Vincent Guedj

Henri Guenancia

Ahmed Zeriahi

Diameter of Kähler currents

Keywords: Kähler manifolds, Monge-Ampère equations, singular Kähler-Einstein metrics, pluripotential theory

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Introduction

The classical Bonnet-Myers theorem ensures that a complete n-dimensional Riemannian manifold (M, g) with positive Ricci curvature Ric(g) ≥ (n -1)K, K > 0, has finite diameter diam(M, g) ≤ π √ K .

This estimate breaks down if the Ricci curvature takes zero or negative values, and one then needs to add extra assumptions on the manifold. Several works have been devoted to establish diameter upper bounds in the last two decades for Kähler manifolds, in connection with the study of degenerate families of Kähler-Einstein metrics (see notably [Pȃu01, Tos09, RZ11, Li21, GPTW21, GPSS22]). The main purpose of this article is to extend these estimates in several directions.

We start by generalizing works of Fu-Guo-Song [START_REF] Fu | Geometric estimates for complex Monge-Ampère equations[END_REF] and Guo-Song [START_REF] Guo | Local noncollapsing for complex Monge-Ampère equations[END_REF], establishing a diameter upper bound under a Ricci lower bound for any metric whose potential is uniformly bounded, and whose cohomology class belongs to a compact subset of the Kähler cone.

Theorem A. -Let (X, ω X ) be a compact Kähler manifold and let K ⊂ H 1,1 (X, R) be a compact subset of the Kähler cone of X. Fix A, B, C ∈ R and let ω be a Kähler form such that [ω] ∈ K, ϕ ω ∞ ≤ C and Ric(ω) ≥ -Aω -Bω X .

Then diam(X, ω) ≤ D, where D depends only on A, B, C and K.

Besides providing a quite general statement (twisted Ricci curvature lower bound, uniform bound on potentials instead of an Orlicz condition on the density), our proof is completely different than previous ones. It is purely complex analytic, using as an essential tool the semicontinuity properties of complex singularity exponents [START_REF] Demailly | Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds[END_REF] and the resolution of the openness conjecture [START_REF] Guan | A proof of Demailly's strong openness conjecture[END_REF].

Combining our techniques with the recent results of Guo-Phong-Song-Sturm [START_REF] Guo | Diameter estimates in Kähler geometry[END_REF], we can actually extend the result above to the case where K is merely a bounded subset of the Kähler cone such that its closure K is contained in the big cone, i.e. the cone of classes represented by Kähler currents. We refer to Theorem 2.3 for this more general version along with the precise meaning of ϕ ω .

Poincaré type metrics have constant Ricci curvature and infinite diameter; we exhibit well chosen smooth Kähler approximants of the latter in Example 2.5, showing that the uniform bound on ϕ ω ∞ is necessary. On the other hand, K-semistable Fano manifolds which are not K-stable illustrate that one cannot expect a complex proof of Myers theorem along these lines (see Example 2.6).

It is however difficult in general to guarantee a uniform Ricci lower bound on natural approximants of canonical Kähler currents, even when the Ricci curvature of the limiting object is bounded below. We illustrate this principle in Section 5 where we give a systematic treatment of Kähler metrics having one isolated singularity which is U(n, C)-invariant (radial symmetry): we show that the Ricci curvature of U(n, C)-invariant smooth approximants is (almost) never bounded below.

Let us now go back to the recent breakthrough result of Guo-Phong-Song-Sturm [START_REF] Guo | Diameter estimates in Kähler geometry[END_REF] mentioned above. It establishes a uniform upper bound on diameters of Kähler metrics ω, which only involves

• an upper-bound on the coholomogy class of [ω] in H 1,1 (X, R);

• an upper-bound on X f ω (log f ω ) p , where f ω = ω n /dV X and p > n;

• a uniform lower bound f ω ≥ γ, where (γ = 0) has small Hausdorff dimension.

We refer the reader to [GPSS22, Theorem 1.1] for a precise statement which, moreover, contains several extra pieces of information on the Riemannian Green's function associated to ω, as well as a non-collapsing result.

The main result of this article is the following estimate which does not involve any uniform lower bound on f ω , the latter being somehow replaced by the positivity of the cohomology class [ω].

Theorem B. -Let X be a compact Kähler manifold of complex dimension n and let H ⊂ C ∞ (X, Ω 1,1 X ) be the set of Kähler forms on X. Let dV X be a smooth volume form and let K ⊂ H 1,1 (X, R) be a compact subset of the Kähler cone of X. Given ω ∈ H, we set f ω = ω n /dV X and consider, for A > 0 and p > 2n fixed,

H A,p,K := {ω ∈ H; [ω] ∈ K and X f ω | log f ω | n (log • log( f ω + 3)) p dV X ≤ A}.
Then there exists a uniform constant C = C(K, dV X , A, p) > 0 such that for all ω ∈ H A,p,K , diam(X, ω) ≤ C. More precisely, given any γ < p -2n, there exists a constant C = C(K, dV X , A, p, γ) > 0 such that for any ω ∈ K A,p and any two points x, y ∈ X, we have

d ω (x, y) ≤ C (log | log d ω X (x, y)|) γ 2 .
We provide several explicit radial examples in Section 5 (more precisely the third example there), showing that the assumptions made here on the density f ω are close to being sharp .

All these results rely on fine continuity properties of Monge-Ampère potentials. If f ω satisfies Condition (K) -introduced by Kolodziej in [START_REF] Kołodziej | The complex Monge-Ampère operator[END_REF]-, then its Monge-Ampère potential is uniformly bounded. Extending results of several authors [Koł98, DDG + 14, GPTW21], we show in Theorem 1.6 that one can obtain a precise control on its modulus of continuity which is, moreover, uniform with respect to the cohomology class [ω] (see Remark 1.7). The diameter estimate in Theorem B is thus a particular case of the following general estimate.

Theorem C. -Let (X, ω X ) be a compact Kähler manifold and let E be a divisor with simple normal crossings. Let X • := X \ E and let T = θ + dd c ϕ be a closed positive (1, 1)-current, where θ is smooth. We assume that ω := T| X • is a Kähler form and that the modulus of continuity m ϕ of ϕ satisfies m ϕ (r) ≤ C (log(log r)) 1+δ for some C, δ > 0. Then we have diam(X • , ω) < +∞.

In connection with the Minimal Model Program, singular Kähler-Einstein metrics ω KE have been constructed in [EGZ09, BBE + 19]. These are Kähler forms on the regular locus V reg of a Kähler variety V, whose local potentials ϕ KE are bounded near the singular locus V sing . Our method applies equally well to this singular context. Indeed we finally show that the finiteness of the diameter of these metrics follows from the conjectural Hölder continuity of their Monge-Ampère potentials and Theorem C. (1) Corollary D. -If the Kähler-Einstein potentials ϕ KE are Hölder continuous on V, then diam(V reg , ω KE ) < +∞.

As we explain in Section 4.3.1, Hölder continuity is the best regularity that makes sense intrinsically in this context. It is known to hold on smooth manifolds [Koł08, DDG + 14], as well as in some singular settings [START_REF] Hein | Calabi-Yau manifolds with isolated conical singularities[END_REF][START_REF] Chiu | Higher regularity for singular Kähler-Einstein metrics[END_REF].

A new difficulty that occurs in the singular case is that we need some coarse control of the distance function near the singularities. This is provided by a generalization of a classical L 2integrability result of Demailly-Peternell-Schneider (see Lemma 4.2). Method of proof and comparison with other works. The starting point of the proof is the the observation made by Y.Li in [START_REF] Li - | On collapsing Calabi-Yau fibrations[END_REF] that the distance function

f (•) = d ω (x, •) is 1-Lipschitz, hence 0 ≤ d f ∧ d c f ≤ ω.
Assuming an L p bound on the densities f ω , Y.Li then uses Hölder regularity of the Monge-Ampère potentials [DDG + 14] to establish a uniform upper bound on diameters (see [START_REF] Li - | On collapsing Calabi-Yau fibrations[END_REF]Corollary 4.2] and Proposition 1.4).

Guo-Phong-Song-Sturm develop in [START_REF] Guo | Diameter estimates in Kähler geometry[END_REF] a systematic study of the fine properties of the Laplace-Green function G ω of ω. Using the representation formula for G ω , the inequality d f ∧ d c f ≤ ω, and ingenious comparisons with various solutions of Monge-Ampère equations associated to ω, they obtain uniform upper bounds on diameters, as well as non-collapsing results. This requires them to impose a uniform non-vanishing condition on the densities f ω ≥ γ, but the range of Kähler forms considered is large.

Our proof follows an analogous path, using a known ambient Monge-Ampère Green function rather than the detailed study of G ω . A drawback is that we need fine estimates on the trace of ω, which require a control on the modulus of continuity of its Monge-Ampère potentials; the latter is only known when the reference cohomology class is Kähler. An advantage is that we can directly deal with singular situations, while it is usually quite difficult to obtain global metric information by approximation.

Acknowledgements. -This work has benefited from State aid managed by the ANR-11-LABX-0040, in connection with the research project HERMETIC, as well as by the ANR projects KARMAPOLIS and PARAPLUI and the Institut Universitaire de France.

Modulus of continuity of Monge-Ampère potentials

In the whole article we let (X, ω X ) denote a compact Kähler manifold of complex dimension n. We set

d = ∂ + ∂ and d c = i 2π (∂ -∂) so that dd c = i π ∂∂.
1.1. Quasi-plurisubharmonic functions. -Recall that a function is quasi-plurisubharmonic if it is locally given as the sum of a smooth and a psh function. Quasi-psh functions ϕ : X → R ∪ {-∞} satisfying ω X + dd c ϕ ≥ 0 in the weak sense of currents are called ω X -psh functions.

Definition 1.1. -We let PSH(X, ω X ) denote the set of all ω-plurisubharmonic functions which are not identically -∞.

The set PSH(X, ω X ) is a closed subset of L 1 (X), for the L 1 -topology.

Demailly has produced various methods of regularization of quasi-psh functions. We recall one, together with precise estimates on the loss of positivity along the smoothing. Consider the exponential mapping with respect to the Riemannian metric induced from ω. It is defined on the tangent space of a given point z ∈ X

exp z : T z X ζ → exp z (ζ) ∈ X, by exp z (ζ) = γ(1)
where γ being the geodesic starting from z with initial velocity γ (0) = ζ. Given any function u ∈ PSH(X, ω X ), we define its τ-regularization

(1.1) ρ τ u(z) = 1 τ 2n ζ∈T z X u(exp z (ζ))ρ |ζ| 2 ω X,z τ 2 dV ω X,z (ζ), τ > 0.
Here ρ is a smoothing kernel, |ζ| 2 ω X,z stands for ∑ n i,j=1 g i j(z)ζ i ζj , and dV ω X,z (ζ) is the induced measure ω n X,z /n!. While ρ τ u already provides a quasi-psh regularization of u, the loss of positivity in dd c ρ τ u is too large for applications. Demailly combines this smoothing with a technique introduced by Kiselman, setting U(z, w) := ρ τ u(z) for w ∈ C, |w| = τ and considering

(1.2) u c,τ (z) := inf 0≤t≤τ U(z, t) + Kt -Kτ -c log t τ .

