Junyan Cao

Henri Guenancia

Mihai Pȃun

Valentino Tosatti

VARIATION OF SINGULAR KÄHLER-EINSTEIN METRICS: KODAIRA DIMENSION ZERO by

Keywords: Classification. -14J10, 14J32, 32Q20

We study several questions involving relative Ricci-flat Kähler metrics for families of log Calabi-Yau manifolds. Our main result states that if p : (X, B) → Y is a Kähler fiber space such that (X y , B| X y ) is generically klt, K X/Y + B is relatively trivial and p * (m(K X/Y + B)) is Hermitian flat for some suitable integer m, then p is locally trivial. Motivated by questions in birational geometry, we investigate the regularity of the relative singular Ricci-flat Kähler metric corresponding to a family p : (X, B) → Y of klt pairs (X y , B y ) such that κ(K X y + B y ) = 0. Finally, we disprove a folkore conjecture by exhibiting a one-dimensional family of elliptic curves whose relative (Ricci-) flat metric is not semipositive.

Introduction

In this article we continue our study of fiber-wise singular Kähler-Einstein metrics started in [START_REF] Cao | Variation of singular Kähler-Einstein metrics: positive Kodaira dimension[END_REF] in the following context.

Let p : (X, B) → Y be a Kähler fiber space, where B is an effective divisor such that (X y , B| X y ) is klt for all y ∈ Y in the complement of some analytic subset of the base Y. We are interested here in the curvature and regularity properties of the metric induced on K X/Y + B by the canonical metrics on fibers X y under the hypothesis κ(K X y + B y ) = 0. The far reaching goal we are pursuing here is a criteria for the birational equivalence of the fibers (X y , B| X y ) of p in a geometric context inspired by results due to E. Viehweg, Y. Kawamata and J. Kollár in connection with the C nm conjecture. To this end, the fiber-wise Kähler-Einstein metrics are playing a crucial role. Due to some technical difficulties -which we hope to overcome in a forthcoming paper-our most complete results are obtained under the more restrictive hypothesis c 1 (K X y + B y ) = 0, i.e. in the absence of base points of log-canonical bundle of fibers.

Main results. -Let p : (X, B) → Y be a proper, holomorphic fibration between two Kähler manifolds, where B = ∑ b i B i is an effective Q-divisor on X whose coefficients b i ∈ (0, 1) are smaller then one. We assume that there exists Y • ⊂ Y contained in the smooth locus of p such that B| X y has snc support and set X • := p -1 (Y • ). The fibers of p are assumed to satisfy c 1 (K X y + B| X y ) = 0 for any y ∈ Y • .

If we fix a reference Kähler form ω on X, then we can construct a fiberwise Ricci-flat conic Kähler θ y metric, i.e. a solution of the equation

Ric θ y = [B y ] θ y ∈ [ω y ]
.

There exists a unique function ϕ ∈ L 1 loc (X • ) such that θ y = ω y + dd c ϕ| X y X y ϕ ω n y = 0 .

The closed (1, 1)-current θ • KE := ω + dd c ϕ on X • is called relative Ricci-flat conic Kähler metric in [ω]. As we shall soon see, the current θ • KE is not positive in general, which marks an important difference with the case of Kähler fiber spaces whose generic fiber is of (log) general type.

Nevertheless, we establish here the following result (cf. Theorem 1.2 for a complete version).

Theorem A. -Let p : (X, B) → Y be a map as above, and let ω be a fixed Kähler metric on X. Assume that the following conditions are satisfied.

(i) For y ∈ Y • , the Q-line bundle K X y + B y is numerically trivial.

(ii) For some m large enough, the line bundle p * (m(K X • /Y • + B)) is Hermitian flat with respect to the Narasimhan-Simha metric h on Y • , cf. (1.7).

Then we can construct a (1, 1)-current θ • KE such that the restriction θ y of θ • KE to X y is a representative of {ω}| X y and solves Ric θ y = [B y ]. Moreover we have

( †) θ •

KE is positive and it extends canonically to a closed positive current θ KE ∈ {ω} on X. ( ‡) The fibration (X, B) → Y is locally trivial over Y • . Moreover, if p is smooth in codimension one and codim X (B X • ) > 1, then p is locally trivial over the whole Y.

The result above has many geometric applications, like for instance a Kähler version of a theorem of Ambro [START_REF] Ambro | The moduli b-divisor of an lc-trivial fibration[END_REF], cf Corollary 1.3 and its proof given in page 19. Another striking consequence is the following positivity property of direct images of pluri-log canonical bundles cf page 19 for a proof. It can be seen as a logarithmic version of Viehweg's Q n,m -conjecture for families of log Calabi-Yau manifolds, cf [START_REF]Weak positivity and the additivity of the Kodaira dimension. II. The local Torelli map[END_REF].

Corollary B. -Let p : (X, B) → Y be a fibration between two compact Kähler manifolds such that c 1 K X y + B| X y = 0 for a generic y ∈ Y. Assume moreover that the logarithmic Kodaira-Spencer map (0.1)

T Y → R 1 p (T X/Y (-log B))
is generically injective. Then the bundle p (m(K X/Y + B)) is big.

The example we exhibit is constructed from a special K3 surface admitting a non-isotrivial elliptic fibration as well as another transverse elliptic fibration. The construction is detailed in Section 3.

Previously known results. -In connection with Theorem A, the statements obtained so far are based on two different type of techniques arising from algebraic geometry and complex differential geometry, respectively. One can profitably consult the articles [START_REF]Weak positivity and the additivity of the Kodaira dimension. II. The local Torelli map[END_REF], [START_REF] Kollár | Subadditivity of the Kodaira dimension: fibers of general type[END_REF] and [START_REF] Kawamata | Minimal models and the Kodaira dimension of algebraic fiber spaces[END_REF] for results aimed at the Iitaka conjecture. From the complex differential geometry side we refer to [START_REF]Strict and nonstrict positivity of direct image bundles[END_REF], [START_REF] Hein | Remarks on the collapsing of torus fibered Calabi-Yau manifolds[END_REF], [START_REF] Berndtsson | Algebraic fiber spaces and curvature of higher direct images[END_REF] and the references therein.

The folklore conjecture that we disprove in Theorem D arose from a result of Schumacher [START_REF] Schumacher | Positivity of relative canonical bundles and applications[END_REF] who proved the semipositivity of the relative Kähler-Einstein metric for families of canonically polarized manifolds (see also the related works [START_REF] Berman | Relative Kähler-Ricci flows and their quantization[END_REF], [START_REF] Tsuji | Canonical singular Hermitian metrics on relative canonical bundles[END_REF]). He also implicitly conjectured that an analogous semipositivity result should hold for families of Calabi-Yau manifolds [Sch12, p.7], and this was explored in the thesis of Braun [START_REF] Braun | Positivität relativer kanonischer Bündel und Krümmung höherer direkter Bildgarben auf Familien von Calabi-Yau-Mannigfaltigkeiten[END_REF] and in the papers [START_REF] Braun | Positivity of direct images of fiberwise Ricciflat metrics on Calabi-Yau fibrations[END_REF][START_REF] Braun | Kähler forms for families of Calabi-Yau manifolds[END_REF] where positive partial results were obtained. The semipositivity question of θ KE also appeared in the work [START_REF] Eyssidieux | Convergence of weak Kähler-Ricci flows on minimal models of positive Kodaira dimension[END_REF] on the Kähler-Ricci flow.

Main steps of the proof. -We will describe next the outline of the proof of Theorems A, C and D above.

• The first item of Theorem A is established by using two ingredients. The first one consists in showing that the conic Ricci-flat metric in {ω X y } on each fiber X y is the normalized limit of the unique solution of the family of equations of type

(0.2) Ric ρ ε = -ρ ε + εω + [B]
on X y where ρ ε ∈ ε{ω X y }. We show that ω • KE | X y is obtained as limit of 1 ε ρ ε as ε → 0. On the other hand, the main result of [START_REF] Guenancia | Families of conic Kähler-Einstein metrics[END_REF] shows that the family ρ ε has psh variation for each positive ε > 0 and the result follows (the flatness of the direct image is crucial in order to be able to use [START_REF] Guenancia | Families of conic Kähler-Einstein metrics[END_REF]).

The arguments for the second item of Theorem A is more involved. We use a different type of approximation of the conic Ricci-flat metric, by regularizing the volume element. Let τ δ be the resulting family of metrics. The heart of the matter is to show that the horizontal lift with respect to τ δ of any local holomorphic vector field on the base has a holomorphic limit as δ → 0. This is a consequence of the estimates in [START_REF] Guenancia | Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors[END_REF] combined with the PDE satisfied by the geodesic curvature of τ δ , cf. [START_REF] Schumacher | Positivity of relative canonical bundles and applications[END_REF]. Then we show that the geodesic curvature tends to a (positive) constant and as a consequence we finally infer that the horizontal lift of holomorphic vector fields with respect to ω • KE is holomorphic and tangent to B.

• The equation Ric ω = -[E] + [B] translates into an Monge-Ampère equation where the righthand side has poles and zeros. Poles are relatively manageable in the sense that they induce conic metrics, that is we know relatively precisely the behavior of the complex Hessian of the solution. Zeros, however, are much more complicated to deal with for several reasons. First, it seems hard to produce a global degenerate model metric that should encode the behavior of the solution. Next, the regularized solutions of the Kähler-Einstein equation do not satisfy a Ricci lower bound, hence it seems difficult to estimate their Sobolev constant. In Proposition 2.1, we establish a uniform (weak) Sobolev inequality where the measure in the right-hand side picks up zeros. Then, we get onto studying the regularity of families of such metrics. Despite having a rather poor understanding of the fiberwise metrics, we are still able to analyze the first order derivatives of the potentials in the transverse directions, leading to an L 2 estimate, yet with respect to a more degenerate volume form, cf Proposition 2.6. This is however enough to deduce the Lipschitz variation of the potentials away from Supp(B + E).

• The counterexample provided by Theorem D is built from an elliptic fibration p : X → P 1 where X is a K3 surface. In the Appendix, it is showed that one can find such a fibration with the following properties: its singular fibers are irreducible and reduced, it is not isotrivial and it admits another transverse elliptic fibration. These properties allow to find a semiample, pample line bundle L → X with numerical dimension one. Then, the relative Ricci-flat metric θ ∈ c 1 (L)| X • cannot be semipositive, for otherwise one can show that it would extend to a positive current θ ∈ c 1 (L) and as L is not big, results of Boucksom show that θ 2 ≡ 0 on X • . Using horizontal lifts of θ, one can finally conclude that the foliation ker(θ) is holomorphic, induced by a local trivialization of the family This contradicts the non-isotriviality of p. Passing from the relative Ricci-flat metric in c 1 (L) to one in a Kähler class can be done using a limiting process.

Organization of the paper. -

• §1 We prove Theorem 1.2, and then derive successively Corollary 1.3 and Corollary B.

• §2: We obtain transverse regularity results for families of Monge-Ampère equations corresponding to adjoint linear systems having basepoints. This leads to Theorem C. • §3: We prove Theorem 3.1 using results from the Appendix. discussions about the topics of this paper. This work has been initiated while H.G. was visiting KIAS, and it was carried on during multiple visits to UIC as well as to IMJ-PRG; he is grateful for the excellent working conditions provided by these institutions. During the preparation of this project, the authors had the opportunity to visit FRIAS on several occasions and benefited from an excellent work environment.

H.G. is partially supported by NSF Grant DMS-1510214, and M.P is partially supported by NSF Grant DMS-1707661 and Marie S. Curie FCFP.

Relative Ricci-flat conic metrics

1.1. Setting. -Let p : X → Y a holomorphic proper map of relative dimension n between Kähler manifolds. We denote by Y • ⊂ Y the set of regular values of p, and let

X • := p -1 (Y • ) so that p |X • : X • → Y • is a smooth fibration. For y ∈ Y • ,
one writes X y := p -1 (X y ) the fiber over y. Let B be an effective Q-divisor on X that has coefficients in (0, 1) and whose support has snc. Our assumption throughout the current section will be that for each y ∈ Y • we have

(1.1) c 1 (K X y + B y ) = 0 ∈ H 1,1 (X y , Q).
Thanks to the log abundance in the Kähler setting, cf. Corollary 1.18 on page 21, we know that K X y + B y is Q-effective. Combining this with Ohsawa-Takegoshi extension theorem in its Kähler version, cf. [START_REF] Cao | Ohsawa-Takegoshi extension theorem for compact Kähler manifolds and applications[END_REF], one can assume that there exists m

1 such that m(K X y + B y ) O X y for all y ∈ Y • .
In this context the main result we obtain here shows that the flatness of the direct image p (mK X/Y + mB) implies the local isotriviality of the family p : (X, B) → Y. By this we mean that there exists a holomorphic vector field v on X • whose flow identifies the pairs (X y , B y ) and (X w , B w ) provided that y, w ∈ Y • are close enough. This is the content of Theorem 1.2 below. Prior to stating our theorems in a formal manner, we need to recall a few notions and facts.

Given a point y ∈ Y • , there exists a coordinate ball U ⊂ Y • containing y and a nowhere vanishing holomorphic section

(1.2) Ω ∈ H 0 (X U , m(K X/Y + B)| X U )
by our assumption (1.1), where

X U := p -1 (U). If f B is a local multivalued holomorphic function cutting out the Q-divisor B, then the form (Ω y ∧ Ω y ) 1 m | f B | 2
induces a volume element on the fibers of p over U. We fix a Kähler class {ω} ∈ H 1,1 (X, R). Up to renormalizing ω, one can assume that the constant function

Y • y → X y ω n
is identically equal to 1. We also define V y :=

X y (Ω y ∧ Ω y ) 1 m | f B | 2 ; this is a Hölder continuous function of y ∈ Y • .
Let ρ y be the unique positive current on X y which is cohomologuous to ω y and satisfies

ρ n y = (Ω y ∧ Ω y ) 1 m V y | f B | 2 ,
cf. [START_REF] Yau | Harmonic functions on complete Riemannian manifolds[END_REF]. One can write ρ y = ω| X y + dd c ϕ y , where the function ϕ y is uniquely determined by the normalization (1.3)

X y ϕ y (Ω y ∧ Ω y ) 1 m | f B | 2 = 0.
For each y ∈ U ⊂ Y • , the current ρ y is reasonably well understood: it has Hölder potentials, and it is quasi-isometric to a metric with conic singularities along B, cf. [START_REF] Guenancia | Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors[END_REF].

We analyze next its regularity properties in the "base directions"; this will allow us to derive a few interesting geometric consequences. The function ϕ defined on X • by ϕ(x) := ϕ p(x) (x) is a locally bounded function on X • (by the family version of Kołodziej's estimates cf. [DDG + 14]) hence it induces a (1, 1) current

(1.4) ρ := ω + dd c ϕ on X • . Let ∆ ⊂ Y • be a small, 1-dimensional disk.
If ∆ is generic enough, then the inverse image X := p -1 (∆) is non-singular, and the restriction map p : X → ∆ is a submersion. We denote by t a holomorphic coordinate on the disk ∆. Following [START_REF] Siu | Curvature of the Weil-Petersson metric in the moduli space of compact Kähler-Einstein manifolds of negative first Chern class[END_REF] we recall next the expression of the horizontal lift of the local vector field ∂ ∂t . For the moment, this is a vector field v ρ with distribution coefficients on the total space X given by the expression

(1.5) v ρ := ∂ ∂t -∑ α ρ βα ρ tβ ∂ ∂z α ,
where the notations are as follows. We denote by (z 1 , . . . , z n , t) a co-ordinate system centered at some point of X , and ρ tβ is the coefficient of dt ∧ dz β . We denote by ρ βα the coefficients of the inverse of the matrix ρ αβ .

The reflexive hull of the direct image

(1.6) F m := p (m(K X/Y + B))
plays a key role in study of the geometry of algebraic fiber spaces. It admits a positively curved singular metric whose construction we next recall, cf. [BP08, PT18] and the references therein.

Let σ ∈ H 0 (U, F m | U ) be a local holomorphic section of the line bundle F m defined over a small coordinate set U ⊂ Y • . The expression

(1.7) σ 2 y := V m-1 y X y |σ| 2 |Ω y | 2 m-1 m e -φ B defines a metric h on F m | Y • .
It is remarkable that this metric extends across the singularities of the map p, and it has semi-positive curvature current, see loc. cit. for more complete statements.

Main results

. -This sub-section aims to the proof of the following results.

Theorem 1.1. -Let p : (X, B) → Y be a proper holomorphic map between Kähler manifolds as in (1.1). We assume moreover that the curvature of F m with respect to the metric in (1.7) equals zero when restricted to Y • . Then the (1, 1)-current ρ defined on X • by (1.4) is semipositive and it extends canonically to a closed positive current on X in the cohomology class {ω}.

For example, if we assume that Y is compact, then the curvature of F m will automatically be zero if c 1 (F m ) = 0 thanks to the properties of the metric (1.7) discussed above, cf [CP17, Thm. 5.2].

What we mean by the word canonical in the Theorem 1.1 above is that the local potential ϕ of ρ are locally bounded above across X X • .

We equally prove the next statement.

Theorem 1.2. -We assume that the hypothesis in Theorem 1.1 are satisfied. Then, p is locally trivial over Y • , that is, for every y ∈ Y • , there exists a neighborhood U ⊂ Y • of y such that

(p -1 (U), B) (X y , B| X y ) × U.
Moreover, if p is smooth in codimension one, then p is locally trivial over the whole Y provided that

codim X X • (B X • ) > 0.
In particular, under the assumptions in the second part of Theorem 1.2 the map p is automatically a locally isotrivial submersion.

As an application, we establish the following result; it partially generalizes to the Kähler case a theorem of F. Ambro [START_REF] Ambro | The moduli b-divisor of an lc-trivial fibration[END_REF].

Corollary 1.3. -Let p : X → Y be a fibration between two compact Kähler manifolds. Let B be a Q-effective klt divisor on X with snc support.

(

1.3.8) If -(K X + B) is nef, then -K Y is pseudo-effective.
(1.3.9) Moreover, if c 1 (K X + B) = 0 and c 1 (Y) = 0, then p is locally trivial, that is, for every y ∈ Y, there exists a neighborhood U ⊂ Y of y such that

(p -1 (U), B) (X y , B| X y ) × U.
In particular, if c 1 (K X + B) = 0, the Albanese map p : X → Alb(X) is locally trivial.

1.3. Proof of Theorem 1.1. -We will proceed by approximation, mainly using the following lemma combined with the results in [START_REF] Guenancia | Families of conic Kähler-Einstein metrics[END_REF].

The next statement will enable us to reduce the problem to the canonically polarized pairs.

Lemma 1.4. -Let X be a compact Kähler manifold and let B be an effective divisor such that (X, B) is klt. We assume that c 1 (K X + B) = 0. Let ω be Kähler form on X. For every ε > 0, let ρ ε ∈ ε{ω} be the unique twisted conic Kähler-Einstein metric such that

(1.10) Ric ρ ε = -ρ ε + εω + [B].
Let ρ ∈ {ω} be the unique conic Kähler-Einstein metric such that Ric

ρ = [B]. Then lim ε→0 1 ε ρ ε = ρ
where the convergence is smooth outside Supp(B).

Proof. -Let m ∈ N such that m(K X + B) is effective. Let Ω ∈ H 0 (X, m(K X + B)) be a holo- morphic section normalized such that (1.11) X (Ω ∧ Ω) 1 m | f B | 2 = 1.
There exists a unique function ϕ ε on X such that (1.4.12)

ρ ε = εω + dd c ϕ ε (1.4.13) ρ n ε = ε n e ϕ ε (Ω∧Ω) 1 m | f B | 2 Now, let us set ψ ε := 1 ε ϕ ε . One has 1 ε ρ ε = ω + dd c ψ ε and (1.14) (ω + dd c ψ ε ) n = e εψ ε (Ω ∧ Ω) 1 m | f B | 2 As (Ω∧Ω) 1 m | f B | 2
and ( 1 ε ρ ε ) n are probability measures and ψ ε is ω-psh, Jensen inequality yields

X (εψ ε ) (Ω∧Ω) 1 m | f B | 2
0, and therefore (1.15)

X ψ ε (Ω ∧ Ω) 1 m | f B | 2 0.
As the measure

(Ω∧Ω) 1 m | f B |
1 m | f B | 2
and ( 1 ε ρ ε ) n are probability measures again, (1.14) shows that inf

X ψ ε 0 sup X ψ ε .
Combining this information with (1.17), we obtain the inequality

(1.18) ||ψ ε || L ∞ (X) C.
Moreover, Jensen inequality applied to the equation

(Ω∧Ω) 1 m | f B | 2 = e -εψ ε ( 1 ε ρ ε ) n yields (1.19) X ψ ε (ω + dd c ψ ε ) n 0
From (1.14) and (1.18), we get uniform estimates at any order for ψ ε outside B. If ψ is a subsequential limit of the family (ψ ε ) ε>0 , it will satisfy (Ω ∧ Ω)

(ω + dd c ψ) n = (Ω ∧ Ω) 1 m | f B |
1 m | f B | 2 = 0
Therefore ψ is uniquely determined, and the whole family (ψ ε ) ε>0 converges to ψ. The lemma is thus proved. Now we can prove Theorem 1.1.

Proof of Theorem 1.1. -We fix a reference Kähler form ω on X, and let U be some small topological open set of Y • . By hypothesis, the curvature of the bundle F m | U is identically zero. By using parallel transport, this is equivalent to the existence of a section

(1.20) s ∈ H 0 (X U , mK X/Y + mB| X U )
whose norm is a constant function on U, namely s h (y) = 1 for every y ∈ U. Let Ω y := s| X y ∈ H 0 (X y , mK X y + mB y )

be the restriction of s to the fibers of p. Since c 1 (K X y + B y ) + εω |X y is a Kähler class for each ε > 0 and for each y ∈ Y • , there exists a unique ϕ ε such that

(εω + dd c ϕ ε ) n = ε n e ϕ ε (Ω y ∧ Ω y ) 1 m | f B | 2 on X y .
Since y ∈ U is a regular value, this is equivalent to

Ric ρ ε,y = -ρ ε,y + εω + [B y ] on X y ,
where

ρ ε,y = εω + dd c ϕ ε | X y .
Next, the section s is holomorphic hence the relative B-valued volume forms (Ω y ∧ Ω y )

1 m induce a metric with zero curvature on K X/Y + B over p -1 (U). Because of that,

ρ ε := εω + dd c ϕ ε
coincides with the current studied in [START_REF] Guenancia | Families of conic Kähler-Einstein metrics[END_REF] and the content of the main theorem in loc. cit. is that ρ ε is positive on p -1 (U). Thanks to Lemma 1.4, ρ is the fiberwise weak limit on p -1 (U) of the fiberwise twisted Kähler-Einstein metrics 1 ε ρ ε ; moreover, the estimate (1.18) is uniform over U, so that ρ is actually the global weak limit of the metrics 1 ε ρ ε on p -1 (U). In particular, ρ 0 on p -1 (U), hence on X • .

As for the extension property, it is proved in [START_REF] Guenancia | Families of conic Kähler-Einstein metrics[END_REF] that ρ ε extends canonically to the whole X as a positive current in {εω}. This means that given any small neigborhood U of a point x ∈ X X • , one has sup U∩X • ψ ε < +∞. In other words, ψ ε extends to an ω-psh function on X. Now, let us fix U as above. The family of ω-psh functions ( ψ ε ) ε>0 on U defined by ψ ε := ψ εsup U ψ ε is relatively compact. In particular one can find a sequence ε k → 0 and an ωpsh function ψ on U such that ψ ε k → ψ a.e. in U. Moreover, we know that ψ ε k = ψ ε k + sup U ψ ε k converges to the ω-psh function ϕ a.e. in U ∩ X • . This implies that sup U ψ ε k converges when k → +∞. By Hartogs lemma, this implies that sup U∩X • ϕ < +∞, which had to be proved.

1.4. Proof of Theorem 1.2. -We will proceed in a few steps, roughly as follows.

• We start by approximating ρ by smoothing the volume element. Let τ δ be the resulting C ∞ form. Then we have lim δ τ δ = ρ in weak sense.

• We analyze next the behavior of the geodesic curvature of τ δ . The main tools are the Laplace equation satisfied by this quantity, cf. [START_REF] Schumacher | Positivity of relative canonical bundles and applications[END_REF], and the C 2 -estimates for conic Monge-Ampère equations, cf. [START_REF] Guenancia | Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors[END_REF]. As a consequence, we first show that we can extract a limit of the horizontal lift v δ (corresponding to τ δ ) which is holomorphic on the fibers of p. Afterwards we show that the geodesic curvature of τ δ converges (on X \ Supp(B)) to a constant as δ → 0. Finally, we infer that v δ converges to v ρ uniformly on the complement of the divisor B.

• After completing the previous steps, we show that v ρ is in fact holomorphic on the total space X by using a few arguments borrowed from [START_REF] Berndtsson | Positivity of direct image bundles and convexity on the space of Kähler metrics[END_REF].

• Finally, we show that v ρ extends across the singular locus of p provided that X is compact and p is smooth in codimension one.

1.4.1.

Approximation. -This is a fairly standard and widely used procedure, so we will be very brief. By hypothesis, we have B = ∑ a j B j where a j ∈ (0, 1) and ∪B j has simple normal crossings. We consider a smooth metric e -φ j on the bundle associated to B j ; it induces a smooth metric e -φ B := e -∑ a j φ j on the Q-line bundle associated to B. For any δ 0 we define the quantity C δ,y as follows

e -C δ,y = X y (Ω y ∧ Ω y ) 1 m ∏ j | f j | 2 + δ 2 e φ j a j .
Here Ω is a section of F m | ∆ whose norm is equal to one at each point, and f j is a local holomorphic function cutting out B j . The expression ∏ j | f j | 2 + δ 2 e φ j -a j is then a globally defined smooth metric on the Q-line bundle associated to B. Finally, we let s j be the canonical section of O X (B j ), and we will denote by |s j | 2 the squared norm of s j with respect to e -φ j . Let us further define the smooth (1, 1)-form (1.21)

τ δ = ω + dd c u δ on X • such that u δ | X y is solution of the following system of equations (1.22)        (ω + dd c u δ ) n = e C δ,y (Ω y ∧Ω y ) 1 m ∏ j | f j | 2 +δ 2 e φ j a j X y u δ (Ω y ∧Ω y ) 1 m ∏ j | f j | 2 +δ 2 e φ j a j = 0 .
By the family version of Kołodziej's estimates [DDG + 14], one can easily see that for any rel-

atively compact subset U Y • , there exists a constant C > 0 independent of δ ∈ (0, 1) such that (1.23) sup y∈U ||u δ || L ∞ (X y ) C
As a consequence, we get the following easy result, cf (1.4) for the definition of ρ and ϕ.

Lemma 1.5. -When δ approaches zero, τ δ converges weakly to ρ on X • . More precisely, one has

u δ → ϕ in L 1 loc (X • ). Proof. -The convergence u δ → ϕ in L 1
loc (X y ) follows from Kołodziej's stability theorem [START_REF] Odziej | The set of measures given by bounded solutions of the complex Monge-Ampère equation on compact Kähler manifolds[END_REF]Thm. 4.1] (one even gets uniform convergence). The convergence on the total space then follows from Lebesgue's dominated convergence theorem coupled with (1.23).

1.4.2.

Uniformity properties of (τ δ ) δ>0 . -In this subsection we will only consider the restriction of our initial family of manifolds above a disk in the complex plane (1.24) p : X → ∆ where we recall that ∆ ⊂ Y • is generic and X = p -1 (∆).

The coordinate on ∆ will be denoted by t. We recall that the geodesic curvature of the form τ δ is the function defined by the equality

(1.25) τ n+1 δ = c(τ δ )τ n δ ∧ √ -1dt ∧ dt If v δ is the horizontal lift of ∂ ∂t
with respect to τ δ , then it is easy to verify that we have

(1.26) c(τ δ ) = v δ , v δ τ δ .
For each δ > 0, the form τ δ induces a metric say h δ on the relative canonical bundle K X /∆ as follows. Let z 1 , . . . , z n , z n+1 be a coordinate system defined on the set W ⊂ X . Recall that t is a coordinate on ∆. This data induces in particular a trivialization of K X /∆ , with respect to which the weight of h δ is given as follows

(1.27) e Ψ δ (z,t) dz 1 ∧ • • • ∧ dz n+1 = τ n δ ∧ √ -1dt ∧ dt.
The curvature of (K X /∆ , h δ ) is the Hessian of the weight

(1.28) Θ δ (K X /∆ )| W = dd c Ψ δ .
We have the following result, relating the various quantities defined above.

Lemma 1.6. -Let ∆ δ be the Laplace operator corresponding to the metric τ δ | X t . Then we have the equality

(1.29) -∆ δ (c(τ δ )) = | ∂v δ | 2 -Θ δ (K X /∆ )(v δ , v δ ).
We will not prove Lemma 1.6 in detail because this type of results appear in many articles, cf. [START_REF] Schumacher | Positivity of relative canonical bundles and applications[END_REF] or [START_REF] Ȃun | Relative adjoint transcendental classes and Albanese map of compact Kähler manifolds with nef Ricci curvature[END_REF]. The main steps are as follows: we have Ψ δ = log det(g αβ ) where we denote g αβ := τ δ,α β and a few simple computations show that the Hessian of Ψ δ evaluated in the v δ -direction equals

∂ ∂ log det(g αβ )(v δ , v δ ) = g αβ g tt,αβ -g αγ g δβ g γδ,t g αβ,t
g αβ g αβ,γt g γµ g tµg αβ g αβ,tγ g µγ g µt (1.30) + g αβ g αβ,γτ g γµ g ρτ g tµ g ρt .

In the rhs term in (1.30) we recognize the beginning of ∆ δ (c(τ δ )) (cf. the 1st term), and in the end this gives (1.29). Again, we refer to [START_REF] Chen | Approximation of weak geodesics and subharmonicity of Mabuchi energy[END_REF], pages 18-19 for a detailed account of these considerations.

Remark 1.7. -The equation (1.29) can be seen as the analogue of the usual C 2 estimates in "normal directions". By this we mean the following: the C 2 estimates are derived by evaluating the Laplace of the (log of the) sum of eigenvalues of the solution metric with respect to the reference metric. Vaguely speaking, in (1.29) we compute the Laplace of the normal eigenvalue.

The following result is an important step towards the proof of Theorem 1.2. Proposition 1.8. -Let t ∈ ∆ be fixed. For any sequence δ j → 0, there exists a holomorphic vector field w on X t Supp(B) such that, up to extracting a subsequence, the sequence (v δ j | X t ) j 0 converges locally smoothly outside Supp(B) to the vector field w.

Remark 1.9. -At this point, it is not obvious that w is independent of the sequence δ j and that it should coincide with to be the lift v of ∂ ∂t with respect to ρ| X • Supp(B) . Before giving the proof of Proposition 1.8 we collect here a few results concerning the family of forms (τ δ ) δ>0 taken from [START_REF] Guenancia | Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors[END_REF] and [START_REF] Guenancia | Families of conic Kähler-Einstein metrics[END_REF].

(a) It follows from [GP16, Sect. 5.2] that τ δ | X y has "uniform regularized conic singularities" in the sense that if on a small coordinate open set Ω ⊂ X , the divisor B is given be B = ∑ r 1 a j B j where B j is defined by {z j = 0}, then there is a constant C independent of δ such that for any y ∈ U, we have

(1.31) C -1 r ∑ k=1 idz k ∧ dz k (|z k | 2 + δ 2 ) a k + ∑ k r+1 idz k ∧ dz k τ δ | X y ∩Ω C r ∑ k=1 idz k ∧ dz k (|z k | 2 + δ 2 ) a k + ∑ k r+1 idz k ∧ dz k (b)
The estimates [Gue20, (3.13), Prop. 4.1&4.2] go through for u δ , that is, for any integer k 0, there exists C k > 0 independent of δ ∈ (0, 1) such that

(1.32) sup t∈∆ ||∂ t u δ || C k (Ω∩X t ) C k
and there exists a constant C > 0 such that the following global estimate holds:

(1.33) sup

t∈∆ X t |v δ | 2 ω τ n δ C
One also gets

(1.34) lim δ→0 sup t∈∆ X t ∩(∪{|s j | 2 <δ}) |v δ | 2 ω τ n δ = 0
Again, we will not reproduce here the arguments for (1.32)-(1.34), but let us comment e.g. (1.33) for the comfort of the reader/referee. The main observation is that in local coordinates this amounts to obtaining a bound of ∇ δ (∂ t u δ ) 2 with respect to the volume element τ n δ on X t . Here | • | 2 is measured with respect to the reference metric ω, and ∇ δ is the gradient corresponding to τ δ . By (1.31) this is smaller than ∇ δ (∂ t u δ ) 2 δ up to a uniform constant. This new quantity is controlled by taking the derivative of the Monge-Ampère equation verified by τ δ in normal directions and integration by parts. Of course, the real proof is much more involved and we refer to loc. cit. for the details.

We see immediately that (1.33)-(1.34) imply the next statement.

Lemma 1.10. -One has the following

lim δ→0 sup t∈∆ X t ∑ j δ 2 |s j | 2 + δ 2 |v δ | 2 ω τ n δ = 0.
The proof of Lemma 1.10 is very elementary and we skip it. We present next the arguments for Proposition 1.8.

Proof. -Recall that in local coordinates,

v δ = ∂ ∂t -∑ α,β τ βα δ τ δ,t β ∂ ∂z α By (1.32), the family (v δ | X t ) δ>0 is relatively compact in the C ∞ loc (X t Supp(B)) topology.
Let δ j a sequence converging to zero such that (v δ j | X t ) j 0 converges locally smoothly outside Supp(B) to a vector field w. Now, the geodesic curvature c(τ δ ) of τ δ satisfies the following equation

(1.31) -∆ τ δ c(τ δ ) = ∂v δ 2 -Θ δ (K X /∆ )(v δ , vδ )
by Lemma 1.6. In our setting (cf. (1.21) and the definition of τ δ ) the curvature term in (1.31) becomes

(1.32) ∂ 2 C δ (t) ∂t∂t -∑ j a j δ 2 √ -1 ∂s j , ∂s j (v δ , vδ ) (|s j | 2 + δ 2 ) 2 + ∑ a j δ 2 Θ j (v δ , vδ ) |s j | 2 + δ 2
where Θ j above is the curvature of the hermitian line bundle (O X (B j ), e -φ j ).

Integrating (1.31) against τ n δ yields (1.33) lim δ→0 sup t∈∆ X t ∂v δ 2 τ n δ + ∑ j a j X t δ 2 √ -1 ∂s j , ∂s j (v δ , vδ ) (|s j | 2 + δ 2 ) 2 τ n δ = lim δ→0 sup t∈∆ ∂ 2 C δ (t) ∂t∂t .
Indeed, thanks to Lemma 1.10 the third term in (1.32) vanishes as δ → 0. We show next that we have

(1.34) lim δ→0 sup t∈∆ ∂ 2 C δ (t) ∂t∂t = 0
and this will end the proof of Proposition 1.8. Recall that the expression of the function in (1.34) is

(1.35) C δ (t) = -log X t (Ω y ∧ Ω y ) 1 m ∏ j | f j | 2 + δ 2 e φ j a j
and given that the norm of Ω is equal to one at each point of ∆, we have

(1.36) C δ (t) = -log 1 - X t ∏ j | f j | 2 + δ 2 e φ j a j -∏ j | f j | 2a j ∏ j | f j | 2a j ∏ j | f j | 2 + δ 2 e φ j a j (Ω y ∧ Ω y ) 1 m .
With the same notations as in (1.31), the restriction of the function under the sum sign in (1.36) on a coordinate set W α reads as

(1.37) F α,δ (z, t) := ∏ j |z j | 2 + δ 2 e φ j a j -∏ j |z j | 2a j ∏ j |z j | 2a j ∏ j |z j | 2 + δ 2 e φ j a j
and then the integral in (1.36) becomes

(1.38) ∑ α W α ∩X t θ α F α,δ (z, t)e f α ω n
where θ α is a partition of unit and the f α are given smooth functions. If v is the horizontal lift of ∂ ∂t with respect to the reference metric ω, then we have the usual formula

(1.39) ∂ ∂t ∑ α X t θ α F α,δ (z, t)e f α ω n = ∑ α X t v θ α F α,δ (z, t)e f α ω n .
The formula (1.35) shows that ∂F α,δ ∂t converges to zero as δ → 0 because only the weights φ j depend on t and the coefficients a j are strictly smaller than 1. Indeed, we have

(1.40) ∂F α,δ ∂t = ∑ j δ 2 e φ j ∂ t φ j |z j | 2 + δ 2 e φ j 1+a j a j ∏ i =j (|z i | 2 + δ 2 e φ i ) a i
and our claim follows since

(C,0) δ 2 (|z| 2 + δ 2 ) 1+a dλ(z) → 0 as δ → 0 for any a < 1.
As for terms involving ∂F α,δ ∂z i we infer the same conclusion (i.e. they tend to zero) by using integration by parts as we explain next. The corresponding terms in (1.39) have the following shape (1.41)

X t ∂F α,δ ∂z i (z)τ α (z)dλ(z)
where τ α is a smooth function with compact support in W α ∩ X t . The integral (1.41) is equal to

(1.42) - X t ∂τ α ∂z i (z)F α,δ (z)dλ(z)
and this tends to zero by dominated convergence. The same type of arguments apply for the second order derivatives of C δ (t); the claim (1.34) follows.

As v δ j → w in the C ∞ loc (X t Supp(B)) topology when j → +∞, it follows from the identity (1.33) above that w |X t Supp(B) is holomorphic.

The next proposition is equally very important in the analysis of the uniformity properties of (v δ ) δ>0 .

Proposition 1.11. -Let t ∈ ∆ be fixed. Then the identity

(1.43) lim δ→0 c(τ δ ) - X t c(τ δ )τ n δ = 0 holds on X t Supp(B).
Proof. -Let G δ : X t × X t → R be the Green function of (X t , τ δ ). Let x ∈ X t Supp(B); by definition, one has

(1.44) c(τ δ )(x) - X t c(τ δ )τ n δ = X t -∆ τ δ c(τ δ )• G δ (x, • ) τ n δ Clearly, Vol(X t , τ δ ) = X t τ n δ = X t ω n = 1 is independent of δ. Moreover, by (1.31), there exists a constant C 1 > 0 independent of δ such that diam(X t , τ δ ) C 2 . Therefore, it follows from [Siu87, A.2] that (1.45) G(x, y) -C 2 for some C 2 > 0 independent of δ. Now recall that G δ (x, y) = +∞ 0 G δ (x, y, s)ds where G δ (x, y, s) satisfies G δ (x, y, s) C 3 s -n e -d τ δ (x,y)/5s if 0 < s < 1 C 4 s -n for any 0 < s < +∞
where d τ δ is the geodesic distance induced by τ δ on X t . This follows respectively by [START_REF] Davies | Gaussian upper bounds for the heat kernels of some second-order operators on Riemannian manifolds[END_REF]Thm. 16] and [Siu87, p.139] -recall that the Ricci curvature of τ δ is uniformly bounded below thanks to (1.31). Integrating the above inequalities, one gets

(1.46) G(x, y) C 3 d τ δ (x, y) 2-2n
for some uniform C 3 > 0. Let I δ (x) := c(τ δ )(x) -X t c(τ δ )τ n δ , and let C 4 > 0 be large enough so that ±Θ δ C 4 ω. One has successively:

|I δ (x)| = X t -∆ τ δ c(τ δ )• (G δ (x, • ) + C 2 ) τ n δ X t ∂v δ 2 + C 4 ( ∑ j δ 2 |s j | 2 + δ 2 )|v δ | 2 ω • (G δ (x, • ) + C 2 ) τ n δ + X t ∑ j a j δ 2 √ -1 ∂s j , ∂s j (v δ , vδ ) (|s j | 2 + δ 2 ) 2 • (G δ (x, • ) + C 2 ) τ n δ C 5 X t ∂v δ 2 + ( ∑ j δ 2 |s j | 2 + δ 2 )|v δ | 2 ω • d τ δ (x, • ) 2-2n τ n δ + X t ∑ j a j δ 2 √ -1 ∂s j , ∂s j (v δ , vδ ) (|s j | 2 + δ 2 ) 2 • d τ δ (x, • ) 2-2n τ n δ
We claim that the right hand side converges to 0 when δ → 0, uniformly on x belonging to a fixed compact subset of X t Supp(B). To see this, it is enough to check that out of any sequence δ j → 0, one has lim j→+∞ I δ j (x) = 0 uniformly on x, up to extracting a subsequence. Thanks to Lemma 1.8, one can assume that v δ j converges locally smoothly to a holomorphic vector field w on X t Supp(B). Let us pick ε > 0.

By the estimates and observations above, one can find a small neigborhood U x X t Supp(B) and a constant C = C(x) > 0 such that:

(i) |v δ | 2 ω C, ∂v δ j 2 ε, and |s j | 2 C -1 hold on U x for any index j; (ii) U x d τ δ (z, • ) 2-2n τ n δ C; (iii) d τ δ (z, w) 2-2n C for any w / ∈ U x .
The rest of the proof is easy: we split the integral into two pieces on U x and its complement.

• On the complement of U x we use the item (iii) so that we can replace the function d τ δ (x, • ) 2-2n in the inequalities above by a constant independent of δ. The proof of Proposition 1.8 shows that the integral of the remaining terms tends to 0 and δ → 0.

• On the set U x we are 'far' from the support of B. Combined with the items (i) and (ii) above, this finishes the proof of Proposition 1.11.

In fact, Proposition 1.11 shows that the limit (1.43) is uniform on compact sets contained in the complement of the divisor B. We intend to couple this with the elliptic equation satisfied by c(τ δ ) in order to obtain bounds for the derivatives of this function in the fiber directions. To this end, we need the following statement.

Proposition 1.12. -There exists a constant C > 0 independent of δ > 0 such that

X t c(τ δ )τ n δ C
Proof. -This statement can be seen as a by-product of the considerations in the article [Gue20, (5.3) & Prop. 5.4]. Therefore we will content ourselves to highlight the main steps. To start with, we recall that the normalization of u δ is as follows (1.47)

X t u δ (Ω y ∧ Ω y ) 1 m ∏ j | f j | 2 + δ 2 e φ j a j = 0
and this can be re-written as (1.48)

X t u δ e F δ ω n δ = 0
where ω δ is a metric with conic singularities on X, whose multiplicities along the components of B are 1 > b j max(a j , 1/2) (notations as in (1.31)). Note that F δ in (1.48) has an explicit expression, being the log of the ratio

τ n δ ω n δ .
Let V δ be the horizontal lift of ∂ ∂t with respect to ω δ . By applying the

∂ 2 ∂t∂t operator in (1.48)
we obtain

X t V δ V δ (u δ ) e F δ ω n δ = - X t V δ (u δ )V δ (F δ )e F δ ω n δ - X t V δ (u δ )V δ (F δ )e F δ ω n δ - X t u δ V δ V δ (F δ ) e F δ ω n δ - X t u δ |V δ (F δ )| 2 e F δ ω n δ (1.49)
Now the point is that, up to terms for which we have a uniform estimate already, the function

V δ V δ (u δ )
is "the same" as c(τ δ ). Hence the absolute value of the lhs of (1.49) is equivalent to

X t c(τ δ )τ n δ .
The terms on the rhs of (1.49) are uniformly bounded, as it is proved in the reference indicated at the beginning of the proof.

We can now prove that the vector field v ρ is holomorphic when restricted to the fibers of p.

Corollary 1.13. -Let t ∈ ∆ be fixed. The family (v δ | X t ) δ>0 converges locally smoothly outside Supp(B) to the lift v of ∂ ∂t with respect to ρ| X • Supp(B) . In particular, v| X t Supp(B) is holomorphic. Proof. -Combining Propositions 1.11 and 1.12, one sees that c(τ δ ) is locally uniformly bounded on X t Supp(B). Given the elliptic equation satisfied by c(τ δ ), it implies local bound at any order (in the fiber directions).

Let W ⊂ X be a coordinate open subset of X such that W ∩ Supp(B) = ∅. In local coordinates, this implies that (1.50) ∂ 2 u δ ∂t∂t is bounded on W by a constant independent of δ. Since we already dispose of this type of bounds for any other mixed second order derivatives of u δ , we infer that we have

(1.51) ∆ u δ C W
where ∆ is the Laplace operator corresponding to the flat metric on W and C W is a constant independent of δ. This implies that the global function u δ admits C 1,α bounds locally on X Supp(B) for any α < 1. By Arzela-Ascoli theorem and Lemma 1.5, it implies that u δ converges to ϕ in C 1,α loc (X Supp(B)). In particular, ϕ is differentiable in the t variable outside Supp(B), and on this locus, ∂ t ϕ t = lim ∂ t u δ in the C α loc topology. Now, (1.32) shows that the convergence actually takes places in C ∞ loc (X t Supp(B)). In particular, outside Supp(B), v ρ | X t is the smooth limit of v δ | X t when δ → 0. Corollary 1.13 is now a consequence of Proposition 1.8.

Corollary 1.14. -Let t ∈ ∆ be fixed. Then dc(τ δ )| X t converges locally uniformly to 0 on the compact subsets of X t Supp(B).

Proof. -Let K X t Supp(B). By the proof of Corollary 1.13 and given (1.29), c(τ δ )| K is bounded in L ∞ norm hence in any C k loc norm on K. This implies that family dc(τ δ )| K is relatively compact in the smooth topology, and the claim follows from Proposition 1.11.

Lemma 1.15. -The vector field v on X Supp(B) is holomorphic and extends across Supp(B).

Proof. -This first assertion follows from a simple computation in [Ber09, Lem. 2.5]. In our setting, this yields on X t Supp(B t ):

(1.52) ∂t v δ ¬ τ δ = ∂c(τ δ ) -iτ δ ( ∂v δ , vδ )
As on X t Supp(B t ), τ δ and v δ converge locally smoothly to ρ and v respectively, one deduces from Corollary 1.14 above that v is holomorphic (hence smooth, too) in the t variable as well, outside Supp(B).

For the second assertion, let us first observe that τ δ n ∧ idt ∧ d t dominates a smooth volume form dV on X . Therefore, it follows from (1.33) that

p -1 (U) Supp(B) |v δ | 2 ω dV C
An application of Fatou lemma gives:

p -1 (U) Supp(B)
|v| 2 ω dV < +∞ By Hartog's theorem, it follows that v extends to a holomorphic vector field across Supp(B).

Lemma 1.16. -The vector field v preserves ρ, hence its flow preserves B.

Proof. -On X Supp(B), we obtain the equality (1.53) L v ρ = 0 as a consequence of (1.52). We show next that (1.53) extends in the sense of currents on X . Indeed, if so then we claim that the flow of v produces the biholomorphic maps F t = X 0 → X t such that F 0 is the identity and such that F * t ω t = ω 0 . It is for this equality that we need (1.53) to hold on X in the sense of currents: it gives (1.54) d dt F t ω t = 0 in weak sense on X , but this is enough to conclude that F * t ω t = ω 0 . If one pulls back the Kähler-Einstein equation satisfied by ω t by F t , one gets

Ric F * t ω t = -F * t ω t + F * t [B t ] where [B t ] = ∑ k a k [B t,k ] if B t,k are the irreducible components of Supp(B). Because F * t ω t = ω 0 , we obtain F * t [B t ] = [B 0 ]
In particular, the local flow of v preserves Supp(B).

Let us now prove that v ¬ ρ is zero on X . First, let us observe that ρ being a positive current, its coefficients are locally defined complex measures. We claim that these measures put no mass on Supp(B). Indeed, by e.g. [Dem12, Proposition 1.14] the "mixed terms" of ρ are dominated by the trace of ρ (the sum of the diagonal coefficients). Therefore everything boils down to showing that if ω is a given smooth Kähler form on X , then the positive measure ρ ∧ ω n does not charge Supp(B).

But it is easy to produce a family of cut-off function χ δ such that χ δ tends to the characteristic function of Supp(B) and such that ||∇ωχ δ || L 2 (ω n+1 ) and ||∆ ω χ δ || L 1 (ω n+1 ) tends to 0. We refer to e.g. [START_REF] Campana | Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields[END_REF]§9] for this classic construction. Finally, let us introduce η a smooth positive function with compact support on X . One can assume that on Supp(η), ρ = dd c ψ admits a local (bounded) potential. Performing an integration by parts, one obtains:

X ηχ δ ρ ∧ ω n = X ηχ δ dd c ψ ∧ ω n = X ηψ dd c χ δ ∧ ω n + X χψ dd c η ∧ ω n + X ψ dη ∧ d c χ δ ∧ ω n ||ψ|| ∞ ||η|| ∞ • ||∆ ω χ δ || L 1 + ||∆η|| ∞ Supp(χ δ ) ω n+1 + ||∇η|| L 2 • ||∇χ δ || L 2
which tends to 0. In conclusion, the coefficients of ρ and hence those of v ¬ ρ are complex measures which do not charge B. As v ¬ ρ = 0 outside Supp(B), this identity extends across Supp(B), which is what we wanted to prove.

If we sum up the results obtained so far, we can find near any y ∈ Y • a sufficiently small polydisk U ⊂ Y • with coordinates (t 1 , . . . , t m ) centered around y as well as holomorphic vector fields v 1 , . . . , v m on p -1 (U) lifting ∂ ∂t 1 , . . . , ∂ ∂t m which are tangent to Supp(B). Up to shrinking U, one can assume that the flow of the vector fields v a := ∑ a i v i for a = (a 1 , . . . , a m ) ∈ D m exists at least up to time one. Here D is the unit disk in C. Then one has a holomorphic map f : X y × D m → p -1 (U) which sends (x, a) to φ a 1 (x) where (φ a t ) t is the flow of v a . It is easy to see that f is an isomorphism onto its image, cf e.g. [START_REF] Morrow | KODAIRA -Complex manifolds[END_REF].

To conclude the proof of Theorem 1.2, we need to show that v ρ extends across the singular locus of p provided that X is compact and p is smooth in codimension one. The argument goes as follows.

End of the proof of Theorem 1.2. -Let n be the relative dimension of p and let m := dim Y. Let Y • ⊂ Y be the smooth locus of p, and let Let x ∈ X be a generic point of p -1 (y). Take a small neighborhood U of x, and set D := p(U). As p is smooth on codim 1, p is smooth on U. We can thus fix a coordinate system (t, z 1 , . . . , z n ) of U, such that t represents the horizontal directions and ∂ ∂z i is in the fiber direction. The notation t means that t = (t 1 , . . . , t m ). There is a slight abuse of notation: the coordinate of the base is also t. But as p is smooth on U, we just mean that p * ( ∂ ∂t i ) = ∂ ∂t i , where the former is on X and the later is on Y. Finally, we set p * (idt ∧ dt) := m k=1 idt k ∧ dt k . Let v k be the holomorphic vector field on X • ∩ p -1 (D) constructed in the proof of Theorem 1.2, attached to ∂ ∂t k , where 1 k m.

X • := p -1 (Y • ). Let Ω ∈ H 0 (X, m(K X/Y + B)). Let ρ = ω + dd c ψ
(1.55)

ρ n ∧ p * (idt ∧ dt) = (Ω ∧ Ω) 1 m | f B | 2 ∧ p * (idt ∧ dt) on U.
We know that ι v k ρ is proportional to d tk , from which it follows that

(1.56)

ι v 1 , v1 • • • ι v m , vm (ρ n ∧ p * (idt ∧ dt)) = ρ n
Combining (1.55) and (1.56), one gets

ι v 1 , v1 • • • ι v m , vm (Ω ∧ Ω) 1 m | f B | 2 ∧ p * (idt ∧ dt) = ρ n
One can find a Kähler form ω X on X such that (Ω∧Ω)

1 m | f B | 2 ∧ p * (idt ∧ dt) ω n+m X . Given that ω m X ∧ [ι v 1 , v1 • • • ι v m , vm (ω n+m X )] = (∏ k |v k | 2 ω X )• ω n+m X
(maybe up to some constant), we eventually get that

U∩X • m ∏ k=1 |v k | 2 ω X • ω n+m X U∩X • ρ n ∧ ω m X
and the right hand side is finite, dominated by

X ρ n ∧ ω m X {ω} n • {ω X } m by [BEGZ10, Prop. 1.6 & 1.20],
given that ρ is a closed, positive current on X in the cohomology class {ω}.

As |v k | 2 ω X is uniformly bounded below by a positive constant on p -1 (D)

∩ X • , one deduce that v k ∈ L 2 (p -1 (D) ∩ X • , ω X )
. By Riemann extension theorem the holomorphic vector fields v k extend to holomorphic vector fields on p -1 (D) whose flow provide the expected trivialization. Indeed, the v k are tangent to B on X • , hence they are tangent to B everywhere by the assumptions in 1.2.

As application of Theorem 1.2 we can prove Corollary B.

Proof of Corollary B. -Our proof follows the same line of arguments as in [START_REF] Kollár | Subadditivity of the Kodaira dimension: fibers of general type[END_REF].

We proceed by contradiction: assume that F m is not big. In any case, this bundle can be endowed with a metric (used several times in the current subsection) with semi-positive curvature form denoted by θ, and smooth on a Zariski open subset V ⊂ Y as B is generically transverse to the fibers. Then we claim that we have

(1.57) θ| dim(Y) V = 0
at each point of V. Indeed, if (1.57) is not true, then there exists a point y 0 ∈ V such that all the eigenvalues of θ y 0 are strictly positive. By the singular version of holomorphic Morse inequalities (cf. [Bou02, Cor. 3.3]) this implies that F m is big, and we have assumed that this is not the case. It follows that the kernel of θ is non-trivial at each point of V. Since θ| V is smooth and closed, locally near each point of V its kernel defines a foliation whose leaves are analytic sets, cf [START_REF] Kollár | Subadditivity of the Kodaira dimension: fibers of general type[END_REF] and the references therein. We choose a smooth holomorphic disk ∆ contained in such a leaf; the restriction of p to p -1 (∆) := X ∆ is a submersion, and the curvature of the direct image of the relative pluricanonical bundle is identically zero. By Theorem 1.2 we infer that the vector v ρ is holomorphic. On the other hand, ∂v ρ is a representative of the image of the tangent vector ∂ ∂t ∈ T ∆ by the map (0.1). Since by hypothesis this map is injective, we obtain a contradiction.

We finish the current section with the proof of Corollary 1.3.

Proof of Corollary 1.3. -The statement (1.3.8) is a direct consequence of [START_REF] Guenancia | Families of conic Kähler-Einstein metrics[END_REF] applied to the right hand side term of the equality

(1.58) -p * (K Y ) = K X/Y + (-K X -B) + B.
By hypothesis the class -c 1 (K X + B) is in the closure of the Kähler cone of X and one can use loc. cit. Given Theorem 1.2, it would be enough to prove that p is smooth in codimension one. We use the following elegant argument due to Q. Zhang, cf. [START_REF] Zhang | On projective varieties with nef anticanonical divisors[END_REF]. Assume that there exists some codimension one subvariety D ⊂ X such that p * (D) is of codimension at least two. Let τ : Y → Y be the composition of the blow-up of the closed analytic set p * (D) with a resolution of singularities of the resulting complex space. There exists an effective divisor E Y whose support is contained in the τ-exceptional locus such that we have

K Y ∼ E Y .
Let p : X → Y be a resolution of indeterminacies of X Y . As c 1 (K X + B) = 0, we have

(p ) * (-K Y ) + E X ≡ Q K X /Y + B ,
where E X is supported in exceptional locus of π : X → X. By [START_REF] Guenancia | Families of conic Kähler-Einstein metrics[END_REF],

K X /Y + B is pseudo- effective. Therefore the direct image π * ((p ) * (-K Y ) + E X ) = π * (-E Y ) is pseudo-effective as well.
However by construction we have π * (E Y ) [D], and we obtain a contradiction. We prove next that the map p is reduced in codimension one. Let E ⊂ Y be a divisor. Its p-inverse image can be written as

p -1 (E) = ∑ i a i [D i ]
where D i ⊂ X are irreducible divisors. It is well know that (cf. [CP17, Thm. 2.4] or also [START_REF] Takayama | Singularities of Narasimhan-Simha type metrics on direct images of relative pluricanonical bundles[END_REF])

K X/Y + B ∑ i (a i -1) + • [D i ],
where (a i -1) + := max{a i -1, 0}.

Therefore we must have a i = 1 for every i, since by assumption K X/Y + B ≡ Q 0. Corollary 1.3 is proved.

1.5. Log abundance in the Kähler setting. -In this section, we briefly explain how to prove the log abundance for klt Kähler pairs (X, B) such that B has snc support. This is based on the following lemma, which is a consequence of [START_REF] Budur | Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers[END_REF] and [Wan16, Cor 1.4] (cf. also [CKP12, Lem.1.1] and [START_REF] Campana | Geometric stability of the cotangent bundle and the universal cover of a projective manifold[END_REF] and the references therein). For the reader's convenience, we recall briefly the proof here. ( * ) After this paper was written, J. Wang [Wan19, Thm. D] proved a slightly more general case of Corollary 1.18 below using similar arguments.

Lemma 1.17. -Let X be a compact Kähler manifold and let ∆ = ∑ a i B i be an effective klt Q-divisor with simple normal crossing support. Assume that ∆ ∼ Q L 1 for some L 1 ∈ Pic(X). For each integer k 0, define L k := kL 1 -k∆ . Then for each k, i and q, the set

V q i (L k ) = {λ ∈ Pic • (X); h q (X, K X + L k + λ) i}
is a finite union of translates of complex subtori of Pic • (X) by torsion points.

Proof. -Let N be the minimal number such that N • a i ∈ N for every i. Let σ : X → X be the N-cyclic cover of L 1 along the canonical section of NL 1 . One can check that X has analytic quotient singularities [Vie77, Lem. 2], hence rational singularities by e.g. [Bur74, Prop. 4.1]. This implies in turn that for any resolution π : X → X, one has π * O X (K X ) = O X (K X ) thanks to e.g. [KM98, Thm. 5.10] and R i π * O X (K X ) = 0 for i > 0 by Grauert-Riemenschneider vanishing. Moreover, one has

σ * O X (K X ) = O X (K X ) ⊗ N-1 k=0 L k and R i σ * O X (K X ) = 0 for i > 0 since σ is finite. Therefore, if we define f := σ • π : X → X, we have (1.59) H q ( X, K X + f * λ) N-1 k=0 H q (X, K X + L k + λ)
for any line bundle λ on X.

( * ) We would like to thank Botong Wang for telling us the following nice application of his result.

Let g : Pic • (X) → Pic • ( X) be the natural morphism induced by f and set

V q i ( f ) := {ρ ∈ Pic • (X); h q ( X, K X + f * ρ) i} and V q i := {ρ ∈ Pic • ( X), h q ( X; K X + ρ) i}. Then we have (1.60) V q i ( f ) = g -1 (V q i
). Thanks to [START_REF] Wang | Torsion points on the cohomology jump loci of compact Kähler manifolds[END_REF], V q i is a finite union of torsion translates of complex subtori of Pic • ( X). Together with (1.60), this shows that V q i ( f ) has the same structure. Thanks to (1.59), we have (1.61)

V q i ( f ) = ∪ i 0 +•••+i N-1 =i ∩ N-1 k=0 V q i k (L k ) , where V q i (L k ) := {ρ ∈ Pic • (X), h q (X, K X + L k + ρ) i}. As V q i ( f )
is the finite union of torsion translates of complex subtori, we get from (1.61) that

V q i (L k ) has the same structure, cf [CKP12, Lemma 1.1].
Corollary 1.18. -Let (X, ∆) be a klt pair where X is compact Kähler and ∆ = ∑ a i B i is an effective

Q-divisor. If c 1 (K X + ∆) = 0 ∈ H 1,1 (X, Q), then K X + ∆ is Q-effective. Proof. -Let π : X → X be a log resolution of (X, ∆). Since Pic • (X ) is a torus and c 1 (K X + ∆) = 0, we can find L ∈ Pic • (X ) such that π * (K X + ∆) ∼ Q L.
We can also find a klt divisor ∆ on X with normal crossing support such that

K X + ∆ ≡ Q π * (K X + ∆) + E
for some Q-effective divisor E supported in the exceptional locus of π having no common component with ∆ . Let m 1 be the smallest integer such that mE has integral coefficients. In particular, m(K X + ∆ ) is equivalent to some line bundle on X by the formula above. Using the identity

m(K X + ∆ ) = K X + ∆ + m -1 m {E} =:∆ + +(m -1)(L + E )
we get a pair (X , ∆ + ) such that • ∆ + has snc support and coefficients in (0, 1) ∩ Q.

• ∆ + ∼ M for some line bundle M on X .

• K X + ∆ + + ρ is effective for some ρ ∈ Pic • (X ). The first two properties are obvious, and the third follows from the identity K X + ∆ + -L = mE -(m -1) E . By applying Lemma 1.17 to K X + ∆ + , we can assume that ρ is torsion hence h 0 (X , r(K X + ∆ + )) 1 for some integer r 1 that we can choose so that m|r. By doing so, one can ensure that r(K X + ∆ + ) = π * (r(K X + ∆)) + F for some effective, integral π-exceptional divisor F. This implies that h 0 (X, r(K X + ∆)) = 0. The Corollary is proved.

Transverse regularity of singular Monge-Ampère equations

In this section our main goal is to prove Theorem C. This will be achieved as a consequence of a few intermediate results which we state in a general setting.

The main source of difficulties in the proof of C arise from the fact that the set of base points of pluricanonical sections may be non-zero. The determinant of the metric adapted to this geometric setting vanishes along the said base points so in particular the Ricci curvature of this metric is not bounded from below. Unfortunately under these circumstances we were not able to obtain a complete analogue of the Sobolev and Poincaré inequalities (which are needed for the study of the regularity properties of Monge-Ampère equations). We will therefore start this section with a weak version of these results.

Weak Sobolev and Poincaré inequalities.

-In this section we will derive a version of the usual Poincaré and Sobolev type inequalities which are needed in our context. As it is well known, they are playing a crucial role in the regularity questions for the Monge-Ampère equations. The set-up is as follows: let (X, ω) be a compact Kähler manifold of dimension n, and let

(2.1)

E := ∑ α∈I e α E α B := ∑ β∈J b β B β be two effective divisors on X without common components, such that e α ∈ Q + , b β ∈ [0, 1[
and such that the support of E + B is snc. We assume that the manifold X is covered by a fixed family of coordinate sets (Ω j ) j such that (2.2)

Ω j ∩ Supp(E + B) = (z 1 j . . . z d j = 0)
where (z j ) are coordinates on Ω j . Let σ i , s i be the canonical section of the Hermitian bundle (O(E i ), h i ) and (O(B i ), g i ) respec- tively, where h i and g i are non-singular reference metrics. For each positive ε 0 and each multi-index q we introduce the following volume element

(2.3) dµ (ε) q := ∏ α∈I (ε 2 + |σ α | 2 ) q α ∏ β∈J (ε 2 + |s β | 2 ) b β dV ω
where dV ω is the volume element corresponding to the reference metric ω. Also, for each positive real number p 2 we define the multi-index q p whose components are

(2.4) 1 - p 2 q α .
Then we have the following statements.

Proposition 2.1. -There exists a constant C > 0 independent of ε (but depending on everything else) such that for every smooth function f on X we have

(2.5) X | f | 2np 2n-p dµ (ε) q 2n-p 2np C X |∇ ε f | p dµ (ε) q p + X | f | p dµ (ε) q p 1 p
where 1 p < 2 is any real number, and the gradient ∇ ε corresponds to the ε-regularization of a fixed metric with conic singularities along the divisor

∑ β∈J b β B β .
As we can see, there is an important difference between the Proposition 2.1 and the standard weighted Sobolev inequalities: the volume element in the left hand side of (2.5) is not the same as the one in the right hand site term. In a similar vein, we have the next version of the Poincaré inequality.

Proposition 2.2. -There exists a constant C > 0 as above such that for any smooth function f on X we have

(2.6) X f -VM µ ( f ) p dµ (ε) q C X |∇ ε f | p dµ (ε) q p
where p 1 is a real number, and where we use the notation

(2.7) VM µ ( f ) := X f dµ (ε) q .
We first prove the statement 2.1; the arguments which will follow have been "borrowed" from the book [HKM06, Chap. 15].

Proof of Proposition 2.1. -We first assume that B = 0 because the arguments for the general case are practically identical.

A first remark is that it is enough to consider the local version of the statement, as follows. Let Ω be one of the domains covering (X, E) as mentioned in (2.2); we denote by (z 1 , . . . , z n ) the corresponding coordinate system. We will assume that we have (2.8)

Ω = ∏ j (|z j | < 1)
and that the function f has compact support in Ω.

In terms of this local setting, the quantity to be evaluated becomes (2.9)

Ω | f | 2np 2n-p d ∏ α=1 (ε 2 + |z α | 2 ) q α dλ
(since b i = 0). Let B := (|t| < 1) ⊂ C be the unit disk in the complex plane. We consider the function (2.10)

F ε (t) = (ε 2 + |t| 2 ) q/2 (1 + ε 2 ) q/2 t
where q > 0 is a real number. It turns out that F ε is a diffeomorphism and the square of the absolute value of its Jacobian dF ε ∧ dF ε verifies the inequality

(2.11) C -1 (ε 2 + |t| 2 ) q dF ε ∧ dF ε dt ∧ dt C(ε 2 + |t| 2 ) q
where C is a constant independent of ε (it can be explicitly computed). Let G ε be the inverse of F ε . The implicit function theorem shows that we have

(2.12) |dG ε (t)| C (ε 2 + |t| 2 ) q/2 .
By the change of variables formula we have (2.13)

Ω | f (z)| 2np 2n-p d ∏ α=1 (ε 2 + |z α | 2 ) q α dλ C Ω | f (w)| 2np 2n-p dλ(w)
where by definition we set

(2.14) f (w) := f G ε (w 1 ), . . . , G ε (w d ), w d+1 , . . . , w n ;
it is a function defined on the "same" poly-disk Ω, and it has compact support. Therefore, by the usual version of the Sobolev inequality we obtain (2.15)

Ω | f (w)| 2np 2n-p dλ(w) 2n-p 2n C Ω |∇ f (w)| p dλ(w).
We use the relation (2.14), together with the change of coordinates w α = F ε (z α ) for α = 1, . . . d and we infer that we have (2.16)

Ω |∇ f (w)| p dλ(w) C Ω |∇ f (z)| p d ∏ α=1 (ε 2 + |z α | 2 ) q α (1-p 2 ) dλ.
In conclusion we have

(2.17)

Ω | f (z)| 2np 2n-p d ∏ α=1 (ε 2 + |z α | 2 ) q α dλ 2n-p 2n C Ω |∇ f (z)| p d ∏ α=1 (ε 2 + |z α | 2 ) q α (1-p 2 ) dλ.
that is to say, we have established the local version of the inequality 2.1. The general case follows by a partition of unit argument which we skip.

The same scheme of proof applies to Proposition 2.2: we will first show that the local version of this statement holds by using a change of coordinates and the classical version of Poincaré inequality, and then we show that the global version (2.6) is true by a well-chosen covering of X.

Proof. -The inequality (2.6) is easily seen to follow provided that we are able to establish the following relation

(2.18) X×X | f (x) -f (y)| p dµ (ε) q (x)dµ (ε) q (y) C X |∇ f | p µ (ε) q p
for any 1 p 2. This is very elementary and we will not provide any additional explanation. Assume that we have a covering of X

(2.19)

X = i U i
where each U i is a coordinate open set. In order to obtain a bound as in (2.18), it would be enough to analyze the quantities (2.20)

U i ×U j | f (x) -f (y)| p dµ (ε) q (x)dµ (ε) q (y)
for each couple of indexes i, j which is what we do next.

To start with, let Ω be one of the coordinate sets U i ; we will show next that the following local version of (2.6) holds true

(2.21) Ω×Ω | f (x) -f (y)| p dµ (ε) q (x)dµ (ε) q (y) C Ω |∇ f | p µ (ε) q α .
We proceed as in the previous proof: we have

(2.22) Ω×Ω | f (x) -f (y)| p dµ (ε) q (x)dµ (ε) q (y) C Ω×Ω f (z) -f (w) p dλ(z, w)
by a change of coordinates as indicated in (2.10). Now we have

(2.23) f (z) -f (w) = 1 0 d dt f ((1 -t)z + tw)) dt
and it follows that we have

(2.24) Ω×Ω f (z) -f (w) p dλ(z, w) C 1 0 dt Ω×Ω |∇ f ((1 -t)z + tw)) | p dλ(z, w),
where the constant C > 0 in (2.24) depends on the diameter of Ω measured with respect to the Euclidean metric.

Then we invoke the usual trick: we split the integral above in two -the first part is as follows

(2.25) 1/2 0 dt Ω×Ω |∇ f ((1 -t)z + tw)) | p dλ(z, w) C Ω |∇ f (z)) | p dλ(z)
where up to a numerical constant, C in (2.25) only depends on the volume of Ω. We have a similar estimate for the integral corresponding to the interval [1/2, 1], so all in all we infer

(2.26)

Ω×Ω f (z) -f (w) p dλ(z, w) C Ω |∇ f (z)) | p dλ(z).
Changing the coordinates back, together with the considerations in the proof of weak Sobolev inequality show that (2.21) is proved. The general case follows by choosing a covering (U j ) of X, such that the following properties are satisfied.

(1) If U p ∩ U q = ∅ and if at least one of them intersects the support of the divisor E, then the union U p ∪ U q is contained in a coordinate set endowed with coordinates adapted to (X, E) (as in the beginning of this section). (2) If U p ∩ U q = ∅ and if neither of U i or U j intersects Supp(E), then the union U p ∪ U q is contained in a coordinate ball which is disjoint of Supp(E).

(3) The dµ

q -volume of the coordinate sets containing U i ∪ U j in (1) and (2) is bounded from above and below by constants which are independent of ε. It is clear that such cover exists, and we fix one denoted by Λ for the rest of the proof. Note that this cover is independent of ε. Next, given any couple U i , U j of sets belonging to Λ, we consider a collection (2.27)

Ξ ij = (Ω 1 , Ω 2 , . . . , Ω N )
of elements of Λ such that the following properties are verified.

(a) We have Ω 1 := U i and Ω N := U j , and all of the intermediate Ω's are elements of Λ.

(b) For any r = 1, . . . N -1 we have Ω r ∩ Ω r+1 = ∅. Again, there are many choices for such Ξ ij , but we just pick one of them for each pair of indexes (i, j).

We are now ready to analyze the quantities (2.20): for each couple (i, j) we consider the collection

Ξ ij . Given (2.28) (x 1 , . . . , x N ) ∈ Ω 1 × • • • × Ω N we have (2.29) | f (x 1 ) -f (x N )| α C ∑ q | f (x q ) -f (x q+1 )| α
for some numerical constant C > 0.

We consider now the following expression

(2.30)

Ω 1 ו••×Ω N | f (x 1 ) -f (x N )| p dµ (ε) q (x 1 ) . . . dµ (ε) q (x N );
on one hand, up to a constant this is simply (2.20). One the other hand (2.30) is bounded from above by

(2.31)

C ∑ q Ω 1 ו••×Ω N | f (x q ) -f (x q 1 )| p dµ (ε) q (x 1 ) . . . dµ (ε) q (x N )
The last observation is that each term of the sum (2.31) is of type (2.21) -here we are using the properties (1)-( 3) and (a), (b) above-for which we have already shown the desired Poincaré inequality. This ends the proof of the case B = 0.

We will not detail the proof of the general statement, because the arguments are identical to the ones already given. The only change is that we will work with geodesics with respect to the model conic metric

(2.32) √ -1 ∑ α∈J dz α ∧ dz α (ε 2 + |z α | 2 ) b α + √ -1 ∑ α ∈J dz α ∧ dz α instead of straight lines (1 -t)x + ty.
The same proof works because the Ricci curvature of the metric (2.32) is bounded from below by some constant independent of ε. For a complete treatment of this point we refer to [START_REF] Saloff | COSTE -Aspects of Sobolev-type inequalities[END_REF], pages 177-179.

Lie derivative of fiberwise

Monge-Ampère Equations. -In this subsection we consider the restriction of our initial family p to a generic disk contained in the base, together with a family of Monge-Ampère equations of its fibers. Let D ⊂ Y be a one-dimensional germ of submanifold contained in a coordinate set of Y, and let X := p -1 (D) (notations as in Theorem C). The resulting map p : X → D will be a proper submersion, provided that D is generic. We recall that the total space (X , ω) of p is a Kähler manifold. We denote by t a coordinate on the unit disk D, and let

(2.33) v = ∂ ∂t + v α ∂ ∂z α
be the local expression of a smooth vector field which projects into ∂ ∂t .

Another piece of data is the following fiberwise Monge-Ampère equation

(2.34) (ω + dd c ϕ) n = e λϕ+ f ω n
on each X t . Here λ 0 is a positive real number, and f is a smooth function on X . We can write this globally as follows

(2.35) (ω + dd c ϕ) n ∧ √ -1dt ∧ dt = e λϕ+ f ω n ∧ √ -1dt ∧ dt on X
, where the meaning of dd c and of ϕ is not the same as in (2.34), but... We take the Lie derivative L v of (2.35) with respect to the vector field v, and then restrict to a fiber X t . The Lie derivative of the left-hand side term of (2.35) equals (2.36)

nL v (ω + dd c ϕ) ∧ (ω + dd c ϕ) n-1 ∧ √ -1dt ∧ dt because we have L v √ -1dt ∧ dt = 0,
given the expression (2.33). The form ω + dd c ϕ is closed, hence by Cartan formula we have

(2.37) L v (ω + dd c ϕ) = d i v • (ω + dd c ϕ)
where i v • ω is the contraction of ω with respect to the vector field v. We evaluate next the quantity

(2.38) d i v • dd c ϕ ∧ √ -1dt ∧ dt
by a point-wise computation, as follows. With respect to the local coordinates as in (2.33), we write

(2.39)

dd c ϕ = ϕ tt √ -1dt ∧ dt + ϕ tα √ -1dt ∧ dz α + ϕ βt √ -1dz β ∧ dt + ϕ βα √ -1dz β ∧ dz α ;
in the expression above we are using the Einstein convention. Then we have (2.40)

d i v • dd c ϕ ≡ ϕ tβα + ϕ γβα v γ + ϕ γα v γ β dz β ∧ dz α
where ≡ means that we are only consider the terms of (1,1)-type which do not contain dt or its conjugate.

On the other hand, the coefficients of the Hessian of the function

(2.41) v(ϕ) = ϕ t + ϕ γ v γ
in the fibers direction are equal to

(2.42) v(ϕ) βα = ϕ tβα + ϕ γβα v γ + ϕ γα v γ β + ϕ γβ v γ α + ϕ γ v γ βα .
The first three terms in the expression (2.42) are identical to those in (2.40). As for the last two terms, they can be expressed intrinsically as follows

(2.43) (ϕ γβ v γ α + ϕ γ v γ βα )dz β ∧ dz α = ∂ ∂v • ϕ .
Here ∂v is a (0, 1)-form with values in T X t and then ∂v • ϕ is a form of (0,1) type on X t .

On the other hand, if we denote by ∆ ϕ = Tr ϕ √ -1∂ ∂ the Laplace operator corresponding to the metric ω ϕ := ω + dd c ϕ on the fibers of p, then we can rewrite the equation (2.37) as follows (2.44)

∆ ϕ v(ϕ) -Tr ϕ ∂ ∂v • ϕ + Ψ ϕ,v ω n ϕ ∧ √ -1dt ∧ dt.
In the expression (2.44) we denote by Tr ϕ the trace with respect to ω ϕ on X t , and we denote by Ψ ϕ,v the function on X such that the equality

(2.45) Ψ ϕ,v ω n ϕ ∧ √ -1dt ∧ dt = L v (ω) ∧ ω n-1 ϕ ∧ √ -1dt ∧ dt holds on X .
As for the right hand side of (2.35), the expression of the Lie derivative reads as follows

(2.46) (λv(ϕ) + v( f ) + Ψ v ) ω n ϕ ∧ √ -1dt ∧ dt
where -as before-the function Ψ v is defined by the equality (2.47)

Ψ v ω n ∧ √ -1dt ∧ dt = L v (ω) ∧ ω n-1 ∧ √ -1dt ∧ dt.
In conclusion, for each t ∈ D we obtain the equality (2.48)

∆ ϕ v(ϕ) -Tr ϕ ∂ ∂v • ϕ + Ψ ϕ,v = λv(ϕ) + v( f ) + Ψ v
which is the identity we intended to obtain in this subsection.

Regularity in transverse directions.

-In this section we will apply the results above in order to analyze the transversal regularity of the solution of the equation

(2.49) (ω + dd c ϕ t ) n = e λϕ+ f ∏ i∈I |σ i | 2e i ∏ j∈J |s j | 2b j ω n on X t .
Here λ 0 is a positive real, and the parameters e i , b j are chosen as above. In case we have λ = 0, the normalization we choose for the solution is

(2.50)

X t ϕ t ω n ϕ t = 0.
The function f in (2.49) is supposed to be smooth on the total space X . We consider the family of approximations of (2.49)

(2.51) (ω + dd c ϕ ε ) n = e λϕ ε + f ∏ i∈I (ε 2 + |σ i | 2 ) e i ∏ j∈J (ε 2 + |s j | 2 ) b j ω n
on X t . By general results in MA theory, the function ϕ ε obtained by glueing the fiberwise solutions of (2.51) is smooth. We will analyze in the next subsections the uniformity with respect to ε of several norms of ϕ ε . We recall the following important result whose origins can be found in [START_REF]On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I[END_REF].

Theorem 2.3. -For any strictly smaller disk D ⊂ D there exists a constant C > 0 such that we have

(2.52) ϕ ε C 1 (X t ) C
for all t ∈ D , where the C 1 norm above is with respect to a fixed metric which is quasi-isometric to (2.32).

If b j = 0, this is a consequence of [START_REF]On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I[END_REF], cf. also the version established in [START_REF] Ȃun | Regularity properties of the degenerate Monge-Ampère equations on compact Kähler manifolds[END_REF], stating that (2.53) ω + dd c ϕ ε Cω| X t . The conic case is much more involved and we refer to Theorem 2.7 and the few lines following that statement, on page 32. Note that inequality (2.53) is still true provided that we replace the RHS with Cω B,ε | X t , where ω B,ε is the regularization of a conic metric corresponding to (X, B) which is quasi-isometric with (2.32). During the rest of the current subsection we assume that λ = 0, which is anyway what we need for the proof of Theorem C. We will explain along the way how to adapt our method to the case λ > 0.

2.3.1.

Mean value of the t-derivative. -Let v be a smooth vector field on X of (1,0)-type, which has the following properties. |v α (z j )| C|z α j | (we use the notations/conventions as in (2.2)) for all α = 1, . . . , d. This means that v is a smooth section of the logarithmic tangent space of (X, E red + B red ).

(i)

Such a vector field v is easy to construct, by a partition of unit of local lifts of ∂ ∂t . We consider the coordinate sets Ω j and the z j adapted to the pair (X , B + E). Then the particular form of the transition implies (ii).

In this context we have the following statement.

Lemma 2.4. -There exists a constant C > 0 independent of ε such that we have (2.54)

X t v(ϕ ε )ω n ϕ ε C for any t ∈ D .
Proof. -We consider a covering of X by coordinate sets (U i , (z i , t)) i where the last coordinate t is given by the map p. The normalization condition (2.50) can be written as (2.55)

∑ i z i <1 θ i (z i , t)ϕ ε (z i , t) ∏ α∈I ε 2 + |z α i | 2 e φ α (z i ,t) e α ∏ β∈J ε 2 + |z β i | 2 e ψ β (z i ,t) b β e F i (z i ,t) dλ(z i )
where θ i is a partition of unit, I ∩ J = ∅ and e F i (z i ,t) dλ(z i ) is the volume element ω n restricted to X t . We take the t-derivative of (2.55) and we have (2.56)

∑ i z i <1 θ i (z i , t) ∂ϕ ε (z i , t) ∂t ∏ α∈I ε 2 + |z α i | 2 e φ α (z i ,t) e α ∏ β∈J ε 2 + |z β i | 2 e ψ β (z i ,t) b β e F i (z i ,t) dλ(z i ) = O(1)
where O(1) above is uniform with respect to t, ε by the C 0 estimates for ϕ ε . Now by the construction of the vector v above the LHS of (2.56) is precisely (2.54), so the lemma follows.

L

∆ ϕ ε τ -Tr ϕ ε ∂ ∂v • ϕ ε + Ψ ϕ ε ,v = λτ + v( f ) + ∑ j e j v log(ε 2 + |σ j | 2 ) -∑ i b i v log(ε 2 + |s i | 2 ) + Ψ v .
The equality (2.58) will be used in order to establish the following statement.

Proposition 2.5. -There exists a constant C > 0 such that we have (2.59)

X t |∇ ε τ| 2 ε ω n ϕ ε C 1 + X t |τ| ω n ϕ ε
for any ε > 0. The operator ∇ ε is the gradient corresponding to the metric ω ϕ ε .

Proof. -In order to establish (2.59) we multiply the relation (2.58) with τ and then we integrate the result on X t against the measure ω n ϕ ε . A few observations are in order. • We have (2.60) sup

X t |v( f )| + ∑ j e j v log(ε 2 + |σ j | 2 ) + ∑ i b i v log(ε 2 + |s i | 2 ) + |Ψ v | C
uniformly with respect to ε, by the property (ii) of the vector field v and the definition (2.47) of the function Ψ v .

• Since the constant λ is positive, the L 2 norm of √ λτ will be on the left-hand side part of (2.59), hence the presence of a strictly positive λ would reinforce the inequality we want to obtain.

The terms (2.60)

Tr ϕ ε ∂ ∂v • ϕ ε , Ψ ϕ ε ,v
are kind of troublesome, because we do not have a L ∞ bound for them. Nevertheless, we recall that we only intend to establish an inequality between L p norms, and we will use integration by parts to deal with (2.60).

For the first term in (2.60) we argue as follows: integration by parts gives (2.61)

X t τ∂ ∂v • ϕ ε ∧ ω n-1 ϕ ε = - X t ∂τ ∧ ∂v • ϕ ε ∧ ω n-1 ϕ ε
and then we use Cauchy-Schwarz: the L 2 norm of ∂τ is what we are after, but on the right hand side term we have it squared. The L 2 norm of ∂v • ϕ ε is completely under control, because it only involves the fiber-direction derivatives of ϕ ε .

The second term is tamed in a similar manner. By definition of Ψ ϕ ε ,v we have (2.62)

X t τΨ ϕ ε ,v ω n ϕ ε = X t τL v (ω) ∧ ω n-1 ϕ ε
and by Cartan formula this is equal to (2.63)

X t τd(i v • ω) ∧ ω n-1 ϕ ε = X t τ∂(i v • ω) ∧ ω n-1 ϕ ε .
By Stokes formula the term (2.63) is equal to (2.64)

X t ∂τ ∧ (i v • ω) ∧ ω n-1 ϕ ε
and now things are getting much better, in the sense that the (0, 1)-form i v • ω is clearly smooth, so its L 2 norm with respect to ω ϕ ε is dominated by

C X t ω ∧ ω n-1 ϕ ε
C and we use the Cauchy-Schwarz inequality.

All in all, we infer the existence of two constants C 1 and C 2 such that we have (2.65)

X t |∇ ε τ| 2 ω n ϕ ε C 1 X t |τ|ω n ϕ ε + C 2 X t |∇ ε τ| 2 ω n ϕ ε 1/2
for any positive ε > 0. The inequality (2.59) follows.

We infer the following statement.

X t |τ|dµ (ε) e C 1 + X t |∇τ|dµ (ε) e/2 .
On the other hand we have

X t |∇τ|dµ (ε) e/2 C X t |∇ ε τ| ε dµ (ε) e/2 C X t |∇ ε τ| 2 ε dµ (ε) e C + C X t |τ|dµ (ε) e 1/2
where we have used Proposition 2.5 for the last inequality. When combined with (2.69), this implies (2.70)

X t |τ|dµ (ε) e C
for any ε > 0.

We define next the sequence of rational numbers (2.71) p 1 = 1, p k+1 := 2np k 2np k as well as the sequence (2.72) q 1 = e, q k+1 := 2 2p k q k .

One can actually find a closed formula for p k = 2n 2n-k+1 holding for 1 k 2n. It also follows that p k < 2 as long as 1 k n which is thus the range of integers for which q k+1 is defined; one can also check the formula q k+1 = (2n)!(n-k)! n!(2n-k)! • q. In particular q n+1 = (2n)! n! 2 • q. This is the factor N in the statement of the proposition.

We observe that for k = 1, . . . n the components of q k are positive rational numbers, greater than the respective components of q. The Sobolev inequality 2.1 gives (2.73)

X t |τ| p k+1 dµ (ε) q k+1 1 p k+1 C X |∇ ε τ| p k ε dµ (ε) q k + X t |τ| p k dµ (ε) q k 1 p k
We iterate (2.64) for k = 1, . . . n and the Proposition 2.6 is proved by observing that the following holds. • For each k = 1, . . . n we have (2.75)

X t |∇ ε τ| p k ε dµ (ε) q k C X t |∇ ε τ| 2 ε dµ (ε) 2 p k q k p k /2 C
where the first inequality is simply Cauchy-Schwarz, and the second one is due to the fact that we have

(2.76) dµ (ε)

2 p k q k Cω n ϕ ε because q k p k q
. This last inequality follows by induction given that

q k+1 p k+1 = 2n-p k 2n-np k • q k p k .

2.4.

A gradient estimate in the conic case. -Theorem 2.7. -Let (X, ω) be a compact Kähler manifold, and let ω ϕ := ω + dd c ϕ be a Kähler metric satisfying ω n ϕ = e λϕ+F ω n for some F ∈ C ∞ (X) and λ ∈ R. We assume that there exists C > 0 and a smooth function Ψ, Φ such that:

(i) sup X |ϕ| C (ii) sup X |Ψ| C and for any δ > 0, there exists C δ such that a. dd c Ψ δ -1 dΨ ∧ d c Ψ -C δ ω b. ∆ ω Ψ δ -1 |∇F| ω -C δ (iii) iΘ ω (T X ) -(Cω + dd c Ψ) ⊗ Id (iv) ω ϕ Cω
Then there exists a constant A > 0 depending only on C and n such that |∇ϕ| ω C.

As a corollary of this result, the gradient estimate (2.52) in Theorem 2.3 holds.

Proof of Theorem 2.3. -Let us rewrite the equation (2.51) as

(ω ε + dd c u ε ) n = e λu ε + f ε ∏ i∈I (ε 2 + |σ i | 2 ) e i ω n ε
where the reference metric ω ε ∈ {ω} is an approximate conic metric along the divisor B, and u ε differs from ϕ ε by a function whose L ∞ norm as well as gradient and complex Hessian are uniformly bounded with respect to ω ε . Therefore it is sufficient to establish (2.52) for u ε . We check successively that conditions (i) -(iv) are satisfied.

The bound (i) follows from Kołodziej's estimate. It is straighforward when λ = 0, and when λ > 0, it requires an additional step easily achieved with Jensen inequality. Next, we choose

Ψ ε := C(∑ i (|σ i | 2 + ε 2 ) ρ + ∑ j (|s j | 2 + ε 2 ) ρ )
for C large enough and ρ > 0 small enough. Condition (ii).a can be checked independently for each summand of Ψ α ε of Ψ ε in which case if follows from the fact that Ψ α ε is uniformly quasi-psh (hence Cω ε -psh). Condition (ii).b is an easy computation combined with [GP16, Sect. 5.2]. Condition (iii) is showed in [GP16, Sect. 4], while (iv) is the content of [GP16, Prop. 1]. To be more precise, op. cit. assumes an upper and lower bound on f ε + ∑ e i log(|σ i | 2 + ε 2 ) in order to get a two-sided inequality for ω ϕ ; however one only needs an upper bound for the previous quantity if one only wishes to prove the onesided inequality (iv).

Proof of Theorem 2.7. -Let β := |∇ϕ| 2 (computed with respect to ω) and α := log βγ • ϕ where γ is a function to specify later. Without loss of generality, one can assume inf ϕ = 0, and we set sup ϕ =: C 0 . We use the local notation (g i j) for ω. We work at a point y ∈ X where α + 2Ψ attains its maximum, and we choose a system of geodesic coordinates for ω such that g i j(y) = δ i j, dg i j(y) = 0, and ϕ i j is diagonal. We set u i j = g i j + ϕ i j the components of the metric

ω ϕ . As α p = β p β -γ • ϕ ϕ p and α p (y) = -2Ψ p (y), one has (2.77) β p β (y) = (γ • ϕ(y))ϕ p (y) -2Ψ p (y)
Moreover, some computations show that

α p p = 1 β R j kp p ϕ j ϕ¯k + 2Re ∑ j u p pj ϕ¯j + ∑ j |ϕ jp | 2 + ϕ 2 p p - |β p | 2 β 2 -2λ -γ |ϕ p | 2 -γ ϕ p p
Therefore at y, one gets from (2.77) the following inequality:

α p p 1 β R j kp p ϕ j ϕ¯k + 2Re ∑ j u p pj ϕ¯j + ∑ j |ϕ jp | 2 + ϕ 2 p p - (2.78) 2λ -γ |ϕ p | 2 -γ ϕ p p -γ ϕ p -2Ψ p 2 so at y, the RHS is non-positive.
Step 1. The curvature term By the assumption (iii), we have for all a, b: R j kp q a j āk b p bq -(C|a j | 2 + Ψ j k a j āk )|b| 2 and by symmetry of the curvature tensor, we get R j kp q a j āk b p bq (γ -C)tr ω ϕ ωγ |∇ ω ϕ| 2 ω ϕnγ + ∆ Ψ -C|∇Ψ| 2 ω ϕ -2|∇F| ω -C As Ψ is quasi-psh and ω ϕ Cω, we have ∆ Ψ C -1 ∆Ψ -Ctr ω ϕ ω so by (ii).b, ∆ Ψ 4|∇F| ω -C(1 + tr ω ϕ ω). Using (ii).a, one ends up with the following inequality at y:

(γ -C)tr ω ϕ ω -γ |∇ ω ϕ| 2 ω ϕ -nγ C
Choosing γ(t) = (C + 1)t -||ϕ|| -1 ∞ t 2 enables to conclude just as in [START_REF] Błocki | A gradient estimate in the Calabi-Yau theorem[END_REF]. Proof of Theorem C. -It is a combination of our preceding considerations. The equation which gives ω KE fiberwise is of the same type as (2.49) (with λ = 0). We conclude by Theorem 2.3 and Theorem 2.6.

Existence of non-semipositive relative Ricci-flat Kähler metrics

Let p : X → Y be a holomorphic fibration between projective manifolds of relative dimension n 1. Let Y • be the set of regular values, and let X • := p -1 (Y • ). We assume that for y ∈ Y • , c 1 (K X y ) = 0, where X y := p -1 (y). Let L be a pseudoeffective, p-ample Q-line bundle on X. One can write L = H + p * M for some ample line bundle H on X and for some line bundle M on Y. In particular, one can find a smooth (1, 1)-form ω ∈ c 1 (L) on X such that for any y ∈ Y • , ω y := ω| X y is a Kähler form on X y . By Yau's theorem, there exists for any y ∈ Y • a unique function ϕ y ∈ C ∞ (X y ) such that:

(i) θ y := ω y + dd c ϕ y is a Kähler form (ii) X y ϕ y ω n y = 0 (iii) Ric θ y = -dd c log ω n y = 0 Moreover, one can use the implicit function theorem to check that the dependence of ϕ y in y is smooth, so that the form θ := ω + dd c ϕ is a well-defined smooth (1, 1)-form on X • which is relatively Kähler. It is a folklore conjecture that the form θ is semipositive on X, say when L is globally ample. Building on the results in the Appendix on page 37, we are able disprove this conjecture.

Theorem 3.1. -There exists a projective fibration p : X → Y as in the setting above and an ample line bundle L on X such that the relative Ricci-flat metric θ on X • associated with L is not semipositive.

Remark 3.2. -The counter-example is actually pretty explicit: X is a K3 surface and p is an elliptic fibration onto Y = P 1 .

Proof of Theorem 3.1. -We proceed in three steps, arguing by contradiction. That is, we assume that the folklore conjecture recalled above is true for any such fibration p : X → Y.

Step 1. Choice of the fibration. We consider a K3 surface X provided by Proposition A.3. Its (singular) fibers are irreducible and reduced. Moreover, X admits a semi-ample line bundle L which is p-ample and has numerical dimension one. Indeed, L can be chosen as the pull-back of O P 1 (1) by another elliptic fibration q : X → P 1 . Moreover, one knows that p is not isotrivial, in the sense that two general fibers X y , X y of p are not isomorphic.

Step 2. Reduction to the semi-ample case. Let us pick A an ample line bundle on X, ω A ∈ c 1 (A) a Kähler form, and let us consider the relative Ricci-flat form θ ε on X • associated with the the pair (L + εA, ω + εω A ). The line bundle L ε is ample, hence it follows from our assumption that for any ε > 0, the relative Ricci-flat metric satisfies θ ε 0 on X • . We are going to show that θ ε converges weakly on X • to the current θ := θ 0 . As a result, this will force θ to be semipositive on X • .

Let us write θ ε = ω + εω A + dd c ϕ ε where ϕ ε is normalized such that for each y ∈ Y • , one has

X y ϕ ε (ω + εω A ) = 0.
If C ε is the constant (converging to 0) defined by

e C ε = [X y ]• c 1 (L) [X y ]• c 1 (L + εA)
for any y ∈ Y • , then one has on X y the following equation:

ω + εω A + dd c ϕ ε = e C ε • (ω + dd c ϕ)
The family of potentials (ϕ ε | X y ) ε,y is normalized in a smooth way with respect to ε and y, and satisfies linear equations depending smoothly on the parameters as well. It is not difficult to see that the standard estimates hold uniformly in ε and y (as long as y evolves in compact subsets of Y • ), hence uniqueness imposes that ϕ ε → ϕ smoothly in each X y , locally uniformly in y ∈ Y • . In particular, ϕ ε converges weakly to ϕ in L 1 loc (X • ).

Step 3. End of the proof.

Thanks to

Step 2, the relative Ricci-flat metric θ = ω + dd c ϕ is semipositive on X • . Moreover, it follows from Proposition A.1 that ϕ is bounded above near X \ X • , hence θ extends to a semipositive current θ ∈ c 1 (L) on the whole X. Let F ⊂ T X be the holomorphic foliation induced by the fibration q : X → P 1 . As the semi-positive current θ is in the class of c 1 (L) = q * (c 1 (O P 1 (1))) and q has connected fiber, it follows that there exists a positive current γ ∈ c 1 (O P 1 (1)) such that θ = q * γ. In particular, if X 1 ⊂ X denotes the locus where q is smooth and if Ω := X • ∩ X 1 , then F | Ω is contained in the kernel Ker θ on Ω. As both foliations are smooth and have rank one on Ω, one has One knows that in a trivializing chart (z, t) defined on a subset of p -1 (U) such that p(z, t) = t, the vector field V can be written as

V = ∂ ∂t
+ a(z, t) ∂ ∂z for some smooth function a. The function c := θ(V, V) satisfies the identity θ 2 = c θ ∧ idt ∧ d t, hence it vanishes identically on p -1 (U), that is, V ∈ C ∞ (p -1 (U), Ker θ).

Thanks to (3.1), this shows that for any x ∈ p -1 (U) ∩ Ω, one has C• V(x) = F x . In particular, there exists a non-vanishing, smooth function f on p -1 (U) ∩ Ω such that f V is holomorphic on p -1 (U) ∩ Ω. Now in local coordinates, this means that

0 = ∂( f V) = ∂ f ⊗ ∂ ∂t + ∂( f a) ⊗ ∂ ∂z hence ∂ f = 0.
As a result, the smooth vector field V on p -1 (U) is holomorphic on p -1 (U) ∩ Ω, hence on the whole p -1 (U). Therefore, its flow induces a local biholomorphism between any two near fibers. In particular, any two smooth fibers over U would be isomorphic, which contradicts the non-isotriviality of p.

  be the positive current constructed in Theorem 1.1, and let y ∈ Y Y • .

(3. 1 )

 1 F | Ω = Ker θ| Ω . Next, let us pick a trivializing open set U ∆ ⊂ Y • , and let V ∈ C ∞ (X • , T 1,0 X ) be the lift of ∂ ∂t with respect to θ over U, cf e.g. [Gue20, Sect. 1.1].

  Cω B,ε on each X t for some constant C which is uniform with respect to ε and with respect to t ∈ D . On the RHS of (2.67) we have ω B,ε which stands for any metric quasi-isometric with (2.32). In particular, for any function f we have(2.68) |∇ f | C|∇ ε f | εwhere the symbols | • |, ∇ and | • | ε , ∇ ε correspond to the metric ω B,ε and ω ε respectively.

	Now, Poincaré inequality 2.2 applied for α = 1 combined with Lemma 2.4 gives
	(2.69)
	Theorem 2.6. -There exists a positive integer N ∈ Z + and a positive constant C such that we have
	(2.66)	X t	|τ| 2 dµ	(ε) Ne	C
	for every positive ε.

Proof. -The arguments which will follow are absolutely standard, by combining the Sobolev and Poincaré inequalities with (2.59). Prior to this, we recall that we have (2.67) ω ε

  -(C|b p | 2 + Ψ p qb p bq )|a| 2 . We apply that to a = ∇ϕ and b the vector with only non-zero component the p-th one, equal to √ u p p, we get: u p p R j kp p ϕ k ϕ¯l -(Cu p p + u p pΨ p p)|∇ ϕ| 2 . As a consequence, (α + Ψ)(γ -C)tr ω ϕ ω + ∑ |ϕ jp | 2γ ϕ p -2Ψ p 2 -(2.82) γ |∇ ω ϕ| 2 ω ϕnγ -2|∇F| -CStep 3. Using the second derivatives Recall that β p = ∑ j ϕ jp ϕ¯j + ϕ p (u p p -1). At y,β p βγ ϕ p = -2Ψp so that at this point, one has∑ j ϕ jp ϕ¯j = (γ β + 1u p p) ϕ p -2βΨ p hence ∑ j ϕ jp ϕ¯j = β (γ ϕ p -2Ψ p ) + β -1 (1u p p) ϕ p . By Schwarz inequality, ∑ j ϕ jp ϕ¯j 2 β ∑ j |ϕ jp | 2 and therefore 1 | 2γ ϕ p -2Ψ p 2 (γ ϕ p -2Ψ p ) + β -1 (1u p p) ϕ p 2 γ ϕ p -2Ψ p 2 -2β -1 |1u p p|• γ ϕ p -2Ψ p • |ϕ p |and by (iv), |1u p p| C, so that we get:

						p	u p p 1 β ∑
	∑ p	u p p 1 β ∑
	(2.79)	1					u p pΨ p	p
							p
	Therefore, Equation (2.78) becomes, at y ∈ X:
	∆ (α + Ψ)	(γ -C)tr ω ϕ ω +	1 β ∑ p	u p p 2Re ∑
	(2.80)						u p p γ ϕ p -2Ψ p	2 -C
							p
	Step 2. The gradient term
	The next term to analyze is
	(2.81)		1 β ∑ p	u p p 2Re ∑ j	u p pj ϕ¯j =	2 β	Re ∑

β ∑ u p p R j kp p ϕ j ϕ¯k -C ∑ u p p -∑ j u p pj ϕ¯j + ∑ j |ϕ jp | 2γ |∇ ω ϕ| 2 ω ϕ -nγ -∑ j F j ϕ¯j

by

[START_REF] Błocki | A gradient estimate in the Calabi-Yau theorem[END_REF] 1.13]

, and this term is dominated (in norm) by 2|∇F|β -1/2 and at the point y, β can always be assumed to be larger than 1 so that our term is bigger that -2|∇F|. In particular, one gets at y:

∆ j β ∑ j |ϕ jp j |ϕ jp | 2γ ϕ p -2Ψ p 2 -C(tr ω ϕ ω + |∇Ψ| 2 ω ϕ )

Combining this last inequality with (2.82), we get at y:

0 ∆ (α + 2Ψ)

Acknowledgments. -We would like to thank Sébastien Boucksom, Tristan Collins, Vincent Guedj, Christian Schnell, Song Sun, Valentino Tosatti and Botong Wang for numerous useful

Appendix by Valentino Tosatti

Let (X n , ω X ) be a compact Kähler manifold, Y a compact Riemann surface, and f : X → Y a surjective holomorphic map with connected fibers. Let Y 0 be the locus of regular values for f , whose complement in Y is a finite set, and X 0 = f -1 (Y 0 ), which is Zariski open in X, so that f : X 0 → Y 0 is a proper holomorphic submersion. We will call the fibers over points in Y\Y 0 the singular fibers of f . Suppose that for every y ∈ Y 0 we have a smooth function ρ y on the fiber X y = f -1 (y) which satisfies

-If all the singular fibers of f are reduced and irreducible, then there is a constant C such that sup

holds for all y ∈ Y 0 .

Proof. -Let ω y = ω X | X y , and g y be its Riemannian metric, where in the following we fix any y ∈ Y 0 . Thanks to (3.2), on X y we have

We have that Vol(X y , g y ) = c, a constant independent of y, and that the Sobolev constant of (X y , g y ) has a uniform upper bound independent of y thanks to the Michael-Simon Sobolev inequality [START_REF] Michael | Sobolev and mean-value inequalities on generalized submanifolds of R n[END_REF], see the details e.g. in [Tos10, Lemma 3.2]. Furthermore, diam(X y , g y ) C, a constant independent of y, thanks to [Tos10, Lemma 3.3]. So far we have not used the assumptions that all singular fibers are reduced and irreducible. This is used now to prove that the Poincaré constant of (X y , g y ) also has a uniform upper bound independent of y, as shown by Yoshikawa [START_REF] Yoshikawa | Degeneration of algebraic manifolds and the spectrum of Laplacian[END_REF] (see also the much clearer exposition in [RZ11, Proposition 3.2]).

At this point we can use a classical argument of Cheng-Li [START_REF] Cheng | Heat kernel estimates and lower bound of eigenvalues[END_REF], which is clearly explained in [Siu87, Chapter 3, Appendix A, pp.137-140], to deduce that the Green's function G y (x, x ) of (X y , g y ), normalized by

for all y ∈ Y 0 and for all x, x ∈ X y , with a uniform constant A. The point of that argument is that A only depends on the constant in the Sobolev-Poincaré inequality, that here as we said we control uniformly, on the dimension and on bounds for the volume and diameter, which we all have. We can now apply Green's formula on X y . Choose a point x ∈ X y such that ρ y (x) = sup X y ρ y , and then, using that ρ y has average zero, together with (3.3) and (3.4), we obtain

(n -1)AVol(X y , g y ).

We now specialize to the setting where X is a K3 surface, Y = P 1 and f : X → P 1 is an elliptic fibration. We further assume that ρ y is chosen so that ω X | X y + √ -1∂ ∂ρ y > 0 is the unique flat metric on X y cohomologous to ω X | X y (and we still assume that ρ y has fiberwise average zero). In this case ρ y varies smoothly in y ∈ Y 0 , and so it defines a smooth function ρ on X 0 . Thanks to Proposition 1.44, we conclude that sup

This, together with the Grauert-Remmert extension theorem, immediately gives:

Corollary A.2. -In this setting, if we have that ω X + √ -1∂ ∂ρ 0 on X 0 , then this extends to a closed positive current on all of X, in the class [ω X ].

Lastly, we need the following examples:

Proposition A.3. -There exists a complex projective K3 surface X which admits two elliptic fibrations, one of which is non-isotrivial and has only reduced and irreducible singular fibers.

Proof. -Let X ⊂ P 2 × P 1 be a general hypersurface of degree (3, 2). It is known that X has Picard number 2 [vG05, Section 5.8]. The projection to the P 1 factor gives an elliptic fibration on X, which is clearly not isotrivial provided X is general.

To obtain the other fibration we compose the first fibration with the automorphism σ of X obtained as follows. Projecting X to the P 2 factor shows that X is a double cover of P 2 ramified along a sextic, and the covering involution of this cover is the σ that we want.

Explicitly, if we let L = O P 2 (1)| X , M = O P 1 (1)| X , the the first elliptic fibration is defined by |M| and the second elliptic fibration by |3L -M| (since σ * M = 3L -M).

Lastly, we show that every elliptic fibration on X has only reduced and irreducible singular fibers. Given an elliptic fibration f : X → P 1 , let j : J → P 1 be its Jacobian family [START_REF] Huybrechts | Lectures on K3 surfaces[END_REF]Section 11.4]. Then J is also an elliptic K3 surface, every fiber of j is isomorphic to the corresponding fiber of f , J has the same Picard number as X, but j always has a section. We can then apply the Shioda-Tate formula [START_REF] Huybrechts | Lectures on K3 surfaces[END_REF]Corollary 11.3.4] to j to obtain 2 = ρ(J) = 2 + ∑ t∈P 1 (r t -1) + rank MW(j), where r t is the number of irreducible components of the fiber J t and MW(j) is the Mordell-Weil group of j. In particular we conclude that r t = 1 for all t, i.e. all fibers of j (and therefore all fibers of f ) are irreducible. Lastly, all fibers of f are reduced by [Huy16, Proposition 3.1.6 (iii)].