L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Eleonora DI NEZZA & Charles FAVRE

Regularity of push-forward of Monge–Ampère measures Tome 68, n^o7 (2018), p. 2965-2979.

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REGULARITY OF PUSH-FORWARD OF MONGE–AMPÈRE MEASURES

by Eleonora DI NEZZA & Charles FAVRE

Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday

Abstract. — We prove that the image under any dominant meromorphic map of the Monge–Ampère measure of a Hölder continuous quasi-psh function still possesses a Hölder potential. We also discuss the case of lower regularity.

Résumé. — Nous démontrons que l’image par une application méromorphe dominante d’une mesure de Monge–Ampère d’une fonction quasi-psh et hölderienne possède aussi un potentiel hölderien. Nous discutons aussi le cas de régularité plus basse.

1. Introduction

Let (X, ω_X) be a compact Kähler manifold of dimension n normalized by the volume conditionR

Xω_Xⁿ = 1. We say that a potential ϕ∈ L¹(X) is ωX-plurisubharmonic (ωX-psh for short) if locally ϕ is the sum of a plurisubharmonic and a smooth function, andω_X+ dd^cϕ>0 in the weak sense of currents, where d =∂+ ¯∂ and d^c= _2πⁱ ( ¯∂−∂) so that dd^c= _πⁱ∂∂.¯ We denote by PSH(X, ω_X) the set of all ω_X-psh functions on X. Recall from [13, Section 1] that the non-pluripolar Monge–Ampère measure of a functionϕ∈ PSH(X, ω_X) is a positive measure defined as the increasing limit

(ωX+ dd^cϕ)ⁿ= lim

j→+∞1{ϕ>−j} (ωX+ dd^cmax{ϕ,−j})ⁿ

where the right hand side is defined using Bedford–Taylor intersection the- ory of bounded psh functions, see [2]. By construction this measure does not charge pluripolar sets.

Keywords:Kähler manifolds, meromorphic map, Monge–Ampère measures.

2010Mathematics Subject Classification:32Q15, 32W20, 32Uxx.

One of the main result of [13] states that if µ is a probability measure onX which does not charge pluripolar sets, then there exists a unique (up to a constant)ω_X-psh functionϕsuch thatR

X(ω_X+ dd^cϕ)ⁿ = 1 and

(1.1) µ= (ωX+ dd^cϕ)ⁿ.

We denote byE(X, ωX) the set of allωX-psh functions whose non-pluripolar Monge–Ampère measure has full mass so that any solution to (1.1) belongs toE(X, ωX).

In the same paper, Guedj and Zeriahi introduced for any p > 0 the subsetE^p(X, ωX) ofE(X, ωX) consisting of allωX-psh functions satisfying the integrability conditionϕ∈L^p((ωX+ dd^cϕ)ⁿ). SinceωX-psh functions are bounded from above it follows that

E^p(X, ω_X)⊂ E^q(X, ω_X), for allp > q.

Observe also that any ωX-psh function lying in L^∞ belongs to the inter- section of allE^p(X, ω_X).

We shall say that a probability measure which does not charge pluripolar setsµ= (ω_X+ dd^cϕ)ⁿis a Monge–Ampère measure having Hölder, contin- uous,L^∞orE^ppotential for somep >0 wheneverϕis Hölder, continuous, L^∞or belongs to the energy classE^p(X, ωX) respectively.

Let us now consider any dominant meromorphic mapf :X 99KY where (Y, ωY) is also a compact Kähler manifold of volume 1, and denote by m its complex dimension. Let Γ be a resolution of singularities of the graph of f. We obtain two surjective holomorphic mapsπ1: Γ→X andπ2: Γ→Y whereπ₁is bimeromorphic so that Γ is a modification of a compact Kähler manifold. By Hironaka’s Chow lemma, see e.g. [17, Theorem 2.8] we may suppose thatπ1is a composition of blow-ups along smooth centers so that Γ is itself a compact Kähler manifold of complex dimensionn. We fix any Kähler formωΓ on it.

One defines the push-forward under f of a measure µ not charging pluripolar sets as follows. Since π1 is a bimeromorphism, there exist two closed analytic subsetsR⊂Γ andV ⊂X such thatπ₁: Γ\R→X\V is a biholomorphism. One may thus setπ₁^∗µto be the trivial extension through R of (π1)|^∗_Γ\Rµ. This measure is again a probability measure which does not charge pluripolar sets.

We then define the probability measure f_∗µ := (π2)_∗π₁^∗µ. We observe that since f is dominant then π₂ is surjective hence the preimage of a pluripolar set inY byπ2 is again pluripolar. By the preceding discussion, there existsψ∈ E(Y, ω_Y) such thatf_∗µ= (ω_Y + dd^cψ)^m.

Our main goal is to discuss the following question.

Problem 1.1. — Suppose µ is a Monge–Ampère measure having Hölder, continuous,L^∞orE^ppotential. Is it true thatf_∗µis also a Monge–

Ampère measure of a potential lying in the same class of regularity?

This problem is hard for Monge–Ampère measures having either contin- uous or L^∞ potentials since there is no known intrinsic characterization of these measures. For these classes of regularity even the case f is the identity map andX =Y is still open (see for example [7, Question 15]).

Problem 1.2. — Supposeµis a probability measure onXnot charging pluripolar sets and writeµ= (ω+ dd^cϕ)ⁿ= (ω⁰+ dd^cϕ⁰)ⁿwhereω, ω⁰ are two Kähler forms of volume1. Is it true thatϕis continuous (resp.L^∞) iff ϕ⁰ is?

Remark. — A variant of Problem 1.1 has been recently investigated in [1, 18]. In particular, one can find in these papers a criterion on the singularities of an algebraic mapf :X→Y which ensures that the push-forward of any continuous volume form remains continuous. We refer to these articles for the precise statements and for some far-reaching generalizations of them over any local fields.

Intrinsic characterizations of Monge–Ampère measures of Hölder func- tions are given by [4] and [9], and in the context of Hermitian compact manifolds by [15]. A characterization of Monge–Ampère measures of func- tions in the energy class E^p is also obtained in [13, Theorem C] so that Problem 1.2 has a positive answer for these two classes of regularity, see [5, Theorem 4.1]. Problem 1.1 remains though quite subtle. If we restrict our attention to the regularity in theE^p energy classes, then the answer is no in general. Suppose thatπ:X →P²is the blow-up at some point p∈P², and let E = π⁻¹(p). It was observed by the first author in [6, Proposi- tion B] that there exists a probability measure µ = (ωX + dd^cϕ)² with ϕ ∈ E¹(X, ωX) but π_∗µ = (ωF S+ dd^cψ)² with ψ /∈ E¹(P², ωF S), where ω_{F S} denotes the Fubini Study metric onP² andω_X is a Kähler form.

In this note we answer Problem 1.1 in two situations. We first treat the caseµis the Monge–Ampère of a Hölder function.

Theorem 1.3. — Letf :X 99KY be any dominant meromorphic map between two compact Kähler manifolds. Ifµis a Monge–Ampère measure having a Hölder potential with Hölder exponentα, thenf_∗µis a Monge–

Ampère measure having a Hölder potential with Hölder exponent bounded byCα^dim(X)for some constantC >0 depending only onf.

We expect that the technics developed in the paper of Kołodziej–

Nguyen [15] in the present volume allows one to extend the previous result to arbitrary compact hermitian manifolds.

Next we treat the case the image of the map has dimension 1.

Theorem 1.4. — Letf :X 99KY be any dominant meromorphic map from a compact Kähler manifold to a compact Riemann surface. If µ is a Monge–Ampère measure having a Hölder,C⁰, L^∞, E^p potential respec- tively, then f∗µ is a Monge–Ampère measure having a potential lying in the same regularity class.

Motivations for studying this question come from the analysis of degen- erating measures on families of projective manifolds developed in [11]. Let us briefly recall the setting of that paper. LetX be a smooth connected complex manifold of dimensionn+ 1, andπ:X →Dbe a flat proper ana- lytic map over the unit disk which is a submersion over the punctured disk and has connected fibers. We assume thatX is Kähler so that each fiber X_t=π⁻¹(t) is also Kähler.

A tame family of Monge–Ampère measures is by definition a family of positive measures{µ_t}_t∈Deach supported onX_tthat can be written under the form

µt=p∗(T|ⁿ_Xt),

whereT is a positive closed (1,1)-current having local Hölder continuous potentials and defined on a complex manifoldX⁰ which admits a proper bimeromorphic morphismp:X⁰ → X which is an isomorphism overX :=

π⁻¹(D^∗). It follows from [3, Corollary 1.6] that the family of measures µ⁰_t:=T|ⁿ_Xt inX⁰ is continuous so thatµ⁰_tconverges to a positive measure µ⁰₀supported onX₀⁰ ast→0. It follows that the convergence lim_t→0µt=µ0

also holds inX.

As a corollary of the previous results we show the limiting measure µ0

is of a very special kind:

Corollary 1.5. — Let{µt}_t∈Dbe any tame family of Monge–Ampère measures, so thatµt→µ0 ast→0.

Then there exist a finite collection of closed subvarieties {Vi}i=0,...,N of X₀ and for each indexi a positive measureν_i supported onV_i such that

µ₀=

N

X

i=1

ν_i

andνi is a Monge–Ampère measure onVi having a Hölder potential.

In the previous statement, it may happen that V_i is singular, in which case it is understood that the pull-back ofνi to a (Kähler) resolution ofVi

is a Monge–Ampère measure having a Hölder continuous potential.

Acknowledgement. We thank Ahmed Zeriahi for useful discussions on these problems.

2. Images of Monge–Ampère measures having a Hölder potential: proof of Theorem 1.3

As already mentioned, a dominant meromorphic map f: X 99K Y can be decomposed as f = π₂◦π₁⁻¹, where π₁: Γ → X is holomorphic and bimeromorphic and π2: Γ → Y is a surjective holomorphic map. Recall that one can assume Γ to be Kähler, and thatf_∗µ:= (π₂)_∗π₁^∗µ.

We first claim that if µ is the Monge–Ampère of a Hölder continuous function thenπ₁^∗µtoo. Let ϕ∈PSH(X, ωX) be the Hölder potential such that µ = (ωX + dd^cϕ)ⁿ. It then follows from Bedford and Taylor theory thatπ₁^∗µ= (π^∗₁ωX+ dd^cπ^∗₁ϕ)ⁿ. Sinceπ^∗₁ωX is a semipositive smooth form, there exists a positive constantC >0 such that π₁^∗µ6(Cω_Γ+ dd^cπ^∗ϕ)ⁿ whereωΓis a Kähler form on Γ, and [4, Theorem 4.3] implies that ˜µ:=π₁^∗µ is the Monge–Ampère measure of a Hölder continuousCω_Γ-psh function.

This proves the claim. We are then left to prove that (π2)∗µ˜is the Monge–

Ampère measure of a Hölder potential. This will be done in Lemma 2.4.

We first show that the push-forward of a smooth volume form has density inL^1+ε, for some constantε >0 depending only onf.

Proposition 2.1. — Letf:X →Y be a surjective holomorphic map.

Thenf∗ω_Xⁿ =gω_Y^mwithg∈L^1+ε(ωⁿ_Y), for some ε >0.

This result is basically [19, Proposition 3.2] (see also [20, Section 2]).

We give nevertheless a detailed proof for reader’s convenience. Pick any coherent ideal sheaf I ⊂ OX, and denote by V(I) = supp(OX/I) the closed analytic subvariety of X associated to I. Let {U_i}^N_i=1 be a finite open covering ofX by balls and{Vi}ibe a subcovering such thatVi ⊂Ui. The analytic sheafI is globally generated on eachUi so that we can find holomorphic functions such thatI|_Ui = (h⁽ⁱ⁾₁ , . . . , h⁽ⁱ⁾_k )· O_Ui. Let{ρ_i}be a partition of unity subordinate toVi. We then define

(2.1) Φ_I:=

N

X

i=1

ρ_i





k

X

j=1

|h⁽ⁱ⁾_j |²



.

Then ΦI: X →R+ is a smooth function which vanishes exactly on V(I).

Observe that if Φ_I and Φ⁰_I are defined using two different coverings, then there existsC >0 such that

1

CΦ⁰_I6Φ_I6CΦ⁰_I.

In the sequel we shall abuse notation and not write the dependence of ΦI

in terms of the local generators of the ideal sheaf. The logarithm of the obtained function is then well-defined up to a bounded function so that all statements in the next Lemma make sense.

Lemma 2.2. — Let I,J ⊆ OX be two coherent ideal sheafs. The fol- lowings hold:

(1) there existsε >0such that|ΦI|^−ε∈L¹(X);

(2) ifI ⊆ J thenΦ_J >cΦ_I for some positivec >0;

(3) ifV(J)⊆V(I)then there existsc, θ >0such thatΦJ >cΦ^θ_I; (4) givenf:X →Y a holomorphic surjective map and a coherent ideal

sheafJ ⊆ O_Y, thenΦ_f^∗_J = Φ_J ◦f (for a suitable choice of local generators ofJ andf^∗J).

Proof. — Using a resolution of singularities ofI, one sees that the state- ment in (1) reduces to show that|z1|^−εis locally integrable for someε >0, and this is the case if we choose ε small enough. The statements in (2) and (4) follow straightforward from the definition in (2.1). The state- ment in (3) is a consequence of Łojasiewicz theorem, see e.g. [16, Theo-

rem 7.2].

Lemma 2.3. — Letf:X→Y be a holomorphic surjective map and let I ⊆ O_X be a coherent ideal sheaf. Then there exists a coherent ideal sheaf J ⊆ OY, and constantsc, θ >0 such that for anyy∈Y we have

inf

x∈f⁻¹(y)

Φ_I>cΦ^θ_J

Proof. — LetJ ⊆ OY be the coherent ideal sheaf of holomorphic func- tions vanishing on the setf(V(I)) which is analytic sincef is proper. Ob- serve thatV(f^∗J) =f⁻¹(V(J))⊃V(I), so that Lemma 2.2 (3) and (4) insure that there existc, θ >0 such that

Φ_I >cΦ^θ_f∗J =c(Φ_J ◦f)^θ.

Hence the conclusion.

Proof of Proposition 2.1. — Recall that Sard’s theorem implies the ex- istence of a closed subvarietyS(Y such thatf:X\f⁻¹(S)→Y \Sis a submersion.

We first prove thatf_∗ωⁿ_X is absolutely continuous w.r.t.ω_Y^m. We need to check thatω_Y^m(E) = 0 impliesf_∗ωⁿ_X(E) = 0 for any Borel subsetE ⊂Y. AsSandf⁻¹(S) have volume zero one may assume thatf is a submersion in which case the claim follows from Fubini’s theorem.

Radon–Nikodym theorem now guarantees that f_∗ωⁿ_X = gω^m_Y for some 06g∈L¹(Y). We want to show that the integral

Z

Y

g^1+εω_Y^m= Z

Y

g^εf∗ωⁿ_X= Z

X

(f^∗g)^εω_Xⁿ

is finite for someε >0 small enough. Consider the smooth functionφ(x) :=

f^∗ω^m_Y∧ω^n−m_X

ωⁿ_X (x), and set ˜φ(y) := inf_x∈f⁻¹_(y)φ(x) so thatφ>f^∗φ. We claim˜ that for anyy∈Y

(2.2) g(y)6 c

φ(y)˜ ,

for some constantc >0. Letχbe a test function (i.e. a non negative smooth function) onY, then

Z

Y

χ gωⁿ_Y = Z

X

f^∗χ ω_Xⁿ = Z

X

f^∗χ

φ f^∗ω_Y^m∧ω^n−m_X 6

Z

X

f^∗ χ

φ˜ω^m_Y

∧ω^n−m_X

Fubini

= C(f) Z

Y

χ φ˜ω_Y^m where c := C(f) = R

f⁻¹(y)ω^n−m_X is the volume of a fiber over a generic point y ∈ Y. The claim is thus proved. Lemma 2.2 (1) and (4) combined with Lemma 2.3 then insure that there exists ε > 0 such that (f^∗g)^ε ∈

L¹(ωⁿ_X).

Theorem 1.3 is reduced to the following result which relies in an essential way on Proposition 2.1.

Proposition 2.4. — Suppose f: X → Y is a surjective holomorphic map between compact Kähler manifolds. Ifµ is a positive measure onX with Hölder continuous potentials, then f_∗µ is a positive measure on Y with Hölder potentials.

Observe that by multiplying ωX by a suitable positive constant we may assume thatf^∗ω_Y 6ω_X. The volume normalization is no longer satisfied but a positive multiple of µis still the Monge–Ampère measure of a ωX- psh Hölder continuous function. Write f_∗µ = (ω_Y + dd^cψ)^m with ψ ∈ PSH(Y, ωY).

We claim that there exists C > 0, and ε > 0 such that for all u ∈ PSH(Y, ωY) withR

Xu ωⁿ_X= 0 (2.3)

Z

Y

exp(−εu) d(f_∗µ)6C.

Indeed, for anyu∈PSH(Y, ωY) we have that Z

Y

e^−εud(f∗µ) = Z

X

e^−ε(u◦f)dµ.

Now the integralR

Xe^−ε(u◦f)dµis uniformly bounded by [10, Theorem 1.1]

since:

• µhas Hölder continuous potentials;

• f^∗ω_Y 6ω_X henceu◦f ∈PSH(X, ω_X);

• and the set of functions in PSH(X, ωX) such thatR

Xu ω_Xⁿ = 0 is compact by [12, Proposition 2.6].

This proves our claim. Using the terminology of [9] this means thatf_∗µis moderate. It is worth mentioning that if [7, Question 16] holds true then the conclusion of Proposition 2.4 would follow immediately since any moderate measure would have a Hölder continuous potential. To get around this problem we use the characterization of measures with Hölder potentials given by Dinh and Nguyen.

Proof of Proposition 2.4. — By [9, Lemma 3.3], f_∗µ is the Monge–

Ampère measure of a Hölder potential if and only if there exist ˜c >1 and β˜∈(0,1) such that

(2.4)

Z

Y

|u−v|df_∗µ6˜cmax

ku−vkL¹(ω_Yⁿ),ku−vk^β_L^˜1(ωⁿ_Y)

for allu, v ∈PSH(Y, ωY). By assumption on µwe know there existc >1 andβ∈(0,1) such thatR

Y |u−v|df_∗µ=R

X|f^∗u−f^∗v|dµ, and (2.5)

Z

X

|f^∗u−f^∗v|dµ6cmax

kf^∗u−f^∗vk_L1(ωⁿ_X),kf^∗u−f^∗vk^β_L1(ωⁿ_X)

.

Also, Proposition 2.1 gives (2.6)

Z

X

|f^∗u−f^∗v|ω_Xⁿ = Z

Y

|u−v|g ω_Yⁿ 6kgkL^1+ε(ω_Yⁿ)ku−vkL^p(ωⁿ_Y)

wherepis the conjugate exponent of 1 +ε. SetC_g:=kgk_L1+ε(ω_Yⁿ)<+∞.

Up to replaceCg withCg+ 1 we can assume thatCg>1.

Denote by m_u :=R

Y u ω_Yⁿ and observe that u⁰ :=u−m_u,v⁰ :=v−m_v satisfyR

Xu⁰ωⁿ_X= 0 =R

Xv⁰ω_Xⁿ. Then the triangle inequality gives

(2.7)

ku−vk_Lp(ω_Yⁿ)= Z

Y

|(u⁰−v⁰) + (mu−mv)|^pω_Yⁿ 1/p

6ku⁰−v⁰k_Lp(ωⁿ_Y)+|m_u−m_v| 6ku⁰−v⁰k_Lp(ωⁿ_Y)+ku−vk_L1(ω_Yⁿ).

At this point, we make use of [9, Proposition 3.2] (that holds for normalized potentials) to replace the L^p-norm with the L¹-norm. We then infer the existence of a constantc⁰ >1 such that

ku⁰−v⁰k_Lp(ω_Yⁿ)6c⁰max(1,−logku⁰−v⁰k_L1(ωⁿ_Y))^p−1p ku⁰−v⁰k

1 p

L¹(ωⁿ_Y). Whent:=ku⁰−v⁰kL¹(ωⁿ_Y)>1/ewe clearly have

ku⁰−v⁰k_Lp(ωⁿ_Y)6c⁰ku⁰−v⁰k

1 p

L¹(ω_Yⁿ),

whereas for any integerN ∈N^∗, there exists a constant cN >0 such that

−logt6c_Nt^−1/N whent61/e, hence

ku⁰−v⁰k_Lp(ω_Yⁿ)6c⁰⁰ku⁰−v⁰k

1

p(^1−p−1_N )

L¹(ωⁿ_Y) .

Asku⁰−v⁰k_L1(ω_Yⁿ) 62ku−vk_L1(ω_Yⁿ), combining (2.5), (2.6) and (2.7) we get

kf^∗u−f^∗vk_L1(µ)6Cmax

ku−vk^β_L^˜_1(ωn

Y),ku−vk_L1(ωⁿ_Y)

, with ˜β= ^β_p 1−^p−1_N

. By [9, Lemma 3.3]f∗µ= (ωY + dd^cψ)ⁿ whereψ is a Hölder continuous function.

To get a bound on the Hölder regularity of ψ, one argues as follows.

First ifµ= (ω+ dd^cϕ)ⁿ withϕ aα-Hölder potential, and π: Γ→X is a proper modification, thenπ^∗µis dominated by a Monge–Ampère measure with α-Hölder potential, and [4, Proposition 3.3 (ii)] is satisfied with b = 2α/(α+ 2n) by [4, Theorem 4.3 (iii)]. Hence, following the proof of [4, Theorem], we see that π^∗µ is a Monge–Ampère measure of a α1-Hölder continuous potential with α1 < b/(n+ 1) (see Remark below for more details about the latter statement).

By [9, Proposition 4.1], (2.5) holds with β =αⁿ₁/(2 +αⁿ₁), and (2.4) is then satisfied for any ˜β < β/p so that f_∗µ is a Monge–Ampère measure with ˜α-Hölder potential for any ˜α <2 ˜β/(m+ 1), see the discussion on [9, p. 83]. Combining all these estimates we see that any

˜

α < αⁿ

p(m+ 1)(α/2 +n)ⁿ(n+ 1)ⁿ

works where p is the conjugate of the larger constant ε > 0 for which

Proposition 2.1 holds.

Remark. — We borrow notations from the proof of [4, Theorem A]. Fix α₁< b/(n+ 1) and chooseε >0 such thatα₁6α6α₀6b−α₀(n+ε). By the previous arguments we know that condition (ii) in [4, Proposition 3.3]

holds, i.e. for any φ ∈ PSH(Γ, ω_Γ), we have kρδφ−φk_L1(π^∗µ) = O(δ^b), whereb= 2α/(α+ 2n). In particular, this gives

π^∗µ(E0)6c1δ^b−α0.

Letg∈L¹(π^∗µ) be defined asg= 0 onE0andg=con Γ\E0 wherec is a positive constant such thatπ^∗µ(Γ) =R

Γgd(π^∗µ). An easy computation gives thatc =π^∗µ(Γ)/π^∗µ(Γ\E0). Let v ∈ PSH(Γ, ωΓ) be the bounded solution of the Monge–Ampère equation (ω_Γ+ dd^cv)ⁿ =g·π^∗µ. Observe that

k1−gk_L1(π^∗µ)= Z

E₀

dπ^∗µ+ Z

Γ\E₀

|1−c|dπ^∗µ= 2 Z

E₀

dπ^∗µ62c₁δ^b−α0. Sinceπ^∗µ= (ωΓ+dd^cϕ)˜ ⁿsatisfies theH(∞) property we can still apply [8, Theorem 1.1] and get

kϕ˜−vkL^∞ 6c3δ

b−α0 n+ε .

The exact same arguments as in [4, Theorem A] then insure that the Hölder exponent of ˜ϕisα₁.

3. Over a one-dimensional base: proof of Theorem 1.4 In this section we treat Problem 1.1 in the case the base is a Riemann surface.

We start with the case of a surjective holomorphic mapf:X →Y from a Kähler compact manifold to a compact Riemann surface.

Let µ = (ωX + dd^cϕ)ⁿ be a Monge–Ampère measure of a continuous ω_X-psh functionϕ. Supposev_k, vis a family ofω_X-psh functions such that vk→v in L¹, then

Z

X

v_kdµ

= Z

X

vk(ωX+ dd^cϕ)ⁿ

= Z

X

v_kω_Xⁿ +

n−1

X

j=0

Z

X

ϕdd^cv_k∧ω^j_X∧(ω_X+ dd^cϕ)^n−j−1→ Z

X

vdµ

by [3, Corollary 1.6 (a)]. Observe that in the last equality we used the fact that

(ωX+ dd^cϕ)ⁿ−ωⁿ_X=

n−1

X

j=0

dd^cϕ∧ω^j_X∧(ωX+ dd^cϕ)^n−j−1 and Stokes’ theorem.

Normalize the Kähler form on Y such that R

ω_Y = 1, and pick any sequenceyk →y_∞ ∈Y. Letwk be the solutions of the equations ∆wk = δ_yk−ω_Y with supw_k= 0 so thatw_k(y)−log|y−y_k|is continuous in local coordinates nearyk. Writef∗µ=ωY + dd^cψ so that

Z

Y

w_kd(f_∗µ) = Z

Y

w_kω_Y + Z

Y

ψ∆w_k=ψ(y_k) + Z

Y

(w_k−ψ)ω_Y. Since wk → w_∞ in L^p_loc for all p < ∞, Proposition 2.1 implies that f^∗w_k → f^∗w_∞ in the L¹ topology, so that the argument above gives R

Ywkd(f_∗µ) =R

Xf^∗wkdµ→R

Xf^∗w_∞dµ=R

Y w_∞d(f_∗µ) We then con- clude thatψ(y_k)→ψ(y_∞). Henceψis continuous.

Suppose then that µ is locally the Monge–Ampère of a bounded psh function, and pick any subharmonic functionudefined in a neighborhood of a pointy∈Y. Thenf^∗uis again psh in a neighborhood off⁻¹(y), and the standard Chern–Levine–Nirenberg inequalities imply thatf^∗u∈L¹(µ) so thatu∈L¹(f_∗µ) with a norm depending only on the L¹-norm ofu. It follows thatf∗µis locally the laplacian of a bounded subharmonic function.

Finally, assume µ = (ωX + dd^cϕ)ⁿ for some ϕ ∈ E^p(X, ωX). By [13, Theorem C] this is equivalent to have that E^p(X, ω_X)⊂L^p(µ). Write as usualf_∗µ= (ωY + dd^cψ) withψ∈ E(Y, ωY).

We claim that u∈ E^p(Y, ω_Y) impliesf^∗u∈ E^p(X, ω_X). Indeed, without loss of generality we can assume that Ω :=ωX−f^∗ωY is a Kähler form and by the multilinearity of the non-pluripolar product we have

Z

X

|f^∗u|^p(ωX+ dd^cf^∗u)ⁿ= Z

X

|f^∗u|^p(f^∗ωY + Ω + dd^cf^∗u)ⁿ

= Z

X

|f^∗u|^p Ωⁿ+ (f^∗ωY + dd^cf^∗u)∧Ωⁿ⁻¹ where the last identity follows from the fact that (f^∗ω_Y + dd^cf^∗u)^j = 0 for j > 1. The term R

X|f^∗u|^pΩⁿ is bounded thanks to the integrabil- ity properties of quasi-plurisubharmonic functions w.r.t. volume forms [14, Theorem 1.47]; while the term

Z

X

|f^∗u|^p(f^∗ωY + dd^cf^∗u)∧Ωⁿ⁻¹=C(f) Z

Y

|u|^p(ωY + dd^cu)

is finite sinceu∈ E^p(Y, ω_Y). This proves the claim.

Now, given any u∈ E^p(Y, ωY) we have Z

Y

|u|^pd(f_∗µ) = Z

X

|f^∗u|^pdµ <+∞

sincef^∗u ∈ E^p(X, ωX) ⊂ L^p(µ). The conclusion follows from [13, Theo- rem C].

Consider now any dominant meromorphic mapf:X99KY from a Käh- ler compact manifold to a compact Riemann surface. As above we decom- posef such thatf_∗µ= (π2)_∗π^∗₁µfor any positive measureµonX.

Assume thatµhas continuous potentials. If we writeµ= (ω_X+ dd^cϕ)ⁿ then π₁^∗µ = (π^∗₁ωX + dd^cϕ◦ π)ⁿ 6 (CωΓ + dd^cϕ◦π)ⁿ := ˆµ where ˆµ has a continuous potential. This implies f_∗µ 6 (π2)_∗µ. Observe that byˆ the previous arguments the measure (π₂)∗µˆ has continuous potential. It follows that locallyf_∗µ= ∆v6∆uwhereu, vare subharmonic functions.

It follows that v is the sum of a continuous function and the opposite of a subharmonic (hence u.s.c.) function. Since it is also u.s.c we conclude to its continuity.

Whenµhas bounded potentials, the same argument applies noting that subharmonic functions are always bounded from above which impliesvto be bounded.

Finally, we consider the case where µ is the Monge–Ampère measure of ϕ ∈ E^p(X, ω_X). We first observe that given v ∈ E^p(Γ, ω_Γ) we have (π1)_∗v∈ E^p(X, ωX). Indeed,

Z

X

|v◦π⁻¹|^p(ω_X+ dd^cv◦π⁻¹)ⁿ= Z

Γ

|v|^p(π^∗₁ω_X+ dd^cv)ⁿ 6

Z

Γ

|v|^p(Cω_Γ+ dd^cv)ⁿ<+∞.

This and the previous arguments give that ifu ∈ E^p(Y, ωY) then f_∗u = (π₁)_∗π^∗₂u∈ E^p(X, ω_X), hence

Z

Y

|u|^pdf_∗µ= Z

X

|u◦f|^pdµ <+∞.

It follows from [13, Theorem C] thatf_∗µis the Monge–Ampère measure of a function inE^p(Y, ωY).

4. The case of submersions

In this section we let (X, ωX),(Y, ωY) be two compact Kähler manifolds of dimensionnandm, respectively and normalized such thatR

Xω_Xⁿ = 1 = R

Yωⁿ_Y.

Proposition 4.1. — Let f: X → Y be a submersion. Then, u ∈ E^p(Y, ω_Y) implies f^∗u ∈ E^p(X, ω_X). In particular, if a probability mea- sureµis the Monge–Ampère of a function inE^p then alsof_∗µ has also a potential inE^p.

Proof. — Since f is a submersion we can assume that there is a finite number of open neighbourhoodsUisuch thatX ⊂SN

j=0Uj,f|U_j(z, w) =z wherez = (z₁, . . . , z_m) and w= (z_m+1, . . . , z_n). Moreover we can assume that on eachUj we have

ωX6Cj

i

2(dz∧d¯z+ dw∧d ¯w), i

2dz∧d¯z6Ajf^∗ωY

whereA_j, C_j >1 and dz∧d¯z,dw∧d ¯ware short notations forPm

j=1dz_j∧dz¯_j andPn

k=m+1dz_k∧d ¯z_k, respectively. We then write Z

X

|f^∗u|^p(ω_X+ dd^cf^∗u)ⁿ 6

N

X

j=1

Z

U_j

|f^∗u|^p

C_ji

2dz∧d¯z+C_ji

2dw∧d ¯w+ dd^cf^∗u n

6

N

X

j=1

Z

U_j

|f^∗u|^p

A⁰_jf^∗ω_Y +C_ji

2dw∧d ¯w+ dd^cf^∗u ⁿ

=

N

X

j=1 n

X

`=0

Z

Uj

|f^∗u|^p A⁰_jf^∗ωY + dd^cf^∗u^`

∧

Cj

i

2dw∧d ¯w n−`

=

N

X

j=1

Z

Uj

|f^∗u|^p A⁰_jf^∗ωY + dd^cf^∗u^m

∧

Cj

i

2dw∧d ¯w n−m

.

The above integral is then finite because by assumption u∈ E^p(Y, Aω_Y) for anyA>1.

The last statement follows from the same arguments in the last part of

the proof in the previous section.

5. Tame families of Monge–Ampère measures: proof of Corollary 1.5

Recall the setting from the introduction: X is a smooth connected com- plex manifold of dimensionn+ 1, andπ:X →Dis a flat proper analytic map over the unit disk which is a submersion over the punctured disk and has connected fibers. We letp:X⁰ → X be a proper bi-meromorphic map from a smooth complex manifoldX⁰which is an isomorphism overπ⁻¹(D^∗).

We let T be any closed positive (1,1)-current on X⁰ admitting local Hölder continuous potentials. Observe that by e.g. [3, Corollary 1.6] we have

µ⁰_t= dd^clog|π◦p−t| ∧Tⁿ→µ⁰₀:= dd^clog|π◦p| ∧Tⁿ.

Let us now analyze the structure of the positive measureµ0:=p∗µ⁰₀. First observe that µ⁰₀ can be decomposed as a finite sum of positive measures µ⁰_E := (T|E)ⁿ where the sum is taken over all irreducible components E of X₀⁰. Each of these measures is locally the Monge–Ampère of a Hölder continuous psh function.

Write V := (E). SinceE is irreducible,V is also an irreducible (possi- bly singular) subvariety of dimension`. To conclude the proof it remains to show thatp∗(µ<sup>0</sup><sub>E</sub>) is the Monge–Ampère measure of Hölder continuous function that is locally the sum of a smooth and psh function. More pre- cisely, one needs to show thatp∗(µ<sup>0</sup><sub>E</sub>) does not charge any proper algebraic subset ofV, and given any resolution of singularities$:V<sup>0</sup>→V the pull- back measure $<sup>∗</sup>(p<sub>∗</sub>(µ<sup>0</sup><sub>E</sub>)) can be locally written as (dd<sup>c</sup>u)<sup>`</sup> where uis a Hölder psh function onV⁰.

This follows from Theorem 1.3 applied to any resolution of singularities V⁰ ofV and to anyE⁰ which admits a birational morphismE⁰ →E such that the mapE⁰→V⁰ induced by pis also a morphism.

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Eleonora DI NEZZA Sorbonne Université 4 Place Jussieu 75005 Paris (France) [email protected] Charles FAVRE

CMLS, École polytechnique, CNRS, Université Paris-Saclay

91128 Palaiseau Cedex (France) [email protected]