Lemma 1.2. -[BD12]

There exists K > 0 such that the function U(z, t) + Kt is increasing in t and one has the following estimate,

(1.3) ω X + dd c u c,τ ≥ -(Ac + Kτ) ω X , where A ≥ 0 is such that Θ(T X , ω X ) ≥ -Aω X ⊗ Id T X .
[BD12, Lemma 1.12] claims a slightly finer control, with τ 2 instead of τ. This requires to efficiently control the mixed terms |w||dz dw| in [BD12, Equation (1.10)]. These can be absorbed by Cauchy-Schwarz inequality at the cost of losing one power of τ, which is how [BD12, Equation (1.10)] allows one to obtain (1.3). The latter is sufficient to deal with sub-Lipschitz regularity of Monge-Ampère potentials, as in Section 1.3 to follow. A more precise control has been obtained by E.DiNezza and S.Trapani in [START_REF] Dinezza | The regularity of envelopes[END_REF]. This approach has been extended to more general modulus of continuity, we shall need the following generalization due to Kovats [START_REF] Kovats | Dini-Campanato spaces and applications to nonlinear elliptic equations[END_REF].

Lemma 1.3. -Let (X, ω X ) be a Kähler manifold of complex dimension n. We let g denote the associated Riemannian metric. Let m : R + → R + be an increasing subadditive continuous function such that m(0) = 0 which satisfies the Dini condition m 1 (r) = r 0 m(t) t dt < ∞. Let u : X → R be a measurable function. Assume that for each compact set K ⊂ X there exists C K > 0 such that for all p ∈ K, for all 0 < r < inj(X, g),

B g (p,r) u(x) - 1 VolB g (x, r) B g (x,r) u dV g 2 dV g (x) ≤ C K r 2n m 2 (r).
Then u is continuous and its modulus of continuity satisfies m u (r) = O(m 1 (r)).

Proposition 1.4. -Let (X, ω X ) be a Kähler manifold of complex dimension n ∈ N * . Assume ϕ ∈ PSH(X, ω X ) is continuous in an open set Ω ⊂ X, with modulus of continuity m ϕ which satisfies the condition m 1 (r) := r 0 √ m ϕ (t) t dt < +∞. If ω := ω X + dd c ϕ is a Kähler form in Ω, then for each compact set K ⊂ Ω there exists C K > 0 such that for all p, q ∈ Ω ∩ K, d ω (p, q) ≤ C K m 1 • d ω X (p, q).
We let here d ω denote the Riemannian distance associated to the Kähler form ω.

Proof. -Fix p ∈ K and let ρ : x ∈ Ω → d ω (x, p) ∈ R + denote the distance function with respect to the Kähler form ω = ω X + dd c ϕ. The function ρ is 1-Lipschitz with respect to d ω hence ∇ ω ρ is well defined almost everywhere with |∇ ω ρ| ≤ 1. We infer

|∇ ω X ρ| 2 ≤ Tr ω X (ω)|∇ ω ρ| 2 ≤ n ω ∧ ω n-1 X ω n X . Let 0 ≤ χ ≤ 1 be a cut off function such that χ ≡ 1 in a neighborhood of B r := B ω X (p, r), Supp(χ) ⊂ B 2r := B ω X (p, 2r) and dd c χ ≤ Cr -2 ω X . Note that C = C K can be chosen uniformly with respect to p ∈ K. It follows from Chern-Levine-Nirenberg inequalities that B r |∇ ω X ρ| 2 ω n X ≤ n X χω ∧ ω n-1 X ≤ Cr 2n + n X [ϕ -inf B 2r ϕ]dd c χ ∧ ω n-1 X ≤ C m ϕ (2r)r 2n-2 .
Since m ϕ (2r) ≤ 2m ϕ (r), it follows from Poincaré inequality that

B r ρ(x) - 1 Vol(B r (x)) B r (x) ρ ω n X 2 ω X (x) n ≤ C P r 2 B r |∇ ω X ρ| 2 ω n X ≤ C m ϕ (r)r 2n .
Of course X may not satisfy a global Poincaré inequality but it holds in a neighborhood of K. Thus ρ belongs to a generalized Morrey-Campanato space and Lemma 1.3 applied with m = √ m ϕ ensures that for all p, q ∈ K, d ω (p, q) ≤ Cm 1 • d ω X (p, q). Let ω X be a Kähler form and let µ be a positive Radon measure normalized so that µ(X) = Vol ω X (X). If µ satisfies Condition (K) , it has been proved by Kolodziej (see [Koł98, Theorem 2.5.2]) that there exists a unique continuous ω-psh function such that

Condition (K)

(ω X + dd c ϕ) n = µ = f dV X ,
up to an additive constant, together with a uniform a priori bound on Osc X (ϕ).

We now establish a precise control of the modulus of continuity of the solution ϕ, when the density f satisfies various integrability conditions. Theorem 1.6. -Let (X, ω X ) be a compact Kähler manifold of complex dimension n ∈ N * . Let µ = f dV X be a positive Radon measure which satisfies Condition (K) with weight w. Let ϕ ∈ PSH(X, ω X ) ∩ C 0 (X) be a solution of the Monge-Ampère equation

(ω X + dd c ϕ) n = f dV X .
We let m ϕ denote the modulus of continuity of ϕ and fix ε > 0.

1) If w(t) = t 1+ε then m ϕ (r) ≤ C α r α , for all 0 < α < 2ε n(1+ε)+ε . 2) If w(t) = t(log t) n+ε , then m ϕ (r) ≤ C(-log r) -ε/n . 3) If w(t) = t(log t) n (log • log(t + 3)) n+ε , then m ϕ (r) ≤ C(log(-log r)) -ε/n .
The first item has been established by Kolodziej in [START_REF] Kołodziej | Hölder continuity of solutions to the complex Monge-Ampère equation with the right-hand side in L p : the case of compact Kähler manifolds[END_REF] (see also [DDG + 14, Theorem A]).

The condition 0 r -1 m ϕ (r)dr < +∞ is always satisfied in this case.

The second item has been addressed by Guo-Phong-Tong-Wang in [START_REF] Guo | On the modulus of continuity of solutions to complex Monge-Ampère equations[END_REF], by using a new method introduced in [START_REF] Guo | On L ∞ estimates for complex Monge-Ampère equations[END_REF]. The condition 0 r -1 m ϕ (r)dr < +∞ is satisfied iff ε > 2n, while it is never satisfied in the third case.

Proof. -We provide a unified proof of these results, following the approach and notations of [DDG + 14, Theorem D]. Since 1) is treated there, we first treat 2). For w(t) = t(log t) n+ε , it follows from Hölder-Young inequality that

µ = f dV X ≤ C 0 Cap 1+ε/n ω X .
[EGZ09, Proposition 2.6] shows that if ψ is bounded and ω X -psh, then for all 0 < δ < 1,

(1.4) (ψ -ϕ) + L ∞ ≤ δ + C 0 Cap ω X (ϕ -ψ < -δ) ε/n 2 , where (ψ -ϕ) + := max(ψ -ϕ, 0) denotes the positive part of ψ -ϕ. Now Cap ω X (ϕ -ψ < -δ) ≤ C 1 δ -n 1 {ψ-ϕ>δ} L χ * ≤ C 2 δ n [-log Vol(ψ -ϕ > δ)] n+ε ≤ C 3 δ n -log (ψ-ϕ) + L 1 δ n+ε ,
where the latter inequality follows from Chebyshev inequality.

We set Φ(z, w) := ρ τ ϕ(z) for w ∈ C, |w| = τ and consider

ϕ c,τ (z) := inf 0≤t≤τ Φ(z, t) + Kt -Kτ -c log t τ .
Recall from Lemma 1.2 that ω X + dd c ϕ c,τ ≥ -(Ac + Kτ) ω X , thus the function

ψ c,τ := (1 -[Ac + Kτ]) ϕ c,τ is ω X -plurisubharmonic on X.
In what follows we set ψ τ := ψ c(τ),τ , where c = c(τ) is chosen below so that c(τ) = o(1) and τ = o(c(τ)).

Observe that ϕ c,τ ≤ ρ t ϕ. Since ϕ is bounded we can shift by an additive constant and assume that ϕ ≥ 1. For τ small enough this ensures that ϕ c,τ ≥ 0, hence

ψ τ = ϕ c,τ -[Aτ + Kτ]ϕ c,τ ≤ ϕ c,τ ≤ ρ τ ϕ. Recall that ϕ -ρ τ ϕ L 1 = O(τ 2 ). Since (ψ τ -ϕ) + L 1 ≤ ρ τ ϕ -ϕ L 1 we infer (ψ τ -ϕ) + L ∞ ≤ δ + C 4 δ ε n [-log τ 2 δ -1 ] ε(ε+n) n 2 .
We optimize the choices of τ, δ by choosing τ 2 = δ exp -C 4 δ -n/ε , which leads to

(ψ τ -ϕ) + L ∞ ≤ C 5 (-log τ) ε/n .
We choose c(τ) = (log τ) -ε/n . The previous inequality yields

ψ τ ≤ ϕ + C 6 (-log τ) ε/n = ϕ + C 6 c(τ).
This provides a uniform lower bound on t = t(z) which realizes the infimum in the definition of ϕ τ,τ . Recall indeed that ρ t ϕ + Kt ≥ ϕ, hence

ϕ(z) + C 6 c(τ) ≥ ρ t ϕ(z) + Kt -Kτ -c(τ) log t τ ≥ ϕ(z) -Kτ -c(τ) log t τ which leads to log t τ ≥ -Kτ c(τ) -C 6 ≥ -C 7 . Thus exp(-C 7 )τ ≤ t ≤ τ, showing that ρ τ ϕ ≤ ϕ + C 8 c(τ).
Using that ϕ is quasi-plurisubharmonic, we finally observe that ϕ ≤ ρ τ ϕ + Bτ 2 ≤ ϕ + B c(τ), since . Thus for all 0 < τ < τ 0 ,

ϕ -ρ τ ϕ L ∞ ≤ C 9 (-log τ) ε/n .
This provides the desired control on the modulus of continuity of ϕ, as convolutions ρ t ϕ(z) and sup B(z,t) ϕ are uniformly comparable for quasi-psh functions (see [START_REF] Zeriahi | Remarks on the modulus of continuity of subharmonic functions[END_REF]).

We finally take care of the third item. The proof follows the same path, but the stability estimate (1.4) has to be modified as follows :

(1.5) max

X (ψ -ϕ) ≤ B [log (-log( ψ -ϕ 1 ))] ε/n ,
provided that ψϕ 1 < 1/e, where B > 0 is a uniform constant. Indeed for w(t) = t(log t) n (log • log(t + 3)) n+ε , Hölder-Young inequality allows one to show that there exists A > 0 such that for any Borel subset K ⊂ X,

(1.6) µ(K) ≤ A f w Cap ω X (K) -log Cap ω X (K) -1-ε/n .
Arguing as in [EGZ09, Proposition 2.6] we infer that for any bounded ω-plurisubharmonic functions ψ such that ψ ≥ ϕ and any δ ∈]0, 1[,

ψ -ϕ L ∞ ≤ δ + B (-log Cap ω X {ψ -ϕ > δ} ε/n , The comparison principle yields Cap ω X ({ψ -ϕ > δ}) ≤ Cδ -n {ψ-ϕ>δ} (ω X + dd c ϕ) n . Since (ω X + dd c ϕ) n = f dV, it follows from Hölder-Young inequality that Cap ω X ({ψ -ϕ > δ}) ≤ Cδ -n f w 1 {ψ-ϕ>δ} w * ≤ C f w δ n [-log(-log Vol({ψ -ϕ > δ})] ε/n ≤ C f w δ n [-log(-log(δ/ ψ -ϕ 1 )] ε/n . We infer max X (ψ -ϕ) ≤ δ + B [-log (-log(δ/ ψ -ϕ 1 ))] ε/n , provided that ψ -ϕ 1 < 1/e.
Optimizing the right hand side by taking

δ = 1 [log (-log( ψ -ϕ 1 ))] ε/n ,
we obtain (1.5). Using Demailly-Kiselman regularization technique as above, this eventually yields the predicted control on the modulus of continuity.

Remark 1.7. -Observe that the constants C only weakly depend on the reference Kähler form ω X . In particular if there exist Kähler forms ω 1 , ω 2 such that ω 1 ≤ ω X ≤ ω 2 , then these constants can be shown to only depend on the forms ω 1 , ω 2 . This will be useful in proving Theorem B in Section 3.

Diameter control using a Ricci lower bound

In this section we establish diameter upper bounds by requiring a Ricci lower bound.

Good representative of positive cohomology classes.

-Let (X, ω X ) be a compact Kähler manifold and let dV X := ω n X . Pick B = {θ 1 , . . . , θ s } smooth closed differential forms whose cohomology classes constitue a basis of the finite dimensional vector space H 1,1 (X, R). If ω is a Kähler form, it follows from the ∂∂-lemma that there exists unique (t 1 , . . . ,

t s ) ∈ R s and ϕ B ω ∈ PSH(X, ∑ j t j θ j ) such that ω = s ∑ j=1 t j θ j + dd c ϕ B ω , with X ϕ B ω dV X = 0.
The following elementary result is left to the reader.

Lemma 2.1. -A subset K of the Kähler cone is bounded in H 1,1 (X, R) if and only if there exists C > 0 such that for all ω ∈ K, one has [ω] • [ω X ] n-1 = X ω ∧ ω n-1 X ≤ C. Fix B and
B two bases of differential forms, and K a bounded subset of H 1,1 (X, R). The following properties are equivalent.

-there exists C B,K > 0 such that ϕ B ω ∞ ≤ C B,K for all ω ∈ K; -there exists C B ,K > 0 such that ϕ B ω ∞ ≤ C B ,K for all ω ∈ K.
To lighten notations in what follows we get rid of the upper-script and simply write ϕ ω to denote the normalized potential of ω with respect to a given basis B.

We now explain, given particular bounded subsets of the Kähler cone, how to choose a basis B with specific curvature properties.

Lemma 2.2. -Let K ⊂ H 1,1 (X, R) be a bounded subset of the Kähler cone of X.

1. If K is closed, then there exists C = C(K) > 0 such that for each α ∈ K, one can find a Kähler form

ω α ∈ α such that C -1 ω X ≤ ω α ≤ Cω X 2. If K ⊂ H 1,1
big (X, R), then there exists C = C(K) > 0 and finitely many quasi-psh functions with analytic singularities ρ i such that for each α ∈ K, one can find a smooth closed (1, 1)-form

η α ∈ α such that C -1 ω X ≤ η α + dd c ρ i α and η α ≤ Cω X .
Proof. -Let θ 1 , . . . , θ s be smooth closed (1, 1)-forms such that the cohomology classes {θ j } generate the finite dimensional vector space H 1,1 (X, R). We endow the latter with the euclidean structure induced from the choice of the above basis.

Assume first that K is closed. Given α ∈ K, let us pick ω ∈ α an arbitrary Kähler form. For

t ∈ R s , consider ω t := ω + ∑ s k=1 t k θ k . Clearly, there exist ε ω , C ω > 0 such that for |t| < ε ω , we have C -1 ω ω X ≤ ω t ≤ C ω ω X .
By compactness of K, one can find a finite set I and Kähler forms

(ω i ) i∈I along with positive numbers ε i , C i > 0 such that i∈I B([ω i ], ε i ) ⊃ K and, for |t| < ε i , the form ω t,i := ω i + ∑ s k=1 t k θ k satisfies C -1 i ω X ≤ ω t,i ≤ C i ω X .
The first item follows with the choice of the constant C := max i∈I C i .

We now treat the case when K may leave the Kähler cone, but still consists of big classes (which are also nef by definition). Given α ∈ K, we let η + dd c ρ ∈ α be an arbitrary Kähler current with analytic singularities. By definition ρ is smooth in some Zariski open set, and δω X ≤ η + dd c ρ for some δ > 0. Consider η t := η + ∑ s k=1 t k θ k for t ∈ R s . Clearly, there exist ε, C > 0 such that for |t| < ε, we have

C -1 ω X ≤ η t + dd c ρ and η t ≤ Cω X .
The second item follows now by compactness of K.

Exponential integrability of Ricci deviations.

-We now provide a proof of a Theorem A relying on purely complex-analytic methods. Actually, the theorem below allows for the Kähler class to degenerate (unlike in Theorem A) and we will appeal to [START_REF] Guo | Diameter estimates in Kähler geometry[END_REF] in a crucial way to be able to treat the degenerate setting. (i) The assumptions made on K are equivalent to the existence of C > 0 such that for all α ∈ K one has

Theorem 2.3. -Let (X, ω X ) be a compact Kähler manifold and let K ⊂ H 1,1 (X, R) be a bounded subset of the Kähler cone of X such that K ⊂ H 1,1 big (X, R). Fix A, B, C ∈ R and let ω be a Kähler form such that [ω] ∈ K, ϕ ω ∞ ≤ C and Ric(ω) ≥ -Aω -Bω X . Then diam(X, ω) ≤ D,
C -1 ≤ α n and α • [ω X ] n-1 ≤ C. (ii)
The main outcome is that under the assumptions, one can write ω n = f dV X , with f p uniformly bounded, p > 1, i.e. Condition (K) is satisfied in a strong way.

(iii) The a priori bound on ϕ ω ∞ does not follow from the control on the Ricci curvature from below, as Example 2.6 below shows. Moreover, it can happen that the diameter is unbounded in the absence of the a priori bound of ϕ ω , cf Example 2.5.

Proof. -We can assume without loss of generality that the reference Kähler form ω X is scaled so that X ω n X = 1, and that A ≥ 0. Let α := [ω]; we write ω = η α + dd c ϕ ω , where η α is provided by Lemma 2.2. From now on, we simply write ϕ = ϕ ω .

By assumption Ric(ω) = -Aω -Bω X + T, where T ≥ 0 is a semi-positive closed (1, 1)-form. It follows from the ∂∂-lemma that

T = Ric(ω X ) + Bω X + Aη α + dd c ψ,
for some quasi-psh function ψ which is uniquely determined up to an additive constant. We fix the latter by imposing

ω n = (η α + dd c ϕ) n = e Aϕ-ψ ω n X . We easily check that (2.1) X e -ψ ω n X ≤ e AC [ω] n , and sup X ψ ≥ -AC -log[ω] n
The assumptions made on K now ensure that we can find a constant C K > 0 such that one has a uniform upper bound X e -ψ ω n X ≤ C K e AC , as well as a uniform lower bound sup X ψ ≥ -AC -C K .

We set M := sup X ψ. We proceed in two steps. In the first step, we show that there exists a constant D = D(A, B, C, K, M) such that diam(X, ω) ≤ D. In the second step, we show that one can bound M in terms of A, B, C and K.

Step 1. Diameter bound relying on an upper bound for ψ. Given any smooth form χ and any real number E > 0, the set

C χ,M,E = {u ∈ PSH(X, χ), sup X u ≤ M and X e -u ω n X ≤ E}
is a compact subset of L 1 (X), as follows from Fatou's and Hartogs lemmas (see [START_REF] Guedj | ZERIAHI -Degenerate complex Monge-Ampère equations[END_REF]). We use this observation here with (cf Lemma 2.2)

χ := Ric(ω X ) + (B + AC K )ω X ≥ Ric(ω X ) + Bω X + Aη α
and E := e AC C K , and simply write C := C χ,M,E . Thanks to (2.1), we have ψ ∈ C. Any function u in C satisfies X e -u ω n X < +∞, hence its complex singularity exponent is larger than one c(u) ≥ 1. Since quasi-psh functions are integrable at the critical exponent c(u) [Ber15, GZ15], and since u → c(u) is lower semi-continuous [DK01, Theorem 0.2.1], we can find p > 1 such that p ≤ c(u) for all u ∈ C.

It follows now from [DK01, Theorem 0.2.2] that sup u∈C X e -pu ω n X ≤ B,
for some uniform constant B > 0. In particular e -ψ ∈ L p and e -ψ L p (ω n X ) is uniformly bounded. We infer that ω n = f ω n X with f ∈ L p and f L p (ω n X ) ≤ B uniformly bounded. When K is compact it follows from Theorem 1.6 that ϕ is Hölder continuous, with a uniform control on its modulus of continuity as [ω] moves inside K (see Remark 1.7). We conclude with Proposition 1.4 that diam(X, ω) is uniformly bounded from above. When K is merely a compact subset of the big cone, one can invoke [GPSS22, Theorem 1.1] to conclude, noting that f = e Aϕ-ψ ≥ c > 0 is uniformly bounded below.

Step 2. Upper bound for ψ depending only on A, B, C and K. We fix C 2 > 0 such that the holomorphic bisectional curvature Bisec ω X ≤ C 2 is bounded from above. Recall that ω = η α + dd c ϕ and Lemma 2.2 provides us with a function ρ with analytic singularities and δ = δ(K) > 0 such that η α + dd c ρ ≥ δω X .

It follows from Chern-Lu inequality (see e.g. [Rub14, Proposition 7.2]) that

∆ ω log Tr ω (ω X ) ≥ -A -(2B + C 2 )Tr ω (ω X ). Setting H = log Tr ω (ω X ) -C 3 (ϕ -ρ) we infer ∆ ω H ≥ -A -(2B + C 2 )Tr ω (ω X ) -C 3 ∆ ω (ϕ -ρ) = -A -C 3 n -(2B + C 2 )Tr ω (ω X ) + C 3 Tr ω (η α + dd c ρ) ≥ -A -C 3 n + Tr ω (ω X ), by choosing C 3 δ = 1 + 2B + C 2 .
Since ρ is smooth in some Zariski open set Ω and ρ → -∞ on ∂Ω, we can apply the maximum principle in Ω. Together with the uniform bound on ϕ, we obtain

Tr ω (ω X ) ≤ C 4 e -C 3 ρ , hence ω X ≤ C 4 e -C 3 ρ ω. We infer ω n X ≤ C n 4 e -nC 3 ρ ω n = e Aϕ-ψ-nC 3 ρ+C 5 ω n X , hence nC 3 ρ + ψ ≤ C 5 + Aϕ ≤ C 6 . Since X ρω n
X is under control, this yields a uniform upper bound on X ψω n X , hence on sup X ψ (see [GZ17, Proposition 8.5]). Example 2.5. -Let X be a projective manifold on which there exists a smooth ample divisor D such that K X + D is ample (e.g. X = P n and D is a smooth hypersurface of degree d > n + 1). Pick a smooth hermitian metric h with positive curvature on D and a volume form dV X such that ω X := -Ric(dV

Examples and

X ) + iΘ h (D) ≥ 0. Next, let s be a section of D such that (s = 0) = D. Set ψ ε := log(|s| 2 h + ε); one can check that iΘ h (D) + dd c ψ ε ≥ 0.
We consider the solution ϕ ε ∈ PSH(X, ω X ) of the Monge-Ampère equation (ω X + dd c ϕ ε ) n = e ϕ e -ψ ε dV X , and we set ω ε = ω X + dd c ϕ ε . Clearly, one has Ric ω ε ≥ -ω ε but we claim that diam(X, ω ε ) → +∞. Indeed, it is somehow classical (see e.g. the proof of [DNL15, Theorem 4.5]) to see that ω ε converges locally smoothly on X \ D to the Kähler-Einstein metric ω 0 with Poincaré singularities along D constructed by R. Kobayashi [START_REF] Kobayashi | Kähler-Einstein metric on an open algebraic manifolds[END_REF] and Tian-Yau [START_REF] Tian | Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry[END_REF]. The latter metric is complete, hence has infinite diameter. Thus the diameter of (X, ω ε ) cannot be bounded. Of course Osc X ϕ ε → +∞.

We now provide a family of Kähler metrics with uniformly positive Ricci curvature and bounded diameters, whose potentials are not uniformly bounded.

Example 2.6. -Let X be a K-semistable Fano manifold which is not K-stable, hence does not admit a Kähler-Einstein metric. A classic example is a general deformation of the Mukai-Umemura threefold [START_REF] Tian | Kähler-Einstein metrics with positive scalar curvature[END_REF], cf also [START_REF] Li - | Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds[END_REF] for the general picture.

Pick ω X ∈ c 1 (X) a Kähler form, and let h denote the unique smooth function such that Ric ω X = ω X + dd c h and X e h ω n X = X ω n X . It is classical (see [START_REF] Tian | Kähler-Einstein metrics with positive scalar curvature[END_REF]) that one can find, for all 0 ≤ t < 1, a Kähler form ω t = ω X + dd c u t such that Ric ω t = tω t + (1t)ω X , but the situation degenerates as t → 1. More precisely (ω X + dd c u t ) n = e -tu t +h ω n X for an appropriate normalization of u t , and the non-existence of a Kähler-Einstein metric is equivalent to lim t→1 Osc X u t → +∞.

Let us set v t = u t -X u t ω n X ; we claim that lim t→1 v t ∞ = +∞, although it follows from Myers theorem that diam(X, ω t ) ≤ c n when t is away from zero since Ric ω t ≥ tω t . The claim can be checked as follows. Since v t has zero mean, we have sup

X v t ≤ C, hence -inf X v t ≥ Osc(v t ) -C. Since Osc(v t ) = Osc(u t ) diverges,
we obtain the claim.

Orlicz integrability.

-Theorem 2.3 allows one to extend and clarify a method of Fu-Guo-Song which provides an upper bound on diameters under Condition (K) and a lower bound on the Ricci curvature.

Corollary 2.7. -Let (X, ω X ) be a compact Kähler manifold. Assume that a function ϕ ∈ PSH(X, ω X ) ∩ C ∞ (X) satisfies (ω X + dd c ϕ) n = e λϕ f dV X , where dV X is a smooth volume form, λ ∈ R + , and 1. ω X ≥ θ with θ smooth, semi-positive and big (i.e. X θ n > 0);

2. Ric(ω) ≥ -Aω -Bω X with ω := ω X + dd c ϕ; 3. µ = f dV X satisfies Condition (K) with weight w such that X w • f dV X ≤ C.
Then diam(X, ω) ≤ D, where D depends on A, B, C, w, θ, dV X , λ.

This result is established in [START_REF] Fu | Geometric estimates for complex Monge-Ampère equations[END_REF] when B = 0 and X f 1+ε dV X ≤ C. An extension has been proposed in [START_REF] Guo | Local noncollapsing for complex Monge-Ampère equations[END_REF] to the case when B = 0 and X f (log f ) n+ε dV X ≤ C; however, it follows from Theorem 2.3 that the two conditions are equivalent.

Proof. -When µ = f dV X satisfies Condition (K) , it follows from [Koł98, EGZ09] that ϕ ω ∞ is uniformly bounded. Thus Corollary 2.7 is a consequence of Theorem 2.3.

The main result

In this section we prove our main result Theorem B.

> 0 such that if X f ω | log f ω | n (log • log( f ω + 3)) p ω n FS ≤ A,
then diam(X, ω) ≤ C(A, p). More precisely, for any γ < p -2n there exists a constant C = C(A, p, γ) such that for two points x, y ∈ X, we have

d ω (x, y) ≤ C (log | log d ω X (x, y)|) γ 2 .
Remark 3.2. -Yang Li's observation yields a uniform bound on the diameter if X f ω | log f ω | p dV X < +∞ with p > 3n, as follows from Theorem 1.6 and Proposition 1.4. Example 5.2.3 shows that our Orlicz condition on the density is almost optimal.

Remark 3.3. -The proof will show, more precisely, that any function f on X which is 1-Lipschitz with respect to ω satisfies Osc( f ) ≤ C(X, [ω]). This is however equivalent to the above statement: consider indeed γ a geodesic joining two points x and y, then

| f (x) -f (y)| = 1 0 d f (γ (t))dt ≤ 1 0 |∇ f | ω |γ (t)| ω dt ≤ d ω (x, y).
Proof.

-Let x = [x 0 : . . . :

x n ] denote the homogeneous coordinates of X := CP n and set

g x (y) = g(x, y) = log |x ∧ y| |x| • |y| ∈ PSH(X, ω FS ).
This "Green function" g x is non-positive, smooth away from x and its complex Monge-Ampère measure is well-defined, satisfying (3.1) (ω FS + dd c y g x ) n = δ x = Dirac mass at x.

Step 1. The diameter bound. Fix x 0 ∈ X and set f (x) = d ω (x, x 0 ). This is a 1-Lipschitz function with respect to d ω , hence it is differentiable a.e. with norm at most one; in particular 0 ≤ d f ∧ d c f ≤ ω. Multiplying (3.1) by f and integrating, we obtain for all x ∈ X,

f (x) = X f (ω FS + dd c g x ) n and 0 = X f (ω FS + dd c g x 0 ) n , so that f (x) = X f [(ω FS + dd c g x ) n -ω n FS ] - X f [(ω FS + dd c g x 0 ) n -ω n FS ] . It thus suffices to establish bounds on X f (ω FS + dd c g x ) n -ω n FS which are uniform in x ∈ X. Observe that (ω FS + dd c g x ) n -ω n FS = dd c g x ∧ ∑ n-1 k=0 ω k g x ∧ ω n-1-k

FS

, where ω g x := ω FS + dd c g x . Stokes theorem and Cauchy-Schwarz inequality yield

X f dd c g x ∧ ω k g x ∧ ω n-1-k FS = X d f ∧ d c g x ∧ ω k g x ∧ ω n-1-k FS ≤ X χ • g x dg x ∧ d c g x ∧ ω k g x ∧ ω n-1-k FS 1/2 X (χ • g x ) -1 d f ∧ d c f ∧ ω k g x ∧ ω n-1-k FS 1/2 .
Here χ : R -→ R is a smooth convex increasing function to be chosen below, whose growth is slightly smaller than t at -∞ so that χ (-∞) = 0. We normalize χ so that χ (0) ≤ 1 and observe that χ • g x ∈ PSH(X, ω FS ), with

ω FS + dd c χ • g x = χ • g x dg x ∧ d c g x + χ • g x ω g x + [1 -χ • g x ]ω FS ≥ χ • g x dg x ∧ d c g x ≥ 0.
We infer, using Stokes theorem,

0 ≤ X χ • g x dg x ∧ d c g x ∧ ω k g x ∧ ω n-1-k FS ≤ X ω χ•g x ∧ ω k g x ∧ ω n-1-k FS = X ω n FS = 1.
To bound the second integral we recall that

d f ∧ d c f ≤ ω so that for all 0 ≤ k ≤ n -1, X (χ • g x ) -1 d f ∧ d c f ∧ ω k g x ∧ ω n-1-k FS ≤ X (χ • g x ) -1 ω ∧ ω k g x ∧ ω n-1-k FS We set h(t) := χ (t) -1 , ψ x := h • g x , I k,x := X ψ x ω ∧ ω k g x ∧ ω n-1-k

FS

and we choose χ(t) = t log(B-t) γ , where γ > 0 is small enough to be determined later and B > 0 is a uniform constant large enough so that χ is convex increasing on R -with χ (0) ≤ 1. We have

(3.2) dd c ψ x = h • g x dd c g x + h • g x dg x ∧ d c g x
A straightforward computation shows that when t ∼ +∞, one has

(3.3) h(-t) ∼ t(log t) 1+γ , h (-t) ∼ (log t) 1+γ , h (-t) ∼ (log t) γ t A direct computation (see [DNGG23, Lemma 2.8]) shows that ω g x ≤ e -2g x ω FS .
Thanks to the first estimate in (3.3), we see that the integral X ψ x ω k g x ∧ ω n-k FS is uniformly bounded in x, since k ≤ n -1. Thus everything comes down to estimating

I k,x := X ψ x dd c ϕ ∧ ω k g x ∧ ω n-1-k

FS

Clearly, the integral above is invariant if we translate ϕ by a constant so we can freely normalize the potential ϕ so that ϕ(x) = 0. In particular, Theorem 1.6 implies that there exist constants δ, C > 0 independent of x, ω such that

(3.4) |ϕ| ≤ C log(-g x ) 1+δ .
We are now ready to bound I k,x ; we are going to distinguish two cases.

Case 1. Assume k < n -1. We claim that under this assumption, we have

I k,x = X ϕdd c ψ x ∧ ω k g x ∧ ω n-1-k FS .
Fix ε > 0 any (small) positive number. It follows from (3.2) and (3.3) that there exists C ε > 0 such that the absolute value of the integral above is bounded by

ϕ ∞ X (h • g x )e -2(k+1)g x ω n FS ≤ C ε ϕ ∞ X e -2(n-1+ε)g x ω n FS ≤ C ε 2n 1 -ε ϕ ∞ ,
as follows from [DNGG23, Lemma 2.8.ii]. Now let us justify the integration by parts that we claimed. Set T k := ω k g x ∧ ω n-1-k

FS

; we have the identity

(3.5) (ϕ∂ ∂ψ x -ψ x ∂ ∂ϕ) ∧ T k = d (ϕ ∂ψ x + ψ x ∂ϕ) ∧ T k .
The current whose differential we are taking in the RHS above is well-defined, as a wedge product of currents whose coefficients are functions whose product is L 1 . Indeed, (3.2) and (3.3) show that ψ x ∂ϕ ∧ T k has coefficients dominated by e -2(k+ε)g x while ∂ψ x ∧ T k has coefficients dominated by e -2(k+1+ε)g x , and both are dominated by e -2(n-1+ε)g x . The claim follows from integrating (3.5) and applying Stokes theorem.

Case 2. Assume k = n -1. Let g x,C := max{g x , -C}; we have g x,C ∈ PSH(X, ω FS ). Given the definition of the Monge-Ampère operator, it is sufficient to bound (3.6)

I C := X ψ x dd c ϕ ∧ ω n-1 g x,C = X ϕdd c ψ x ∧ ω n-1 g x,C
independently of C, x and ϕ. Given (3.2), we can split this integral as follows

I C = X ϕ(h • g x )ω g x ∧ ω n-1 g x,C =:I C,1 + X ϕ(h • g x )dg x ∧ d c g x ∧ ω n-1 g x,C =:I C,2 - X ϕ(h • g x )ω FS ∧ ω n-1 g x,C =:I C,3
. Now, we choose γ < δ so that, we have

(3.7) |ϕ h • g x | 1 log(-g x ) δ-γ ≤ M,
where M > 0 is some uniform constant. In particular, (3.8)

I C 1 + I C 3 ≤ M X ω g x ∧ ω n-1 g x,C + X ω FS ∧ ω n-1 g x,C = 2M.
As for the second integral, it follows from (3.3) and (3.4) that the density of the integrand with respect to ω n FS is dominated by (3.9) 1 e 2ng x (-g x ) log(-g x ) 1+δ-γ .

Using polar coordinates we are reduced to showing that the integral 0 dr r log r(log(log r)) 1+η converges for any η > 0, which is indeed the case (set u = log(-log r) and perform the change of variable).

All in all, we have showed that I C is uniformly bounded independently of C, hence the first part of the theorem.

Step 2. Controlling the distance function.

We now fix a point x ∈ X and look at f (y) = d ω (x, y) for y close to x. We introduce the function ρ = ρ x,y defined by ρ := g yg x so that dd c ρ = ω g yω g x . Observe that |z -y| 2 (z i -

x i ) -|z -x| 2 (z i -y i ) = |z -x| 2 (y i -x i ) + (z i -x i )(|x -y| 2 + 2Re z -x,
xy , so that using the triangle inequality |z -x| ≤ |z -y| + |x -y|, we obtain

z i -y i |z -y| 2 - z i -x i |z -x| 2 ≤ 2|x -y| |z -x||z -y| • 1 + |x -y| |z -y| . hence (3.10) dρ ∧ d c ρ ≤ C|x -y| 2 |z -x| 2 |z -y| 2 • 1 + |x -y| |z -y| 2 ω FS .
We are going to use the pluricomplex Green function to estimate the distance function f . Since f (x) = 0, we have from (3.1)

f (y) = X f (ω n g y -ω n g x ) = n-1 ∑ k=0 X f dd c ρ ∧ ω k g y ∧ ω n-1-k g x = - n-1 ∑ k=0 X d f ∧ d c ρ ∧ ω k g y ∧ ω n-1-k g x . We define ψ 0 = ψ x = h • g x , ψ n-1 = ψ y = h • g y and for 1 ≤ k ≤ n -1, we set ψ k = ψ x,y := h • (g x + g y )
where h(t) = |t|(log |t|) 1+γ for γ > 0 small to be determined later (cf (3.3)). By Cauchy-Schwarz inequality, we have

(3.11) f (y) ≤ n-1 ∑ k=0 X ψ -1 k dρ ∧ d c ρ ∧ ω k g y ∧ ω n-1-k g x =:J k 1 2 X ψ k ω ∧ ω k g y ∧ ω n-1-k g x =:I k 1 2
2.1. Estimating the J k integrals. We split the computation of the integral into two zones. First, we look outside a small neighborhood of x (and y, which is assumed to be close to x), and then we look at what happens near x. We next use a local coordinate system (z i ) such that x = 0 hence g x = log |z| + O(1) and g y = log |z -y| + O(1). Let ε := |y| and let ȳ := ε -1 y, which satisfies ȳ ∈ S 2n-1 . Finally, we introduce the variable w = ε -1 z.

• The first, easiest, case is when we look at J k on X \ (|z| ≤ C -1 ) for C > 0 large but fixed. There, the function ρ is a smooth function of z, x, y with uniformly bounded derivatives (in all variables). In particular dρ ∧ d c ρ ≤ C|x -y| 2 ω FS and ω g x + ω g y ≤ Cω FS for some uniform C hence J k on the considered zone is less than C|x -y| 2 X ω n FS = Cε 2 .

• Next, we estimate J k over (|z| ≤ Cε) for C > 0 large.

We have to estimate

|z|≤Cε ψ k (z) -1 1 |z| 2 + 1 |z -y| 2 1 |z -y| 2k |z| 2(n-1-k) |dz| 2 which is equal to |w|≤C ψ k (εw) -1 1 |w| 2 + 1 |w -ȳ| 2 1 |w -ȳ| 2k |w| 2(n-1-k) |dw| 2 .
By symmetry, it is enough to consider the two cases k = 0 and 0 < k < n -1. There are two meaningful zones which are near w = 0 and w = ȳ. Near w = 0, we have g x = log ε|w| + O(1) and g y = log ε + O(1) hence ψ k (z) ≥ h(log ε|w|) + O(1) no matter the value of k. Therefore the integral behaves (uniformly in ȳ ∈ S 2n-1 ) like

|w|≤ 1 2 |dw| 2 |w| 2(n-k) (-log |εw|)(log(-log |εw|)) 1+γ .
Clearly, the most singular case is when k = 0 where the integral becomes (3.12)

|w|≤ ε 2 |dw| 2 |w| 2n (-log |w|)(log(-log |w|)) 1+γ = O 1 log(-log ε) γ . Near w = ȳ, ψ k (z) ≥ -log ε + O(1) hence the integral is dominated by |w-ȳ|≤ 1 2 |dw| 2 |w -ȳ| 2(k+1) (-log ε)(log(-log ε)) 1+γ = O 1 log ε since k + 1 < n.
• Finally, we analyze J k in the zone (Cε ≤ |z| ≤ C -1 ), or equivalently (C ≤ |w| ≤ (Cε) -1 ).

Here again, we assume that k < n -1. Then we have ψ k ≥ h(log ε|w|) + O(1). Moreover, it follows from (3.10) that in the above zone, J k is dominated by

C≤|w|≤(Cε) -1 1 ψ k (εw) ε ε 2 |w||w -ȳ| 2 1 + 1 |w -ȳ| 2 1 (ε|w -ȳ|) 2k (ε|w|) 2(n-1-k) ε 2n |dw| 2 which is controlled by C≤|w|≤(Cε) -1 |dw| 2 |w| 2n+2 (-log |εw|)(log(-log |εw|)) 1+γ .
Reversing to the variable z = εw and using polar coordinates, this integral becomes

ε 2 Cε≤r≤C -1 dr r 3 (-log r)(log(-log r)) 1+γ .
Setting u =log r, the above quantity can be rewritten as

ε 2 -log C≤u≤-log Cε e 2u du u(log u) 1+γ .
For u large enough, the integrand is increasing hence the integral is dominated by

ε 2 (-log ε) e -2 log ε (-log ε)(log(-log ε)) 1+γ 1 log(-log ε) 1+γ
hence the result.

Estimating the I k integrals.

When bounding the diameter, we have already proved that I 0 and I n-1 are bounded uniformly, so from now on we assume that 0 < k < n -1.

• On X \ (|z| ≤ 1), ψ k is bounded hence on that zone I k is less than

C X ω ∧ ω k g y ∧ ω n-1-k g x = C.
From now on, we look at

I k := |z|≤1 ψ k ω ∧ ω k g y ∧ ω n-1-k g x
and one can assume without loss of generality that ψ k has compact support in (|z| ≤ 1). This will allow us to perform integration by parts without having to deal with boundary terms.

I k = |z|≤1 ψ k ω FS ∧ ω k g y ∧ ω n-1-k g x =:I k,1 + |z|≤1 ψ k dd c ϕ ∧ ω k g y ∧ ω n-1-k g x =:I k,2 • The integral I k,1 . It is dominated by |z|≤1 | log 2 (|z||z -y|)|dz| 2 |z -y| 2k |z| 2(n-1-k) = |w|≤ε -1 ε 2 log 2 (ε 2 |w||w -ȳ|)|dw| 2 |w -ȳ| 2k |w| 2(n-1-k) .
Near w = 0, we have for any small δ > 0 that the integrand is dominated by

ε 2 (ε 2 |w|) -δ |w| -2(n-1-k) ≤ |w| -2n+1
, and convergence follows. The behavior near w = ȳ is dealt with similarly. Finally, near infinity, the integrand is dominated by

ε 2+2δ |w| -2(n-1-δ) while ε -1 ε 2+2δ |dw| 2 |w| 2(n-1-δ) = ε -1
ε 2+2δ r 1+2δ dr 1; we are done with I k,1 .

• The integral I k,2 . One can assume that ϕ(x) = 0, hence

(3.13) |ϕ(z)| ≤ C (log(-log |z|)) 1+δ
for some δ > 0. An integration by parts shows that

I k,2 = |z|≤1 ϕdd c ψ k ∧ ω k g y ∧ ω n-1-k g x
. Next, if we set σ := g x + g y , (3.3) and (3.13) show that if γ < δ, one has

(3.14) |ϕh (σ)(εw)| ≤ C • log(-log |w-ȳ|) 1+γ log(-log ε) 1+δ if |w -ȳ| ≤ 1 2 ; C otherwise for |w| ≤ ε -1 .
as well as

(3.15) |ϕh (σ)(εw)| ≤ C if |w| ≤ C; C (-log ε|w|)(log(-log ε|w|)) 1+δ-γ if C ≤ |w| ≤ ε -1 .

Now, let us write

dd c ψ k = h (σ)(dd c g x + dd c g y ) + h (σ)d(g x + g y ) ∧ d c (g x + g y ) = h (σ)(ω g x + ω g y -2ω FS ) + h (σ)d(g x + g y ) ∧ d c (g x + g y ).
As in the previous computations, we split the integral I k,2 into the zones (|w| ≤ C) and (C ≤ |w| ≤ ε -1 ). On (|w| ≤ C) we have by (3.15),

|ϕh (σ)|dσ ∧ d c σ ∧ ω k g y ∧ ω n-1-k g x ≤ C 1 |w| 2 + 1 |w -ȳ| 2 • |dw| 2 |w -ȳ| 2k |w| 2(n-1-k)
whose integral on the considered zone converges since k > 0 and k + 1 < n. Next, by (3.14) we have on

(|w| ≤ C) ∩ (|w -ȳ| ≥ 1 2 ) the following inequality |ϕh (σ)|(ω g x + ω g y + ω FS ) ∧ ω k g y ∧ ω n-1-k g x ≤ C|dw| 2 |w| 2(n-k)
and the integral of the RHS is clearly convergent on the latter domain. On (|w -ȳ| ≥ 1 2 ), (3.14) shows that we have

|ϕh (σ)|(ω g x + ω g y + ω FS ) ∧ ω k g y ∧ ω n-1-k g x ≤ C log(-log |w -ȳ|) 1+γ |dw| 2 |w -ȳ| 2(k+1)
and the convergence follows since k + 1 < n. Combining those three inequalities shows that I k,2 is uniformly convergent over (|w| ≤ C).

For the zone (C ≤ |w| ≤ (Cε) -1 ) we use (3.15) again to obtain

|ϕh (σ)|dσ ∧ d c σ ∧ ω k g y ∧ ω n-1-k g x ≤ C|dw| 2 |w| 2n (-log ε|w|)(log(-log ε|w|)) 1+δ-γ .
The integral of the RHS is the same as

εC≤|z|≤C -1 |dz| 2 |z| 2n (-log |z|)(log(-log |z|)) 1+δ-γ which is uniformly bounded as ε → 0 since γ < δ.
|ϕh (σ)|(ω g x + ω g y + ω FS ) ∧ ω k g y ∧ ω n-1-k g x ≤ C(ω g x + ω g y + ω FS ) ∧ ω k g y ∧ ω n-1-k g x
and the integral of the RHS can be computed in cohomology; it is equal to 3. This ends to show that I k,2 is uniformly bounded, hence the theorem is proved.

General case.

-We now deal with the general case whose statement we recall:

Theorem 3.4. -Let (X, ω X ) be a compact Kähler manifold of complex dimension n and let dV X :=

ω n X n!
the associated smooth volume form. Let K ⊂ H 1,1 (X, R) be a compact subset of the Kähler cone of X.

Given ω a Kähler form in K, we set f ω = ω n /dV X and consider, for A > 0 and p > 2n fixed,

K A,p := {ω ∈ K such that X f ω | log f ω | n (log(log f ω + 3)) p dV X ≤ A}.
For any γ < p -2n, there exists a constant C = C(K, dV X , A, p, γ) > 0 such that for all ω ∈ K A,p and any two points x, y ∈ X, we have

d ω (x, y) ≤ C (log | log d ω X (x, y)|) γ 2 .
The proof is similar to that of Theorem 3.1, but it requires to control extra terms due to the fact that we can only construct an approximate Green function for the complex Monge-Ampère operator (see [START_REF] Coman | Quasiplurisubharmonic Green functions[END_REF]).

Proof. -We use a double cover of X by finitely many open charts U j ⊂ V j and fix

χ j ∈ C ∞ c (V j ) such that 0 ≤ χ j ≤ 1 with χ j ≡ 1 near U j .
For x ∈ X, we let j,x (z) denote the function log |z -x| -C in the chart V j , where z denotes holomorphic coordinates, and set g x (z) = χ j j,x (z). If x ∈ X belongs to several charts, we choose one index j = j(x) so that g x is uniquely determined. The constant C > 0 is chosen so that g x ≤ -1 on X.

The function g x is psh near x and smooth in X \ {x}, hence it is quasi-psh. Moreover

dd c g x ≥ dχ j ∧ d c j,x + d j,x ∧ d c χ j + j,x dd c χ j ≥ -C 2 ω X for a constant C 2 > 0 independent of x.
Rescaling ω X we can assume that C 2 = 1, so that each function g x belongs to PSH(X, ω X ). We set ω g x := ω X + dd c g x . The complex Monge-Ampère measure of this approximate Green function is well-defined and satisfies

(ω X + dd c g x ) n = ω n X + (dd c g x ) n + n-1 ∑ k=1 n k ω k X ∧ (dd c g x ) n-k (3.16) = ω n X + δ x + Θ x + n-1 ∑ k=1 n k ω k X ∧ (ω g x -ω X ) n-k = ω n X + δ x + Θ x + dd c g x ∧ ω X ∧ S x ,
where

• δ x is the Dirac measure at point x, • Θ x is a smooth measure on X, • and S x = ∑ n-2 =0 c (ω X + dd c g x ) ∧ ω n-2- X , for some c j ∈ R independent of x.
As in the proof of Theorem 3.1, we consider f (x) = d ω (x, x 0 ), where x 0 is a fixed base point. This is a 1-Lipschitz function with respect to d ω , hence it s differentiable almost everywhere with 0 ≤ d f ∧ d c f ≤ ω. Observe that for all x ∈ X,

(3.17) f (x) = X f (ω X + dd c g x ) n - X f ω n X - X f Θ x - X f dd c g x ∧ ω X ∧ S x . For x = x 0 we obtain f (x 0 ) = 0, hence f (x) = X f [(ω X + dd c g x ) n -ω n X ] - X f [(ω X + dd c g x 0 ) n -ω n X ] + X f dd c g x 0 ∧ ω X ∧ S x 0 - X f dd c g x ∧ ω X ∧ S x + X f [Θ x 0 -Θ x ].
We are going to bound each of these integrals from above, this will provide a uniform bound on f hence on the diameter of (X, ω). Decomposing

(3.18) (ω X + dd c g x ) n -ω n X = dd c g x ∧ n-1 ∑ k=0 ω k g x ∧ ω n-1-k X ,
controlling the first four integrals reduces to establish a uniform bound on integrals of the form

X f dd c g x ∧ ω k g x ∧ ω n-1-k X
. This can be treated as in the proof of Theorem 3.1 by integrating by parts and using a twisted Cauchy-Schwarz inequality.

By comparison with the proof of Theorem 3.1, we need to see that the bounds on all integrals involved are uniform with respect to ω ∈ K. Using Lemma 2.2 we decompose ω = ω α + dd c ϕ, where C -1 ω X ≤ ω α ≤ Cω X for some uniform constant C = C(K) > 0. The salient point is then to establish a uniform bound on the modulus of continuity of ϕ. The latter follow from Theorem 1.6 and Remark 1.7.

We finally focus on the last term

X f [Θ x -Θ x 0 ]. The smooth (n, n)-form Θ x -Θ x 0 is dd c - cohomologous to δ x 0 -δ x ,
hence it can be written as dd c η(x, x 0 ) for some smooth form η of bidegree (n -1, n -1) defined by the Green operator of Hodge theory (see [START_REF] Griffiths | Principles of algebraic geometry[END_REF]). In particular η depends smoothly on x, x 0 . Using a partition of unity, we can decompose η = ∑ ,I,J η ,I,J β ,I,J , where η ,I,J are smooth functions and β ,I,J is a local basis of smooth forms of bidegree (n -1, n -1) which are closed and positive. Integrating by parts we thus obtain

X f [Θ x -Θ x 0 ] = ∑ ,I,J X f dd c η ,I,J ∧ β ,I,J = ∑ ,I,J - X d f ∧ d c η ,I,J ∧ β ,I,J ≤ ∑ ,I,J X d f ∧ d c f ∧ β ,I,J 1/2 X dη ,I,J ∧ d c η ,I,J ∧ β ,I,J 1/2 ≤ C ∑ ,I,J X ω ∧ β ,I,J 1/2 ≤ C X ω ∧ ω n-1 X 1/2
, bounding each form β ,I,J by a uniform multiple of the reference form ω n-1 X . Observe finally that X ω ∧ ω n-1 X remains uniformly bounded from above as the cohomology class [ω] evolves in a bounded subset of H 1,1 (X, R).

The more precise bound

d ω (x, y) ≤ C(log | log d ω X (x, y)|) -γ
2 is proved similarly.

Diameter bounds for currents

In this section, we prove Theorem C and adopt the following setup.

Setup 4.1. -Let (X, ω X ) be a compact Kähler manifold and let E be a divisor with simple normal crossings. Let X • := X \ E and let T = θ + dd c ϕ be a closed positive (1, 1)-current where θ is a smooth closed (1, 1)-form. We assume that ω := T| X • is a Kähler form. Given x ∈ X • , we define ρ x (y) = d ω (x, y) for any y ∈ X • .

Integrability properties of the distance. -

The following result builds upon [DPS93, Lemma 1.3].

Lemma 4.2. -In the Setup 4.1, the function ρ x belongs to W 1,2 (X, ω n X ) for any x ∈ X • . Proof. -We proceed in two steps.

Step 1: ρ x ∈ L 2 (X). This step is very similar to [DPS93, Lemma 1.3], and we will explain how to deal with the singularities near E. We pick two open neighborhoods E ⊂ U U of E ⊂ X such that U = ∪ N i=1 U i is covered by finitely many coordinate charts U i ∆ n where ∆ := {z ∈ C; |z| < 2} and E ∩ U i is a union of hyperplanes. Next, on can assume that each

U i := U ∩ U i is the convex set K := [-1, 1] 2n under the natural embedding [-1, 1] 2 ⊂ ∆. We let C > 0 be such under the suitable identifications, we have C -1 ω eucl ≤ ω X | U i ≤ Cω eucl where ω eucl is the standard euclidean metric on C n . Now, let us pick x 0 ∈ X • .Clearly, ρ x 0 is bounded on X \ U, so it is enough to show that ρ x 0 | U i is L 2 for each i.
From now on, we fix i and we work exclusively on

U i K . Given x, y ∈ K, consider the line L x,y = {tx + (1 -t)y, t ∈ [0, 1]}. We claim that the set S := {(x, y) ∈ K × K; L x,y ∩ E = ∅} is included in a finite union of proper real analytic subvarieties of K × K. Indeed, if we write x = (u 1 , v 1 , . . . , u n , v n ) and let k be such that E ∩ U i = (z 1 • • • z k = 0), then S ⊂ (E × K) ∪ (K × E) ∪ k j=1 {(x, x )|u j v j -v j u j = 0},
which shows the claim.

Let dV ⊗2 = p * 1 ω n eucl ⊗ p * 2 ω n eucl (and similarly for dV X ) be the product volume form on K × K where p i are the two projections onto K. We have S dV ⊗2 = 0. For x, y ∈ K × K \ S, we have d ω (x, y) ≤ ω (L x,y ) where ω means the length with respect to ω. Therefore, we have

(4.1) K×K d 2 ω (x, y)dV ⊗2 (x, y) ≤ K×K 2 ω (L x,y )dV ⊗2 (x, y)
where both integrals are unambiguously defined thanks to the previous observation. As a side note, we should say that the choice of S as before makes the presentation a bit cleaner but we could just as well have worked with S = E × E since for (x, y) / ∈ E × E, the line L x,y meets E finitely many times so we can still define ω (L x,y ).

Using Cauchy-Schwarz inequality it suffices to bound from above the integral

I = K×K 1 0 ω tx+(1-t)y (x -y)dtdV ⊗2 (x, y),
where xy is seen as a (constant) tangent vector. Let h = tr ω eucl ω. Using the elementary inequality ω ≤ tr ω eucl ω • ω eucl , we get

I ≤ 1 0 dt K×K x -y 2 eucl h(tx + (1 -t)y)dV ⊗2 (x, y) ≤ 4 1 0 dt K×K h(tx + (1 -t)y)dV ⊗2 (x, y). Now, given x ∈ K and t ∈ [0, 1 2 ], consider the linear isomorphism u : C n → C n defined by u(y) = tx + (1 -t)y. We have u(K) ⊂ K hence the change of variable yields 1/2 0 dt y∈K h(tx + (1 -t)y)ω n eucl (x) ≤ 2 2n-1 K h(u)ω n eucl (u).
The symmetric operation yields

I ≤ 2 2n vol eucl (K) K h ω n eucl when t ∈ [ 1 2 , 1]. Now, since tr ω eucl ω • ω n eucl = nω ∧ ω n-1 eucl , we obtain I ≤ n2 4n K ω ∧ ω n-1 eucl ≤ n2 4n C n-1 X T ∧ ω n-1 X .
By (4.1), this implies that

U×U d 2 ω (x, y)dV ⊗2 X (x, y) ≤ n2 4n C 3n-1 N X T ∧ ω n-1 X , hence ρ x is in L 2 (ω n X ) for almost every x ∈ X • . Now, given two points x, x ∈ X \ E, the triangle inequality yields |ρ x -ρ x | ≤ d ω (x, x ) hence ρ x is in L 2 (ω n X ) for every x ∈ X • .
Step 2: dρ x ∈ L 2 (X). This amounts to showing that the current dρ x does not charge E. It is standard (see e.g. [CGP13, §9] ) to construct a family of cut-off functions (χ δ ) with values in [0, 1] such that • χ δ ≡ 1 near E and Supp(χ δ ) converges to E when δ → 0.

• dχ δ ∧ d c χ δ ≤ ω P where ω P is a Kähler metric on X \ E with Poincaré type growth near E. Let S δ be the support of dχ δ . Let α be any smooth (2n -1)-form on X; up to replacing ω X with a large multiple, we have

X χ δ dρ x ∧ α ≤ S δ ρ x ω n X + X ρ x dχ δ ∧ α ≤ S δ ρ x ω n X + S δ ρ x (tr ω X ω P ) 1 2 ω n X ≤ S δ ρ x ω n X + S δ ρ 2 x ω n X + S δ nω P ∧ ω n-1 X .
Since ρ x ∈ L 2 (ω n X ) by the first step and ω P ∧ ω n-1 X puts no mass on E, the RHS converges to zero when δ → 0, hence the lemma.

Combining Lemma 4.2 and Yang Li's results [Li21, Theorem 4.1], we can drastically improve the integrability properties of ρ x : Lemma 4.3. -In the Setup 4.1, there exists α > 0 such that for any x ∈ X • , e αρ x ∈ L 1 (X, ω n X ). Proof. -The proof is identical to that of [Li21, Theorem 4.1], so we'll only briefly sketch it. Given a small ball of radius r in X, it follows from Lelong's results that the masses 1 r 2n-2 B(r) T ∧ ω n-1 X increase with r so that

B(r) T ∧ ω n-1 X ≤ Cr 2n-2
where C depends only on X and the cohomology class of T. On X

• , we have |dρ x | 2 ω X ≤ tr ω X ω • |dρ x | 2 ω = tr ω X ω hence B(r) |dρ x | 2 ω X ω n X ≤ Cr 2n-2 . Since dρ x ∈ L 2 (X)
by Lemma 4.2, the previous inequality holds for any balls (even those centered on E) and we can appeal to the Poincaré inequality for small balls on (X, ω X ) to infer that ρ x has bounded mean oscillation, hence John-Nirenberg inequality yields α > 0 such that

X e α(ρ x -X ρ x ω n X ) ω n X < +∞,
which concludes the proof.

Control on the diameter.

-So far we have assumed nothing on the regularity of T near E. If the potential ϕ of T is continuous and its modulus of continuity m ϕ with respect to d ω X satisfies the condition m 1 (r) := r 0 t -1 m ϕ (t)dt < +∞ (e.g. if ϕ is Hölder continuous), then the proof of Proposition 1.4 goes through verbatim and shows that

d ω ≤ m 1 • d ω X on X • × X • . In particular, diam(X • , ω) < +∞.
The next result provides a sharp criterion to guarantee that this diameter is finite.

Theorem 4.4. -In the Setup 4.1, assume that there exists δ > 0 such that the modulus of continuity m ϕ of ϕ satisfies m ϕ (r) ≤ C (log(log r)) 1+δ for some C > 0. Then we have diam(X • , ω) < +∞.

Proof. -The proof is essentially the same as that of Theorems 3.1 and 3.4 but we need to treat the integrations by parts carefully here. Let us fix a point x 0 ∈ X • and consider f := ρ x 0 . We know that f is Lipschitz on X • and satisfies d f ∧ d c f ≤ ω on that open set. Thanks to Lemma 4.2, we know that f ∈ W 1,2 (X); in particular, the previous inequality extends to the inequality

(4.2) d f ∧ d c f ≤ T weakly on X.
Given (3.17) and (3.18), we have for all x ∈ X •

(4.3) f (x) = n-1 ∑ k=0 c k X f dd c g x ∧ ω k g x ∧ ω n-1-k X - X f Θ x
where c k ∈ R and Θ x is a smooth (n, n)-form varying continuously with x. In particular, there exists C > 0 independent of x such that ±Θ x ≤ Cω n V so that X f Θ x ≤ C f L 1 (ω n X ) . We are left to controlling the integrals

J k,x := X f dd c g x ∧ ω k g x ∧ ω n-1-k X .
Note that the current

S k := f d c g x ∧ ω k g x ∧ ω n-1-k X is well-defined for all k. Now X dS k = 0, hence (4.4) J k,x = - X d f ∧ d c g x ∧ ω k g x ∧ ω n-1-k X .
Since d f ∈ L 2 we can use Cauchy-Schwarz and (4.2) to obtain

|J k,x | ≤ X χ • g x dg x ∧ d c g x ∧ ω k g x ∧ ω n-1-k X 1/2 X (χ • g x ) -1 T ∧ ω k g x ∧ ω n-1-k X 1/2
for any smooth convex function χ. The proof of Theorem 3.1 then goes through without a change, once we observe that one can perform the integration by parts (3.5) and (3.6) just like in the smooth case. Indeed, with the notation of (3.5), T k and ψ x are smooth away from x (hence near E, too) and ϕ ∈ L 1 (X, ω n X ) is smooth near x so that the current (ϕ ∂ψ x + ψ x ∂ϕ) ∧ T k is well-defined globally on X.

d V (x, y) = inf { (γ), γ ∈ Γ(V) with γ(0) = x and γ(1) = y} . It is a classical fact that d V is a distance such that d P N ≤ d V .
It is perhaps less known that one can also bound d V from above as follows.

Proposition 4.6. -There exists C > 0 and α ∈ (0, 1] such that

d P N ≤ d V ≤ Cd α P N .
We thank Philippe Eyssidieux for providing the references. Of course one can take α = 1 if V is smooth, while one necessary has α < 1 already for cusps in dimension 1.

Proof. -Consider the U(N + 1)-equivariant real-algebraic embedding

f : L ∈ P N (C) → P L ∈ M N+1 (C),
which sends a complex line L to P L , the orthogonal projection to L. By equivariance we have f * ds 2 M N+1 (C) = Ads 2 FS , hence d P N and the pull-back of the euclidean metric f * d eucl are biLipschitz equivalent.

Since f (V) is a compact connected semialgebraic subset of M N+1 (C), the result therefore follows from a result of Lojasiewicz that we state below.

Theorem 4.7 (Lojasiewicz 65). -Let W ⊂ R N be a compact connected semialgebraic set. There exists K > 0 and α > 0 such that for every (x, y) ∈ W, one can find a piecewise analytic arc drawn on W and joining x to y, whose length is less than K xy α .

We refer the reader to [START_REF] Lojasiewicz | Ensemble semi-analytiques[END_REF] for the original statement, and to [START_REF] Kurdyka | Distance géodésique sur un sous-analytique[END_REF] for a more recent treatment.

Let ϕ : X → R be a continuous quasi-psh function. It follows from Proposition 4.6 that one can measure its modulus of continuity equivalently by using d V or d P n . In particular being Hölder continuous is a notion that is intrinsically well defined, although the exponent of Hölderianity is not.

4.3.2.

Singular Kähler-Einstein metrics. -In this section, (V, ω V ) is a compact Kähler space, and V reg (resp. V sing ) denotes its regular (resp. singular) locus.

Kähler-Einstein currents ω KE have been constructed on mildly singular compact Kähler spaces in [EGZ09, BBE + 19]. Their two defining features are as follows.

-They are Kähler forms on V reg such that Ric (ω KE ) is proportional to ω KE , -They extend to positive currents with bounded local potentials across V sing . The asymptotic behavior of ω KE near the singularities remains to be understood.

Let us now be a bit more precise and write down the general setting in which the aforementioned objects are well-defined. We assume that V has log terminal singularities; in particular K V is a Q-Cartier divisor. If h is a smooth hermitian metric on K V and σ is a local generator of mK V , then

µ h := i n 2 (σ ∧ σ) 1 m |σ| 2/m h ⊗m
defines a positive measure on V reg with finite mass (by the log terminal condition), independent of the choice of m or σ. We extend it to V trivially. Finally, let ω V be a Kähler metric on V, and consider the Monge-Ampère equation

(4.5) (ω V + dd c ϕ) n = e λϕ µ h , for λ ∈ R and ϕ ∈ PSH(V, ω V ) ∩ L ∞ (V).
We scale h so that V dµ h = V ω n V . If c 1 (K V ) has a sign (i.e. it contains a multiple of a Kähler metric), then one can choose λ, h, ω V such that iΘ h (K V ) = λω V . Then, if ϕ is a solution of (4.5), the current ω ϕ := ω V + dd c ϕ satisfies Ric ω ϕ = -λω ϕ in the weak sense. Definition 4.8. -Let V be a compact Kähler space with log terminal singularities such that c 1 (K V ) has a sign. A Kähler-Einstein metric is a positive current of the form ω KE = ω V + dd c ϕ KE where ϕ KE ∈ PSH(V, ω V ) ∩ L ∞ (V) solves (4.5) and where (λ, h, ω V ) satisfies iΘ h (K V ) = λω V .

If λ ≥ 0, the equation (4.5) admits a unique (normalized) solution [START_REF] Eyssidieux | Singular Kähler-Einstein metrics[END_REF], but as in the smooth case the situation in the case λ < 0 is more complicated [BBE + 19]. In particular, if K V is ample of numerically trivial, then V admits a Kähler-Einstein metric. Moreover, it is showed in loc. cit. that any Kähler-Einstein metric ω KE induces a smooth Kähler metric on V reg .

The resolution of (4.5) goes as follows. Let π : X → V be a resolution of singularities of V and let dV X be a smooth volume form on X. The pull back measure π * µ h is well-defined and satisfies π * µ h = f dV X where f ∈ L p (dV X ) for some p > 1. Solving (4.5) is then equivalent to solving the degenerate complex Monge-Ampère equation

(4.6) (ω + dd c ψ) n = e λψ f dV X ,
where ω = π * ω V is semipositive but degenerate along the exceptional locus of π. An immediate corollary of Lemma 4.3 is the following.

Lemma 4.9. -Let V be a compact Kähler space of dimension n with log terminal singularities admitting a Kähler-Einstein metric ω KE . Let d KE be the geodesic distance induced by the Kähler metric ω KE | V reg . There exists α > 0 such that for any x ∈ V reg , we have

V reg e αd KE (•,x) ω n KE < +∞.

Proof. -Let π : X → V be a resolution of singularities of V, let E be the exceptional divisor of π and let ω X be a Kähler metric on X. The current T := π * ω KE induces a smooth Kähler metric on X \ E. We can see the function ρ x := d KE (•, x) as a Lipschitz function on X \ E which extends to L 1 function on X by Lemma 4.2. Moreover, Lemma 4.3 implies that e βρ x ∈ L 1 (X, ω n X ) for some β > 0. and

(dd c v) n = c n (χ • L) n-1 χ • L |z| 2n dV eucl = c n χ n-1 χ e -n• ) • L dV eucl
Diameter. -One can rewrite the Kähler form ω using the usual coordinates on C n as

ω = χ • √ -1 |z| 2 ∑ i,j δ ij - zi z j |z| 2 dz i ∧ d zj + χ • √ -1 |z| 2 ∑ i,j zi z j |z| 2 dz i ∧ d zj = ω s + ω r
where the composition with L has been omitted. The subscripts s (resp. r) stand for spherical (resp. radial). Indeed, the kernel of ω s in T 1,0 C n is generated by the holomorphic radial vector field ξ := ∑ n i=1 z i ∂ ∂z i while ω r has rank one and ξ 2 ω r = 1.

We claim that the rays through the origin of C n are geodesics for ω. Indeed, if x ∈ C n say of norm one and γ r (t) = tx for t ∈ [ε, 1], then we have γ r (t

) = t -1 (ξ + ξ) γ r (t) hence γ r (t) 2 ω = t -2 χ (L(γ r (t))
) and the length of γ r with respect to ω is

(γ r ) = 0 log ε χ (s)ds.
Now, let γ be any path connecting εx and x. Since dL ⊗ d c L = (d log r) 2 , we have

(γ) = 1 ε γ (t) ω dt ≥ 1 ε γ (t) ω r dt = 1 ε χ (L(γ(t)) |dr(γ (t))| r(γ(t)) dt ≥ 0 log ε χ (s)ds
where in the last line we have used |dr| ≥ dr and performed the change of variable s := r(γ(t)).

In particular, we find that (γ) ≥ (γ r ) which proves that γ r is a geodesic. Tracing back the inequalities, we actually see that any geodesic connecting εx and x must have γ (t) ω s = 0, hence it must coincide with γ r .

Next, if x, y ∈ B * , we can connect x to y using a radial ray, a spherical geodesic on S 2n-1 and another radial ray. Since ω is smooth away from the origin, this shows that the Riemannian metric associated to ω on B * has finite diameter if and only if

diam(B * , d ω ) < +∞ ⇐⇒ -∞ χ (s)ds < +∞.
Modulus of continuity. -We restrict ourselves to continuous potentials, i.e. we assume that χ(-∞) > -∞. We can normalize v without loss of generality so that v(0) = χ(-∞) = 0. Since v is smooth outside of the origin, its modulus of continuity is

m v (r) := sup t<log r χ(t).
Integrability condition (K). -The Monge-Ampère measure (dd c v) n = f dV satisfies Condition (K) if there exists an increasing function h such that +∞ h(t) -1 dt < +∞ and

B * f (log f ) n (h • log • log f ) n dV < +∞.
In order to simplify the next computation, we will make the assumption that χ(t) does not go to zero too fast at infinity. More precisely, we impose that χ (t), χ (t) ≥ e Ct near t = -∞ for some C > 0. This guarantees that log f ∼ log |z|, so that v satisfies Condition (K) iff

-∞ χ (t)(χ (t)) n-1 |t| n (h • log |t|) n dt < +∞.
Ricci lower bound. -As the computations are involved, we analyze whether the Ricci curvature of the above radial metrics is bounded below in a separate section 5.3. We will actually be interested in the refined question of whether the approximants ω ε =: dd c χ • log(|z| 2 + ε 2 ) have a lower bound on their Ricci curvature independent on ε.

Explicit examples. -

In what follows, we give four families of examples and check whether the various conditions mentioned above are satisfied.

1. Consider χ α (t) = exp(αt) for some α > 0.

We have Ric(ω α ) = -dd c log f = n(1α)dd c log |z| 2 . In particular, Ric ω α ≥ 0 if and only if α ≤ 1. Moreover, since χ tends to zero at -∞, one can easily see that the Ricci curvature of ω α is unbounded below when α > 1. Thus

• Condition (K) is satisfied; X f α | log f α | log(log( f α + 3)) p dV < +∞ for all p; • m v α (r) ∼ r α and 0 t -1 m v α (t)dt < +∞; • diam(B * , d ω α ) < +∞ for all α > 0; • Ric ω α ≥ -Aω α if and only if 0 < α ≤ 1.
2. Consider χ α (t) = (-t) -α for some α > 0.

We have

m v α (r) = (-log r) -α , hence 0 √ m v (r) r dr = 0 dr r(-log r) α/2 is finite if and only if α > 2. One has χ (k) (t) = k-1 ∏ j=0 (α + j) • (-t) -α-k so that with the notation from Section 5.3, λ = n -nα+n+1 (-log z 2 ) ≥ 0 for |z| 1 but µ χ = - n(α + 1) + 1 α(α + 1) • (-log |z| 2 ) α
which goes to -∞ when z → 0, hence Ric ω α is unbounded from below. Thus

• Condition (K) is satisfied; X f α | log f α | log(log( f α + 3)) p dV < +∞ for all p; • 0 t -1 m v α (t)dt < +∞ if and only if α > 2; • diam(B * , d ω α ) < +∞ for all α > 0;
• Ric ω α is unbounded from below.

3. Consider χ α (t) = (log(-t)) -α , where α > 0.

One computes χ α

(t) = α (-t)(log(-t)) 1+α and χ α (t) ∼ α t 2 (log(-t)) 1+α , so diam(B * , d α ) < +∞ ⇐⇒ -∞ dt |t|(log(-t)) 1+α 2 dt < +∞ ⇐⇒ α > 1. We obtain m v α (r) = (log(-log r)) -α hence 0 √ m vα (t) t dt = +∞. Observe that -∞ χ α (t)(χ α (t)) n-1 |t| n h(log |t|) n dt ∼ -∞ h(log |t|) n dt |t|(log |t|) n(1+α) = +∞ h(s) n ds s n+nα .
If nα > 1, then this integral converges for h(s) = s 1+ε as long as 0 < ε 1. Moreover, Hölder inequality with p = n+1 n and q = n + 1 yields:

+∞ = ∞ ds s = +∞ h(s) -n n+1 h(s) n n+1 s ds ≤ +∞ ds h(s) n n+1 +∞ h(s) n ds s n+1 1 n+1 hence +∞ h(s) n ds s n+nα is divergent as soon as nα ≤ 1: Condition (K) is satisfied iff α > 1/n.
A computation shows that for every k ≥ 1, one has

χ (k) α (t) = (k -1)! • α (-t) k (log(-t)) α+1 1 + O 1 log(-t) when t → -∞. The eigenvalue µ of Ric ω α satisfies µ χ α ∼ z →0 - n + 1 α • (log(-log |z| 2 )) α+1
which is unbounded below. In particular Ric ω α is unbounded from below. Thus • Ric ω α is unbounded from below. In particular we obtain X f 1 | log f 1 | n log(log( f 1 + 3)) p dV < +∞ for all p < 2n -1, while diam(B * , d ω 1 ) = +∞ and Condition (K) is satisfied (if n ≥ 2).

4.

Consider χ α (t) = -(log(-t)) α , where α > 0.

Of course in that case the potential is unbounded but there is interesting geometric behavior associated to these metrics. We have χ α (t) = α (-t)(log(-t)) 1-α , and χ α (t) ∼ s n(1-α) = +∞ since h ≥ 1 near +∞. Thus Condition (K) is never satisfied, as expected. However, the density f of ω α with respect to a smooth volume form dV X satisfies X f (log f ) n (log log( f + 3)) p dV X ∼ +∞ ds s n-nα-p which converges iff p < n -1nα. Taking 0 < α 1, we get a density such that f (log f ) n (log log( f + 3)) n-1-ε ∈ L 1 , which shows how close to sharp Condition (K) is.

As for the Ricci curvature, we will only treat the case n = α = 1 for simplicity, since we already see an interesting phenomenon appear. More precisely, if n = α = 1, then we are just considering the Poincaré metric on the punctured disk. Thus

• Condition (K) is never satisfied; v α is unbounded;

• diam(B * , d ω α ) = +∞ for all α > 0;

• Ric ω α = -ω α if α = n = 1.
The Ricci curvature of the smoothing ω ε := dd c (log(-log(|z| 2 + ε 2 ))), is however unbounded from below as ε → 0. Indeed set χ(t) =log(-t), F := ε 2 e -t ∈ (0, 1] and s := -t ≥ 0. