Eleonora Di Nezza

Vincent Guedj

Henri Guenancia

FAMILIES OF SINGULAR KÄHLER-EINSTEIN METRICS by

Keywords: 2000 Mathematics Subject Classification. -14D06, 32Q20, 32U05, 32W20 Singular Kähler-Einstein metrics, Families of complex spaces, Stable families, Log Calabi-Yau manifolds

Refining Yau's and Kolodziej's techniques, we establish very precise uniform a priori estimates for degenerate complex Monge-Ampère equations on compact Kähler manifolds, that allow us to control the blow up of the solutions as the cohomology class and the complex structure both vary.

We apply these estimates to the study of various families of possibly singular Kähler varieties endowed with twisted Kähler-Einstein metrics, by analyzing the behavior of canonical densities, establishing uniform integrability properties, and developing the first steps of a pluripotential theory in families. This provides interesting information on the moduli space of stable varieties, extending works by Berman-Guenancia and Song, as well as on the behavior of singular Ricci flat metrics on (log) Calabi-Yau varieties, generalizing works by Rong-Ruan-Zhang, Gross-Tosatti-Zhang, Collins-Tosatti and Tosatti-Weinkove-Yang.

Introduction

Let p : X → Y be a proper, surjective holomorphic map with connected fibers between Kähler varieties. It is a central question in complex geometry to relate the geometry of X to the one of Y and the fibers X y of p. An important instance of such a situation is when one can endow X y with a Kähler-Einstein metric and study the geometry of X induced by the properties of the resulting family of metrics. This is the main theme of this article.

Einstein metrics are a central object of study in differential geometry. A Kähler-Einstein metric on a complex manifold is a Kähler metric whose Ricci tensor is proportional to the metric tensor. This notion still makes sense on midly singular varieties as was observed in [EGZ09, section 7]. The solution of the (singular) Calabi Conjecture [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I[END_REF][START_REF]Singular Kähler-Einstein metrics[END_REF] provides a very powerful existence theorem for Kähler-Einstein metrics with negative or zero Ricci curvature. It is important to study the ways in which these canonical metrics behave when they are moving in families. In this paper we consider the case when both the complex structure and the Kähler class vary and we try and understand how the corresponding metrics can degenerate.

Constructing singular Kähler-Einstein metrics on a midly singular variety V boils down to solving degenerate complex Monge-Ampère equations of the form (ω + i∂∂ϕ) n = f e λϕ dV X , where π : X → V is a resolution of singularities, dV X is a volume form on X, ω = π * ω V is the pull-back of a Kähler form on V, -the sign of λ ∈ R depends on that of c 1 (V), -f ∈ L p (X) with p > 1 if the singularities of V are mild (klt singularities), and ϕ is the unknown. The latter should be ω-plurisubharmonic (ω-psh for short), i.e. it is locally the sum of a psh and a smooth function, and satisfies ω + i∂∂ϕ ≥ 0 in the weak sense of currents. We let PSH(X, ω) denote the set of all such functions. The uniform estimate. -A crucial step in order to prove the existence of a solution to the above equation is to establish a uniform a priori estimate. In order to understand the behavior of the solution ϕ as the cohomology class {ω V } and the complex structure of V vary, we revisit the proof by Yau [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I[END_REF], as well as its recent generalizations [START_REF] Kołodziej | The complex Monge-Ampère operator[END_REF][START_REF]Singular Kähler-Einstein metrics[END_REF], and establish the following (see Theorem 1.1): Theorem A. -Let X be a compact Kähler manifold of complex dimension n ∈ N * and let ω be a semi-positive form such that V := X ω n > 0. Let ν and µ = f ν be probability measures, with 0 f ∈ L p (ν) for some p > 1. Assume the following assumptions are satisfied: (H1) there exists α > 0 and A α > 0 such that for all ψ ∈ PSH(X, ω), X e -α(ψ-sup X ψ) dν A α ;

(H2) there exists C > 0 such that X | f | p dν 1/p C. Let ϕ be the unique ω-psh solution ϕ to the complex Monge-Ampère equation

V -1 (ω + i∂∂ϕ) n = µ,
normalized by sup X ϕ = 0. Then -M ϕ 0 where M = 1 + C 1/n A 1/nq α e α/nq b n 5 + eα -1 C(q!) 1/q A 1/q α , 1/p + 1/q = 1 and b n is a constant such that exp(-1/x) b n n x 2n for all x > 0.

Remark 0.1. -Let us observe that the condition (H1) in Theorem A above guarantees that the measure ν does not charge pluripolar sets, since any such set can be included in the polar locus of a global ω-psh function by [START_REF] Guedj | Intrinsic capacities on compact Kähler manifolds[END_REF]Thm. 7.2]. The existence (and uniqueness) of the solution ϕ in Theorem A follows from [BEGZ10, Thm. A].

We also establish slightly more general versions of Theorem A valid for less regular densities (Theorem 1.5) or big cohomology classes (Theorem 1.9). We then move on to checking hypotheses (H1) and (H2) in various geometrical contexts.

• Hypothesis (H1). If π : X → D is a projective family whose fibers X t = π -1 (t) have degree d with respect to a given projective embedding X ⊂ P N × D, and ω = ω t is the restriction of the Fubini-Study metric, we observe in Proposition 2.5 that

V = X t ω n t = P N ω n FS ∧ [X t ] = d
is independent of t and the following uniform integrability holds.

Proposition B.

-For for all ψ ∈ PSH(X t , ω t ),

X t e -1 nd (ψ-sup X t ψ) ω n t (4n) n • d • exp - 1 nd X t (ψ -sup X t ψ) ω n t .
The hypothesis (H1) is thus satisfied in this projective setting, with α = 1/nd, as soon as we can uniformly control the L 1 -norm of ψ. We take care of this in Section 3. This non-trivial control requires the varieties X t to be irreducible (see Example 3.5).

Bypassing the projectivity assumption, we show that (H1) is actually satisfied for many Kähler families of interest, by generalizing a uniform integrability result of Skoda-Zeriahi [START_REF] Skoda | Sous-ensembles analytiques d'ordre fini ou infini dans C n[END_REF][START_REF] Zeriahi | Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions[END_REF] (see Theorem 2.9). This is the content of Theorem 3.4.

• Hypothesis (H2). We analyze (H2) in section 4. We show that, up to shrinking the base, it is always satisfied if the f t 's are canonical densities associated to a proper, holomorphic surjective map π : X → D from a normal, Q-Gorenstein Kähler space X to the unit disk such that the central fiber has only canonical singularities, cf Lemma 4.4 and its application to families of Calabi-Yau varieties, Theorem F.

While previous works tend to use sophisticated arguments from Variations of Hodge Structures (see e.g. the Appendix by Gross in [START_REF] Rong | Continuity of extremal transitions and flops for Calabi-Yau manifolds[END_REF]), we use here direct elementary computations in adapted coordinates, in the spirit of [EGZ09, section 6].

In the context of families of varieties with negative curvature though, it is essential to allow worse singularities than the ones described above, cf Setting 4.1 for the precise context. The trade-off is that the canonical densities do not satisfy condition (H2) anymore, reflecting the fact that the local potentials of the Kähler-Einstein metrics at stake need not be bounded anymore. This legitimizes the introduction of a weaker condition (H2') (see Theorem 1.5 and Lemma 4.6). This allows us to derive an almost optimal control of the potentials of Kähler-Einstein metrics along a stable family, cf Theorem E below.

Let us end this paragraph by emphasizing that our approach enables us to work with singular families (i.e. families where the generic fiber is singular, cf Theorems E and F) as opposed to all previously known results on that topic, requiring to approximate a singular variety by smooth ones using either a smoothing or a crepant resolution.

We now describe more precisely four independent geometric settings to which we apply the uniform estimate provided by Theorem A.

Ric(T t ) = -T t .

It can be written T t = θ t + dd c ϕ t , where ϕ t is the unique θ t -psh function with minimal singularities that satisfies the complex Monge-Ampère equation (θ t + dd c ϕ t ) n = e ϕ t +h t ω n t on Amp(K X t ), where h t is such that Ric(ω t )dd c h t = -θ t and X t e h t ω n t = vol(K X t ). For x ∈ X , set (0.1) φ(x) := ϕ π(x) (x) and consider (0.2)

V Θ = sup{u ∈ PSH(X , Θ); u 0}.

We prove that conditions (H1) and (H2) are satisfied in this setting. It then follows from Theorem A and the plurisubharmonic variation of the T t 's ([CGP17, Thm. A]) that φ -V Θ is uniformly bounded on compact subsets of X , cf Theorem 5.5 and Remark 5.6:

Theorem C. -Let π : X → D be a smooth Kähler family of manifolds of general type, let Θ ∈ c 1 (K X /D ) be a smooth representative and let φ be the Kähler-Einstein potential as in (0.1). Given any compact subset K X , there exists a constant M K such that the following inequality -M K φ -V Θ M K holds on K, where V Θ is defined by (0.2).

The same results can be proved if the family π : X → D is replaced by a smooth family π : (X , B) → D of pairs (X t , B t ) of log general type, i.e. such that (X t , B t ) is klt and K X t + B t is big for all t ∈ D.

Stable families. -A stable variety is a projective variety X such that X has semi-log canonical singularities and the Q-line bundle K X is ample. We refer to [START_REF] Kovács | Singularities of stable varieties[END_REF][START_REF] Kollár | Book on moduli of surfaces[END_REF] for a detailed account of these varieties and their connection to moduli theory.

In [START_REF] Berman | GUENANCIA -"Kähler-Einstein metrics on stable varieties and log canonical pairs[END_REF], it was proved that a stable variety admits a unique Kähler-Einstein metric ω, i.e. a smooth Kähler metric on X reg such that, if n = dim C X, Ric(ω) = -ω and

X reg ω n = (K n X ).
The metric ω extends canonically across X sing to a closed, positive current in the class c 1 (K X ). It is desirable to understand the singularities of ω near X sing . In [GW16, Thm. B], it is proved that ω has cusp singularities near the double crossings of X. Moreover, it is proved in [START_REF] Song | Degeneration of Kähler-Einstein manifolds of negative scalar curvature[END_REF] that the potential ϕ of ω with respect to a given Kähler form ω X ∈ c 1 (K X ), i.e. ω = ω X + dd c ϕ, is locally bounded on the klt locus of X. We make this assertion more precise by establishing the following (cf. Proposition 5.9). Proposition D. -For any ε > 0, there is a constant C ε such that

(0.3) C 1 ϕ -(n + 1 + ε) log(-log |s|) -C ε
where (s = 0) is any reduced divisor containing the non-klt locus of X.

This estimate is almost optimal. Indeed, if X is the Satake-Baily-Borel compactification of a ball quotient, it is a normal stable variety and it admits a resolution (X, D) which is a toroidal compactification of the ball quotient obtained by adding disjoint abelian varieties. Then, the potential ϕ of the Kähler-Einstein metric on (X, D) with respect to a smooth form in c 1 (K X + D) satisfies

ϕ = -(n + 1) log(-log |s D |) + O(1) if (s D = 0) = D.
A slight refinement of Theorem A (cf. Theorem 1.5) allows us to establish a uniform family version of the estimate (0.3). In order to state it, let X be a normal Kähler space and let π : X → D be a proper, surjective, holomorphic map such that each fiber X t has slc singularities and K X /D is an ample Q-line bundle. If ω X ∈ c 1 (K X /D ) is a relative Kähler form and ω X t := ω X | X t , then the Kähler-Einstein metric of X t can be written as ω X t + dd c ϕ t where ϕ t is uniquely determined by the equation (5.7) from section 5. The behavior of ϕ t is then described by the following (see Theorem 5.11) Theorem E. -Let X be a normal Kähler space and let π : X → D be a proper, surjective, holomorphic map such that

• Each schematic fiber X t has semi-log canonical singularities.

• K X /D is an ample Q-line bundle. In particular, X t is a stable variety for any t ∈ D. Assume additionally that the central fiber X 0 is irreducible.

Let ω X t + dd c ϕ t be the Kähler-Einstein metric of X t and let D = (s = 0) ⊂ X be a divisor which contains Nklt(X , X 0 ), cf (4.4). Fix some smooth hermitian metric | • | on O X (D). Up to shrinking D, then for any ε > 0, there exists C ε > 0 such that the inequality

C 1 ϕ t -(n + 1 + ε) log(-log |s|) -C ε holds on X t for any t ∈ D.
Let us finally mention the very recent results of Song, Sturm and Wang [SSW20, Proposition 3.1] where similar bounds are derived in the context of smoothings of stable varieties over higher dimensional bases, with application towards Weil-Petersson geometry of the KSBA compactification of canonically polarized manifolds.

Families of Q-Calabi-Yau varieties. -A Q-Calabi-Yau variety is a compact, normal Kähler space X with canonical singularities such that the Qline bundle K X is torsion. Up to taking a finite, quasi-étale cover referred to as the index 1 cover (cf e.g. [KM98, Def. 5.19]), one can assume that K X ∼ Z O X . Given any Kähler class α on X, it follows from [START_REF]Singular Kähler-Einstein metrics[END_REF] and [START_REF] Ȃun | Regularity properties of the degenerate Monge-Ampère equations on compact Kähler manifolds[END_REF] that there exists a unique singular Ricci flat Kähler metric ω KE ∈ α, i.e. a closed, positive current ω KE ∈ α with globally bounded potentials inducing a smooth, Ricci-flat Kähler metric on X reg . Now, we can consider families of such varieties and ask how the bound on the potentials vary. This is the content of the following (see Theorem 6.1 and Remark 6.2) Theorem F. -Let X be a normal, Q-Gorenstein Kähler space and let π : X → D be a proper, surjective, holomorphic map. Let α be a relative Kähler cohomology class on X represented by a relative Kähler form ω. Assume additionaly that

• The relative canonical bundle K X /D is trivial.

• The central fiber X 0 has canonical singularities.

• Assumption 3.2 is satisfied. Up to shrinking D, each fiber X t is a Q-Calabi-Yau variety. Let ω KE,t = ω t + dd c ϕ t be the singular Ricci-flat Kähler metric in α t , normalized by X t ϕ t ω n t = 0. Then, given any compact subset K D, there exists C = C(K) > 0 such that one has osc X t ϕ t C for any t ∈ K, where osc X t (ϕ t ) = sup X t ϕ tinf X t ϕ t .

In the case of a projective smoothing (i.e. when X admits a π-ample line bundle and X t is smooth for t = 0), the result above has been obtained previously by Rong-Zhang [RZ11a] by using Moser iteration process.

Log Calabi-Yau families. -Let X be a compact Kähler manifold and let B = ∑ b i B i be an effective R-divisor such that the pair (X, B) has klt singularities and c 1 (K X + B) = 0.

It follows from [Yau78, EGZ09, BEGZ10] that one can find a unique Ricci flat metric in each Kähler class α t . A basic problem is to understand the asymptotic behavior of these metrics as α t approaches the boundary of the Kähler cone. Despite motivations coming from mirror symmetry, not much is known when the norm of α t converges to +∞ (this case is expected to be the mirror of a large complex structure limit, see [START_REF] Kontsevich | Homological mirror symmetry and torus fibrations[END_REF]). We thus only consider the case when α t → α 0 ∈ ∂K X .

The non-collapsing case (vol(α 0 ) > 0) can be easily understood by using Theorem A (see Theorem 6.5). We describe here a particular instance of the more delicate collapsing case vol(α 0 ) = 0. Let f : X → Z be a surjective holomorphic map with connected fibers, where Z is a normal Kähler space. Let ω X (resp. ω Z ) be a Kähler form on X (resp. Z). Set

ω t := f * ω Z + tω X . There exists a unique singular Ricci-flat current ω ϕ t := ω t + dd c ϕ t in { f * ω Z + tω X } for t > 0, where X ϕ t ω n X = 0. It satisfies ω n ϕ t = V t • µ (X,B) , where µ (X,B) = (s ∧ s) 1 m e -φ B .
Here, s ∈ H 0 (X, m(K X + B)) is any non-zero section (for some m 1) and φ B is the unique singular psh weight on O X (B) solving dd c φ B = [B] and normalized by

X (s ∧ s) 1 m e -φ B = 1.
The probability measure f * µ (X,B) has L 1+ε -density with respect to ω m Z thanks to [EGZ18, Lem. 2.3]. It follows therefore from [START_REF]Singular Kähler-Einstein metrics[END_REF] that there exists a unique current ω ∞ ∈ {ω Z } solution of the Monge-Ampère equation

ω m ∞ = f * µ (X,B) .
In the case where X is smooth, B = 0 and c 1 (X) = 0, the Ricci curvature of ω ∞ coincides with the Weil-Petersson form of the fibration f of Calabi-Yau manifolds.

Understanding the asymptotic behavior of the ω ϕ t 's as t → 0 is an important problem with a long history, we refer the reader to the thorough survey [START_REF]Collapsing Calabi-Yau manifolds[END_REF] for references. We prove here the following: Theorem G. -Let (X, B) be a log smooth klt pair such that c 1 (K X + B) = 0 and such that X admits a fibration f : X → Z. With the notations above, the conic Ricci-flat metrics ω ϕ t ∈ { f * ω Z + tω X } converge to f * ω ∞ as currents on X when t goes to 0.

When B = 0 is empty, it has been shown in [Tos10, GTZ13, TWY18, HT18] that he metrics ω ϕ t converge to f * ω ∞ in the C α -sense on compact subsets of X \ S X for some α > 0, where S X = f -1 (S Z ) and S Z denotes the smallest proper analytic subset Σ ⊂ Z such that Σ contains the singular locus Z sing of Z and the map f is smooth on f -1 (Z \ Σ).

The proof of Theorem G follows the strategy developed by the above papers with several twists that notably require the extensive use of Theorem A and conical metrics.

Acknowledgements. -We thank S.Boucksom, M.Pȃun, J.Song and A.Zeriahi for several interesting discussions. We are grateful to the anonymous referees for a remarkably careful reading, for suggesting many improvements as well as for pointing out a gap in §3 of an earlier version. The authors are partially supported by the ANR project GRACK.

Chasing the constants

Our goal in this section is to establish the following a priori estimate which is a refinement of the main result of Kolodziej [START_REF] Kołodziej | The complex Monge-Ampère operator[END_REF] (see also [EGZ09, EGZ08, DP10]):

Theorem 1.1. -Let (X, ω X ) be a compact Kähler manifold of complex dimension n ∈ N * and let ω be a semi-positive form which is big, i.e. such that

V := Vol ω (X) = X ω n > 0.
Let ν and µ = f ν be probability measures, with 0 f ∈ L p (ν) for some p > 1. Assume the following two assumptions are satisfied: (H1) there exists α > 0 and A α > 0 such that for all ψ ∈ PSH(X, ω),

X e -α(ψ-sup X ψ) dν A α ; (H2) there exists C > 0 such that X | f | p dν 1/p C.
Let ϕ be the unique ω-psh solution ϕ to the complex Monge-Ampère equation

V -1 (ω + dd c ϕ) n = µ, normalized by sup X ϕ = 0. Then -M ϕ 0 where M = 1 + C 1/n A 1/nq α e α/nq b n 5 + eα -1 C(q!) 1/q A 1/q α , 1/p + 1/q = 1 and b n is a constant such that exp(-1/x) b n n x 2n for all x > 0.
Here

d = ∂ + ∂ and d c = i 2 (∂ -∂) so that dd c = i∂∂. Recall that a function ϕ : X → R ∪ {-∞} is ω-plurisubharmonic (ω-psh for short) if it
is locally given as the sum of a smooth and a psh function, and such that ω + dd c ϕ 0 in the weak sense of currents. We let PSH(X, ω) denote the set of all ω-psh functions.

The non-pluripolar Monge-Ampère measure of arbitrary ω-psh functions has been defined in [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF]. It follows from assumption (H1) that the measure µ does not charge pluripolar sets, since the latter can be defined by ω-psh functions (as follows easily from [GZ05, Thm. 7.2] since a big class contains a Kähler current). The existence of a unique normalized ω-psh solution to V -1 (ω + dd c ϕ) n = µ follows from [BEGZ10, Theorem A] (the case of Kähler forms had been earlier treated in [START_REF]The weighted Monge-Ampère energy of quasi plurisubharmonic functions[END_REF][START_REF] Dinew | Uniqueness in E (X, ω)[END_REF]).

We will use this result to obtain uniform a priori estimates on normalized solutions ϕ t to families of complex Monge-Ampère equations

V t -1 (ω t + dd c ϕ t ) n = µ t ,
when the hypotheses (H1,H2) are satisfied, i.e. the constants 1/α t , A α t , q t , C t in the theorem are actually bounded from above by uniform constants 1/α, A, q, C independent of t. Here q denotes the conjugate exponent of p > 1, 1/p + 1/q = 1. The assumption on this exponent is thus that p > 1 stays bounded away from 1. The reader should keep in mind that assumption (H1) is the strongest of all. In some applications one can assume f ≡ 1 hence (H2) is trivially satisfied.

We are going to eventually obtain a version of Theorem 1.1 that applies to big cohomology classes, extending [BEGZ10, Theorem B]. The proof is almost identical but explaining the statement requires to introduce various notions and technical notations, so we first treat the case of semi-positive classes and postpone this to section 1.4.

1.1. Preliminaries on capacities. -Let K ⊂ X be a Borel set and consider V K,ω := (sup{ψ | ψ ∈ PSH(X, ω) and ψ 0 on K}) * , where * denotes the upper semi-continuous regularization.

The Alexander-Taylor capacity is the following:

T ω (K) := exp -sup X V K,ω .
It is shown in [GZ17, Lem. 9.17] that If K is pluripolar then V K,ω ≡ +∞ and T ω (K) = 0. When K is not pluripolar then

-0 V K.ω ∈ PSH(X, ω) and V K,ω = 0 on K off a pluripolar set; -the Monge-Ampère measure MA(V K,ω ) is concentrated on E.
We denote here and in the sequel by

MA(u) = 1 V (ω + dd c u) n
the normalized Monge-Ampère measure of a ω-psh function u, where V = X ω n = {ω} n is the volume of the cohomology class {ω}. It is defined for any ω-psh function u, cf. e.g. [GZ07, § 1.1]. For a Borel set K ⊂ X, the Monge-Ampère capacity is

Cap ω (K) := sup K MA(u) ; u ∈ PSH(X, ω) and 0 u 1 .
This capacity also characterizes pluripolar sets, i.e.

Cap * ω (P) = 0 ⇐⇒ P is pluripolar. Here Cap * ω is the outer capacity associated to Cap ω defined for any set E ⊂ X as

Cap * ω (E) := inf{Cap ω (G) ; G open, E ⊂ G}. Moreover, if K ⊂ X is a compact set than Cap * ω (K) = Cap ω (K).
The Monge-Ampère and the Alexander-Taylor capacities compare as follows:

Lemma 1.2. - T ω (K) exp 1 - 1 Cap ω (K) 1/n .
We refer the reader to [GZ05, Proposition 7.1] for a proof which also provides a reverse inequality that is not needed in the sequel. where q is the conjugate exponent, 1/p + 1/q = 1.

Proof of

Let K ⊂ X be a non pluripolar Borel set. Recall that V K,ω (x) = 0 for ν-almost every point x ∈ K. The hypothesis (H1) therefore implies that

ν(K) X e -α V K,ω dν A α T ω (K) α .
Combining previous information we obtain

µ(K) CA 1/q α e α/q exp - α/q Cap ω (K) 1/n D Cap ω (K) 2 ,
where

D = b n n CA
1/q α e α/q , with b n a numerical constant such that exp(-1/x) b n n x 2n for all x > 0. We now need to relate the Monge-Ampère capacity of the sublevel sets of a ω-psh function to the Monge-Ampère measure of similar sublevel sets: Lemma 1.3. -Let ϕ be a bounded ω-psh function. For all s > 0 and 0 < δ < 1,

δ n Cap ω ({ϕ < -s -δ}) MA(ϕ) ({ϕ < -s})
We refer to [EGZ09, Lemma 2.2] for a proof.

1.2.2.

Bounding the solution from below. -Under our assumptions (H1,H2), it follows from general arguments that there is a unique bounded ω-psh solution ϕ of MA(ϕ) = µ normalized by sup X ϕ = 0, cf Remark 0.1. The non-expert reader could even think that ϕ is smooth: the point here is to establish a uniform a priori bound from below.

We let f : R + → R + denote the function defined by

f (s) := - 1 n log Cap ω ({ϕ < -s})
Observe that f is non decreasing and such that f (+∞) = +∞. It follows from our previous estimates that for all s > 0 and 0 < δ < 1,

f (s + δ) 2 f (s) + log δ - log D n .
Our next lemma guarantees that such a function reaches +∞ in finite time:

Lemma 1.4. -f (s) = +∞ for all s 5D 1/n + s 0 , where

s 0 = inf{s > 0 | eD 1/n Cap ω ({ϕ < -s}) < 1}.
Proof. -We define a sequence (s j ) of positive reals by induction as follows, s j+1 = s j + δ j with δ j = eD 1/n exp(-f (s j )).

We fix s 0 large enough (as in the statement of the Lemma) so that δ 0 < 1. It is straightforward to check, by induction, that the sequence (s j ) is increasing, while (δ j ) is decreasing. Thus 0 < δ j < 1 and f (s j+1 ) f (s j ) + 1, hence f (s j ) j.

We infer δ j eD 1/n exp(-j) and

s ∞ = s 0 + ∑ j 0 (s j+1 -s j ) s 0 + ∑ j 0 eD 1/n exp(-j) s 0 + 5D 1/n .
It remains to obtain a uniform bound on s 0 . It follows from Chebyshev inequality and Lemma 1.3 (used with δ = 1) that for all s > 0,

Cap ω ({ϕ < -s -1}) 1 s X (-ϕ)dµ,
since MA(ϕ) = µ. Hölder inequality and (H2) yield

X (-ϕ)dµ C X (-ϕ) q dν 1/q .
Observe that for all t 0, t q q! α q exp(αt) and use (H1) to conclude that Cap ω ({ϕ < -s -1})

C(q!) 1/q A 1/q α αs .

Thus

s 0 = 1 + eD 1/n C(q!) 1/q A
1/q α α is a convenient choice. This yields the desired a priori estimate and concludes the proof.

1.3. More general densities. -The setting of Theorem 1.1 is the most commonly used in geometric applications, as it allows e.g. to construct Kähler-Einstein currents on varieties with log-terminal singularities (see section 6). For varieties of general type with semi log-canonical singularities (see section 5.2), one has to deal with slightly more general densities. The following result is a refinement of [Koł98, Theorem 2.5.2] and [EGZ09, Theorem A].

Theorem 1.5. -Let (X, ω X ) be a compact Kähler manifold of complex dimension n ∈ N * and let ω be a semi-positive form with V := Vol ω (X) = X ω n > 0. Let ν and µ = f ν be probability measures, with 0 f ∈ L 1 (ν). Assume the following assumptions are satisfied: (H1) there exists α > 0 and A α > 0 such that for all ψ ∈ PSH(X, ω), X e -α(ψ-sup X ψ) dν A α ;

(H2') there exists C, ε > 0 such that X | f || log f | n+ε dν C.
Let ϕ be the unique ω-psh solution ϕ to the complex Monge-Ampère equation

V -1 (ω + dd c ϕ) n = µ, normalized by sup X ϕ = 0. Then -M ϕ 0 where M = M(C, ε, n, A α ).
Proof. -The proof follows the same lines as that of Theorem 1.1, so we only emphasize the main technical differences and focus on the case ε = 1. Set, for t ≥ 0,

χ(t) = (t + 1) n+1 ∑ j=0 (-1) n+1-j (n + 1)! j! (log(t + 1)) j .
Observe that χ is a convex function such that χ(0) = 0 and χ (t) = (log(t + 1)) n+1 . Its Legendre transform is

χ * (s) = sup t>0 {s • t -χ(t)} = st(s) -χ(t(s))
,

where 1 + t(s) = exp(s 1 n+1 ) satisfies s = χ (t(s)), thus χ * (s) = P(s 1 n+1 ) exp(s 1 n+1
)s, where P is the following polynomial of degree n,

P(X) = n ∑ j=0 (-1) n-j (n + 1)! j! X j .
We let the reader check that (H2') is equivalent to

|| f || χ ≤ C , where || f || χ denotes the Luxemburg norm of f , || f || χ := inf r > 0, X χ(| f |/r)dν ≤ 1 .
Let K ⊂ X be a non pluripolar Borel set. It follows from Hölder-Young inequality [BBE + 19, Proposition 2.15] that

µ(K) ≤ 2C ||1 K || χ * , where ||1 K || χ * = inf{r > 0, ν(K)χ * (1/r) ≤ 1} = r K , with χ * (1/r K ) = 1 ν(K) .
We are interested in the behavior of this function as ν(K) approaches zero, i.e. for small values of r K . Observe that χ * (s) ≤ exp(2s

1 n+1 ) for s ≥ 1/r n , hence ν(K) ≤ δ n =⇒ µ(K) ≤ 2C r K ≤ 2 n+2 C (-log ν(K)) n+1 .
Recall that (H1) and Lemma 1.2 yield

ν(K) ≤ A α e α exp - α Cap ω (K) 1/n It follows that for ν(K) ≤ δ n , µ(K) ≤ C Cap ω (K) 1+1/n ,
and we can then conclude by reasoning as in Lemma 1.4. This completes the proof when ε = 1. The proof for arbitrary ε > 0 is similar, the crucial point being the domination of µ by a multiple of Cap 1+ε/n ω , with an exponent 1 + ε/n > 1.

1.4. Big cohomology classes. -We now consider a similar situation where the reference cohomology class α is still big but no longer semi-positive. We assume for convenience that the ambient manifold (X, ω X ) is again compact Kähler, but one could equally well develop this material when X belongs to the Fujiki class (i.e. when X is merely bimeromorphic to a Kähler manifold).

By definition α is big if it contains a Kähler current, i.e. there is a positive current T ∈ α and ε > 0 such that T εω X . It follows from [START_REF]Regularization of closed positive currents and intersection theory[END_REF] that one can further assume that T has analytic singularities, i.e. it can be locally written T = dd c u, with

u = c 2 log s ∑ j=1 | f j | 2 + v,
where c > 0, v is smooth and the f j 's are holomorphic functions.

Definition 1.6. -We let Amp(α) denote the ample locus of α, i.e. the Zariski open subset of all points x ∈ X for which there exists a Kähler current in α with analytic singularities which is smooth in a neighborhood of x.

It follows from the work of Boucksom [START_REF]Divisorial Zariski decompositions on compact complex manifolds[END_REF] that one can find a single Kähler current T 0 with analytic singularities in α such that Amp(α) = X \ Sing T 0 .

We fix θ a smooth closed differential (1, 1)-form representing α. Following Demailly, one defines the following θ-psh function with minimal singularities:

V θ := sup{u ; u ∈ PSH(X, θ) and u ≤ 0}.

Definition 1.7. -A θ-psh function ϕ has minimal singularities if for every other θ-psh function u, there exists C ∈ R such that u ϕ + C.

There are plenty of such functions, which play the role here of bounded functions when α is semi-positive. Demailly's regularization result [START_REF]Regularization of closed positive currents and intersection theory[END_REF] insures that α contains many θ-psh functions which are smooth in Amp(α). In particular a θ-psh function ϕ with minimal singularities is locally bounded in Amp(α). The Monge-Ampère measure (θ + dd c ϕ) n is thus well defined in Amp(α) in the sense of Bedford and Taylor [START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF].

Definition 1.8. -It follows from the work of Boucksom [START_REF] Boucksom | On the volume of a line bundle[END_REF] that Amp(α)

(θ + dd c ϕ) n =: V α > 0 is independent of ϕ, it is the volume of the cohomology class α.
One can therefore develop a pluripotential theory in the Zariski open set Amp(α). This was done in [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF], where the following properties have been established:

the class PSH(X, θ) enjoys several compactness properties;

the operator MA(ϕ) = V -1 α (θ + dd c ϕ) n is a well defined probability measure on the set of θ-psh functions with minimal singularities; -the extremal functions V K,θ = sup{u ; u ∈ PSH(X, θ) and u 0 on K} and the Alexander-Taylor capacity T θ (K) = exp (sup X V K,θ ) enjoy similar properties as in the semi-positive case; -in particular it compares similarly to the Monge-Ampère capacity Cap θ (K) := sup K MA(u) ; u ∈ PSH(X, θ) and 0 u -V θ 1 ; -the comparison principle holds so Lemma 1.3 holds here as well.

The same proof as above therefore produces the following uniform a priori estimate, which is a refinement of [BEGZ10, Thm. 4.1]: Theorem 1.9. -Let (X, ω X ) be a compact Kähler manifold of complex dimension n ∈ N * . Let α be a big cohomology class of volume V α > 0 and fix θ a smooth closed differential (1, 1)-form representing α.

Let ν and µ = f ν be probability measures, with 0 f ∈ L p (ν) for some p > 1. Assume the following assumptions are satisfied:

(H1) ∃α > 0, A α > 0 such that ∀ψ ∈ PSH(X, θ), X e -α(ψ-sup X ψ) dν A α ; (H2) there exists C > 0 such that X | f | p dν 1/p C.
Let ϕ be the unique θ-psh function with minimal singularities such that

V α -1 (θ + dd c ϕ) n = µ,
and sup X ϕ = 0. Then -M ϕ -V θ 0 where

M = 1 + C 1/n A 1/nq α e α/nq b n 5 + eα -1 C(q!) 1/q A 1/q α
, where b n is a uniform constant such that exp(-1/x) b n n x 2n for all x > 0.

Remark 1.10. -We also have an analogue of Theorem 1.5 in the big setting.

Uniform integrability

We wish to apply the previous uniform estimates when the complex structure of the underlying manifold is moving. In this section we pay a special attention to assumption Setting 2.1. -Let X be an irreducible and reduced complex Kähler space. We let π : X → D denote a proper, surjective holomorphic map such that each fiber X t = π -1 (t) is a ndimensional, reduced, irreducible, compact Kähler space, for any t ∈ D.

For later purposes, we pick a covering {U α } α of X by open sets admitting an embedding j α : U α → C N for some N n + 1. Moreover, we fix a Kähler form ω on X . Up to refining the covering, the datum of ω is equivalent to the datum of Kähler metrics on open neighborhoods of j α (U α ) ⊂ C N that agree on each intersection

U reg α ∩ U reg β . Equiv- alently, ω is a genuine Kähler metric on X reg such that (j α ) * (ω| U reg α ) is the restriction of a Kähler metric defined on a an open neighborhood of j α (U α ) ⊂ C N .
Let us point out that this definition of a Kähler metric on a singular space X is much more restrictive than merely asking for a Kähler metric on X reg (even say, by requiring that the latter has local potentials near X sing , and that those are bounded). One important property that Kähler metrics satisfy is that their pull back under a modification is a smooth form (i.e. locally the restriction of a smooth form under a local embedding in C N ); in particular, it is dominated by a Kähler form.

For each t ∈ D, we set

ω t := ω |X t .
An easy yet important observation is the following.

Lemma 2.2. -In the Setting 2.1 and using the notation above, the quantity X t ω n t is independent of t ∈ D. We will denote it by V in the following.

Proof. -The function D t → X t ω n t coincides with the push-forward current π * ω n of bidimension (1, 1). Its distributional differential is zero as d commutes with π * and ω is closed.

We fix a smooth, closed differential (1, 1)-form Θ on X and set θ t = Θ |X t . Up to shrinking D, one will always assume that there exists a constant

C Θ > 0 such that (2.1) -C Θ ω Θ C Θ ω.
In particular, one has the inclusion PSH(X t , θ t ) ⊆ PSH(X t , C Θ ω t ). We assume that the cohomology classes {θ t } ∈ H 1,1 (X t , R) are psef, i.e. the sets PSH(X t , θ t ) are nonempty for all t. The notions of (quasi)-plurisubharmonic functions, positive currents and Monge-Ampère measure are well defined on singular spaces [START_REF]Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines[END_REF].

Uniform integrability index

. -Recall from [Dem82, Déf. 3] that if T is a closed,
positive current of bi-dimension (p, p) on a complex space X and if x ∈ X is a closed point, then the Lelong number of T at x is defined as the limit

(2.2) ν(T, x) := lim r→0 ↓ 1 r 2p {ψ<r} T ∧ (dd c ψ) p
where ψ := ∑ i∈I |g i | 2 and (g i ) i∈I is a (finite) system of generators of the maximal ideal m X,x ⊂ O X,x . It is proved in loc. cit. that the limit above is a decreasing limit, independent of the choice of the generators. Moreover, one has the formula

(2.3) ν(T, x) = {x} T ∧ (dd c log ψ) p cf [Dem82, bottom of p. 45].
Finally, if ϕ is a θ-psh function on X for some smooth, closed (1, 1)-form θ, then the Lelong number of ϕ at a given point x ∈ X is defined to be the quantity ν(θ + dd c ϕ, x).

Proposition 2.3. -In the Setting 2.1, let ϕ t ∈ PSH(X t , θ t ) be a collection of θ t -psh functions on X t . Then

sup t∈D 1/2 sup x∈X t ν(ϕ t , x) < +∞.
Proof. -Let U α U α be a relatively compact open subset such that the U α are still a covering of X . Up to adding more elements to the initial covering, one can always assume that one can find such a refinement. One picks cut-off functions χ α such that χ α ≡ 1 on U α and Supp(χ α ) ⊂ U α . Now, let x ∈ X ; there exists α = α(x) such that x ∈ U α . Recall that we have an embedding

j α : U α → C N ; we set x := j α (x) and G x : C N z → log(∑ N i=1 |z i -x i | 2 ).
One can easily check that there exists a constant A > 0, independent of the point x now ranging in the compact set π -1 (D 1/2 ), such that the function

H x := χ α • j * α G
x defines an Aω-psh function on the whole X . By the formula (2.3), one has

ν(ϕ t , x) = {x} (θ t + dd c ϕ t ) ∧ (dd c (j * α G x )| X t ) n-1 U α ∩X t (θ t + dd c ϕ t ) ∧ (dd c H x ) n-1 U α ∩X t (θ t + dd c ϕ t ) ∧ (Aω t + dd c H x ) n-1 X t (C Θ ω t + dd c ϕ t ) ∧ (Aω t + dd c H x ) n-1 = C Θ A n-1 • V.
The conclusion follows.

It follows from Skoda's integrability theorem [START_REF] Skoda | Sous-ensembles analytiques d'ordre fini ou infini dans C n[END_REF] that the Lelong number ν(ϕ t , x) controls the local integrability index α(ϕ t , x) of a θ t -psh function ϕ t ,

α(ϕ t , x) := sup c > 0 ; e -cϕ t ∈ L 2 loc (X t , x) , via 1 ν(ϕ t , x) α(ϕ t , x) n ν(ϕ t , x) .
Proposition 2.3 thus yields:

Corollary 2.4. -In the Setting 2.1, the following quantity

α(Θ) := inf α(ϕ t , x); t ∈ D 1/2 , x ∈ X t , ϕ t ∈ PSH(X t , θ t )
is positive.

Skoda's integrability theorem in families: the projective case. -Zeriahi [Zer01]

has established a uniform version of Skoda's integrability theorem. We now further generalize Zeriahi's result by establishing its family version. We first provide a very explicit result in the projective case which does not rely on Corollary 2.4 unlike its general Kähler analogue that will be given later, cf. Theorem 2.9. This should also help the reader in following the somehow tricky computations in the general Kähler case.

Proposition 2.5. -Let V ⊆ P N be a projective variety of complex dimension n and degree d.

Let ω = ω FS | V and ϕ ∈ PSH(V, ω) be such that sup V ϕ = 0. Then V e -1 nd ϕ ω n (4n) n • d • exp - 1 nd V ϕω n .
To our knowledge, the inequality given in Proposition 2.5 above is new.

Remark 2.6. -When π : X → D is a projective family whose fibers have degree d with respect to a given projective embedding, the above result gives the integrability of e -1 nd ϕ t on V t := π -1 (t). In particular, one gets α(ω FS ) 1 2nd .

Proof. -Embedding P 1 in P 2 if necessary, we assume without loss of generality that N ≥ 2. We first claim that it is enough to prove the Proposition when ϕ is smooth. Indeed, thanks to [CGZ13, Cor. C], there exists a sequence of smooth functions ϕ n ∈ PSH(V, ω FS ) decreasing pointwise to ϕ. Let ε n := sup V ϕ n ; by Hartog's theorem, we have ε n → 0. If the Proposition holds for smooth functions, we will have

V e -1 nd ϕ n ω n e εn •(d-1) nd (4n) n • d • exp - 1 nd V ϕ n ω n
Using Fatou Lemma and the monotone convergence theorem, we deduce the expected inequality for ϕ. From now on, one assumes that ϕ is smooth.

The projective logarithmic kernel on P N × P N is defined by the following formula

G(x, y) := log ||x ∧ y|| ||x|| • ||y|| , x, y ∈ P N ,
writing x, y in homogeneous coordinates. By [AAZ20, Lem. 4.1], for any fixed y, x → G(x, y) is a non positive ω FS -psh function in P N such that (ω FS + dd c x G(•, y)) N = δ y . We set g = G| V and g y = g(•, y). By definition, g y has Lelong number one at y. Therefore, it follows from [Dem85, Cor. 4.8] that ω n g y := (ω + dd c g(•, y)) n δ y . From Stokes formula (cf Lemma 2.11 below) it follows that

ϕ(y) V ϕω n g y = V ϕ(ω + dd c g y ) ∧ ω n-1 g y = V ϕω ∧ ω n-1 g y + V g y (ω + dd c ϕ) ∧ ω n-1 g y - V g y ω ∧ ω n-1 g y V ϕω ∧ ω n-1 g y + V g y ω ϕ ∧ ω n-1 g y ,
using that g y 0. One obtains similarly

V ϕω ∧ ω n-1 g y V ϕω 2 ∧ ω n-2 g y + V g y ω ∧ ω ϕ ∧ ω n-2 g y V ϕω 2 ∧ ω n-2 g y + V g y ω ϕ ∧ ω n-1 g y ,
where the second inequality follows from

V g y ω ∧ ω ϕ ∧ ω n-2 g y = V g y ω ϕ ∧ ω n-1 g y + V dg y ∧ d c g y ∧ ω ϕ ∧ ω n-2 g y V g y ω ϕ ∧ ω n-1 g y .
Iterating the process n times we end up with ϕ(y)

V ϕω n + n V g y ω ϕ ∧ ω n-1 g y . Hence V e -1 nd ϕ ω n exp - 1 nd V ϕω n • I
where

I := y∈V exp - 1 d x∈V g y (x)ω ϕ (x) ∧ ω g y (x) n-1 ω(y) n The (n, n)-form 1 d • ω ϕ ∧ ω n-1 g y induces a probability measure on V given that V ω ϕ ∧ ω n-1 g y = P N ω ϕ ∧ ω n-1 g y ∧ [V] = {ω FS } n • {V} = d.
From Jensen's inequality, one can then derive

I 1 d y∈V x∈V e -g(x,y) ω ϕ (x) ∧ (ω(x) + dd c x g(x, y)) n-1 ∧ ω(y) n . Lemma 2.8 (i) yields ω ϕ (x) ∧ (ω(x) + dd c x g(x, y)) n-1 e -2(n-1)g(x,y) ω ϕ (x) ∧ ω(x) n-1 . Lemma 2.8 (ii) below (for δ = 1/2n) now yields I 1 d y∈V x∈V e (-2n+1)g(x,y) ω ϕ (x) ∧ ω(x) n-1 ∧ ω(y) n = 1 d x∈V y∈V e -2(1-1 2n )g(x,y) ω(y) n ω ϕ (x) ∧ ω(x) n-1 (4n) n x∈V 1 d y∈V (ω + dd c χ 1 2n • g x ) n ω ϕ (x) ∧ ω(x) n-1 = (4n) n x∈V ω ϕ (x) ∧ ω(x) n-1 = (4n) n • d.
Remark 2.7. -The same arguments as above show that for any γ ∈ (0, 2)

V e -γ nd ϕ ω n C γ • d exp - γ nd V ϕω n ,
where C γ > 0 depends on n and γ. We have fixed γ = 1 in the above proposition to simplify the statement.

Lemma 2.8. -With the notations of the proof of Proposition 2.5 above, we fix a point y ∈ V and set g := g y . Moreover, let δ ∈ (0, 1) be a given number. Then, the following set of inequalities hold as currents on V.

(i) ω g e -2g ω (ii) δ 2 e -2(1-δ)g ω ω + dd c χ δ • g Here, χ δ is the function defined on R by the expression χ δ (t) := e 2δt 4δ . It is understood here that we take derivatives w.r.t. x and the estimates are uniform both in x and y.

Proof. -We proceed in three steps.

Step 1. Reduction to a computation on C N . First of all we observe that the function g as well as the (1, 1)-currents ω and ω g are the restriction to V of a function or (1, 1)-currents on P N . As positivity is preserved by restriction to a subvariety, it is enough to prove the inequalities of currents above on the whole P N where they make sense as well. Now, recall that PU(N, C) acts transitively on P N by transformations preserving ω FS and an isometry u sends G y to G u(y) . Therefore it suffices to prove all the inequalities above on P N , for the special point y = [1 : 0 : • • • : 0]. We work in the affine chart (U 1 , z) where U 1 := {x ∈ P N : x 1 = 0} and z := (z j ) j , z j = x j /x 1 . In these coordinates

ω FS | U 1 = 1 2 dd c log(1 + z 2 ).
Note that U 1 is dense in P N and both ω FS , ω G are smooth on the complement P N \ U 1 ; thus it is sufficient to prove the inequalities on U 1 C N .

We actually claim that is is sufficient to prove the inequalities on U 1 \ {y}, where all the currents involved are smooth differential forms. This is because neither of the positive currents e -2G ω FS and ω FS + dd c χ δ • G on P N puts any mass on {y}. This follows from the integrability of e -2G for the first one (recall that N 2) and the boundedness of χ δ • G for the second one.

As observed in [AAZ20, Lem. 4.1], for (x,

y := [1 : 0 • • • : 0]) ∈ U 1 × U 1 we have G(x, y) = N(z, 0) - 1 2 log(1 + z 2 )
where z = z(x) and N(z, 0) := 1 2 log z 2 . Thus in U 1 we have e -2G = 1 + 1 z 2 and

ω(x) + dd c x G y (x) = dd c z N(z, 0) = 1 2 dd c z log z 2 . Let us define β := dd c z 2 = i ∑ N k=1 dz k ∧ d zk and let α 1 := ∑ N k=1 zk dz k .
Step 2. Proof of Item (i).

Standard computations give

(ω FS ) j k = (1 + z 2 )δ j k -zj z k 2(1 + z 2 ) 2 and N j k = 1 2 • z 2 δ j k -zj z k z 4
or equivalently

ω FS = 1 2 1 1 + z 2 β - 1 (1 + z 2 ) 2 iα 1 ∧ ᾱ1 and ω G = 1 2 1 z 2 β - 1 z 4 iα 1 ∧ ᾱ1
The matrix A(z) := (z i zj ) ij is semipositive with rank at most one and trace z 2 . Therefore, if λ, µ ∈ R (they can depend on z), the matrix λId + µA is hermitian with eigenvalues λ (with multiplicity N -1) and λ + z 2 • µ (with multiplicity one). In particular, it is semipositive if and only if λ max(0,z 2 • µ).

The computations above show that the eigenvalues of the (1, 1)-form λβ + µiα 1 ∧ ᾱ1 with respect to β are λ and λ + z 2 • µ. Now, if C is some non-negative constant, the (1, 1)-form Ce -2g ω FSω G can be rewritten as follows

1 2(1 + z 2 ) z 4 • (C -1) z 2 (1 + z 2 ) • β + (1 + z 2 ) -C z 2 • iα 1 ∧ ᾱ1 .
The latter form is semipositive if and only if C 1. This proves (i).

Step 3. Proof of Item (ii).

Observe that χ δ is convex increasing with 0 χ δ 1/2 for t 0. Standard computations give

dd c χ δ • G = χ δ • G dd c G + χ δ • G dG ∧ d c G. Next, we have dd c G = 1 2 z 2 (1 + z 2 ) β - 1 + 2 z 2 z 2 (1 + z 2 ) • iα 1 ∧ ᾱ1
with the notation introduced in Step 1. Similarly, one finds

dG ∧ d c G = 1 4 z 4 (1 + z 2 ) 2 iα 1 ∧ ᾱ1 .
To lighten notation, we will from now on write χ (resp. χ ) to denote χ δ • G (resp. χ δ • G). One has

ω FS + dd c χ δ • G = 1 2(1 + z 2 ) 1 + χ z 2 β + 1 2 χ -χ (1 + 2 z 2 ) z 4 (1 + z 2 ) iα 1 ∧ ᾱ1 .
As a result, the two eigenvalues λ, µ of ω FS + dd c χ δ • G with respect to ω FS are given by

λ = 1 + χ z 2 and µ = (1 + z 2 ) • 1 + χ z 2 + 1 2 χ -χ (1 + 2 z 2 ) z 2 (1 + z 2 ) = (1 + z 2 -χ ) + χ 2 z 2
Using the definition of χ and the fact that e -2G = 1 + 1 z 2 , one easily sees that λ 1 2 e -2(1-δ)G and µ δ 2 e -2(1-δ)G . The conclusion follows.

2.4. Skoda's integrability theorem in families: the general case. -In this section, we bypass the projectivity assumption and establish a quite general family version of Skoda's integrability theorem, valid for families of compact Kähler varieties:

Theorem 2.9. -In Setting 2.1, let us choose a positive number α ∈ (0, α(Θ)), which is possible thanks to Corollary 2.4. Then, there exist constant A α , C > 0 such that for all t ∈ D 1/2 and for all ϕ t ∈ PSH(X t , θ t ) with sup X t ϕ t = 0, (2.4)

X t e -αϕ t ω n t C exp -A α X t ϕ t ω n t .
Proof. -The proof follows the same strategy as in [START_REF] Zeriahi | Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions[END_REF], as presented in [GZ17, Thm. 2.50]. There exists a finite number of trivializing charts {U τ } of X such that π -1 (D 1/2 ) ⊂ ∪ τ U τ . The statement will then follow if we prove the bound for the integral on the left-hand side replacing X t by X t ∩ U τ . Moreover, w.l.o.g we can assume that we have an immersion j τ : U τ → B, where B is the unit ball in C N . Up to shrinking U τ , one can also assume that there exists a smooth function ρ on B such that sup B ρ = -2 and Θ| U τ = dd c j * τ ρ. We define

ρ t := (j * τ ρ)| U τ ∩X t ; this is a potential of θ t | U τ ∩X t . Note that ψ t := ϕ t + ρ t is a non-positive psh function in U τ ∩ X t such that (2.5) ϕ t -2 ψ t ϕ t -C τ for some constant C τ > 0 depending only on U τ . It is also clear that proving (2.4) is equivalent to showing that (2.6) U τ ∩X t e -αψ t ω n t C τ exp -A α,τ U τ ∩X t ψ t ω n t ,
for some constants C τ , A α,τ that do not depend on t.

Claim 2.10. -It is sufficient to prove (2.6) for smooth, non-positive psh functions ψ t on U τ ∩ X t such that (2.7) dd c ψ t (j * τ dd c z 2 )| X t . Proof of Claim 2.10. -Indeed, as

U τ ∩X t e -αψ t ω n t e α U τ ∩X t e -α(ψ t +j * τ z 2 ) ω n t ,
we can replace ψ t by the function ψ t + j * τ z 2 , bounded above by -1. Next, thanks to a result of Fornaess-Narasimhan [FN80, Thm. 5.5], one can write ψ t as a decreasing limit of non-positive, smooth psh functions on U τ ∩ X t (up to shrinking U τ possibly). The combination of the monotone convergence theorem and the integrability of e -αϕ t on X t provided by Corollary 2.4 settles the claim.

From now on, we assume that ψ t is smooth, and we work exclusively on U τ that we view inside the unit ball B of C N . By abuse of notation, we will denote by B ∩ X t the set U τ ∩ X t . In the same vein, we will identify the coordinate functions z = (z 1 , . . . , z N ) on B ⊂ C N with their pull-back by j τ on U τ .

Let us pick some number t ∈ D 1/2 and some point x ∈ B ∩ X t . We denote by Φ x the automorphism of the unit ball B that sends x to the origin and consider

G x (z) := log Φ x (z)
the pluricomplex Green function of the unit ball B. Recall that G x is the unique plurisubharmonic function in B such that (dd c G x ) N = δ x in the weak sense of currents, G x 0 and G x is identically zero on ∂B. Standard computations yield (2.8)

dd c G x C 0 Φ x (z) 2 dd c z 2 on B.
for some dimensional constant

C 0 = C 0 (N) > 0. Since [X t | B ] is a positive (N -n, N -n)-current on B and the singular set of the re- striction of the Green function G x | X t is compact (it is indeed equal to {x}), the mixed Monge-Ampère measure (dd c G x ) n ∧ [X t ] is well defined [GZ17, Prop. 3.15]
and it has a Dirac mass with coefficient 1 at the point x. Since ψ t 0 we then have

ψ t (x) B ψ t (dd c G x ) n ∧ [X t ] = B∩X t ψ t (dd c G x ) n .
Now, we have the following result, which is Stokes' formula in a context of isolated singularities.

Lemma 2.11. -Let X ⊂ B C N (0, 2) be a a proper, n-dimensional complex subspace of the ball of radius 2 in C N , center at the origin. Let u, v, w be psh functions on B C N (0, 2) with isolated singularities, i.e. they are smooth outside a discrete set of points in B C N (0, 2) which we assume does not meet ∂B C N (0, 1). Finally, let B := B C N (0, 1) ∩ X. Then, we have

(2.9) ∂B (ud c v -vd c u) ∧ (dd c w) n-1 = B (udd c v -vdd c u) ∧ (dd c w) n-1
We include a proof for the reader's convenience.

Proof of Lemma 2.11. -By using a (regularized) maximum operation, we can find a family of smooth psh functions u ε (resp. v ε , w ε ) decreasing to u (resp. v, w) and which coincide with their limit outside a compact set K ε B which collapses onto a finite set S B. By the usual Stokes' formula, one has

∂B (u ε d c v ε -v ε d c u ε ) ∧ (dd c w ε ) n-1 = B (u ε dd c v ε -v ε dd c u ε ) ∧ (dd c w ε ) n-1
The left-hand side of the formula above is identical to the left-hand side of (2.9). To prove that the right-hand side above converges to the right-hand side of (2.9), we prove that the current (u

ε dd c v ε -v ε dd c u ε ) ∧ (dd c w ε ) n-1 on B converges to (udd c v -vdd c u) ∧ (dd c w) n-
= X t , u = ψ t , v = w = G x (recall that G x | ∂B ≡ 0), we get B∩X t ψ t (dd c G x ) n = B∩X t G x dd c ψ t ∧ (dd c G x ) n-1 =:I t + ∂B∩X t ψ t d c G x ∧ (dd c G x ) n-1 =:J t
By Lemma 2.12, in order to get a lower bound for J t , it is enough to bound from above the quantity

∂B∩X t (-ψ t ) d c z 2 ∧ (dd c z 2 ) n-1 . Applying (2.9) to u = -ψ t , v = w = z 2 -1, we find ∂B∩X t (-ψ t ) d c z 2 ∧ (dd c z 2 ) n-1 = B∩X t (-ψ t ) (dd c z 2 ) n + B∩X t ( z 2 -1) dd c ψ t ∧ (dd c z 2 ) n-1 B∩X t (-ψ t ) (dd c z 2 ) n C n 1 X t (-ϕ t ) ω n t + C τ • V ,
where C 1 is such that dd c z 2 C 1 ω on B and C τ is given in (2.5).

We now take care of the most singular term I t . Set

γ t (x) := B dd c ψ t ∧ (dd c G x ) n-1 ∧ [X t ] so that µ := γ -1 t dd c ψ t ∧ (dd c G x ) n-1 ∧ [X t
] is a probability measure on B (depending on x). We claim that for any x ∈ B there exists a constant ν > 0 independent of t and x such that 1 γ t < ν. The uniform upper bound follows from the same computations in the proof of Proposition 2.3. By (2.7) we can infer that

B dd c ψ t ∧ (dd c G x ) n-1 ∧ [X t ] B dd c z 2 ∧ (dd c G x ) n-1 ∧ [X t ] ν((dd c G x ) n-1 ∧ [X t ], x) ν([X t ], x) = m(X t , x) 1
In the second inequality we used the fact that r → 1 r 2 B r dd c z 2 ∧ T is decreasing to ν(T, x) when r ↓ 0 (see (2.2)). The first equality follows from (2.3) while the second one comes from Thie's theorem. Recall that the origin of B is identified with the point x.

We now use Jensen's formula and (2.8) to obtain exp (-αI

t (x)) = exp z∈B -αγ t G x dµ 1 γ t z∈B e -αγ t G x dd c ψ t ∧ (dd c G x ) n-1 ∧ [X t ] = 1 γ t z∈B dd c ψ t ∧ (dd c G x ) n-1 ∧ [X t ] Φ x (z) αγ t C 0 z∈B dd c ψ t ∧ (dd c z 2 ) n-1 ∧ [X t ] Φ x (z) αν+2n-2 ,
where we can assume w.l.o.g. that αν < 2. By Fubini's theorem, we have

x∈B 1/2 e -αψ t ω n ∧ [X t ] x∈B 1/2 e -α(I t +J t ) ω n ∧ [X t ] K • x∈B 1/2 e -αI t ω n ∧ [X t ] C 0 • K • x∈B 1/2 z∈B dd c ψ t ∧ (dd c z 2 ) n-1 ∧ [X t ] Φ x (z) αν+2n-2 ω n ∧ [X t ] C 0 • K • z∈B x∈B 1/2 (dd c x 2 ) n ∧ [X t ] Φ x (z) αν+2n-2 dd c ψ t ∧ (dd c z 2 ) n-1 ∧ [X t ],
where K := exp{-α C n 1 X t ψ t ω n t }. Moreover, using the same computation as in the proof of Lemma 2.13 below, one can check that if β := 2-αν 2n > 0, there exists a constant C β > 0 such that the inequality of (n, n)-currents below holds on B (2.10)

C -1 β (dd c x Φ x (z) 2β ) n 1 Φ x (z) αν+2n-2 (dd c x 2 ) n C β (dd c x Φ x (z) 2β ) n
Fix z ∈ B and for any x ∈ B let f x (z) := Φ x (z) . We define an extension of f x to X by

F x (z) := χ • f x (z) if x ∈ B 0 else.
Here, χ is a smooth cut-off function such that Supp(χ) ⊂ B and χ ≡ 1 on B 1/2 . It is easy to check that F x is an Aω-psh function on X for some A = A τ big enough (that a priori depends on U τ but can be chosen independently of x ∈ B 1/2 ). Thus

x∈B 1/2 1 Φ x (z) αν+2n-2 (dd c x 2 ) n ∧ [X t ] C β x∈B 1/2 (dd c x Φ x (z) 2β ) n ∧ [X t ] C β x∈X (Aω + dd c x F x (z) 2β ) n ∧ [X t ] C β • A n • V := C 2 .
It then follows that

x∈B 1/2 e -αψ t ω n ∧ [X t ] C 0 • C 2 • K • z∈B dd c ψ t ∧ (dd c z 2 ) n-1 ∧ [X t ] C 3 • K, where C 3 := C 0 C 2 C Θ C n-1 1 • V.
The last inequality follows from the fact that on B t , we have dd c ψ t ∧ (dd c z 2 ) n-1 (θ t + dd c ϕ t ) ∧ (C 1 ω) n-1 , and one can dominate the integral of the right-hand side on B t by its integral on X t and use (2.1). This is the conclusion.

Lemma 2.12. -With the notations introduced at the beginning of the proof of Theorem 2.9, there exists a constant C = C(n) > 0 such that for all x ∈ B 1/2 ⊂ C N and z ∈ X t ∩ S 2N-1 ,

(2.11) 1 C d c z 2 ∧ (dd c z 2 ) n-1 d c G x ∧ (dd c G x ) n-1 Cd c z 2 ∧ (dd c z 2 ) n-1
Proof. -One knows that there exists a neighborhood U of S 2N-1 ⊂ C N not containing x such that dd c Φ x 2 defines a Kähler form ω x on U. This follows for instance from the fact that Φ x can be extended as an holomorphic map to an open neighborhood of the closed ball -and that neighborhood can be chosen to be independent of x ∈ B 1/2 . On U, ω x is comparable to the euclidean metric on C N and therefore, ω x and ω eucl induce uniformly equivalent Riemannian metrics g x and g eucl on U ∩ X t first, and then as well on the real hypersurface X t ∩ S 2N-1 ; we denote them by g x and g eucl respectively. In particular their volume forms dV g x , dV g eucl are equivalent too. One has dV g eucl = ι v dV g eucl where v is the restriction to X t of the unit outward radial vector

n+k ∑ j=1 z j ∂ ∂z j + zj ∂ ∂ zj .
Hence, on X t ∩ S 2N-1 one has

dV g eucl = ι v (dd c z 2 ) n = 2 i π n-1 d c z 2 ∧ (dd c z 2 ) n-1 .
In the same way, dV g x = ι v x dV g x where v x is the restriction to X t of the unit outward vector with respect to dd c Φ x 2 , hence v x = Φ * x v. Therefore one has on X t ∩ S 2N-1 ,

dV g x = ι v x (dd c Φ x 2 ) n = Φ * x (ι v (dd c z 2 ) n ) = 2 i π n-1 d c Φ x 2 ∧ (dd c Φ x 2 ) n-1 = 2 n+1 i π n-1 d c G x ∧ (dd c G x ) n-1 .
given that G x = 1 2 log Φ x 2 vanishes on the sphere and that d c log u ∧ (dd c log u) n-1 = 1 u n d c u ∧ (dd c u) n-1 for any smooth function u. This shows that the above two volume forms on X t ∩ S 2N-1 are uniformly equivalent on X t ∩ S 2N-1 hence it ends the proof.

Lemma 2.13. -Let β > 0 and B ⊂ C n be the unit ball. Then z 2β is psh on B and there exists a constant C β > 0 (that depends only on β) such that

C -1 β z 2(1-β) • dd c z 2 dd c z 2β C β z 2(1-β) • dd c z 2 .
Proof. -Let χ : R + → R + be defined as χ(t) := t β and u := z 2 . One has

dd c χ • u = βu β-1 dd c u -(1 -β)u -1 du ∧ d c u . Note that min{1, β} • dd c u dd c u -(1 -β)u -1 du ∧ d c u max{1, β} • dd c u.
Observe that he hermitian matrix associated to the (1, 1)-form du ∧ d c u is ( zi z j ) i j. The latter has rank one and its non-zero eigenvalue coincides with its trace, i.e. u. Therefore the eigenvalues of the hermitian matrix A := I n -(1β)u -1 ( zi z j ) i j are 1 (with multiplicity n -1) and β (multiplicity 1). This ends the proof.

Normalization in families

Previous section allows us to check hypothesis (H1), as soon as the mean value of sup-normalized θ t -psh functions is uniformly controlled. It is classical that one can compare the supremum and the mean value of θ-psh functions on a fixed compact Kähler variety (see [GZ17, Prop. 8.5]). We conjecture that the following results holds Conjecture 3.1. -In the Setting 2.1, there exists a constant C > 0 such that: the inequality sup

X t ϕ t -C 1 V X t ϕ t ω t n sup X t ϕ t
holds for all t ∈ D 1/2 and for every function ϕ t ∈ PSH(X t , θ t ).

In a preprint version of this paper, we claimed a proof of the conjecture above but a referee, whom we thank, pointed out a gap. In this section, we propose a large class of families for which the conjecture holds. More precisely, let us consider the following Assumption 3.2. -In Setting 2.1, we assume additionnally that one of the following conditions is satisfied by the family π : X → D.

1. The map π is projective. 2. The map π is locally trivial. 3. The fibers X t are smooth for t = 0. 4. The fibers X t have isolated singularities for every t ∈ D.

Recall that π is said to be projective if we have a commutative diagram as below

X P N × D D ι π pr 2
locally trivial if, up to shrinking D, there exists a euclidean open cover (U α ) α of X and a collection of isomorphisms

F α : X | U α -→ (U α ∩ X 0 ) × D such that the following diagram is commutative (3.1) X | U α (U α ∩ X 0 ) × D D π F α pr 2
For instance, if X is smooth and if the map π is a holomorphic submersion, then π is automatically locally trivial. The main result in this section is the following Proposition 3.3. -In the Setting 2.1 and if Assumption 3.2 is satisfied, then Conjecture 3.1 holds. That is, there exists a constant C > 0 such that: the inequality sup

X t ϕ t -C 1 V X t ϕ t ω t n sup X t ϕ t
holds for all t ∈ D 1/2 and for every function ϕ t ∈ PSH(X t , θ t ).

We will prove Proposition 3.3 in several independent steps.

• In § 3.2, we prove the locally trivial case.

• In § 3.3, we treat the case of isolated singularities.

• In § 3.4-3.5-3.6 we introduce the material (Sobolev and Poincaré inequalities, heat kernels and Green's functions) that we will use in the final section. • In § 3.7, we establish at the same time the projective case and the case of a smoothing, thereby completing the proof of Proposition 3.3. By combining the above result with Theorem 2.9, we get the following Theorem 3.4. -In Setting 2.1, let us choose a positive number α ∈ (0, α(Θ)), which is possible thanks to Corollary 2.4. If Assumption 3.2 is satisfied, there exists a constant C α > 0 such that for all t ∈ D 1/2 and for all ϕ t ∈ PSH(X t , θ t ), we have

X t e -α(ϕ t -sup X t ϕ t ) ω n t C α .

Irreducibility of the fibers. -

The irreducibility of all the fibers is a necessary assumption for the left-hand-side inequality in Conjecture 3.1 to hold as the following example shows:

Example 3.5. -Consider X ⊂ P 2 × C where

X := {([x : y : z], t) ; xy -tz 2 = 0}.
The variety X is smooth and comes equipped with the proper morphism π : X → C induced by the second projection

P 2 × C → C. Set X t = {[x : y : z] ∈ P 2 : xy = tz 2 }.
Note that X t is a smooth conic for t = 0 while X 0 = {[x : y : z] ∈ P 2 : xy = 0} is the union of two lines. The quasi-psh function ϕ on P 2 defined by

ϕ([x : y : z]) = 1 4 log(|x| 2 + |z| 2 ) + log |y| 2 - 1 2 log(|x| 2 + |y| 2 + |z| 2 ) + log 2 2 clearly induces a ω-psh function Φ on X , where ω = ω FS + dd c |t| 2 , Φ([x : y : z], t) = ϕ([x : y : z]).
We set ϕ t := Φ| X t and ω t := ω| X t . A simple computation shows that sup X Φ = 0 and it is attained at points ([x : y : z], t) such that |y| 2 = |x| 2 + |z| 2 . We also find that sup X t ϕ t = 0 and the supremum is attained on the set

S t := [x : 1 : z] : |x| = 1 2|t| • 4|t| 2 + 1 -1 , z 2 = xt -1 .
As t → 0, S t becomes the circle C := {[0 : 1 : e iθ ]; θ ∈ R} ⊂ X 0 . Note also that X 0 = ∪ , where := {[0 : y : z]} and := {[x : 0 : z]} and C ⊂ . The open annulus

U t := {[z 2 : t : z]; 1 < |z| 2 < 2} ⊂ X t satisfies U t ω t δ
for some δ > 0 independent of t as well as

ϕ t | U t 1 2 (log |t| + 1)
from which it follows that lim

t→0 X t ϕ t ω t = -∞.
3.2. The locally trivial case. -In this section, we prove Proposition 3.3 under the assumption that π is locally trivial; we borrow the notations from Diagram (3.1).

One can reduce the problem to showing that there exists a constant C > 0 depending only on π such that given any sequence of complex numbers t k → 0 and any functions ϕ k ∈ PSH(X t k , θ t k ) such that sup X t k ϕ k = 0, one has

X t k ϕ k ω n t k -C.
By compactness of π -1 (D 1/2 ), one can assume that α ranges among the finite set {1, . . . , r} and without loss of generality, one can assume that U α+1 ∩ U α = ∅, for any α ∈ {1, . . . , r -1}. Up to splitting the sequence (ϕ k ) into (at most) r subsequences, we can assume that for every k, ϕ k attains its maximum in the same set U α 0 for some fixed α 0 ∈ {1, . . . , r}. By simplicity, we assume that α 0 = 1.

Let G α,k : U α ∩ X 0 → U α ∩ X t k be the biholomorphism defined as the inverse of the restriction of F α to U α ∩ X t k and let us analyze the sequence of functions

ψ α,k := G * α,k ϕ k . As F * α (ω 0 + idt ∧ d t) is commensurable to ω, there exists C > 0 depending only on π such that (3.2) C -1 ω 0 G * α,k ω t k Cω 0 .
In particular, up to increasing C, one can assume that G * α,k θ t k Cω 0 . As a result, one has ψ α,k ∈ PSH(U α ∩ X 0 , Cω 0 ).

The family (ψ 1,k ) k is a family of non-positive Cω 0 -psh functions on the complex space U 1 ∩ X 0 attaining the value zero there, so it is relatively compact for the L 1 loc topology, cf e.g. [GZ17, Proposition 8.5]. In particular, given any compact subset U 1 U 1 , the integral U 1 ψ 1,k ω n 0 admits a lower bound depending only on U 1 but not on k. Next, the family (ψ 2,k ) k is a family of non-positive Cω 0 -psh functions on U 2 ∩ X 0 . Therefore, either it converges locally uniformly to -∞ or it is relatively compact on each compact subset. From (3.2), it follows that the family of automorphisms

H k := (G -1 2,k )| U 1 ∩U 2 ∩X t k • G 1 | U 1 ∩U 2 ∩X 0 of U 1 ∩ U 2 ∩ X 0 satisfies C -1 ω 0 H * k ω 0 Cω 0 and ψ 2,k = H * k ψ 1,k .
One deduces then easily that for any compact subset U 12 U 1 ∩ U 2 , the integral

U 12
ψ 2,k ω n 0 admits a lower bound independent of k. In turn, this implies that (ψ 2,k ) k is relatively compact for the L 1 loc topology on the whole U 2 ∩ X 0 . By iterating the argument, one finds that for any α, the family (ψ α,k ) k is relatively compact for the L 1 loc (U α ∩ X 0 ) topology and using the estimate (3.2) as above, one concludes easily that X t k ϕ k ω n t k admits a uniform lower bound as claimed. This shows that Proposition 3.3 holds whenever π is locally trivial. An easy consequence is the following Corollary 3.6. -In Setting 2.1, there exists a discrete set Z ⊂ D such that for every compact subset K D \ Z, there exists a constant C K such that X t ϕ t ω n t -C K for any collection of functions ϕ t ∈ PSH(X t , θ t ) such that sup X t ϕ t = 0. Moreover, one can take Z = ∅ provided that the family π : X → D admits a simultaneous resolution of singularities, i.e. a proper, surjective holomorphic map f : Y → X from a Kähler manifold Y such that for any t ∈ D, the induced morphism f | Y t : Y t → X t is a resolution of singularities, where Y t := f -1 (X t ).

Proof. -Let f : Y → X be a resolution of singularities of X . One can assume that Y is a Kähler manifold; let us pick ω Y a Kähler form on Y. The induced map ρ := π • f : Y → D is surjective, hence by generic smoothness, it is smooth over the complement of a proper analytic subset Z of D. In particular, Z is discrete. Note that over Z, the fibers of ρ may have several irreducible components.

We denote by f t the restriction f | Y t : Y t → X t of f to the fiber Y t , where Y t := ρ -1 (t). For any t ∈ D \ Z, the map f t is bimeromorphic, i.e. it is a resolution of singularities of X t . Let us choose K D a compact subset. There exists a constant

C K such that f * ω C K ω Y on ρ -1 (K). In particular, for any t ∈ K, one has f * t ϕ t ∈ PSH(Y t , C K C Θ ω Y ) and sup Y t f * t ϕ t = 0. Now, if
we additionally assume that K D \ Z, we can apply the result above to the smooth family ρ| ρ -1 (K) :

ρ -1 (K) → K to find another constant C K > 0 satisfying Y t ( f * t ϕ t ) ω n Y -C K for any t ∈ K. As ω n Y C -n K f * t ω n t , we deduce that X t ϕ t ω n t -C K • C n K
which concludes the first part of the proof. The second statement is an immediate consequence of the proof of the first one. Indeed, if Y t is smooth (as an analytic space), then π • f is smooth in a neighborhood of Y t and the argument above can be run over a neighborhood of t.

3.3. The case of isolated singularities. -In this section, we prove Proposition 3.3 in the case where all fibers X t , t ∈ D, have isolated singularities.

Remark 3.7. -We would like to start with two observations.

• This case includes the case where n = dim X t = 1.

• If one only assumes that X 0 has isolated singularities, then it is easy to check that there exists ε > 0 such that X t has isolated singularities for any t satisfying |t| < ε. This is because the locus Z ⊂ X where π is not smooth is an analytic set such that dim(Z ∩ X 0 ) = 0 and by upper semi-continuity, Z has relative dimension 0 over a neighborhood of 0 ∈ D.

We now proceed to proving Proposition 3.3 in several steps.

Step 1. Localization of the problem at t = 0. Let f : Y → X be a resolution of singularities X . The induced family π • f : Y → D is generically smooth over D so for r > 0 small enough, the restriction of π • f to the inverse image of D r has a most one singular fiber, corresponding to t = 0. In particular, the family Y → D r is locally trivial away from Y 0 . Applying the result in the locally trivial case (cf. § 3.2) to the collection of f * θ t -psh functions f * ϕ t , we see that for every compact subset K D * r , there exists a constant C K independent of the chosen family such that sup t∈K X t (-ϕ t )ω n t C K , cf also Corollary 3.6. This shows that it is enough to prove that for any sequence t k → 0 and any collection of sup-normalized θ t k -psh functions ϕ t k , one has sup

k 1 X t k (-ϕ t k )ω n t k < +∞.
Step 2. Choice of a good covering.

As the fibers are reduced, it follows from the jacobian criterion for smoothness that the smooth locus of π coincides with the union of the smooth loci of X t when t ranges in D.

Recall that Z, the singular locus of π, is an analytic space of relative dimension at most zero. It has finitely many irreducible components (say when restricted to π -1 (D 1/2 )) and we can assume without loss of generality that this number is equal to the cardinality of Z ∩ X 0 . Let (V α ) α be a finite collection of (small) open balls in X centered at the (finitely many) singular points of X 0 . Up to adding a finite amount of balls to the collection, one can assume that (i) The reunion

V := ∪ α V α is an open neighborhood of X 0 ⊂ X .
(ii) Each point of Z ∩ X 0 belongs to exactly one element V α of the covering.

(iii) For all α, there exists

ρ α ∈ C ∞ (V α , R) such that ω| V α = dd c ρ α .
(iv) There exists r > 0 such that for all α, one has

Z ∩ ∂V α ∩ π -1 (D r ) = ∅.
Up to substracting a constant to ρ α , one can assume that ρ α is non-negative. Moreover, there exists a constant C 1 > 0 such that ρ α C 1 on V α , for any α. Let (χ α ) α be a partition of unity associated to the covering (V α ) α . That means that ∑ α χ α ≡ 1 and Supp(χ α ) ⊂ V α . Finally, let ρ := ∑ χ α ρ α . If follows from (ii) that one has ω = dd c ρ in some neighborhood W α of each point of Z ∩ X 0 . We pick a relatively compact open subset W α W α and set W := ∪W α . Up to decreasing r a little, one can assume that Z ∩ ∂W ∩ π -1 (D r ) = ∅. In particular, there exists δ > 0 such that for any t ∈ D r , one has d ω (∂W ∩ X t , Z) δ. In summary

(3.3) 0 ρ C 1 , ω = dd c ρ on W, d ω (∂W ∩ X t , Z) δ for all t ∈ D r .
Step 3. Weak compactness locally outside Z. Let t k be a sequence of numbers converging to zero, and let ϕ t k ∈ PSH(X t k , θ t k ) such that sup X t k ϕ t k = 0. We claim that there exists a sequence of points

x k ∈ X t k and a constant C 2 > 0 such that (i) ϕ t k (x k ) -C 2 . (ii) d ω (x k , Z) δ/2. Indeed, let y k ∈ X t k be such that ϕ t k (y k ) = 0. If d ω (y k , Z)
δ/2, then we are done. Otherwise, it means that we have y k ∈ W by the third item of (3.3). Now, the function C Θ ρ + ϕ t k is psh on W so by the maximum principle, there exists

x k ∈ ∂W such that (C Θ ρ + ϕ t k )(x k ) (C Θ ρ + ϕ t k )(y k ) 0.
By the first item of (3.3), we deduce ϕ t k (x k ) -C 2 where we set C 2 := C 1 C Θ .

Let U := {x ∈ π -1 (D r ); d(x, Z) > δ/2}. The map π is smooth on U and one can cover U by finitely many open subsets (U j ) 1 j p isomorphic to (U j ∩ X 0 ) × D r over D r . Because of (i), we can argue as in the locally trivial case (cf. § 3.2) by exporting the functions ϕ t k | U j ∩X t k to the fixed space U j ∩ X 0 and get relative compactness there. In particular, one can find a constant C 3 > 0 independent of k such that (3.4)

U∩X t k (-ϕ t k )ω n t k C 3 .
Step 4. The integral bound. On W, one has ω = dd c ρ. This implies that ω n = (dd c ρ) n + T for some smooth, closed

(n, n)-form T on π -1 (D r ) such that T| W ≡ 0. Let us introduce constants C 4 , C 5 such that -C 4 ω n-1 (dd c ρ) n-1 C 4 ω n-1 and T C 5 ω n . As the complement of W in π -1 (D r ) is included in U, it follows from (3.4) that (3.5) X t k (-ϕ t k ) T C 5 C 3 .
Moreover, one has

X t k (-ϕ t k )(dd c ρ) n = X t k -ρdd c ϕ t k ∧ (dd c ρ) n-1 = - X t k ρ(θ t k + dd c ϕ t k ) ∧ (dd c ρ) n-1 + X t k ρθ t k ∧ (dd c ρ) n-1 C 4 C 1 X t k (θ t k + dd c ϕ t k ) ∧ ω n-1 + C Θ C 4 C 1 • V 2C 1 C 4 C Θ • V.
All in all, one finds

X t k (-ϕ t k )ω n t k C 6 where C 6 = C 3 C 5 + 2C 1 C 4 C Θ • V.

Sobolev and Poincaré inequalities. -

In this section, we work in the Setting 2.1 above and we assume from now on that the relative dimension n = dim C X t satisfies n > 1, since the case n = 1 has already been dealt with in § 3.3, cf. Remark 3.7.

For t ∈ D, we set X t := π -1 (t) and denote by X reg t the regular locus of X t . We fix a Kähler form ω on X and set

ω t := ω |X t . Proposition 3.8. -Let K D. There exists C S = C S (K) such that ∀t ∈ K, ∀ f ∈ C ∞ 0 (X reg t ), X t | f | 2n n-1 ω n t n-1 n C S X t (| f | 2 + |d f | 2 ω t ) ω n t .
Remark 3.9. -The inequality above extends immediately to the functions f ∈ W 1,2 (X

= C P (K) such that ∀t ∈ K, ∀ f ∈ W 1,2 0 (X reg t ), X t | f | 2 ω n t C P X t |d f | 2 ω t ω n t .
In the statement above, the space W 1,2 0 (X reg t ) is defined as the space of functions f on

X reg t such that f , d f ∈ L 2 (X reg t , ω t ) and X t f ω n t = 0.
Proof. -First, we claim that for each t ∈ D, there exists such a Poincaré constant C P,t . Indeed, thanks to [Bei19, Thm. 0.2], the Laplacian ∆ ω t is positive, self-adjoint and its spectrum is discrete. It remains to show that its kernel is one-dimensional. Now, if f ∈ W 1,2 (X reg t ) is such that ∆ t f = 0, it means that for every u ∈ W 1,2 (X reg t ), we have ∇u, ∇ f = 0. In particular, taking u = f shows that f is locally constant on X reg t . As X t is irreducible, X reg t is connected and the result follows. Given the absolute case explained above, the family version of Poincaré inequality follows from Proposition 3.8 and the irreducibility of the fibers: we refer the reader to [RZ11b, Prop. 3.2] for a detailed argument (the projectivity assumption made by Ruan-Zhang being unnecessary for this part of the argument).

Heat kernels and

Green's functions. -In this section as well as in the following section 3.6, we go back to the absolute case and consider an irreducible and reduced Kähler space (X, ω)

of dimension n = dim C X satisfying n > 1.
When X is smooth, it is well-known (cf e.g. [Cha84, § VI]) that there exists a smooth, positive function H : X × X × (0, +∞), symmetric in its space variable and such that if ∆ := tr ω dd c , one has

• (-∆ y + ∂ t )H(x, y, t) = 0. • For every x ∈ X, one has weak convergence H(x, •, t)ω n -→ t→0 δ x .
In the general case where X may have singularities, one can introduce X ε = X \ V ε where V ε is a closed ε-neighborhood of X sing with smooth boundary. Then, there exists a unique smooth, positive function

H ε on X ε × X ε × (0, +∞) such that • (-∆ y + ∂ t )H ε (x, y, t) = 0.
• H ε (x, y, t) → 0 whenever x or y approaches ∂X ε .

• For every x ∈ X ε , one has weak convergence

H ε (x, •, t)ω n -→ t→0 δ x .
Moreover, given (x, y, t) ∈ X ε 0 × X ε 0 × (0, +∞), the function (0, ε 0 ) ε → H ε (x, y, t) is decreasing. Using [Cha84, VIII.2 Thm. 4] and its proof, we additionally see that the limit H := lim ε H ε is everywhere finite and satisfies

• H is positive and smooth on X reg × X reg × (0, +∞).

• (-∆ y + ∂ t )H(x, y, t) = 0.
• For all x, y ∈ X reg and t, s > 0, one has

(3.6) H(x, y, t + s) = X H(x, •, t)H(•, y, s)ω n .
• For any x ∈ X reg , one has H(x, •, t)ω n -→ t→0 δ x weakly.

When X ⊂ P N is projective and ω = ω FS | X , Li and Tian have showed in [START_REF] Li | On the heat kernel of the Bergmann metric on algebraic varieties[END_REF] that there is an absolute inequality (3.7) H(x, y, t) H P n (d P N (x, y), t)

for any x, y ∈ X reg and t ∈ (0, +∞), where H P n is the heat kernel of (P n , ω FS ), whose dependence in the space variables x, y is known to reduce to a single real variable, namely the distance between those two points.

In particular, H(x, •, t) is bounded on X reg for any x ∈ X reg and t > 0. Since X sing has real codimension at least two, it admits cut-off functions whose gradient converges to zero in L 2 , and this allows one to perform integration by parts as in the compact case for bounded functions in W 1,2 . We refer to [LT95, Lem 3.1] for more details; we will also rely on the latter result which states that H(x, •, t) ∈ W 1,2 and that is satisfies the conservation property

∀t > 0, X H(x, •, t)ω n = 1.
Below are a few more properties that will be useful later, which are certainly standard in the smooth case. For this purpose, one introduces the function G(x, y, t) := H(x, y, t) -1 V where V := X ω n . The key information for us will be given by the fourth item, for which the arguments are borrowed from [START_REF] Cheng | LI -kernel estimates and lower bound of eigenvalues[END_REF], see also [Siu87, App. A].

Lemma 3.11. -Assume either that X is smooth or that X ⊂ P N is projective and ω = ω FS | X . Let x, y ∈ X reg . We have

1. G(x, y, t) > -1 V , X G(x, •, t)ω n = 0 and X |G(x, •, t)|ω n 2. 2. |G(x, y, t)| 2 G(x, x, t)G(y, y, t). 3. H(x, x, t) → +∞ when t → 0.
4. There exists a constant C 0 depending only on the Sobolev and Poincaré constant of (X reg , ω) such that |G(x, y, t)| C 0 t -n for any x, y ∈ X reg and any t > 0.

Proof. -Under the assumptions on X, we know that H(x, •, t) is bounded on X reg , in W 1,2 and satisfies the conservation property. We will only rely on these non-quantitative properties to establish the items below, and not on the more precise inequality (3.7) which certainly does not hold if X is not projective.

(1) a trivial consequence of the positivity of H and the fact that X H(x, •, t)ω n = 1.

(2) is classical when X is smooth, so we assume for the time being that X is projective.

Let K ε be the Neumann heat kernel on X ε , let V ε := X ε ω n and let G ε := K ε -1 V ε . Then we have K ε (x, y, t) = ∑ i 0 e -λ i,ε t φ i,ε (x)φ i,ε (y) where (φ i,ε ) is an orthonormal basis of L 2 (X ε ) consisting of Neumann eigenfunctions of -∆ with eigenvalues λ i,ε . Note that φ 0,ε = 1 √ V ε
. By Cauchy-Schwarz, we find that

| G ε (x, y, t)| 2 G ε (x, x, t) • G ε (y, y, t).
Thanks to [LT95, Lemma 3.2], K ε converges to H locally smoothly on X 2 reg × (0, +∞) when ε → 0, hence G ε → G in the same way and we get the second item.

(3) Since H H ε , It is enough to show the third claim for H ε . We consider a Sturm-Liouville decomposition as before [START_REF]CHAVEL -Eigenvalues in Riemannian geometry[END_REF]VII (31)]. The sought property now follows since

H ε (x, y, t) = ∑ i 0 e -µ i,ε t ψ i,ε (x)ψ i,ε (y) but now, (ψ i,ε ) is an orthonormal basis of L 2 (X ε ) consisting of Dirichlet eigenfunctions of -∆ with eigenvalues µ i,ε , cf
∑ ψ i,ε (x) 2 is the norm of the unbounded functional L 2 ∩ C ∞ (X ε ) f → f (x).
(4) We start from the identity (3.6), which holds for G as well as one checks easily. Taking y = x and differentiating with respect to s and eventually setting s := t, one finds

-G (x, x, 2t) = dG(x, •, t) 2 L 2 (C S (C P + 1)) -1 G(x, •, t) 2 L 2n n-1
since integration by parts is legitimate as we explained above and X G(x, •, t)ω n = 0. Moreover, the interpolation inequality gives

G(x, x, 2t) = G(x, •, t) 2 L 2 G(x, •, t) 2 n+1 L 1 • G(x, •, t) 2n n+1 L 2n n-1 hence G(x, •, t) 2 L 2n n-1 2 -2 n G(x, x, 2t) n+1 n and - 1 n G (x, x, t)G(x, x, t) -1-1 n C -1 for C = n4 1 n • C S (C P + 1)
. Integrating this inequality w.r.t. t and using the second item, we get the fourth item -recall that G(x, x, t) > 0 for any x ∈ X reg given its expansion as power series, cf (2).

Under the assumptions of Lemma 3.11 above, the integral

G(x, y) := +∞ 0 G(x, y, t)dt is convergent whenever x = y and defines a function G on X reg × X reg such that G(x, •) ∈ L 1 (X reg ). Moreover, since (-∆ + ∂ t )G(x, •, t) = 0, G(x, •, t) → t→+∞ 0 and G(x, •, t)ω n → t→0 δ x -1 V , we have dd c G(x, •) ∧ ω n = ω n V -δ x , i.e. for all f ∈ C ∞ 0 (X reg ), we have (3.8) X ∆ f • G(x, •) ω n = 1 V X f ω n -f (x).
Finally, the first and fourth item of Lemma 3.11 enable us to find a lower bound of the Green function as follows

G(x, y) = 1 0 G(x, y, t)dt + +∞ 1 G(x, y, t)dt (3.9) - 1 V - C n -1
where C only depends on the Sobolev and Poincaré constants of (X reg , ω).

3.6. Green's inequality for general psh functions. -In this section, we assume that the assumptions of Lemma 3.11 are satisfied.

Let us first generalize Formula (3.8) to some functions f ∈ C ∞ (X reg ) that are not necessarily compactly supported. For that purpose, let p : Y → X a log resolution of singularities, let D be the exceptional divisor of p and let Y • := p -1 (X reg ) = Y \ D. We claim that for any f ∈ C ∞ (X reg ) such that p * f extends smoothly across D, the formula (3.10)

X reg ∆ f • G(x, •) ω n = 1 V X reg f ω n -f (x)
holds. First observe that all the terms are well-defined as one sees by pulling back by p, which is an isomorphism over X reg . Indeed, recall that x ∈ X reg and that G(x, •) is locally bounded near X sing so that p

* G(x, •) is in L 1 (Y • , ω Y ) for any Kähler form ω Y on Y.
Next, we choose a family (χ δ ) δ of cut-off functions for D. As they are identically 0 on D, they come from X under p and one can see them either as functions on X or Y interchangeably. It is classical (cf e.g. [CGP13, Sect. 9]) that one can choose χ δ such that both dχ δ ∧ d c χ δ and ±dd c χ δ are dominated by some fixed Poincaré metric ω P (independently of δ). In particular, using Cauchy-Schwarz and the dominated convergence theorem, one finds

(3.11) lim δ→0 X reg G(x, •) f dd c χ δ + d f ∧ d c χ δ + dχ δ ∧ d c f ∧ ω n-1 = 0
by the dominated convergence theorem. Formula (3.10) is now a direct application of (3.8). The next result is the key for the proof of Proposition 3.3.

Claim 3.12. -Under the assumptions of Lemma 3.11, let ϕ ∈ PSH(X, ω), V = X ω n and let x ∈ X reg . Then, one has

1 V X ϕω n -ϕ(x) nV • inf X reg G(x, •).
Proof. -Replacing ϕ by max(ϕ, -j) and letting j → +∞, one sees that it is enough to prove the claim for bounded functions ϕ. Next, thanks to Demailly's regularization theorem, one can write p * ϕ as a pointwise decreasing limit of smooth function ψ ε satisfying p * ω + εω Y + dd c ψ ε 0 for some fixed Kähler metric ω Y on Y. Using (3.10) and setting

G x := G(x, •), one finds 1 V X ϕω n -ϕ(x) = lim ε→0 Y • np * G x dd c ψ ε ∧ p * ω n-1 .
Moreover, as G x have zero mean value, one has

Y • p * G x dd c ψ ε ∧ p * ω n-1 = Y • (p * G x -inf X reg G x )dd c ψ ε ∧ p * ω n-1 = Y • (p * G x -inf X reg G x )(p * ω + εω Y + dd c ψ ε ) ∧ p * ω n-1 - Y • p * G x ∧ (p * ω + εω Y ) ∧ p * ω n-1 + inf X reg G x • V + ε Y ω Y ∧ p * ω n-1 inf X reg G x • V + ε • inf X reg G x • Y ω Y ∧ p * ω n-1 - Y • p * G x ω Y ∧ p * ω n-1
Taking the limit as ε → 0, we get the expected result.

Proof of Proposition 3.3. -

We can now finish the proof of Proposition 3.3. We are left to treating the cases where π is projective or X t is smooth for t = 0. Moreover, we can assume that n = dim X t 2 since otherwise, X t would have at most isolated singularities and we could then appeal to § 3.3, cf. Remark 3.7.

Moreover, the content of Proposition 3.3 is insensitive to replacing ω by another Kähler metric on X . In the case where π is projective, i.e. if we have X ⊂ P N × D such that π commutes with the second projection, then we will assume that ω = ω FS | X .

Finally, in the case where X t is smooth for t = 0, it is sufficient to prove Proposition 3.3 for t = 0 since it is already well-known that the L 1 -sup comparison holds on the fixed irreducible complex space X 0 . We know from § 3.4 that the Kähler manifolds (X reg t , ω t ) admit uniform Poincaré and Sobolev constants. As the volume V of (X t , ω t ) is constant, it follows from (3.9) that there exists

C G > 0 independent of t such that ∀x, y ∈ X reg t , G t (x, y) -C G ,
where G t (•, •) is the Green function of (X t , ω t ). As ϕ t is sup-normalized and upper semi-continuous, there exists x t ∈ X reg t such that ϕ t (x t ) -1. Applying Claim 3.12 to ϕ := ϕ t and x := x t , we find 1

V X t (-ϕ t ) ω n t nVC G + 1.
The Proposition is proved.

Densities along a log canonical map

We now pay attention to hypotheses (H2) and (H2'). We focus in this section on the integrability properties of some canonical densities.

Semi-stable model. -

Setting 4.1. -Let π : X → D be a proper, holomorphic surjective map from a Kähler space X with connected fibers to the unit disk of relative dimension n. We make the following assumption

(4.1)
For each t ∈ D, the pair (X , X t ) has log canonical singularities where X t = π -1 (t) is the schematic fiber at t ∈ D, cf [KM98, Def. 7.1].

About the singularities. In Setting 4.1, the following properties hold 1. Every fiber is reduced, 5. By loc. cit., the condition (4.1) is equivalent to asking X to be normal, Q-Gorenstein, and that each fiber X t has semi-log canonical singularities.

K X /D is Q-Cartier
By [START_REF] Kempf | SAINT-DONAT -Toroidal embeddings. I[END_REF], one can find a semi-stable model of π (1) . More precisely, up to shrinking D, there exists a finite cover ϕ : t → t k of the disk for some integer k 1 and a proper, surjective birational morphism

f : X → X × ϕ D (4.2) X X × ϕ D X D D π f g pr 2 π
ϕ such that X is smooth, f is isomorphic over the smooth locus of π and such that around any point x ∈ X 0 , there exists an integer p n + 1 and a system of coordinates (z 0 , . . . , z n ) centered at x and such that π (z 0 , . . . ,

z n ) = z 0 • • • z p .
Additional assumption. Up to shrinking D, one will assume that π is smooth away from 0 so that for any t = 0, the induced morphism (g • f )| X t : X t → X t is a resolution of singularities. Note that X t need not be connected.

Let m

1 be an integer such that mK X /D is a Cartier divisor. We can cover X with open sets U i such that U i ∩ X reg admits a nowhere vanishing section

Ω U i ∈ H 0 (U i ∩ X reg , mK X /D ). For any t ∈ D, the restriction Ω U i | X reg t defines a nowhere vanishing section Ω U i | X reg t ∈ H 0 (U i ∩ X reg t , mK X t ).
In particular, mK X t is a Cartier divisor for all t. We want to understand the behavior of the volume forms

(Ω U i ∧ Ω U i )| 1 m X reg t
when t → 0. In order to do so, it is enough to work on X × ϕ D (1) The reference [START_REF] Kempf | SAINT-DONAT -Toroidal embeddings. I[END_REF] deals with the case of a proper morphism between algebraic varieties but the construction extends to the analytic case mutatis mutandis, as stated in e.g. [KM98, Thm. 7.17] directly as explained below.

Reduction step. The finite map g induces an isomorphism of Q-line bundles K X × ϕ D/D g * K X /D . In particular, one can replace X by X × ϕ D in the following, or equivalently assume that ϕ = Id D ; i.e k = 1. By what was said above, the "new" family still satisfies the condition (4.1).

Analytic expression of the densities in

a semi-stable model. -Let us start with some notation. Once and for all, we fix an open set U := U i 0 for some i 0 . We set Ω := Ω U and Ω t := Ω| X reg t . One can cover f -1 (U) by a finite number of open subsets V j ⊂ X isomorphic to the unit polydisk of C n+1 and endowed with a system of coordinates as above. We let V := V j 0 be one of them. The goal is to understand f * Ω when restricted to V, using our preferred set of coordinates. Finally, we set U t := U ∩ X t and V t := V ∩ X t .

Next, we write (4.3)

K X + Y 0 = f * (K X + X 0 ) + ∑ i a i E i
where the E i 's are f -exceptional divisors with a i -1 for all i and Y 0 is the strict transform of X 0 . Note that some of the divisors E i 's may be irreducible components of X 0 . The others surject onto D thanks to the additional assumption made in the previous section. The divisor E := ∑ i E i is the exceptional locus of f and E + Y 0 has simple normal crossing support. Under our assumptions, the analytic set

(4.4) Nklt(X , X 0 ) := f a i =-1 E i
contains the non-klt locus of every fiber X t , t ∈ D. This is an easy consequence of the adjunction formula, at least when the X t 's are normal.

We now let x ∈ Y 0 and we assume that the coordinates mentioned above are chosen such that Y 0 = (z 0 • • • z r = 0) locally for 0 r p being the number of irreducible components of Y 0 minus one on that chosen open set. On V t , t = 0, the functions (z 1 , . . . , z n ) induce a system of coordinates and the form f * Ω on V can be seen as a collection of m-th powers of holomorphic n-forms

f * Ω t = g t (z 1 , . . . , z n )(dz 1 ∧ • • • ∧ dz n ) ⊗m
for some holomorphic function g t on V t \ E, with poles of order at most (-ma i ) + along

E i ∩ X t . The form Ω ∧ π * dt t ⊗m is trivialisation of m(K X + X 0 ) over U reg . The pull- back f * (Ω ∧ π * dt t ⊗m ) is a well-defined m-th power of a (n + 1)-form on f -1 (U reg )
with logarithmic poles along Y 0 that extends meromorphically to f -1 (U) with poles of order at most (-ma i ) + along E i . As

f * π * dt t = (π ) * dt t = p ∑ i=0 dz i z i on V, the form f * (Ω ∧ π * dt t ⊗m ) is equal on that set to (-1) mn (z 1 • • • z r ) m g π (z) (z 1 , . . . , z n ) dz 0 z 0 ∧ dz 1 z 1 ∧ • • • ∧ dz r z r ∧ dz r+1 ∧ • • • ∧ dz n ⊗m so that the function (V \ E ∪ Y 0 ) z → (-1) nm (z 1 • • • z r ) m g π (z) (z 1 , .
. . , z n ) extends to a meromorphic function h on V, holomorphic along Y 0 and with poles of order at most (-ma i ) + along E i and satisfying (4.5)

f * Ω t = (-1) mn h(z) (z 1 • • • z r ) m (dz 1 ∧ • • • ∧ dz n ) ⊗m
on V t , for t = 0. When t = 0, one can also obtain a formula as above for f * Ω 0 but it requires to first choose a component

Y (k) 0 of Y 0 . Let 0 i r such that Y (k) 0 ∩ V 0 = (z i = 0)
. On that set (say after removing E), one has

(4.6) f * Ω 0 = (-1) i+mn h(z) (z 1 • • • z i • • • z r ) m dz 0 ∧ • • • ∧ dz i ∧ • • • ∧ dz n ⊗m .
Note that if X 0 (or equivalenty, Y 0 ) is irreducible, then r = 0 in the formula above.

Claim 4.2. -If X 0 has canonical singularities, then r = 0 and the meromorphic function V z → h(z) is holomorphic on V.

Proof. -As X 0 is normal, it is irreducible, hence Y 0 is smooth and irreducible. In particular, the map f | Y 0 : Y 0 → X 0 induces a resolution of singularities.

As X 0 has canonical singularities, the pull-back f * Ω 0 of the form Ω 0 on X reg 0 ∩ U extends holomorphically across Y 0 ∩ E. Given (4.6), it means that h|V ∩ Y 0 extends holomorphically along each E i ∩ Y 0 . As h is holomorphic on V and does not vanish outside V 0 , its divisor is an n-dimensional variety supported on V ∩ E, therefore div(h) = ∑ b i E i for some integers b i . As E + Y 0 is snc, the decomposition div(h|

Y 0 ) = ∑ b i (E i ∩ Y 0 )
is the decomposition into irreducible components. As h| Y 0 is holomorphic along the nonempty set Y 0 ∩ E i , we have necessarily b i 0 for any i. The claim is proved.

Integrability properties of the canonical densities. -

Definition 4.3. -In Setting 4.1, let ω be a Kähler form on X . We define the function γ on U ∩ X reg by

(Ω ∧ Ω) 1 m = e -γ ω n .
We want to analyze the integrability properties of e -γ . Arguing as in the proof of [RZ11a, Thm. B.1(i)] (see also [EGZ09, Lem. 6.4]), it is easy to infer from the normality of X that given any small open set U ⊂ U, there exist bounded holomorphic functions

( f 1 , • • • , f ) on U such that V( f 1 , . . . , f ) ⊂ U sing and (4.7) γ| U reg = 1 m log ∑ i | f i | 2 .
Let us pick a section s E ∈ H 0 (X , O X (E)) cutting out the exceptional divisor E and let us choose | • | a smooth hermitian metric on O X (E). Given (4.7), there exists a constant A > 0 such that (4.8)

f * γ A log |s E | 2 .
We have the following Lemma 4.4. -Assume that X 0 has canonical singularities and set ω t := ω| X t . Then up to shrinking D, there exists p > 1 and a constant C > 0 such that for any t ∈ D, one has U t e -pγ ω n t C.

Proof. -We set p := 1 + δ for some δ > 0 small enough to be chose later. Given (4.8), we have

U t e -pγ ω n t = f -1 (U t ) e -δ f * γ f * (Ω t ∧ Ω t ) 1 m f -1 (U t ) |s E | -2δA f * (Ω t ∧ Ω t ) 1 m .
Now, one can cover f -1 (U t ) by finitely many open sets V t = V ∩ X t as above. On V, the system of coordinates (z 0 , . . . , z n ) induces a system of coordinates (z 1 , . . . , z n ) such that we have

|s E | -2δA f * (Ω t ∧ Ω t ) 1 m C p ∏ i=1 |z i | -2δA idz 1 ∧ d z1 ∧ • • • ∧ idz n ∧ d zn
for some uniform constant C thanks to (4.5) and Claim 4.2. Recall that

V = ∏ n i=0 {|z i | < 1} ⊂ C n+1 and V t = V ∩ {z 0 • • • z p = t} → {(z 1 , . . . , z n ) ∈ C n ; t |z i | < 1} ⊂ D n .
where the injective map is given by pr z 1 ,...,z n | V t , i.e. the restriction to V t of the projection map onto the last n coordinates in C n+1 . For δ small enough, the function D z → |z| -δA is integrable with respect to the area measure; this concludes the proof.

For the next lemma, we come back to the general case. We start by choosing a component Y (k 0 ) 0 of Y 0 , and we denote by X (k 0 ) 0 the irreducible component of X 0 birational to Y (k 0 ) 0 via f . Next, we consider the reduced divisor F on X whose support consists of the union of the other components Y (k) 0 , k = k 0 , along with the divisors E i whose discrepancy a i is equal to -1, cf (4.3).

Let h F be a smooth hermitian metric on O X (F) and let s F ∈ H 0 (X , O X (F)) such that div(s F ) = F. We let (4.9)

ψ F := -log(-log |s F | 2 h F ). Similarly, let F klt := E -F ∩ E, and let ψ klt := log |s F klt | 2 .
Claim 4.5. -There exists δ > 0 small enough such that for any ε > 0, there exists a constant C ε such that for any t ∈ D,

f -1 (U t ) e (1+ε)ψ F -δψ klt f * (Ω t ∧ Ω t ) 1 m C ε .
Proof. -The statement is local on X , so it is enough to control the integrals over V t . Up to relabelling, one can assume that Y

(k 0 ) 0 ∩ V = (z 0 = 0), F ∩ V = (z 1 • • • z s = 0) so that for s + 1 i p, f
* Ω t has a pole of order at most m -1 along (z i = 0). We implicitly assumed that V meets Y (k 0 ) 0 ; it actually does not matter much for the computation which is insensitive to whether that condition is fulfilled or not. Using (4.5), our integral is bounded by the following one

V t s ∏ i=1 1 |z i | 2 (-log |z i |) 1+ε • p ∏ i=s+1 1 |z i | 2(δ-a i ) • dλ C n where -1 < a i < 0 and V = ∏ n i=0 {|z i | < 1} ⊂ C n+1 and V t = V ∩ {z 0 • • • z p = t}.
By Fubini theorem, one can reduce the integral to V p t := V t ∩ C p+1 (i.e. fixing z p+1 , . . . , z n ). There is no harm in assuming that δ < min i 1+a i 2 so that the integral is bounded by

V p t s ∏ i=1 1 |z i | 2 (-log |z i | 2 ) 1+ε • p ∏ i=s+1 1 |z i | 2(1-δ/2) • dλ C p
Using polar coordinates, one can assume that t is real (in (0, 1)) and the integral becomes over

W t := {(r i ) 1 i p ∈ [0, 1/2] p ; r 1 . . . r p t} W t s ∏ i=1 1 r i (-log r i ) 1+ε • p ∏ i=s+1 1 r 1-δ i • dλ R p As W t ⊂ ∏ p i=1 {t
r i 1/2} and the functions r → 1 r(log r) 1+ε and r → 1 r 1-δ are integrable on [0, 1/2], the conclusion follows from Fubini's theorem.

The result above allows us to generalize Lemma 4.4 when no assumption on the central fiber is made. To do so, we first need some notation. The function ψ F is well defined on X but it does not necessarily come from X . Given that Nklt(X , X 0 ) is an analytic set in X and up to shrinking D a little, one can construct a function ρ such that

• ρ -1 on X .

• ρ is quasi-psh and has analytic singularities along Nklt(X , X 0 ); in particular, it is identically -∞ on that set. We set ψ :=log(-ρ) on X . Up to scaling ρ, one can assume that Proof. -In order to compute the integral, we pull it back by f and work on V t . We have successively

| f * γ ε | -log |s E | + log(-log |s F |) -log |s F | -log |s F klt |.
The first inequality is a combination of (4.8) and (4.10). To obtain the second inequality, we use the fact that E = F ∪ F klt to split the term log |s E | while log(-log |s F |) can be absorbed by the more singularlog |s F |. The integral to bound becomes

V t (-log |s F |) n+ε + (-log |s F klt |) n+ε e (n+1+2ε)ψ F f * (Ω t ∧ Ω t ) 1 m
which itself is controled by

V t e (1+ε)ψ F f * (Ω t ∧ Ω t ) 1 m + V t e 2ψ F -δψ klt f * (Ω t ∧ Ω t ) 1 m
for any given δ > 0. The lemma now follows from Claim 4.5.

Negative curvature

In this section we apply our previous results to the study of families of varieties with "negative canonical bundle": we consider families of manifolds of general type, as well as families of "stable varieties".

Families of manifolds of general type. -

Setting 5.1. -Let X be an irreducible and reduced complex space endowed with a Kähler form ω and a proper, holomorphic map π : X → D. We assume that for each t ∈ D, the (schematic) fiber X t is a n-dimensional Kähler manifold X t of general type, i.e. such that its canonical bundle K X t is big. In particular, X is automatically non-singular and the map π is smooth. One can view the fibers X t as deformations of X 0 .

We fix Θ a closed differential (1, 1)-form on X which represents c 1 (K X /D ) ∈ H 1,1 ∂ ∂ (X ) and set θ t = Θ |X t . Shrinking D if necessary and rescaling, we can assume without loss of generality that -ω Θ ω.

Lemma 5.2. -In the Setting 5.1, the quantity vol(K X t ) is independent of t ∈ D.

Proof. -We work in two steps. First, we assume that the family π : X → D is projective, i.e. there exists a positive line bundle L over X . In that case, we know that the invariance of plurigenera holds [START_REF] Siu - | Invariance of plurigenera[END_REF][START_REF] Ȃun - | Siu's invariance of plurigenera: a one-tower proof[END_REF] in that the function t → h 0 (X t , mK X t ) is constant on D, without even assuming that X t is of general type for all t. In particular, it would even be enough to assume that only X 0 is of general type from which it results that X t is of general type for all t and that the volume vol(K X t ) is independent of t.

Coming back to the general case, we know that K X /D is big. Thanks to Demailly's regularization theorem, there exists a Kähler current T ∈ c 1 (K X /D ) with analytic singularities along V(I ) for some ideal sheaf I ⊂ O X . Let f : X → X be a log resolution of (X , I). By Hironaka's theorem, we know that one can construct such a morphism f by a sequence of blow-ups along smooth centers only. We write f * T = T + [F] for some smooth semipositive form T on X and some effective divisor F. Remark that this sequence may be infinite; however, the centers project onto a locally finite family of subsets of X . Up to co-restricting f to π -1 (K) for some compact subset K D, one can assume that f is a finite composition of blow-ups and that T δπ * ω for some δ > 0 small enough.

Let E be the exceptional divisor of f , with irreducible components E = ∑ N k=1 E k . A classical argument (cf e.g. [DP04, Lem. 3.5]) allows one to find smooth (1, 1)-forms θ E k ∈ c 1 (E k ) with support in an arbitrarily small neighborhood of E k along with positive numbers (a k ) such that the sum θ = ∑ k a k θ k defines a (1, 1)-form on X which is negative definite along the fibers of f . It follows that for ε > 0 small enough, the smooth form π * ωεθ E is Kähler. In particular, Tδεθ is a Kähler form whose cohomology class belongs to NS R (X ). This implies that the Kähler cone of X meets NS Z (X ), i.e. π • f is projective.

Let X t := f -1 (X t ) and let K • ⊂ K be the set of regular value of π • f . For any t ∈ K • , the map f | X t : X t → X t is birational hence vol(K X t ) = vol(K X t ). By the first step, the volume vol(K X t ) is independent of t ∈ K • , hence the same holds for vol(K X t ). The set K \ K • is finite and without loss of generality, one can assume that it consists of the single element {0}. The fiber X 0 can be decomposed as

X 0 = Y 0 + ∑ E i where f | Y 0 : Y 0 → X 0 is birational and E i is contracted by f | X 0 . Let Y 0 → Y 0 be
a resolution of singularities. By [Tak07, Thm. 1.2], we have vol(K Y 0 ) vol(K X t ) for t = 0. As X 0 and Y 0 are smooth and birational, we have vol(K X 0 ) = vol(K Y 0 ) vol(K X t ). Finally, as the function t → vol(K X t ) is upper semi-continous, we have vol(K X 0 ) = vol(K X t ) for any t ∈ K. The lemma is proved.

Remark 5.3. -In the last step of the proof of Lemma 5.2, we could also use the existence of relative minimal models, provided D is replaced by a quasi-projective smooth curve C. The general fiber of the projective morphism X → C is a projective variety of general type, hence it admits a good minimal model over C by [START_REF] Birkar | Existence of minimal models for varieties of log general type[END_REF]. By [START_REF] Fujino | Direct images of relative pluricanonical bundles[END_REF]Thm. 3.3] and [Tak19, Cor. 1.2], it follows that X → C admits a birational model φ : X X over C such that: φ -1 does not contract any divisor, every fiber X t of X → C has canonical singularities and satisfies that K X t is semiample and big. For any t ∈ C, one has vol(K X t ) = (K n X t

). By flatness, this quantity does not depend on t. Finally, we claim that X 0 is birational to X 0 . This is a combination of the following two facts. First, the variety X 0 has canonical singularities and K X 0 is big hence it is of general type and, in particular, it is not uniruled. Next, X 0 is birational to a component of X 0 and all of them but the strict transform of X 0 by f are covered by rational curves as f is a composition of blow-ups of smooth centers from a smooth manifold.

The positive (n, n)-forms (ω n t ) t∈D induce a smooth hermitian metric on -K X /D . Since

[Θ] = c 1 (K X /D ) ∈ H 1,1 ∂ ∂ (X ); there exists a smooth function h on X such that -dd c X log ω n t = -Θ + dd c X h
We will denote by h t := h| X t the restriction to the fiber X t . The function h becomes unique (and remains smooth) after one imposes the following normalization

X t h t ω n t = 0.
We define a function h on X by imposing that h t := h| X t satisfies

h t = h t -log 1 V t X t e h t ω n t .
In particular, one has (5.1)

X t e h t ω n t = V t := vol(K X t ).
As h is smooth on X , one has the following obvious consequence.

Lemma 5.4. -Given any compact subset K D, one has

sup t∈K h t L ∞ (X t ) < +∞.
It follows from [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF], a generalization of the Aubin-Yau theorem [Aub78, Yau78], that there exists a unique Kähler-Einstein current on X t . This is a positive closed current T t in c 1 (K X t ) which, by [EGZ09, BCHM10], is a smooth Kähler form in the ample locus Amp(K X t ), where it satisfies the Kähler-Einstein equation Ric(T t ) = -T t .

It can be written T t = θ t + dd c ϕ t , where ϕ t is the unique θ t -psh function with minimal singularities that satisfies the complex Monge-Ampère equation (θ t + dd c ϕ t ) n = e ϕ t +h t ω n t on Amp(K X t ).

The minimal singularity assertion is equivalent to the following uniform bound: for all x ∈ X t , -M t (ϕ t (x)sup

X t ϕ t ) -V θ t (x) 0,
where V θ t (x) = sup{u t (x); u t ∈ PSH(X t , θ t ) and u t 0}. We can choose M t independent of t by using Theorem 1.9: Theorem 5.5. -In Setting 5.1, let K D be a compact subset. There exists a constant M K such that for all x ∈ π -1 (K), one has

-M K ϕ t (x) -V θ t (x) M K where t = π(x).
Proof. -From Lemma 5.2, it follows that the volume V t of K X t is independent of t. We denote it by V. Set µ t = e h t ω n t /V and recall that this is a probability measure, by our choice of normalization. We first observe that (5.2) 0 sup

X t ϕ t -inf π -1 (K) h C K .
Let us first prove the left-hand side inequality.

(ω t + dd c ϕ t ) n (θ t + dd c ϕ t ) n = e ϕ t +h t ω n t , while the constant function u t (x) = -inf π -1 (K) h is a supersolution of the same equa- tion,
(ω t + dd c u t ) n = ω n t e u t +h t ω n t . It follows from the comparison principle [GZ17, Prop. 10.6] that ϕ tinf π -1 (K) h. The rightmost inequality in (5.2) follows from Lemma 5.4 above.

We can thus rewrite the complex Monge-Ampère equation as

1 V (θ t + dd c ψ t ) n = e ψ t +sup X t ϕ t µ t = f t µ t ,
where ψ t = ϕ tsup X t ϕ t and f t = exp(ψ t + sup X t ϕ t ). Combining the inequalities ψ t 0 and (5.2), it follows that the densities f t are uniformly bounded.

Recall that π is smooth so, in particular, it is locally trivial. Therefore, Theorem 3.4 applies and we can now appeal to Theorem 1.9 with p = +∞ and 0 < α < α(Θ, X ) and obtain -M K ψ t -V θ t 0. Note that one used here that the volumes V t stay away from zero. The conclusion follows since ψ tϕ t is uniformly bounded by (5.2).

Remark 5.6. -Set

V Θ (x) = V θ π(x) (x). and φ(x) := ϕ π(x) (x). It is tempting to compare φ to VΘ = sup{u ∈ PSH(X , Θ); u 0}. Clearly VΘ V Θ hence VΘ -M K φ. It follows from [CGP17, Thm. A] that φ is Θ-psh on X , thus φ -sup π -1 (K) φ VΘ and -M K φ -VΘ M K .
Remark 5.7. -The same results can be proved if the family π : X → D is replaced by a smooth family π : (X , B) → D of pairs (X t , B t ) of log general type, i.e. such that (X t , B t ) is klt and K X t + B t is big for all t ∈ D.

Stable varieties.

-A stable variety is a projective variety X such that 1. X has semi-log canonical singularities. 2. The Q-line bundle K X is ample. We refer to [KSB88, Ale96, Kar00, Kov13, Kol] for a detailed account of these varieties and their connection to moduli theory.

In [START_REF] Berman | GUENANCIA -"Kähler-Einstein metrics on stable varieties and log canonical pairs[END_REF], it was proved that a stable variety admits a unique Kähler-Einstein metric ω. There are several equivalent definitions for such an object, but the simplest is probably the following: Definition 5.8. -A Kähler-Einstein metric ω on a stable variety is a smooth Kähler metric on X reg such that Ric(ω) = -ω and

X reg ω n = (K n X ) if n = dim C X.
It is proved in loc. cit. that ω extends canonically across X sing to a closed, positive current in the class c 1 (K X ). It is desirable to understand the singularities of ω near X sing . In [GW16, Thm. B], it is proved that ω has cusp singularities near the double crossings of X. Moreover, it is proved in [START_REF] Song | Degeneration of Kähler-Einstein manifolds of negative scalar curvature[END_REF] that the potential ϕ of ω with respect to a given Kähler form ω X ∈ c 1 (K X ), i.e. ω = ω X + dd c ϕ, is locally bounded on the klt locus of X. More precisely, given any divisor D = (s = 0) ∼ Q K X containing the non-klt locus of X and given any ε > 0, there exists a constant C ε > 0 such that (5.3) ϕ ε log |s| 2 -C ε , where | • | is some smooth hermitian metric on O X (D). We wish to refine that estimate and obtain a version for families of canonically polarized manifolds degenerating to a stable variety. Proposition 5.9. -Let X be a stable variety of dimension n and let ω X ∈ c 1 (K X ) be a Kähler metric. Next, let ω = ω X + dd c ϕ be the Kähler-Einstein metric of X. Let D = (s = 0) be a divisor containing the non-klt locus of X and let | • | be some smooth hermitian metric on O X (D). For any ε > 0, there is a constant C ε such that

(5.4) ϕ -(n + 1 + ε) log(-log |s|) -C ε .
Remark 5.10. -The estimate (5.4) is an important refinement of (5.3), as it insures that ϕ belongs to the finite energy class E 1 (X, ω X ), cf [START_REF]The weighted Monge-Ampère energy of quasi plurisubharmonic functions[END_REF] or [BEGZ10, Sect. 2] for the definitions and main properties of these classes. This estimate is almost optimal. Indeed, if X is the Satake-Baily-Borel compactification of a ball quotient, it is a normal stable variety and it admits a resolution (X, D) which is a toroidal compactification of the ball quotient obtained by adding disjoint abelian varieties. The, the potential ϕ of the Kähler-Einstein metric on (X, D) with re- spect to a smooth form in c

1 (K X + D) satisfies ϕ = -(n + 1) log(-log |s D |) + O(1) if (s D = 0) = D.
Proof. -Let f : Y → X be a resolution of singularities of X such that f induces an isomorphism over X reg . The complex Monge-Ampère equation satisfied by ϕ pulls back to Y and reads (5.5)

( f * ω X + dd c f * ϕ) n = e f * ϕ dµ Y
where dµ Y := ∏ r i=1 |t i | 2a i ω n Y is a positive measure with possibly infinite mass. Here, ω Y is a Kähler form on Y, and (t i = 0) are divisors sitting over X sing (they need not be exceptional though, as X may have singularities in codimension one). Finally, one has a i -1 for all i, and any divisor (t i = 0) such that a i = -1 sits above the non-klt locus of X. Now, let F be an effective divisor on X and let σ X ∈ H 0 (X, O X (F)) be a section cutting out F. Let h be a smooth hermitian metric on O X (F); there exists a constant

C F such that Θ h (F) C F ω X . One can scale h such that |σ X | 2 h < e -2(n+2)C F on X.
Finally, let σ Y := f * σ X and and let ψ :=log(-log |σ Y | 2 ). We have

dd c ψ = Dσ Y , Dσ Y |σ Y | 2 (-log |σ Y | 2 ) - 1 (-log |σ Y | 2 ) • f * Θ h (F).
By our choice of scaling, the function Aψ is f * ω X -psh for any 0 A 2(n + 2). Moreover, it belongs to the class E (Y, f * ω X ) thanks to e.g. [Gue14, Prop. 2.3] and [DDNL18, Thm. 1.1(ii)].

We apply this construction to F some (very ample, say) divisor containing the nonklt locus of X. This yields a section σ Y of f * F that vanishes at order at least one along the (t i = 0) for which a i = -1. As a result, the measure

e (n+1+2ε)ψ dµ Y ∏ a i =-1 1 |t i | 2 (-log |t i | 2 ) n+1+2ε ∏ a i >-1 |t i | 2a i • ω n Y has a density g ε with respect to ω n Y that satisfies Y g ε | log g ε | n+ε ω n Y < +∞
for any ε > 0. By Theorem 1.5, this implies that the unique solution

u ε ∈ E (Y, 1 2 f * ω X ) of the Monge-Ampère equation ( 1 2 f * ω X + dd c u ε ) n = e u ε +(n+1+2ε)ψ dµ Y is bounded, i.e. there exists a constant C ε > 0 such that (5.6) u ε L ∞ (Y) C ε . Now, the function v ε := u ε + (n + 1 + 2ε)ψ ∈ E (Y, f * ω X ) satisfies the inequality ( f * ω X + dd c v ε ) n ( 1 2 f * ω X + dd c u ε ) n = e v ε dµ Y ,
i.e. v ε is a subsolution of (5.5). By the comparison principle, we obtain that f * ϕ v ε and it follows from (5.6) that

f * ϕ (n + 1 + 2ε)ψ -C ε ,
from which the conclusion follows.

Stable families.

-Now one can establish a family version of the previous estimate, i.e. Proposition 5.9. In Setting 4.1, let us assume additionally that K X /D is ample. We let h be a smooth hermitian metric on K X /D whose curvature is a Kähler form ω • The central fiber X 0 is irreducible. Let ω X t + dd c ϕ t be the Kähler-Einstein metric of X t , solution of (5.7) and let D = (s = 0) ⊂ X be a divisor which contains Nklt(X , X 0 ), cf (4.4). Fix | • | a some smooth hermitian metric on O X (D). Up to shrinking D, then for any ε > 0, there exists C ε > 0 such that the inequality (5.8) ϕ t -(n + 1 + ε) log(-log |s|) -C ε holds on X t for any t ∈ D. This estimate improves an interesting control obtained previously by J.Song (see [START_REF] Song | Degeneration of Kähler-Einstein manifolds of negative scalar curvature[END_REF]Lem. 4.2]).

X := Θ h (K X /D ); we set ω X t := ω X | X t . If Ω is a local trivialization of mK X /D , then the quantity µ X /D,h := i n 2 (Ω ∧ Ω) 1/m |Ω| 2/
Proof. -Let f : X → X be a semi-stable model as in (4.2). The first observation is that the behavior of f * (Ω t ∧ Ω t ) 1/m and f * µ X t ,h on X t is the same, uniformly in t, because there exists a constant C > 0 such that for any trivializing open set, one has C |Ω| 2 h C -1 , where Ω ranges among the finitely many trivializations of mK X /D . This follows from the fact h is a smooth hermitian metric on mK X /D .

We set ψ := f * (log(-log |s| 2 )); it is a quasi-psh function on X satisfying ψ ψ F + O(1) where ψ F is defined in (4.9).

By scaling the metric | • | on O X (D), one can assume that Aψ is f * ω X -psh for any 0 A 2(n + 2). For any t ∈ D * , the function ψ t := ψ| X t belongs to E (X t , f * ω X t ) by the same argument as in the proof of Proposition 5.9.

Let u ε,t ∈ E (X t , 1 2 f * ω X t ) be the unique solution of the Monge-Ampère equation (5.9) ( 1 2 f * ω X t + dd c u ε,t ) n = e u ε,t +(n+1+2ε)ψ t f * µ X t ,h .

bounded potentials, solution of the Monge-Ampère equation ω m ∞ = f * µ (X,B) , cf [START_REF]Singular Kähler-Einstein metrics[END_REF]. In the case where X is smooth, B = 0 and c 1 (X) = 0, the Ricci curvature of f * µ X (or, equivalently, ω ∞ ) coincides with the Weil-Petersson form of the fibration f of Calabi-Yau manifolds. We propose the following problem.

Problem 1. -Let f : X → Z be a surjective holomorphic map with connected fibers between compact, normal Kähler spaces. Assume that there exists an effective divisor B on X such that (X, B) is klt and κ(K X + B) = 0. Let ω X (resp. ω Z ) be a Kähler form on X (resp. Z) and let ω ϕ t be the unique singular Ricci-flat current in { f * ω Z + tω X } for t > 0. Then, the currents ω ϕ t converge weakly to f * ω ∞ when t → 0, where ω ∞ ∈ {ω Z } solves

ω dim Z ∞ = f * µ (X,B) .
The Problem above is motivated by a string of papers (cf below) where the expected result is proved along with some additional information on the convergence. Theorem 6.6. -[Tos10, GTZ13, TWY18, HT18] Assume that X is smooth, B = 0 and c 1 (K X ) = 0. Then, the metrics ω ϕ t converge to f * ω ∞ in the C α loc -sense on X \ S X , for some α > 0.

In this section, we aim at providing a positive answer to Problem 1 whenever X is smooth, B has simple normal crossings support and c 1 (K X + B) = 0. We will follow the strategy of Tosatti [START_REF]Adiabatic limits of Ricci-flat Kähler metrics[END_REF] rather closely. However, some adjustments need to be made, requiring the use of conical metrics and the results of the present paper. Theorem 6.7. -In the Setting of Problem 1, assume furthermore that X is smooth, B has snc support and c 1 (K X + B) = 0. Then, ω ϕ t → f * ω ∞ as currents on X, when t goes to 0.

Proof. -We will proceed in several steps, similarly to [START_REF]Adiabatic limits of Ricci-flat Kähler metrics[END_REF]. In order to simplify some computations to follow, one will assume that S Z is contained in a divisor D Z , cut out by a section σ Z ∈ H 0 (Z, O Z (D Z )). If Z is projective, this is not a restriction. The general case requires to follow Tosatti's computations more closely but does not present significant additional difficulties.

Step 1. Choice of some suitable conical metrics We list in the Proposition below the properties of the conical metric that will be important for the following. It is mostly a recollection of well-known results, cf e.g. [START_REF] Guenancia | Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors[END_REF]. By abuse of notation, we will not distinguish between B and Supp(B). Proposition 6.8. -There exists a Kähler current ω B ∈ {ω X } on X such that 1. ω B is a smooth Kähler form on X \ B and has conical singularities along B.

2. There exists a constant C > 0 and a quasi-psh function Ψ ∈ C ∞ (X \ B) ∩ L ∞ (X) such that the following inequalities of tensors hold in the sense of Griffiths on X \ B -(Cω B + dd c Ψ) ⊗ Id T X Θ ω B (T X ) Cω B ⊗ Id T X .

3. Let h := ω n B /ω n X . There exists p > 1 such that for any K Z • , one has sup z∈K h| X z L p (ω k Xz ) < +∞.

there exists a constant C t > 0 such that dd c χ -C t ω ϕ t . Therefore, for any δ ∈ (0, C -1 t ), one has an inequality ∆ ω ϕ t (log u -Aϕ t + δχ) u -C.

As ω ϕ t is a conical metric for t > 0, the function u is bounded above on X \ B and therefore, H t,δ := log u -Aϕ t + δχ attains its maximum at a point x t,δ ∈ X \ B such that u(x t,δ ) C. As a result, the estimate obtained in 1. allows one to show that for any x ∈ X \ B, one has log u(x) = H t,δ (x) + Aϕ t (x)δχ(x) H t,δ (x t,δ ) + Cδχ(x) Cδχ.

As this holds for any δ > 0 small enough, we can pass to the limit and conclude that u e C on X \ B, hence everywhere.

3.

The equation solved by ϕ t can be rewritten as (6.5)

( f * ω Z + tω X + dd c ϕ t ) n = t k e F t ω n B
where F t is uniformly bounded independently of t. Next, one has on X z (z

∈ Z • ) (6.6) (ω ϕ t | X z ) k ω k X z = ω k ϕ t ∧ f * ω m Z ω k X ∧ f * ω m Z Cg • ω n ϕ t ω n X
thanks to 2. Observing that ω ϕ t | X z = (ω ϕ tdd c ϕ t )| X z , one sees from Eq. (6.5) that (ϕ tϕ t )| X z satisfies

(ω X z + dd c ( 1 t (ϕ t -ϕ t )| X z )) k gh| X z • ω k X z
where h = ω n B /ω n X . Thanks to the third item of Proposition 6.8, Theorem 3.4 and Theorem 1.1, we can derive 3. Actually, we used a version of Theorem 3.4 for higherdimensional bases, but only for smooth morphisms, in which case the proofs in the one-dimensional case go through without any change.

4.a

We first prove the right-hand side inequality. Let us start by writing ω B = ω X + dd c ψ B where ψ B ∈ L ∞ (X) ∩ C ∞ (X \ B). From the second item of Proposition 6.8 and Siu's Laplacian inequality (cf [GP16, (2.2)]), one concludes that ∆ ω ϕ t (log tr ω B ω ϕ t + Ψ) -C(1 + tr ω ϕ t ω B ). We want to bound from below the term dd c ϕ t . In order to achieve this, one writes dd c ϕ t = dd c f * (ϕ t ω k X ) = f * (dd c ϕ t ∧ ω k X ) (6.9)

-f * ( f * ω Z ∧ ω k X + tω k+1 X ) -ω Z -t f * ω k+1 X -g • ω Z
given that f * ω k X = 1. In particular, one has (6.10)

∆ ω ϕ t ϕ t -g where χ is defined in the proof of 2 and h D Z is a smooth hermitian metric on the divisor D Z (containing S Z ). As ρ is quasi-psh on X, there exists C t > 0 such that (6.13) dd c ρ -C t ω ϕ t .

In particular, one has (6.14) ∆ ω ϕ t (log tr ω B ω ϕ t + F + δρ) tr ω ϕ t ω B -g t .

as soon as δ ∈ (0, C -1 t ). We choose such a δ for the following. As the quantity log tr ω B ω ϕ t + F is globally bounded on X and smooth on X • \ B, the function log tr ω B ω ϕ t + F + δρ attains it maximum at a point y t,δ ∈ X • \ B such that tr ω ϕ t ω B (y t,δ ) g t thanks to the maximum principle. Combining this with 2, one finds (6.15)

tr ω ϕ t ( f * ω Z + tω B )(y t,δ ) g

Using the standard inequality tr ω ω ω n ω n (tr ω ω ) n-1 valid for any two positive (1, 1)-forms, one gets from (6.15) tr f * ω Z +tω B (ω ϕ t )(y t,δ ) g since ω n ϕ t t k ω n B is uniformly comparable to ( f * ω Z + tω B ) n by Claim 6.10 below. As ω B dominates f * ω Z + tω B , we infer from the inequality above the following (6.16)

tr ω B ω ϕ t (y t,δ ) g.

Given the definition of y t,δ , the boundedness of F and that δ > 0 is arbitrary, we find as in the proof of 2. above that (6.16) actually implies tr ω B ω ϕ t g on X • \ B hence on the whole X • .

To conclude the proof of the RHS inequality in 4., it remains to prove the following Claim 6.10. -We have (6.17

ω n B = K • ω n C n and ( f * ω Z + tω B ) n = t k (1 + t) m K • ω n C n
which gives the expected result.

4.b

We now move on to the LHS inequality in 4. Let us set v := tr ω ϕ t (tω B ). Remember from Proposition 6.8 2. that ω B has holomorphic bisectional curvature bounded from above on X \ B. By Chern-Lu's inequality, we get on X \ B ∆ ω ϕ t log v -Ct -1 v.

Combining that inequality with (6.7)-(6.10) and (6.13), one finds, for A = C + 1

∆ ω ϕ t (log v - A t (ϕ t -ϕ t ) + Aψ B + δρ) 1 t (v -g).
Applying the maximum principle and arguing as before, we eventually find v g on X • \ B, hence on X • .

5.

The LHS inequality is a direct consequence of 4, by restriction. As for the RHS, it follows easily from the LHS since

tr ω Bz ω ϕ t | X z (ω ϕ t | X z ) k ω k B z • (tr ω ϕ t | Xz ω B z ) k-1 gt k-(k-1)
thanks to (6.6). This ends the proof of Proposition 6.9.

  Domination by capacity. -It follows from Hölder inequality and (H2) that µ Cν 1/q ,

  (H1), by generalizing an integrability result of Skoda-Zeriahi [Sko72, Zer01]. 2.1. Notations. -In all what follows, given a positive real number r, we denote by D r := {z ∈ C; |z| < r} the open disk of radius r in the complex plane. If r = 1, we simply write D for D 1 .

  (4.10) f * ψ ψ F . Next, we introduce for ε > 0 the function γ ε := γ -(n + 1 + 2ε)ψ defined on U. In other words, one has (4.11) e (n+1+2ε)ψ (Ω ∧ Ω) 1 m = e -γ ε ω n . Lemma 4.6. -With the notation above, there exists a constant C ε such that U t |γ ε | n+ε e -γ ε ω n t C ε for any t ∈ D.

  ) g -1 t k • ω n B ( f * ω Z + tω B ) n gt k • ω n BProof of Claim 6.10. -The statement is local, so one can assume that f : C n → C m is given by the projection onto the last m factors and that B = ∑ r i=1 b i (z i = 0) for some r k. As the inequality is invariant under quasi-isometry, one can choose ω Z = ∑ n j=k+1 idz j ∧ d zj to be the euclidean metric on C m whileω B = r ∑ j=1 idz j ∧ d zj |z j | 2b j + n ∑ j=r+1 idz j ∧ d zjis the standard cone metric. Setting K := ∏ r j=1 |z j | -2b j and ω C n := ∑ n j=1 idz j ∧ d zj , one finds

  1 weakly, globally on B and locally smoothly on B \ S. The local smooth convergence outside S is obvious. As for the global weak convergence, it follows from the convergence u ε dd c v ε ∧ (dd c w ε ) n-1 udd c v ∧ (dd c w) n-1 (and its symmetrical version swapping u and v), proved by Demailly, cf e.g. [Dem85, Thm. 2.6 and Rem. 2.10].

	Applying Lemma 2.11 to X

  Proof. -Because of the existence of partition of unity, the statement above is local. That means that it is enough to show the above inequality for any t ∈ K and anyf ∈ C ∞ 0 (U i ∩ X reg t )where U i ⊂ X are open sets such that ∪U i = X . We fix such an open set U i and we drop the index i in what follows. Without loss of generality, one can assume that there exists an embedding U i → C N and that ω| U and ω C N | U are quasi-isometric. Because Sobolev inequality is essentially insensitive to quasi-isometry, it is enough to show the inequality replacing ω t by ω C N | U t whereU t := U ∩ X t . Now, the isometric embeddings (U reg t , ω C N | U t ) → (C N , ω C N )provide a family of minimal submanifolds (i.e. with zero mean curvature vector) of the euclidean space by virtue of Wirtinger inequality. The expected inequality is now a direct application of Michael-Simon's Sobolev inequality [MS73, Thm. 2.1].

	Proposition 3.10. -Let K D. There exists C P

reg t ), i.e. such that f , d f ∈ L 2 (X reg t , ω t ).

  and X has log canonical singularities. 2. The space X has canonical singularities if and only if the general fiber X t has canonical singularities, cf [KM98, Lem. 7.2]. 3. The condition (4.1) is preserved by finite base change from a smooth curve, cf [KM98, Lem. 7.6]. 4. If (X , X 0 ) has lc singularities, then (X , X t ) has lc singularities for |t| 1, see [Kol13, Cor. 4.10 (2)] and [Kol18, Thm. 2.3].

  As the measures 1 V (θ t + dd c ϕ t ) n = e ϕ t µ t

	have mass one, one has
	1	X t	e	sup X t	ϕ t dµ t = e sup X t ϕ t
	and therefore, sup X t ϕ t 0. To prove the inequality in the middle in (5.2), we observe that, since θ t	ω t , ϕ t is a
	subsolution of the equation

  m h is independent of Ω or m (yet it depends on h) and for any t ∈ D, it restricts to X KE,t ∈ c 1 (K X t ) which solves the Monge-Ampère equation (5.7)(ω X t + dd c ϕ t ) n = e ϕ t µ X t ,h on X t . This is due to [Aub78, Yau78] when X t is smooth and to[START_REF] Berman | GUENANCIA -"Kähler-Einstein metrics on stable varieties and log canonical pairs[END_REF] in general. In Setting 4.1, assume that • The relative canonical bundle K X /D is ample.

			reg t	as
	a positive measure
	µ X t ,h := µ X /D | X reg t
	which we extend by zero across X	sing t	. For each t ∈ D, there exists a unique Kähler-
	Einstein metric ω Theorem 5.11. -

  ∆ ω ϕ t (-ϕ t + tψ B ) = tr ω ϕ t (-ω ϕ t + f * ω Z + tω B ) t tr ω ϕ t ω Bn ∆ ω ϕ t (log tr ω B ω ϕ t + Ψ -A t ϕ t + Aψ B ) tr ω ϕ t ω B -C t .

	Next, one has
	(6.7)
	so that
	(6.8)

  thanks to 2. Combining that estimate with (6.8), one finds (6.11)∆ ω ϕ t (log tr ω B ω ϕ t + Ψ -A t (ϕ tϕ t ) + Aψ B ) tr ω ϕ t ω B -g t .We now set F := Ψ -A t (ϕ tϕ t ) + Aψ B ; it is a bounded function on X, smooth on X • \ B such that (6.12)|F| g thanks to 3. Next, we set ρ := χ + f * log |σ Z | 2

	h D Z

One can write e (n+1+2ε)ψ t f * µ X t ,h = e ρ t f * ω n X t where ρ t is the restriction to X t of the difference of quasi-psh functions on X with uniformly bounded L 1 norm on X t . Set V := X t ω n X t . Integrating both sides of (5.9) and using Jensen inequality we have

C for some C > 0 independent of ε, t. Since u ε,t is f * ( 1 2 ω X t )-psh, it is the pull-back of a 1 2 ω X t -psh function on X t to which one can apply Proposition 3.3 since π is projective. To summarize, we get an upper bound (5.10) u ε,t C.

Next, we wish to apply Theorem 1.5; in order to do so, one has to check that hypotheses (H1) and (H2') are satisfied in our situation. For (H1), it is a consequence of Theorem 3.4 -recall that up to shrinking D, all fibers X t are irreducible since so is X 0 . As for (H2'), it follows from Lemma 4.6 that we pull back via f to the smooth Kähler manifold X t . All in all, we can find C ε > 0 independent of t ∈ D such that (5.11)

Now, the function v ε,t := u ε,t + (n + 1 + 2ε)ψ t ∈ E (X t , f * ω X t ) satisfies the inequality

i.e. v ε,t is a subsolution of (5.7). By the comparison principle, we obtain that f * ϕ t v ε,t and it follows from (5.11) that

from which the conclusion follows.

Log Calabi-Yau families

6.1. Families of Calabi-Yau varieties. -In Setting 4.1, let us assume additionally that K X /D is relatively trivial and that X 0 has canonical singularities. For t small enough, X t has canonical singularities as well and K X t is linearly trivial. Let α be a relative Kähler cohomology class on X represented by a relative Kähler form ω. We set α t := α| X t , ω X t := ω| X t and V := X t ω n t ; it does not depend on t, cf Lemma 2.2. Let Ω be a trivialization of K X /D , so that the quantity . We set c t := log X t dµ X t . For each t ∈ D, there exists a unique Kähler-Einstein metric ω KE,t = ω t + dd c ϕ t ∈ α t which solves the Monge-Ampère equation

on X t and that we normalize by sup X t ϕ t = 0. This is due to [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I[END_REF] when X t is smooth and to [START_REF]Singular Kähler-Einstein metrics[END_REF] in general.

Theorem 6.1. -In Setting 4.1, assume that • The relative canonical bundle K X /D is trivial.

• The central fiber X 0 has canonical singularities.

• Assumption 3.2 is satisfied. Let ω t + dd c ϕ t be the Kähler-Einstein metric of X t , solution of (6.1). Up to shrinking D, there exists C > 0 such that one has

A particular case of this result has been obtained previously by Rong-Zhang (see [RZ11a, Lemma 3.1]) by using Moser iteration process. Remark 6.2. -One can replace the first two assumptions in Theorem 6.1 above by the following weaker ones: X is normal, Q-Gorenstein, K X /D is trivial and X 0 has canonical singularities. Indeed, it follows from the inversion of adjunction [Kol18, Thm. 2.3] that (X, X t ) is lc for t close enough to 0. Moreover, an easy computation relying on the adjunction formula shows that X t has canonical singularities for t close to 0.

Proof of Theorem 6.1. -A first observation is that the quantities c t remain bounded when t varies thanks to Lemma 4.2. The result now follows from Theorem 1.1. Indeed, (H1) is satisfied thanks to Theorem 3.4 while (H2) holds thanks to Lemma 4.4 that we pull back to X t via f , with the notation of the Lemma. Setting 6.3. -Let X be an n-dimensional compact Kähler space and let B = ∑ b i B i be an effective R-divisor such that the pair (X, B) has klt singularities. We assume furthermore that the log Kodaira dimension of the pair (X, B) vanishes, i.e.

The log

In what follows, we denote by E the (unique) effective R-divisor in c 1 (K X + B). Thanks to log abundance in numerical dimension zero (see [CGP19, Cor. 1.18]), a particular instance of such pairs is provided by klt pairs (X, B) with rational boundary such that the Chern class c 1 (K X + B) ∈ H 2 (X, Q) vanishes. Definition 6.4. -In Setting 6.3, given a cohomology class α ∈ H 1,1 (X, R) that is nef and big, it follows from [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF] that there exists a unique singular Ricci flat current T ∈ α, i.e. a closed, positive current of bidegree (1, 1) representing α, with the following properties:

(i) T has minimal singularities in α;

(ii) T is a Kähler form on the analytic open set

The current T can be found by solving the Monge-Ampère equation

where θ ∈ α is a smooth representative, ϕ ∈ PSH(X, θ) is the unknown function and

Here, s ∈ H 0 (X, m(K X + B)) is any non-zero section (for some m 1) and φ B is the unique singular psh weight on O X (B) solving dd c φ B = [B] and normalized by

We let K X denote the Kähler cone, i.e. the set of cohomology classes α ∈ H 1,1 (X, R) which can be represented by a Kähler form. We fix (α t ) 0<t 1 ⊂ K X a path of Kähler classes and assume that α t → ∂K X as t → 0.

When X is smooth and B = 0, the existence of a unique Ricci flat Kähler metric ω t in α t for each 0 < t 1 dates back to the celebrated work of Yau [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I[END_REF]. A basic problem is to understand the asymptotic behavior of the ω t 's, as t → 0. This problem has a long history, we refer the reader to [START_REF] Gross | Collapsing of abelian fibered Calabi-Yau manifolds[END_REF] for references.

Despite motivations coming from mirror symmetry, not much is known when the norm of α t converges to +∞ (this case is expected to be the mirror of a large complex structure limit, see [START_REF] Kontsevich | Homological mirror symmetry and torus fibrations[END_REF] or the recent survey [START_REF]Collapsing Calabi-Yau manifolds[END_REF]). We thus only consider the case when α t → α 0 ∈ ∂K X . There are two rather different settings, depending on whether α 0 is big (vol(α 0 ) > 0), or merely nef with vol(α 0 ) = 0.

The non-collapsing case. -

We first consider the case when the volumes of the α t 's are non-collapsing, i.e. vol(α 0 ) > 0. Then, we have the following result, generalizing theorems of Tosatti [START_REF] Tosatti | Limits of Calabi-Yau metrics when the Kähler class degenerates[END_REF] and Collins-Tosatti [START_REF] Collins | TOSATTI -"Kähler currents and null loci[END_REF]. Theorem 6.5. -Let (X, B) be a pair as in Setting 6.3 and let (α t ) 0<t 1 ⊂ K X be a smooth path of Kähler classes such that α t → α 0 ∈ ∂K X as t → 0, with vol(α 0 ) > 0. Then, the singular Ricci-flat currents T t ∈ α t converge to T 0 as t → 0 weakly on X, and locally smoothly on Ω α .

Proof. -One can work in a desingularization p : Y → X of X. The path α t induces a path β t = p * α t of semi-positive and big classes. The currents T t can be decomposed as T t = θ t + dd c ϕ t where θ t ∈ β t is a smooth representative and ϕ t are normalized by sup X t ϕ t = 0 and solve the complex Monge-Ampère equation

where the volumes V t = α n t are bounded away from zero and infinity, C -1 V t C, and µ Y = f dV Y is a fixed volume form, with f ∈ L p (Y) for some p > 1 (because (X, B) has klt singularities, see [EGZ09, Lem. 6.4]).

The hypothesis of Theorem 1.1 (H2) is thus trivially satisfied, while (H1) follows if we initially bound from above α t γ X by a fixed Kähler class. The most delicate C 0 -estimate follows thus here from Theorem 1.9. When X is smooth, the C 0 -estimate in [START_REF] Tosatti | Limits of Calabi-Yau metrics when the Kähler class degenerates[END_REF] is obtained by using a Moser iteration argument as in Yau's celebrated paper [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I[END_REF], but this argument can no longer be applied in the present more singular setting. The rest of the proof is then roughly the same as in the case of smooth manifolds. It consists in adapting Yau's Laplacian estimate by using Tsuji's trick (first used in [START_REF] Tsuji | Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type[END_REF]), the remaining higher order estimates being local ones.

6.4. The collapsing case. -We now consider the case when the volumes of the α t 's are collapsing, i.e. vol(α 0 ) = 0. This case is more involved and only special cases are fully understood.

Suppose there is a surjective, holomorphic map with connected fibers f : X → Z, where Z is a compact, normal Kähler space of positive dimension m. We denote by k := nm = dim Xdim Z the relative dimension of the fiber space f . We let S Z denote the smallest proper analytic subset Σ ⊂ Z such that

• Σ contains the singular locus Z sing of Z,

• The map f is smooth on f -1 (Z \ Σ),

• For any z ∈ Z \ Σ, Supp(B) intersects X z transversally, and we set S X = f -1 (S Z ). Finally, we set

By the last item, each component of B| X • dominates Z • . A general fiber X z satisfies κ(K X z + B z ) 0, but the inequality may be strict. If c 1 (K X + B) = 0, then log abundance implies that K X z + B z ∼ Q O X z for z general. Moreover, Iitaka's conjecture predicts that κ(K X z + B z ) vanishes as soon as κ(Z) 0, which in turn should be equivalent to Z not being uniruled.

Fix ω Z a Kähler form on Z. For simplicity, we assume that Z ω m Z = 1. The form f * ω Z is a semi-positive form such that f * ω p Z = 0 for any p > m. We also choose a Kähler form ω X on X. The quantity X z ω k X = f * ω k X is constant in z ∈ Z; up to renormalizing ω X , we may assume that the constant is 1.

We assume that our path (α t ) t 0 in H 1,1 (X, R) is given by α 0 = { f * ω Z } and α t = α 0 + t{ω X }. As a result, one has (6.4)

We set ω t := f * ω Z + tω X and let ω ϕ t := ω t + dd c ϕ t denote the singular Ricci-flat current in α t , normalized by X ϕ t ω n X = 0. It satisfies

) , cf Eq. (6.3). The probability measure f * µ (X,B) has L 1+ε -density with respect to ω m Z thanks to [EGZ18, Lem. 2.3]. Therefore, there exists a unique positive current ω ∞ ∈ {ω Z } with Sketch of proof of Proposition 6.8. -To construct such a metric ω B , one first chooses smooth metrics h i on B i , sections s i ∈ H 0 (X, O X (B i )) cutting out B i , and one sets ω B := ω X + dd c ∑ i |s i | 2(1-b i ) . Up to scaling down the metrics h i , one can easily achieve the first condition. The third condition also follows easily.

The left-hand side inequality of 2 ("lower bound" on the holomorphic bisectional curvature) follows from [GP16, (4.3)] with ε = 0. As for the right-hand side inequality (upper bound on the holomorphic bisectional curvature), a proof has been given in [JMR16, App. A] in the case where B is smooth but a very simple argument has been found by Sturm, cf [Rub14, Lem. 3.14].

Step 2. Estimates We list in the Proposition below various estimates on ω ϕ t that will be useful for the last step. First, we define for z ∈ Z • the quantity ϕ t (z) := X z ϕ t ω k X z . In the following, we will not distinguish between ϕ t and f * ϕ t . Proposition 6.9. -There exist a constant C > 0 as well as a positive function g ∈ C ∞ (X • ), both independent of t, such that

Proof of Proposition 6.9. -In this proof, C will denote a constant that may change from line to line but is independent of t. In the same way, g will be a smooth, positive function on X • that should be thought as blowing up to +∞ near S X ; it can be assumed to come from Z • via f . 1. This is a consequence of [EGZ08, Thm. A] or [DP10, p. 401].

2.

Let us consider the holomorphic map f : (X \ B, ω ϕ t ) → (Z, ω Z ). Given that Ric(ω ϕ t ) = 0 and that ω Z is a smooth Kähler metric on the compact space Z, Chern-Lu's formula [START_REF] Chern | On holomorphic mappings of hermitian manifolds of the same dimension[END_REF][START_REF] Lu | On holomorphic mappings of complex manifolds[END_REF] provides a constant C > 0 such that the non-negative function

Let τ be a section of O X ( B ) cutting out B and let h B be a smooth hermitian metric on that line bundle. We set χ := log |τ| 2 h B . As ω ϕ t is a Kähler current and χ is quasi-psh,

Step 3. Convergence Thanks to Proposition 6.9 1., the family (ϕ t ) 0<t 1 is relatively compact for the L 1topology. All we have to do is showing that all its clusters values coincide. Let ϕ ∞ be such a cluster value; it is an f * ω Z -psh function but f has connected fibers so that ϕ ∞ is necessarily constant on the fibers. Equivalently, one has ϕ ∞ = f * ϕ ∞ for the (unique)

We want to show that the following equality of measures (6.18)

holds on Z. Given that Eq. 6.18 has a unique normalized bounded solution, this will prove the Theorem. As ϕ ∞ is globally bounded on X thanks to Proposition 6.9 1. and f * µ (X,B) does not charge any pluripolar set, it is actually enough to show that the equality of measures (6.18) holds on Z • . In order to prove (6.18) on Z • , given that f * ω k X = 1, it is enough to prove instead that for any function u ∈ C ∞ 0 (Z • ), one has (6.19)

We start from the identity (6.20)

) when t → 0, cf (6.4). Set ψ t := ϕ tϕ t and decompose ω ϕ t as

By expanding, one obtains

Performing an integration by parts, one gets

for degree reasons. By dominated convergence theorem, we have that ϕ t → ϕ ∞ in the L 1 loc (Z • ) topology. Moreover, as B intersects the fibers of f tranversally over Z • , an easy argument relying on partition of unity shows that f * (ω B ∧ ω k X ) is a smooth (1, 1)-form on Z • . Combining this with Proposition 6.9 4., we find dd

Together with (6.9), this implies (6.21)

By standard result, this shows that ϕ t → ϕ ∞ in C 1,α loc (Z • ) for any α < 1. In particular, the quasi-psh functions ϕ t converge uniformly on Supp(u). By Bedford-Taylor theory, one deduces that

In the end, one has showed that (6.22)

• Case i < m. We expand

From (6.21), we find (6.23) t n-i V t X f * u • f * (ω Z + dd c ϕ t ) i ∧ ω n-i X = O(t m-i ) = o(1).

For the remaining terms, an integration by parts yields

From Proposition 6.9 3., one has |ψ t | gt. Moreover, among the (nij -1) eigenvalues of dd c ψ t involved in the integral, at least (nij -1) -(m -(i + 1)) = kj must come from the fiber. Given Proposition 6.9 4-5., the integrand is a O(t 1+k-j ). As a result,

Combining that result with (6.23), we see that for any i > m, one has (6.24) lim

Putting together (6.20), (6.22) and (6.24), we obtain

In summary, (6.19) is proved, which concludes the proof of the Theorem.