(L)},

and define

P[ψ]φ := limC→∞

Pψ′+Cφ = sup{γ ≤ φ, γ ≤ ψ′ +O(1), γ ∈ PSH(L)}.


The notation is justified by the observationP[ψ]φ is independent of the choice ofφF(because the latter is globally bounded) and thus depends only on the singularity type[ψ]. Sincemin{φ, ψ′} is upper semicontinuous, it follows that the upper semicontinuousregularisation ofPψ′φ is still less thanmin{φ, ψ′}, and thusPψ′φ ∈ PSH(L). HencePψ′(Pψ′φ) = Pψ′φ, i.e.Pψ a projection operator toPSH(L). ClearlyPψφ is monotonewith respect to bothψ andφ. We shall always assume thatL is pseudoeffective, otherwisePSH(L) will be empty and the above envelopes will be identically−∞.

Definition 3.2. Themaximal envelopeof φ with respect to the singularity type[ψ] is

φ[ψ] := (P[ψ]φ)∗

where the star denotes the upper-semicontinuous regularisation. Theequilibrium setasso-ciated toφ andψ is

D = D(φ, ψ) = {x ∈ X : φ[ψ](x) = φ(x)}

Clearly thenφ[ψ] ∈ PSH(L) andφ[ψ] ≤ φ.

Example 3.3 (Trivial Singularities). If ψ is locally bounded thenφ ≤ ψ′ + C for Csufficiently large, and thus

φ[ψ] = Pψ′+Cφ = sup{γ ≤ φ : γ ∈ PSH(L)}.

These are exactly the envelopes considered by Berman in[3].

Remark 3.4. In the locally bounded case, maximal envelopes are examplesof metrics withminimal singularities in that ifγ is any other positive metric onL thenγ ≤ φ[ψ] +O(1).

Example 3.5 (Divisorial singularities). SupposeD is a smooth divisor inX andF =OX(D) with singular metricψ = ln |sD|

2 wheresD is the defining function forD. Then

φ[ψ] = sup{γ ≤ φ : γ ∈ PSH(L), νD(γ) ≥ 1}∗

whereνD denotes the Lelong number alongD. This case is considered by Berman[4, Sec.4].

Example 3.6 (Pluricomplex Green Function). For non-compactX , takingL to be thetrivial bundle andφ = 0 the trivial metric, the maximal envelope becomes the pluricom-plex Green function onX . Whenψ has analytic singularities this has been studied byRashkovskii-Sigurdsson[28]. We remark in passing that the pluricomplex Green functionon compact manifolds with boundary has recently been considered by Phong-Sturm[24],but is more commonly studied on domains inCn along with a boundary condition, forwhich it has a long and rich history (see[9] and the references therein).

Remark 3.7 (Invariance under holomorphic automorphisms). If φ and ψ are invariantunder some groupG of holomorphic automorphism of(X,L) then the same is true ofPψ′φ, P[ψ] andφ[ψ]. The proof is immediate, for ifθ is such an automorphism thenPψ′φ ◦θ ∈ PSH(L) and is bounded bymin{φ, ψ′} and thus also bounded byPψ′φ. Applyingto the inverse ofθ then yieldsPψ′φ ◦ θ = Pψ′φ. Thus there is no loss in replacing theenvelope in(7) with thoseγ that are invariant underG.

Example 3.8(Toric metrics). Consider now the case of a toric varietyX with torus in-variant L which we assume is ample. Let∆ be the associated Delzant polytope inRn.Lettingzi be complex coordinates on the large torus inX , any hermitian metricφ onLdescends to a convex function onRn after the change of variablesxi = ln |zi|

2 which byabuse of notation we denote byφ(x). Moreoverφ is positive if and only ifφ(x) is convex.


Thus ifψ is locally bounded (and so irrelevant) thenφ[ψ](x) is simply the convex hull thegraph ofφ(x) (see[3, 5.2]).

Suppose instead we haveψ = λ ln |z1|2 for some fixedλ > 0. Thenφ[ψ](x) is the

supremum of all convex functionsγ on Rn bounded byφ such that forx1 ≫ 0 we haveγ ≤ λx1 + C for someC.

Remark 3.9. In the first version of this paper it was noted that it is not obvious if the max-imal envelopeφ[ψ] has the same singularity type asψ. Whenψ has analytic singularitiesthis can be shown rather easily by passing to a smooth resolution (see[28] which alsocontains an alternative proof). This topic has since been taken up by Darvas[11, Theorem3] who shows that in general this is not the case, and gives an interesting criterion for it tohold in terms of a certain natural classE(X,L) of positive metrics onL.

4. EXTENSION OF RESULTS OFBERMAN

In this section we extend some results of Berman to the maximal envelopes consideredin this paper. What follows is essentially due to Berman, which in turn is based on the workof Bedford-Taylor [1]. The exposition here follows closely[3] which in fact announces thatsuch an extension should hold [3, Sec 1.3]. The related work [4, Sec 4] deals with the caseof envelopes that appear from order of vanishing along a divisor and [19] proves relatedresults in the case of general graded linear series.

4.1. Logarithmically homogeneous plurisubharmonic functions. We first describe ageneral framework which allows us to pass from metrics over acompact space to metricsover an auxiliary non-compact space. Recalling thatL is a line bundle overX , let Y bethe total space of the dual bundleL∗ andπ : Y → X be the projection.

ConsiderY as a subset of the compactificationY := P(L∗ ⊕ C) whereC denotes thetrivial line bundle overX . Then overY the hyperplane line bundleOY (1) has a sections ∈ H0(Y ,OY (1)) given by the constant section on the factorC. We let

ζ := ln |s|2 ∈ PSH(OY (1))

which is well-defined up to the addition of a constant. We writeOY (1) for the restrictionof OY (1) to Y , and denote the restriction ofζ to Y by the same letter. Thus, concretely, ifw is a local coordinate on the fibre direction ofY , thens is given locally by the equationw = 0 and so

ζ = ln |w|2.

Finally, to any metricγ onL we define

γ = π∗γ + ζ.

Definition 4.1. The set oflogarithmically homogeneous plurisubharmonic functionsonYis defined to be

(8) PSHh(Y ) := {χ ∈ PSH(π∗L⊗OY (1)) : χ(λy) = ln |λ|2 + χ(y) : λ ∈ C∗},

where the multiplication byC∗ is taken in the fibre direction ofY .

Thus the mapγ 7→ γ gives a bijection betweenPSH(L) andPSHh(Y ). Moreoverthis bijection respects taking envelopes, which we make precise in the following lemma.DefineP

ψ′φ andP[ψ]φ andφ[ψ] exactly as in Section 3, where now the supremum in (7) is

taken over allγ in PSHh(Y ).


Lemma 4.2. We havePψ′φ = Pψ′φ andP[ψ]φ = P[ψ]φ andφ[ψ] = φ[ψ].

Proof. Forφ, ψ′ ∈ PSH(L)

Pψ′φ = π∗Pψ′φ+ ζ = supγ∈PSH(L)

{π∗γ + ζ : γ ≤ min{φ, ψ′}}

= supγ∈PSHh(Y )

{γ : γ ≤ min{φ, ψ′}}

which gives the first identity. Moreover

P[ψ]φ = ζ + π∗ limC→∞

Pψ′+Cφ = limC→∞

(π∗Pψ′+Cφ+ ζ)

= limC→∞

Pψ′+Cφ = limC→∞

Pψ′+C

φ = P[ψ]φ

where the penultimate equality uses part (a) andψ + C = ψ + C. Finally

(P[ψ]φ)∗ = (P[ψ]φ)

∗ = (π∗(P[ψ]φ) + ζ)∗

= π∗(P[ψ]φ∗) + ζ = φ[ψ] = φ[ψ] + ζ

where the third inequality uses the fact thatζ is upper-semicontinuous and the local (andelementary) fact that(f(w) + g(z))∗ = f∗(w) + g∗(z). �

4.2. Exponential holomorphic coordinates. Now fix a continuous metricφ on L. Wechoose a smooth metricφF onF and setψ′ := φ− φF + ψ. To ease notation let

τC := Pψ′+Cφ

so by definitionφ[ψ] = ( lim

C→∞τC)

∗.

Our aim is to show regularity around a fixed point

x0 ∈ X − B+(L− F ) ∪ Sing(ψ).

From the assumption thatL − F is big there is, fork ≫ 0, a Kodaira decompositionk(L−F ) = A+E whereA is ample andE is an effective divisor inX . SinceB+(L−F )is the intersection over all suchE ask varies, we can arrange so thatx0 /∈ E. Furthermorethere exists a positive metricφ+ onL− F of the form

φ+ = k−1(φA + ln |sE |2),

whereφA is a smooth positive metric onA andsE is the defining section ofE.

Observe that for anyk ≥ 1.

Pkψ′ (kφ) = kPψ′φ and (kφ)[kψ] = kφ[ψ].(9)

Thus by scalingL,φ andψ we may assume without loss of generality thatk = 1. Moreoversinceψ was assumed to be exponentially Holder continuous,kψ is exponentially Lipschitzfor sufficiently largek. Thus there is no loss in assumingψ is exponentially Lipschitz.Furthermore, by subtracting a constant fromφ+ if necessary we can also arrange

(10) φ+ + ψ ≤ τC for all C ≥ 0.

Now letY0 = Y − (j(X) ∪ π−1(E ∪ Sing(ψ))


wherej : X → Y is the inclusion ofX as the zero section, and define a disc bundle inYby

U = {φ+ + π∗ψ ≤ 1}.

To show regularity ofφ[ψ] nearx0 it is, by Lemma 4.2, sufficient to show regularity ofφ[ψ]near a chosen point in the fibreπ−1(x0). To this end pick such a point,

y0 ∈ π−1(x0) ∩ U − j(X),

so we havey0 ∈ U ∩ Y0. The method of proof now follows the original approach ofBedford-Taylor, using particular holomorphic coordinates aroundy0 that arise from holo-morphic vector fields constructed in the next lemma (in fact this is the reason for passingto the auxiliary spaceY since, in general,X may have no holomorphic vector fields at all).

Lemma 4.3. There exist global holomorphic vector fieldsV1, . . . , Vn+1 onY whose re-striction aty0 spanTy0Y . Moreover, given any positive integerm there is a constantCmso that these vector fields can be made to satisfy

(11) |Vi(z, w)|2 ≤ Cmmin{|w|2m, |sE(z)|

2memψ(z)} onU.

Proof. The spaceY has the compactificationY = P(L∗ ⊕ C), and we shall denote thetautological bundle byOY (−1). Fix a continuous metricφE onE and consider the metriconL given by

φ+,k = φA + (1 + k−1/2)(ln |sE |2 + ψ)− k−1/2σ

whereσ is a fixed choice of smooth metric onE+F . Observe that fork ≫ 0 we will haveddcφ+,k ≥ (1/2)(ddcφ+) sinceψ is positive.

Now define a metric on the line bundleL = π∗Lk ⊗OY (1) overY by

φ := π∗kφ+,k + ln(1 + eφ)

which extends to a metric over all ofY which has strictly positive curvature fork suffi-

ciently large. Then by the Ohsawa-Takegoshi Theorem (2.4) the vector bundleTY ⊗ Ll

is globally generated forl ≫ 0. Fix such anl with kl ≥ m. Thus there are sections

W1, . . . ,Wn+1 whose evaluation aty0 spanTYy0Ll. Thinking ofw as the tautological

section ofπ∗L∗ we letVi be the restriction ofWiwkl to Y . Thus theVi are vector fields

onY and whose evaluation aty0 spanTy0Y .

Now the vector fieldsWi are holomorphic, so have bounded supremum norm overYand so also overY . Thus locally on a fixed neighbourhood inY we have

|Vi(z, w)|2 ≤ C1|w|

2klelφ = C1|w|kleklφ+,kelφ

≤ C2|w|2kl((|sE |

2eψ)k(1+k−1/2)|w|2)l

≤ C3(|w|2|sE |

2eψ)(k+1)l|sE |2memψ

which yields the required bound on|Vi| sinceφ+ + π∗ψ = ln(|w|2|sE |2eψ) + φA, so

‖w‖2‖sE|2eψ is bounded onU = {φ+ + π∗ψ ≤ 1}. �

Remark 4.4. It is the hypothesis thatL−F is big that allows for the fact that the conclu-sion of the previous Lemma is stronger than[3, Lemma 3.6]on which it is based.


Lemma 4.5. Let V be a smooth vector field on a manifoldY , andY0 ⊂ Y be such thatV = 0 onY−Y0. SupposeU ⊂ Y is a closed subset, andα : Y0 → Y0 is a diffeomorphism.

W := α(U ∩Y0) is relatively compact and so that the vector fieldα∗V can be extendedto a vector field onW that vanishes on the boundaryW −W . Then there exists at0 suchthat the flowexp(tV )(y) exists for all|t| ≤ t0 and ally ∈ U .

Proof. Denote the extension ofα∗V by V . By compactness ofW there is at0 such thatflow of V exists for any|t| ≤ t0 and initial pointy ∈ W . SinceV vanishes onW −W , wehave that ify ∈W then this flow remains completely withinW . However this is preciselythe image (underα) of the flow ofV , which proves that the flow ofV exists for all|t| ≤ t0andy ∈ U ∩ Y0. Finally if y ∈ U − Y0 theny is fixed by the flow ofV , and thus the flowexists for all time in this case as well. �

Lemma 4.6. There exists at0 > 0 such that the flowy 7→ exp((∑

i λiVi)y) exists for all(λ, y) such that|λ| ≤ t0 andy ∈ U .

Proof. We apply the previous lemma to the vector fieldV = Vi for i = 1, . . . , n + 1 andY0 = Y − j(X) ∪ π−1(E ∪ Sing(ψ)). Picking smooth metricsφF andφE onF andErespectively, defineα : Y0 → Y0 by

α(ζ) = fζ wheref = |sE |φEe(ψ−φF )/2

and the multiplication is to be understood in the fibre direction ofY , andψ−φF is thoughtof as a function onX . Recall also that included in the assumption thatψ is exponentiallyLipschitz is thateψ is smooth away fromSing(ψ). Thus, by construction,α is a diffeo-morphism. Now a simple calculation reveals

(φ+ + π∗ψ) ◦ α = φA + π∗(φF − φE) onY0,

and thus ifW = α(U ∩ Y0) thenW = α(U ∩ Y0) = {φA + π∗(φF − φE) ≤ 1}

which is compact. Now setV = α∗V and suppose that locally in(z, w) coordinateV =

vz∂∂dz + vw

∂∂dw and similarly for V . Writing α locally as(z, w) 7→ (z, w) we have

vz(z, w) = vz(z, w) and

vw(z, w) =∂f

∂z

vzfw + vw(z, w)f(z).

Now the fact thatψ is exponentially Lipschitz implies∂f∂z is globally bounded. Thus takingm = 2 in the estimate in (11) boundsf−1vz by |sE |e

ψ. HenceV extends overW andvanishes onW −W , so the previous lemma applies to give the result. �

4.3. Proof of Lipschitz regularity. SupposeV1, . . . , Vn+1 are the vector fields providedby Lemma 4.3 whose evaluation aty0 spanTy0Y takingm = 2. Consider the flow

θλ(y) = exp(

n+1∑

i=1

λiVi)(y),

which by Lemma 4.6 is well defined fory ∈ U and|λ| sufficiently small. Then for anyfunctionf onU denote the pullback function by

fλ(y) := f(θλ(y))

Thus to show that a functionf onY is Lipschitz neary0 it is sufficient to prove

|fλ(y)− f(y)| ≤ C|λ|


for some constantC andy in some neighbourhood ofy0.

Lemma 4.7.

(1) Letφ0 be a metric onL such thatφ0− δ1|sE |2− δ2ψ is smooth for some constant

δi ≥ 0. Then there is a constantCα so that

|φ0λ− φ0| ≤ Cα|λ|

onU ∩ Y0.(2) Letφ0 be a metric onL such thatφ0 − δ|sE |

2 is smooth for some constantδ ≥ 0.Then there is a constantCα so that for all multiindicesα of total order at mosttwo

|∂αz,w(φ0λ− φ0)| ≤ Cα|λ|

onU ∩ Y0.

Proof. Fory ∈ U ∩ Y0 let

γ(t) = exp(t

n+1∑

i=1

λiVi)(y),

soγ(0) = y andγ(1) = θλ(y). Now φ0 is smooth away fromj(X)∪ π−1(E ∪ Sing(ψ)),so we can write

(12) φ0λ(y)− φ0(y) =

∫ 1

0

dφ0|γ(t)

(dγ

dt

)dt.

We shall prove the first statement. Pick local coordinatesz onX andw onL aroundsome point inY . Then

(13) dφ0(dγ

dt) =

(w−1dw + dφ0

)(∑

λiVi)

From (11),|Vi| ≤ C1|w|2, so|w−1dw(Vi)| is uniformly bounded and thus

|w−1dw(∑

i

λiVi)| ≤ C2|λ|,

for some constantC2.

If our chart lies outsideπ−1(E ∪ Sing(ψ)) then φ0 is smooth so the second term in(13) is clearly bounded by a constant times|λ|. To deal with the case that the chart meetsE ∪ Sing(ψ), we use the hypothesisφ0 − δ1 ln |sE |

2 − δ2ψ is smooth to deduce

|dφ0(Vi)| ≤ C4 + δ1

∣∣∣∣1

sE

∂sE∂x

(Vi)

∣∣∣∣+ δ2e−ψ

∣∣∣∣∂eψ

∂x(Vi)

∣∣∣∣

where the derivative ofeψ exists weakly (and is globally bounded) by the assumption thateψ is Lipschitz. Now using (11),|Vi|2 ≤ C5 min{|sE |

2, eψ}, so

|(w−1dw + dφ0

)(∑

λiVi)| ≤ C5|λ|

for some constantC5. Putting all of this together with (13) and (12) gives the first statementof the Lemma. The proof of the second is exactly the same, observing that the assumptiononφ0 now means we do not need to take any further derivatives ofψ. �


Corollary 4.8. For |λ| sufficiently small we have

ψλ = ψ +O(1)

onU ∩ Y0.

Proof. Apply the previous Lemma withφ0 = ψ. �

Lemma 4.9. Suppose thatf is anS1-invariant plurisubharmonic function onU that isstrictly increasing on the fibres ofY . Suppose thatE′ ⊂ X is locally pluripolar, and thatthere is a constantc such thatf < c on j(X − E′) andf > c on ∂U ∩ π−1(X − E′)

where∂U = {φ+ + π∗ψ = 1}. Then there exists an extensionE(f) ∈ PSHh(Y ) suchthatE(f) = f on the level set{f = c}.

Proof. This is [3, Lemma 3.8] (thatE′ is allowed to be pluripolar is remarked in the proofof the cited result). �

Now suppose thatg is some function defined on the disc bundleU ⊂ Y . To applythe previous result we need invariant functions, which are obtained easily through the ho-mogenisation operator that takes a functiong onU to

H(g) = ( supθ∈[0,2π]

g(eiθy))∗

where the multiplication is in the fibres ofY = L∗.

Lemma 4.10. Suppose that either

f = H(τCλ)

or

f = H

(1

2(τC

λ+ τC

−λ)

).

Then for|λ| sufficiently small (independent ofC) there is an extensionE(f) ∈ PSHh(Y )such thatE(f) = f on the level setS = {y : f(y) = f(y0)}.

Proof. We shall only considerf = H(τCλ) since the other case is essentially the same. It

is sufficient to construct such an extension onY0, since any such plurisubharmonic functionextends toY . Now exactly as in the first half of the proof of [3, Lemma 3.9] one deducesthatf is strictly increasing along the fibres ofY , sinceτC is plurisubharmonic.

We make the normalisation soτC(y0) = 0, and observe thatf(y0) ≤ 1/2 for λ suffi-ciently small. Then sinceφ+ + ψ ≤ τC

f ≥ τCλ≥ φ+ + ψ

λ≥ φ+ + π∗ψ − C′|λ|

overU ∩ Y0 for someC′ > 0 (independent ofC) where we have used (4.7) withφ0 :=φ+ + ψ soφ0 − ln |sE |

2 − ψ = φA is smooth.

Thus on∂U ∩ Y0 we have

f ≥ 1− C′|λ| ≥ 1/2

for |λ| sufficiently small (independent ofC). Thus the existence of the extensionf isprovided by (4.9) usingE′ = E ∪ Sing(ψ). �

Lemma 4.11. Suppose thatφ is Lipschitz. Thenφ[ψ] is Lipschitz overY0 and thusφ[ψ] isLipschitz overX − B+(L− F ) ∪ Sing(ψ).


Proof. Fix C and letτC = Pψ+Cφ. Then from the definition of the operatorH we have

τCλ(y0) ≤ H(τC

λ)(y0) ≤ EH(τC

λ)(y0)

where the operatorE is the extension coming from Lemma (4.10) from the level setS =

{y : H(τCλ)(y) = H(τC

λ)(y0)}. As τC ≤ φ we have that on the level setS,

EH(τCλ)(y) = H(τC

λ)(y) ≤ supθ∈[0,2π]

φλ(eiθy)(14)

≤ supθ∈[0,2π]

φ(eiθy) + C′|λ| = φ+ C′|λ|(15)

for some constantC′ (independent ofC) where we have used thatφ is Lipschitz in thepenultimate inequality to apply Lemma 4.7(2) withφ0 = φ, and the final equality comesfromS1 invariance ofφ in the fibre directions ofY . Since this holds on the level setS weconclude by homogeneity that

EH(τCλ)− C|λ| ≤ φ

on all ofY .

Now note that by Lemma 4.7(1) applied withφ0 = ψ we see there is a constantC′′ suchthatψλ ≤ ψ + C′′. Thus using the inequalityτC ≤ ψ′ + C we have by similar reasoningto above

EH(τCλ)(y) = H(τC

λ)(y) ≤ sup

θ∈[0,2π]

ψ′ + Cλ(eiθy) = sup

θ∈[0,2π]

ψ′λ(eiθy) + C′′

≤ supθ∈[0,2π]

ψ′(eiθy) + C′|λ|+ C′′

= ψ′ + C′′(y) + C′|λ|

Thus we conclude thatEH(τCλ)−C′|λ| ≤ ψ′ +C′′ onY . HenceEH(τC

λ)−C|λ| is a

candidate for the supremum appearing in the definition ofPψ′+C′′

φ giving

τCλ − C′|λ| ≤ EH(τC

λ)− C′|λ| ≤ Pψ′+C′′

φ = τC′′ ≤ φ[ψ]

by Lemma 4.2. Now taking the limit asC tends to infinity and then the upper-semi-continuous regularisation (which commutes with pulling back byθλ) gives

φλ[ψ]

− C′|λ| ≤ φ[ψ]

Repeating the above withλ replaced with−λ gives the inequality

|φλ[ψ]

− φ[ψ]| ≤ C′|λ|

which proves thatφψ is Lipschitz neary0. Sincey0 was arbitrary inU ∩ Y0, this provesthatφ[ψ] is Lipschitz overX − B+(L− F ) ∪ Sing(ψ) as claimed. �

4.4. Proof of C1,1 regularity.

Theorem 4.12. Suppose thatL − F is big. If φ is Lipschitz (resp.C1,1) overX then thesame is true forφ[ψ] overX − B+(L− F ) ∪ Sing(ψ).


Proof. By the above the Lipschitz statement is proved. Thus the weakderivative ofφ[ψ]exists and is globally bounded. As explained in [3, p 21], to showφ[ψ] isC1,1 it is sufficientto prove the inequality

(16)1

2

(φλ[ψ]

+ φ−λ[ψ]

)− φ[ψ] ≤ C′|λ|2.

on some compact set aroundy0.

To achieve this, argue exactly as above replacingτλC with

gC =1

2(τC

λ + τC−λ)

Now a simple Taylor series show that12 (φ

−λ + φ−λ) ≤ φ+C|λ|2 over all ofX . The fact

thatgC ≤ ψ + C′′ is similarly shown, in fact is easier and only requires the exponentiallyLipschitz hypothesis onψ.

Thus we deduce thatgC − C′|λ|2 is a contender for the supremum definingτC , sogC − C′|λ|2 ≤ τC . Observing thatC′ is independent ofC, we letC tend to infinity andtaking the upper semicontinuous regularisation to give thedesired inequality. �

4.5. Monge Ampere measures.We next consider the Monge Ampere measure of themaximal envelopes. The proof of Theorem 1.2 is unchanged from the non-singular case,and we shall not repeat the full details of the arguments here.

Lemma 4.13. We have

D(φ, ψ) ⊂ {x : ddcφx ≥ 0},

and thus1DMA(φ) = 1D∩X(0)MA(φ).

Proof. This follows by observingD(φ, ψ) ⊂ D(φ, φ) and then using [3, 3.1(iii)] andExample 3.3 to deduceD(φ, ψ) ⊂ {x : ddcφx ≥ 0}. �

Lemma 4.14. We havedet(ddcφ[ψ]) = det(ddcφ) almost everywhere onB+(L − F ) ∪Sing(ψ).

Proof. The proof of this the same as [3, p21]. In fact the proof given there shows thatlocally for any twoC1,1 metricsφ andφ′ one has

∂2

∂z∂z(φ− φ′) = 0

almost everywhere on the set{φ = φ′}. �

The proof of the remaining equalities

µ(φ, ψ) = 1X−B+(L−F )∪Sing(ψ)MA(φ[ψ]) = 1DMA(φ) = 1D∩X(0)MA(φ),

as stated in Theorem 1.2 is now exactly as in [3]; we omit the details.

4.6. The partial Bergman function. We now turn to the partial Bergman function, andfix a smooth metricφ onL and aψ ∈ PSH(F ) that is exponentially Holder continuous.We start by recalling the following upper bound on the Bergman function. RecallX(0) ={x : ddcφx > 0}.


Lemma 4.15. (Local Holomorphic Morse Inequalities) There is a global upper bound

Bk(φ)dV ≤ Ck1X(0)MA(φ)

whereCk is a sequence of numbers that tends to1 ask tends to infinity. Moreover if∆k

denotes the ball of radiusk−1 ln k in some coordinate patchU around a fixed pointx ∈ X ,andfk is a sequence of holomorphic functions onU then

|fk(x)|2

‖fk‖2L2,∆k

≤ 1X(0)MA(φ)kn + o(kn)

where‖fk‖L2,∆k =∫∆k

|fk|2e−kφdV .

Proof. Both of these inequalities follow easily from the submean value inequality for holo-morphic functions (see [3, 4.1] or [5, Sec 2]). �

Recalling thatBk(ψ) denotes the Bergman function ofψ define theBergman metricas

ψk = ψ +1

klnBk(ψ).

Definition 4.16. We say thatψ hastame singularitieswith coefficientc > 0 if

(17) ψ +O(1) ≤ ψk ≤(1−

c

k

)ψ +O(1)

where theO(1) term is independent ofk.

The following condition for tameness is well known (see, forexample [7, 5.10]).

Lemma 4.17. If ψ is exponentially Holder continuous with constantc then it has tamesingularities with constantc−1 dimX .

Proof. Let n = dimX . Following Demailly, the Bergman metrics approximateψ in thefollowing sense [14]: letψ be defined on some open ballB; then there is a constantC > 0that depends only on the diameter ofB, such that for allk,

(18) kψ(x)− C ≤ kψk ≤ supB(x,r)

kψ + C − n log r

whereB(x, r) is the ball of radiusr centered atp with r is small enough so thatB(x, r) ⊂B. From this the lower bound in (17) follows immediately. For the upper bound, observefirst that sinceψ is exponentially Holder continuous with exponentc,

supB(x,r)

eψ ≤ eψ(x) + C1rc

for some constantC1 and thus

supB(x,r)

ψ ≤ ln(1 + C1) + ψ(x).

Moreover, from the same assumption, one sees there is aC2 such that ifr = C2eψ(x)/c

thenB(x, r) ∩ Sing(ψ) is empty. Hence applying (18) with this value ofr yields

kψk(x) ≤ k ln(1 + C1) + C + (k − nc−1)ψ(x)− n logC2

as required. �


Proposition 4.18. Supposeψ ∈ PSH(F ) has tame singularities with constantc. Thereis a constantC depending onφ andψ such that

Bk(φ, ψ) ≤ Ckne−cψek(φ[ψ]−φ)

over all ofX . In particular

limk→∞

k−nBk(φ, ψ) = 0 for x /∈ D(φ, ψ) ∪ Sing(ψ).

Proof. As is easily verified, the partial Bergman kernel has the extremal property

Bk(φ, ψ) = sup{|s(x)|2φ : ‖s‖φ,dV = 1, s ∈ H0(I(kψ)Lk)}.

Thus it is sufficient to prove the existence of aC such that ifs ∈ H0(I(kψ)Lk) and‖s‖φ,dV = k−n then

|s|2φ ≤ Ce−cψek(φ[ψ]−φ).

From the holomorphic Morse inequalities there is aC with k−nBk(φ) ≤ C for all k. Sosince|s|2φ ≤ k−nBk(φ, ψ) ≤ k−nBk(φ) we have

(19) k−1 ln |s|2 − k−1 lnC ≤ φ.

Moreover by the assumptions on the singularities ofψ,

k−1 ln |s|2 ≤ ψk +O(1) ≤ (1 − k−1c)ψ +O(1)

where theO(1) term may depends onk. Without loss of generality supposeψ ≤ 0 globally.Then we seek−1 ln |s|2 + ck−1ψ − k−1 lnC is bounded above by bothφ andψ + O(1),and thus is a candidate for the supremum definingP[ψ]φ, so

k−1 ln |s|2 + ck−1ψ − k−1 lnC ≤ φ[ψ]

and rearranging proves the first statement of the proposition, from which the second fol-lows immediately. �

Before moving on we observe that a slightly more precise statement is possible whenψ has algebraic singularities. Suppose that the singularities ofψ is modeled on(I, c). Wefix a resolutionπ : X → X such thatπ∗I = O(−D) whereD =

∑j αjDj is a normal

crossing divisor (see for example [14, 5.9] for this basic technique).

Definition 4.19. The set ofpotential jumping numbersfor ψ is

J(ψ) = {k : kαj ∈ N for all j}.

In the simplest case,ψ =∑

j αj ln |gj | globally withαj ∈ Q+ in which case it is clearthatJ(ψ) = {k : kαj ∈ N for all j}. It is clear in general that potential jumping numbersexist that are arbitrarily large. The terminology comes from the fact thatJ(ψ) restricts theset on which the multiplier idealsI(tψ) can “jump” ast ∈ R+ varies.

Proposition 4.20. There is a constantC (depending onφ) such that for allψ with alge-braic singularities, and allk ∈ J(ψ) we have

Bk(φ, ψ) ≤ Cknek(φ[ψ]−φ)

Proof. We first consider first the special case thatψ is of the formψ =∑

j αj ln |gj|2 with

Dj = g−1j (0) smooth normal crossing divisors. As in the previous proof wehave to show

that if s ∈ H0(I(kψ)Lk) with k ∈ J(ψ) and‖s‖φ,dV = k−n then|s|2φ ≤ Cek(φ[ψ]−φ).


Just as above, from the holomorphic Morse inequalities we havek−1 ln |s|2−k−1 lnC ≤ φwhereC is independent ofφ. Now ass lies in the multiplier ideal, locally we can write

s = hΠjg2mjj

for some holomorphich andmj ≥ ⌊kαj⌋ = kαj [14, 5.9] (here we have usedk ∈ J(ψ)sokαj ∈ N). Thus

|s|2 ≤ C′Πj |gj |2mj

for some constantC′ (which may depend ons). Therefore we deduce that nearD =Sing(ψ),

k−1 ln |s|2 ≤ k−1 lnC′ + k−1∑

j

mj ln |gj|2 ≤

∑

j

αj ln |gj |2 + C′′ = ψ + C′′.

Thusk−1 ln |s|2 − k−1 lnC is a candidate for the supremum definingφ[ψ] which givesthe result for this particular form ofψ.

Now for generalψ with algebraic we can reduce to the above by passing to log resolu-tion π : X → X . As is easily checked, ifψ =

∑j αj ln |gj |

2 whereDj = g−1j (0) then

π∗Bk(φ, ψ) = Bk(π∗φ, ψ) andπ∗ψ ∼ ψ, givingπ∗(φ[ψ]) = (π∗φ)[ψ]. Then applying the

previous part of the proof gives the result we require. �

We now turn to proving a lower bound for the partial Bergman function. Recall thatφFis a fixed smooth metric onF and we have setψ′ = φ− φF + ψ.

Proposition 4.21. Letω be a Kahler form onX . There exists ak′ andC′ > 0 such thatfor anyk ≥ k′, anyx ∈ X − B+(L− F ) ∪ Sing(ψ) and anyC > 0 there exists a metricσk = σk,C onLk such that

(1) σk,C ≤ kPψ′+Cφ onX(2) There is an neighbourhoodU of x and constantC′

U (both independent ofk andC) such that

kPψ′+Cφ ≤ σk + C′U onU.

(3) σk,C has the extension property with constantC′ (as defined in Section 2.4).(4) ddcσk,C ≥ C′−1ω for all k andC.

Proof. We may as well assumeL−F is big otherwise the statement is trivial. ThusL−Fadmits a metricφ+ with strictly positive curvature that is smooth away fromB+(L − F ).By subtracting a constant fromφ+ we may assume that

(20) φ+ ≤ min{φ− ψ, φ− φF }

which yieldsφ+ + ψ ≤ Pψ′φ ≤ Pψ′+Cφ for all C > 0.

Thus there is ac > 0 with ddcφ+ ≥ cω, so we can takek′ andC′ as in the statement ofthe Ohsawa-Takegoshi Theorem (2.4). Now letk ≥ k′ andx ∈ X−B+(L−F )∪Sing(ψ).Set

φF ′ = (k − k′)Pψ′+Cφ+ k′ψ

which is a metric onF ′ = Lk−k′

⊗ F k′

and let

σk = k′φ+ + φF ′ .

Observe thatddcφ+ ≥ cω andddcφF ′ ≥ 0, so the extension property (3) holds and alsoddcσk,c ≥ k′ddcφ+ so (4) follows immediately. Moreover

σk = k′(φ+ + ψ) + (k − k′)Pψ′+Cφ ≤ kPψ′+Cφ


which gives (1). Finally

kPψ′+C − σk = k′(Pψ′+C − ψ − φ+)

≤ k′(φ− ψ − φ+)

Hence ifU is a small ball aroundx in B+(L−F )∪ Sing(ψ) thenφ−ψ− φ+ is boundedonU by some constantC′

U , sokPψ′+C − σk ≤ C′U onU for all C > 0. �

Theorem 4.22.We havek−1 lnBk → φ[ψ] − φ

uniformly on compact subsets ofX −B+(L−F )∪ Sing(ψ) ask tends to infinity. That is,given a compact subsetK ⊂ X − B+(L− F ) ∪ Sing(ψ) there is aCK > 0 such that

C−1K ek(φ−φ[ψ]) ≤ Bk(φ, ψ) ≤ CKk

nek(φ−φ[ψ])

overK for all k.

Proof. The upper bound forBk comes from Proposition 4.18 sinceψ is bounded on anycompactK outside ofSing(ψ). For the other direction letk′ andC′ be as in the statementof 4.21 andx ∈ X − B+(L − F ) ∪ Sing(ψ). To ease notation letτC = Pψ′+Cφ. Pick ametricσk as in the previous proposition. So by the extension propertyof σk there exists ans ∈ H0(Lk) such that‖s‖σk ≤ C′ and|s(x)|σk = 1. But this implies that‖s‖kτC ≤ C′

and |s(x)|kτC ≥ (C′U )

−1. In particular‖s‖k(ψ′+C) is finite, and sos ∈ H0(I(kψ)L).Thus by the extremal property of the Bergman function we deduce that

Bk(φ, ψ) ≥ C′ek(τC−φ)

on some neighbourhoodU of x, for some constantC′ that is independent ofC andU .LettingC tend to infinity and then taking the upper-semicontinuous regularisation (usingthat the Bergman function is continuous) yields

Bk(φ, ψ) ≥ C′ek(φ[ψ]−φ)

onU as required. �

Lemma 4.23. We have

lim infk→∞

k−nBk(φ, ψ)dV ≥MA(φ)

almost everywhere onD(φ, ψ) ∩X(0).

Proof. It is sufficient to prove the existence of a sequencesk ∈ H0(Lk⊗I(kψ)) such that

limk→∞

|sk(x)|2φ

kn‖sk‖2φdV = MA(φ)x(21)

Sincex is general we may assumex /∈ B+(L − F ) ∪ Sing(ψ). So from Lemma 4.21there is an open setU aroundx and metricsσk,C ∈ PSH(Lk) for k ≥ 0 andC > 0 suchthat

σk,C ≤ kPψ′+C onX,(22)

kPψ′+C ≤ C1 + σk+C onU and,

ddcσk,C ≥ C−11 ω,

where the constantC1 is independent ofk andC andω is some chosen smooth Kahlerform.


Next we consider smooth sections ofLk that are “peaked” and supported inU . Usingcut-off functions, and thatφ[ψ] is C1,1, one can produce, for sufficiently general pointsx

in D(φ, ψ) ∩ X(0), a sequence of smooth sectionsfk of Lk that are supported inU andsatisfy

limk→∞

|fk(x)|2φ

kn‖fk‖2φdV = MA(φ)x

‖∂fk‖kφ[ψ]≤ C2e

−k/C2

for some constantC1 (see [3, Lemma 4.4]).

To perturb these to holomorphic sections we apply the Hormander estimate with themetricσk,C to obtain, fork sufficiently large, smooth sectionsgk,C of Lk with ∂gk,C =∂fk and‖gk,C‖σk,C ≤ C3‖∂fk‖σk,C (observeC3 can be taken independent ofC andkfrom the lower bound forddcσk,C from (22)). Now the first two statements in (22), andthe fact thatfk is supported onU imply

‖gk,C‖kφ ≤ ‖gk,C‖kPψ′+Cφ ≤ ‖gk,C‖σk,C ≤ C3‖∂fk‖σk,C ≤ C4‖∂fk‖kPψ′+C,

whereC4 is also independent ofC andk.

Temporarily fixingk, and recalling thatPψ′+C tends toφ[ψ] pointwise almost every-where asC tends to infinity, an application of the dominated convergence theorem yields

limC→∞

‖gk,C‖kφ ≤ C4‖∂fk‖kφ[ψ]≤ C5e

−k/C2 .

In particular choosingC = C(k) sufficiently large we deduce that for eachk there is anhk = gk,C(k) such that∂hk = ∂fk and

(23) ‖hk‖φ ≤ 2C5e−k/C2 .

Thussk := fk−hk is holomorphic and since‖sk‖kPψ′+Cis finite andPψ′+C ≤ ψ+C

we in fact havesk ∈ H0(Lk⊗I(kψ′)) = H0(Lk⊗I(kψ)). Moreover‖sk‖2φ ≤ ‖fk‖2φ+

O(k−∞) and, sincehk is holomorphic nearx, the local holomorphic Morse inequalityimplies |sk(x)|2φ = |fk(x)|

2φ + O(k−∞) as well. Thus (21) holds for this sequence of

sectionssk, completing the proof of the Lemma. �

Theorem 4.24.SupposeL− F is big. Then there is a pointwise limit

limk→∞

k−nBk(φ, ψ)dV = 1D(φ,ψ)∩X(0)MA(φ)

almost everywhere onX(0). Moreover

limk→∞

k−nBk(φ, ψ)dV → µ(φ, ψ)

weakly in the sense of measures.

Proof. We have shown thatlimk→∞ k−nBk(φ, ψ) = 0 for x /∈ D(φ, ψ) ∪ Sing(ψ), andthus almost everywhere outsideD(φ, ψ) sinceSing(ψ) is pluripolar and thus has measurezero. For generalx ∈ D(φ, ψ) the limit follows by combining the upper bound comingfrom the local holomorphic Morse inequalities, and the lower bound in the previous propo-sition. The statement about the measures now follows from the Dominated ConvergenceTheorem, and the global upper boundk−nBk(φ, ψ) ≤ k−nB(φ) ≤ C. �

In particular, by integrating the previous theorem overX we see the volume of theequilibrium setD(φ, ψ) captures the rate of growth of the filtration of the space of sectionsdetermined by the multiplier ideal ofψ:


Corollary 4.25. ∫

D(φ,ψ)

MA(φ) = limk→∞

k−nh0(Lk ⊗ I(kψ)).

5. MAXIMAL ENVELOPES ONPRODUCTS

In this section we shall consider maximal envelopes on products. Suppose thatXi fori = 1, 2 are smooth complex manifolds on which we have line bundlesLi with smoothmetricsφi. Our aim is to consider maximal envelopes on the productX1×X2 with respectto the product metric

φ := π∗1φ1 + π∗

2φ2

whereπi are the projection maps (for simplicity we shall suppress theπi in what followswhere it cannot cause confusion).

Let Fi be additional line bundles onXi with metricsψi ∈ PSH(Fi). For simplicitywe assume thatLi − Fi are ample, soB+(Li − Fi) is empty. Furthermore set

ψ′ = sup{ψ1, ψ2}

so, recalling the abuse of notation described in (3),ψ′ ∈ PSH(F1 + F2).

Theorem 5.1. Supposeψi have algebraic singularities. Then

φ[ψ′] = sup{(φ1)[λψ1] + (φ2)[(1−λ)ψ2] : λ ∈ (0, 1)}∗(24)

Said another way, such maximal envelopes on products can be calculated by consideringonly metrics whose variables separate, i.e.

φ[ψ′] = sup{γ1 + γ2, γi ∈ PSH(Li), γ1 ∼ λψ1, γ2 ∼ (1 − λ)ψ2, λ ∈ (0, 1)}∗.

As remarked in the introduction, such a formula resembles known results for the Siciakextremal function on products of domains inCn (see, for example, [2, 8, 30]) and for thepluricomplex Green function on products [28] and these results may suggest other methodsof proof. The techniques we employ here have an algebraic flavour, using results from theprevious section to recast the problem in terms of the partial Bergman function and thenapplying a combination of the Kunneth formula and the Mustata summation formula formultiplier ideals.

Proof of Theorem 5.1.Fix λ ∈ (0, 1) and supposeγi ∈ PSH(Li) with γi ≤ φi fori = 1, 2 andγ1 ∼ λψ1 andγ2 ∼ (1 − λ)ψ2. Thus there is a constantC such thatγi ≤ λψ1 + C andγ2 ≤ (1− λ)ψ2 + C, which yields

γ1 + γ2 ≤ λψ′ + C + (1 − λ)ψ′ + C = ψ′ + 2C.

Henceγ1 + γ2 ∼ ψ′ which impliesγ1 + γ2 ≤ φ[ψ′]. Taking the supremum over allγ1 andthen allγ2 yields

(φ1)[λψ1] + (φ2)[(1−λ)ψ2] ≤ φ[ψ′].

Thus taking the supremum over allλ ∈ (0, 1) shows the right hand side of (24) is less thanor equal to the left hand side.

Now asψi have algebraic singularities, it is immediate that (24) holds onSing(φ[ψ′]) =Sing(ψ′) = Sing(ψ1)× Sing(ψ2). So supposex ∈ X1 ×X2 is such thatφ[ψ′](x) 6= −∞and letǫ > 0. Writing L = L1 ⊗ L2, we know from Theorem 1.3 that fork sufficientlylarge there exists ans ∈ H0(I(kψ′)Lk) with ‖s‖φ = 1 and

k−1 ln |s(x)|2φ ≥ φ[ψ′](x)− ǫ.


Consider next the multiplier idealsI(tkψ) for t ∈ R+. These form an nested sequenceof ideal sheaves that induce a finite filtration

0 = H0(I(t1kψLk1) ⊂ H0(I(t2kψ)L

k1) ⊂ · · · ⊂ H0(I(tNkψ)L

k1) = H0(Lk1).

Pick anL2-orthonormal basis{aj} for H0(Lk1) that is compatible with this filtration,soaj ∈ H0(I(kλjψ1)L

k1)). Similarly pick anL2-orthonormal basis{bj} for H0(Lk2) so

bj ∈ H0(I(µjkψ2)Lk2). Then sinceH0(L) = H0(Lk1)⊗H0(Lk2) we can write

s =∑

ij

αijaj ⊗ bj

where∑

ij |αij |2 = 1. Now since all the metrics in question have algebraic singularities,

the Mustata summation formula [22] gives

I(kψ′) ⊂∑

λ+µ=1

I(kλψ′1)I(kµψ

′2).

Thus we deduceαij = 0 unlessλj + µj = 1.

Fix (i0, j0) so |ai0 ⊗ bj0(x)| ≥ |ai ⊗ bj(x)| for all i andj. Hence ifNk = h0(Lk)we have|s(x)|2 ≤ Nk|ai0 |

2|bj0 |2. Defineγ1 = k−1 ln |ai0 |

2 andγ2 = k−1 ln |bj0 |2 so

γi ∈ PSH(Li), and

φ[ψ′](x)− ǫ ≤ k−1 ln |s(x)|2 ≤ k−1 lnNk + γ1(x) + γ2(x).

Asψ1 is algebraic it is certainly tame (with constantc say). Thus from (4.18) we have

γ1(x) ≤ (φ1)[λjψ1](x) + k−1 lnC − k−1cλjψ1(x) ≤ (φ1)[λjψ1](x) + k−1 lnC′

sinceλj ≤ 1 and with a similar expression boundingγ2(x). Now Nk is bounded by apolynomial ink, for k sufficiently large,

φ[ψ′](x) − ǫ ≤ (φ1)[λjψ1] + (φ2)[µjψ2] + ǫ

≤ supλ∈(0,1)

{(φ1)[λψ1] + (φ2)[(1−λ)ψ2]}∗ + ǫ.

Sinceǫ was arbitrary this gives the inequality required. �

We do not expect the previous theorem to be optimal. In addition to the likelihood ofbeing able to relax the assumptions on the singularity type of ψi, it seems reasonable toconjecture that an analogous statement holds for maximal envelopes coming from “testcurves” of singularities. In the simplest case, whereF = L, an example of a test curve is afamily ψλ ∈ PSH(L) for λ ∈ (0, 1) that is concave inλ (see below (6.1) for more generaldefinition).

Conjecture 5.2. Supposeψ1,λ andψ2,λ are test curves forL1 andL2 respectively and setψ = sup{ψ1,λ + ψ2,1−λ}

∗. Then, possibly under some regularity assumptions of the testcurves, the maximal envelope on the product is given by

φ[ψ] = sup{(φ1)[ψ1,λ] + (φ2)[ψ2,1−λ] : λ ∈ (0, 1)}∗.

We have not seriously attempted to prove this conjecture andthus will not discuss itmuch further, but presumably the simplest next case to consider are “piecewise linear” testcurves that are locally inλ of the formψλ = ζ0 +λζ1 for fixed singular metricsζ0 andζ1.It would be interesting also to investigate if this generalisation has an algebraic counterpartbeing related to some kind of “limit” of the Mustata summation formula.


6. THE LEGENDRETRANSFORM AS AMAXIMAL ENVELOPE

Our goal in this section is to show how maximal envelopes on the product ofX witha disc captures the Legendre transform of a test curve of singular metric, as previouslyconsidered by the authors [29]. In the following fix a compactcomplexX and big linebundleL and some smooth positive metricφ onL.

Definition 6.1. SupposeF is a line bundle onX . A test curveonF is a mapλ 7→ ψλ ∈PSH(λF ) for λ ∈ (0, c) for somec such that

(1) ψλ is concave inλ(2) λ−1ψλ is decreasing inλ(3) There is a metricφF onF such thatλ−1ψλ ≤ φF for all λ ∈ (0, c).

Example 6.2. The simplest example is whenF has a holomorphic sections in which caseψλ = λ ln |s|2 for λ ∈ (0, 1) defines a test curve onF .

Remark 6.3. This definition differs slightly from that used[29]. To see the compatibility,suppose thatL is ample and setF = L. Then letψλ ∈ PSH(λL) for λ ∈ (0, 1) be a testcurve onL. By definition, we can pick a positiveφL onL such thatφL ≥ λ−1ψλ for allλ. Now set

ψ′λ =

φL λ ≤ 0ψλ + (1 − λ)φL λ ∈ (0, 1)−∞ λ ≥ 1

Then (1)ψ′λ ∈ PSH(L) and is concave inλ, (2) ψ′

λ is locally bounded forλ < 0 and(3) ψ′

λ = −∞ for λ > 1. If we assume in additionψλ has small unbounded locus forλ ∈ (0, 1) then this is essentially what is called test curve in the sense of [29]. Noticefurthermore that forλ ∈ (0, 1) we have thatL− λF = (1 − λ)L is big (it is even ample)and

φ[ψ′

λ]= φ[ψλ].

Thus we may apply our regularity result, Theorem 1.1, and itsconsequences, to theseenvelopes.

Now fix a test curveψλ, and set

φλ = φ[ψλ] for λ ∈ (0, c).

It is convenient also to defineφλ = φ for λ ≤ 0 andφλ = −∞ for λ ≥ c.

Definition 6.4. TheLegendre transformof a test curveψλ, is defined to be

φt := (supλ∈R

{φλ + tλ})∗,

wheret ∈ [0,∞).

In [29] the authors prove that, whenL is ample, the Legendre transform is a weakgeodesic ray in the space of metrics inPSH(L) emanating fromφ. By this it is meant thatt = − ln |z| wherez is a coordinate on the closed unit discB in C, thenΦ(x, t) := φt(x)defines a positive metric on the pullback ofL toX ×B that isS1 invariant and solves theHomogeneous Monge Ampere equationMA(Φ) = 0.

We proceed now to show how the Legendre transform itself a maximal envelope. Letπ : X ×B → X be the projection and set

ψ′ = supλ∈(0,c)

{ψλ + λt}∗


where thec is as in the definition of the test curve.

Theorem 6.5. Letφ′ = φ+ ct. Then overX ×B we have

φ′[ψ′] = φt

Proof. Supposeγ ∈ PSH(L) with γ ≤ φ andγ ≤ ψλ + C for someλ ∈ (0, c) andconstantC. Then ast ≥ 0, γ + λt ≤ γ + ct ≤ φ+ ct = φ′ and

γ + λt ≤ ψλ + λt+ C ≤ ψ′ + C.

Henceγ + λt is a candidate for the envelopePψ′+Cφ′ giving

γ + λt ≤ Pψ′+Cφ′ ≤ φ′[ψ′].

Taking the supremum over all suchγ gives

Pψλ+Cφ+ λt ≤ φ′[ψ′],

so taking the limit asC tends to infinity and then the upper semicontinuous regularisationyields

φλ + λt ≤ φ′[ψ′]

and then taking the supremum over allλ ∈ (0, c) gives

φt ≤ φ′[ψ′].

For the other inequality, for fixedC > 0 andλ ∈ (0, c) define

γCλ (x) = inft≥0

{Pψ′+Cφ′(x, t) + λt}.

ThusγCλ is an infimum of plurisubharmonic functions onX ×B that depends only on themodulus ofw ∈ B, so by the Kiselman minimum principleγλ is plurisubharmonic. Nowone clearly hasγCλ (x) ≤ φ′(x, 0) = φ(x) and moreover

γCλ ≤ inft≥0

{ψ′ + C + λt} = ψλ + C

as the Legendre transform is an involution. Hence,γCλ ≤ φ[ψλ] and so

γCλ + λt ≤ φt.

Now taking the supremum over allλ ∈ (0, c) and using the involution property of theLegendre transform again yields

Pψ′+Cφ′ ≤ φt.

TakingC to infinity and the upper semicontinuous regularisation gives

φ′[ψ′] ≤ φt

which completes the proof. �

Remark 6.6. As previously remarked, the previous theorem is in fact a special case of thegeneral conjecture we made for the maximal envelope of a product (5.2). Essential in theabove proof is that all quantities defined onB have been taken to beS1 invariant (i.e. theydepend only ont = − ln |w| rather thanw) and so the Kiselman minimal principle can beapplied.


Remark 6.7. As a consequence of the previous theorem, clearly any regularity enjoyed bythe maximal envelopeφ′[ψ′] will be similarly enjoyed by the geodesicφt. A simple modi-fication of the argument in the previous section shows that, as long asψ′ is exponentiallyHolder continuous andφ is smooth, thenφt isC1,1 in the direction ofX (the modificationneeded is simply to replaceY withL∗ ×B and demand that the vector fields in(4.3) lie inthe subbundleTL∗ ⊂ T (L∗ × B)). Moreover one can show with these same techniquesthat φt is Lipschitz in thet variable, but we have not been able to use them to show it isC1,1 in this direction as well.

7. EXHAUSTION FUNCTIONS OFEQUILIBRIUM SETS

We continue to consider the maximal envelopes

φλ := φ[ψλ]

whereψλ is a test curve of singularities, and the associated equilibrium sets

Dλ := D(φ, ψλ) = {x ∈ X : φλ = φ}.

Observe thatφλ are decreasing inλ, and thusDλ are closed and increasing, i.e.

Dλ ⊂ Dλ′ if λ ≥ λ′.

Definition 7.1. Denote the exhaustion functionH : X → R by

H(x) = sup{λ > 0 : x ∈ Dλ}

= sup{λ > 0 : φλ(x) = φ(x)}.

When necessary we shall writeHφ,ψ orHφ to emphasise the dependence on the metricsin question. Note that since eachDλ is closed, it is clear thatHφ is upper-semicontinuous.

We show now that this exhaustion function is the time derivative of the associated geo-desic ray coming from the Legendre transform:

Theorem 7.2. Fix a smooth metricφ ∈ PSH(L) and letφt be the Legendre transformassociated to the test curveψλ. Then

dφtdt

∣∣∣∣t=0+

= Hφ

The proof will rely on an elementary lemma from convex geometry.

Lemma 7.3. Let u = ut be a real valued convex function in the one variablet such thatut = u0 for t ≤ 0 and setvλ = inft{ut − λt}. Then

du

dt

∣∣∣∣t=0+

= sup{λ : vλ = u0}.

Proof. Observe that by convexity and the assumption thatut is constant fort < 0 we haveut is increasing int. Moreovervλ ≤ u0 for all λ and sov0 = inft ut = u0. Thus the setS = {λ : vλ = u0} is a non-empty interval inR. Settingw(ǫ) := ǫ−1(uǫ − u0) for ǫ > 0,the convexity ofut impliesw is non-decreasing and we have to show

limǫ→0+

w(ǫ) = supS.

First supposeλ ∈ S. Thenut−λt ≥ u0 for all t, and sow(ǫ) ≥ λ for all ǫ > 0, and hencelimǫ→0+ w(ǫ) ≥ supS. In the other direction, supposeλ > supS and pick someλ′ ∈ S


with λ′ < λ. Then asλ /∈ S there is at so thatut − tλ < u0. But asλ′ ∈ S we certainlyhaveu0 = vλ′ ≤ ut − λ′t and putting these together showst > 0. Thusw(t) < λ andso limǫ→0+ w(ǫ) ≤ λ by monotonicity ofw. SinceS is an interval and this holds for allλ > supS we concludelimǫ→0+ w(ǫ) ≤ supS as required. �

Proof of Theorem 1.9.Define

γλ = inft{φt − λt}

By the Kiselman minimum principleγλ ∈ PSH(L). We claim that

(25) γλ = φλ.

Assuming this, the result we want follows directly from the previous Lemma (7.3) as

dφtdt

∣∣∣∣t=0+

= sup{λ : γλ = φ0} = sup{λ : φλ = φ} = Hφ.

Thus is remains to prove (25). Note thatφt(x) = supλ{φλ(x)+λt} for almost everyx(the point being that the Legendre transform also requires us to take the upper semicontin-uous regularisation). Thus for suchx an elementary argument (essentially the involution ofthe Legendre transform) yieldsγλ(x) = φλ(x). Thusγλ andφλ are two plurisubharmonicfunctions that agree almost everywhere, and hence are identically equal.

�

8. DIVISORIAL EXHAUSTION MAPS

We now restrict to the special case of the exhaustion map associated to a divisorD. Letλmax = sup{λ : L−λD is big}. We assume thatD does not intersectB+(L−λD) for allλ ∈ (0, λmax). Defineψλ = λ ln |sD|

2 for λ ∈ (0, λmax) wheresD is the defining sectionof D.

As above we setφλ = φ[ψλ] andDλ = {φλ = φ} (and the usual convention thatφλ = φ for λ < 0 soDλ = X , and forλ > λmax we setφλ ≡ −∞ soDλ is empty.). Wewrite the associated exhaustion function asH orHD.

Now the volume of the equilibrium sets measures the rate of growth of the subspaceof sections ofLk contained in the relevant multiplier idealI(kψλ) = I(k ln |sD|

2) =O(−kD) (Corollary 4.25). Ifλ ∈ (0, λmax) thenL− λD is big, and thus

h0(k(xL − λD)) = vol(L− λD)kn +O(kn−1)

wherevol(L − λD) := 1n!

∫X(c1(L − λD))n. Thus we conclude

(26) vol(Dλ,MA(φ)) :=

∫

Dλ

MA(φ) = vol(L − λD) for all λ.

Our goal is to analyse this volume change in terms of the sections ofkL that vanish to acertain order alongD. To do so it is natural to express our results in terms of the Okounkovbody whose construction we briefly recall now (and refer the reader to [21, 29, 20] fordetails).

LetX ⊃ Yn−1 ⊃ · · · ⊃ Y1 be a flag of smooth subvarieties ofX with dimYi = i andn = dimX . Starting withYn−1 we have a valuation

ν1 : H0(X,Lk) → Z given by ν1(s) = ordYn−1(s),


whereordYn−1 is the order of vanishing alongYn−1. If t denotes the defining equation forYn−1, then by definitions := st−ν1(s) restricts to a non-trivial section ofL|Yn−1, and thuswe have a second valuation

ν2(s) = ordYn−2(s|Yn−2).

Proceeding in this way gives a map

ν : H0(Lk) → Zn given by ν(s) = (ν1(s), . . . , νn(s)).

We set∆k = k−1 im(ν : H0(L⊗k) → Zn) and the Okounkov body is defined to be

∆ = ∆(X,L) =⋃

k

Convex(∆k)

whereConvex denotes the taking the convex hull and the bar denotes topological closure.

WhenL is big, the volume of∆ taken with respect to the Lebesgue measure is preciselythe volume ofL taken with respect to the line bundleL. This fundamental property lies atthe cornerstone of the work of Lazarsfeld-Mustata who usethe Okounkov body to studythe volume functional on the space of big line bundles, and ofKaveh-Khovanskii who giveapplications by considering even more general valuations.

Suppose now our flag of smooth subvarieties whose divisorialpart is given byD (i.e.Yn−1 = D in the notation above).

Theorem 8.1. The pushforward of the volume formMA(φ) under the exhaustion functionHD is given by

HD∗(MA(φ)) = p1∗(dσ|∆(X,L))

wheredσ denotes the Lebesgue measure onRn andp1 : Rn → R is the projection to thefirst coordinate.

Proof of 8.1. As we will see, this result follows rather easily from our knowledge of thevolume of the equilibrium sets (26). Forλ ∈ Q let Uλ = (λ,∞). First observe that ifλ < 0 (resp.λ > λmax) thenp−1

1 (Uλ) = ∆(X,L) (resp. is empty) andH−1(Uλ) = X(resp. is empty) and so both measures in question are concentrated on the interval[0, λmax].So now letλ ∈ (0, λmax) ∩Q. Then by constructionDλ ⊂ H−1(Uλ) and so

vol(Uλ, H∗MA(φ)) ≥ vol(Dλ,MA(φ)) = vol(L − λD).

On the other handp−11 (Uλ) ∩∆(X,L)) is (a translate of) the Okounkov body ofX taken

with respect toL− λD [21, 4.24]. Thus

vol(Uλ, H∗MA(φ)) ≥ vol(L − λD) = vol(p−11 (Uλ), dσ).

Since this holds for rationalλ, by continuity it holds for allλ ∈ (0, λmax). But the totalmass of the two measures in question is equal tovol(L), and thus since they are bothpositive measure they must be equal. �

For our second result along these lines notice that by construction |∆k| = h0(Lk).Moreover the points in∆k determine a filtration ofH0(Lk) obtained by the valuation,namely forα ∈ ∆k setFα = {s ∈ H0(Lk) : ν(s) ≥ kα} where the inequality is takenin the lexicographic order. Thus using theL2-inner product onH0(Lk) we see there is auniqueL2-orthonormal basis{sα} for α ∈ ∆k for H0(Lk) with the property

ν(sα) = αk for α ∈ ∆k


Remark 8.2. In the toric case the Okounkov body is nothing other than the usual Delzantpolytope andk∆k is precisely the integral points ink∆ and one can pick torus invariantsections to achieve the same result. Thus what we are doing here can be thought of as ageneralisation of the usual toric picture in which the torusaction has be replaced with thedata of a divisorD in X and a hermitian metricφ onL.

Theorem 8.3. We have

HD = lim supk→∞

∑α α1|sα|

2φ∑

α k|sα|2φ

whereα1 = p1(α).

almost everywhere onX

The proof of this will uses the connection between the partial Bergman function and themaximal envelopes. For fixed rationalλ, we consider the partial Bergman kernel

Bλ,k = B(λψ, φ) =∑

β

|sβ|2φ for kλ ∈ N

where{sβ} is anyL2-orthonormal basis for the subspaceH0(IλkD Lk). Since this definitionis independent of basis chosen, in terms of the notation above it is thus given by

Bλ,k =∑

α1≥λ

|sα|2φ.

(the sum being understood as over allα ∈ ∆k whose first coordinate is at leastλ).

Proof of 1.10.By the standard asymptotic of the Bergman function,k−n∑α |sα|

2φ tends

to 1 uniformly onX ask tends to infinity. Thus if we let

fk = k−n∑

α

α1|sα|2φ

and

f := lim supk

fk

it becomes sufficient to prove thatf = HD almost everywhere onX . To this end, fixx ∈ X and some rationalλ′ > HD(x). Then ifλ′ ≥ λ we haveBλ,k ≤ Cke

φ(x)−φλ(x) ≤

Ckeφ(x)−φλ′(x). Thus

fk(x) = k−n∑

λ∈p1(∆k)

Bλ,k(x)

= k−n∑

λ≤λ′

Bλ,k(x) + k−n∑

λ≥λ′

Bλ,k(x)

= ≤ Ckλ′ + Cknek(φ(x)−φλ′(x)).

whereCk is a sequence of constants that tends to1 ask tends to infinity. Now sinceλ′ > HD(x) we haveφλ′(x) < φ(x). Thus taking the limsup yieldsf(x) ≤ λ′ and lettingλ′ tend toHD(x) we deducef(x) ≤ HD(x) for all x ∈ X .

Now as eachsα has unitL2-norm we clearly have∫

X

fkMA(φ) =∑

α∈∆k

α1 →

∫

∆

x1dσ


wheredσ is the Lebesgue measure andx1 is the first coordinate. But from the pushforwardproperty (Theorem 8.1) this last integral is equal to

∫X HDMA(φ). Thus from Fatou’s

Lemma,∫

X

fMA(φ) =

∫

X

lim sup fkMA(φ) ≥ limk

∫

X

fkMA(φ) =

∫

X

HDMA(φ).

Hence we must in fact have∫XfMA(φ) =

∫XHDMA(φ), and thusf = HD almost

everywhere onX as required. �

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JULIUS ROSS, UNIVERSITY OF CAMBRIDGE, [email protected] .AC.UK

DAVID WITT NYSTROM, UNIVERSITY OF GOTHENBURG, [email protected]

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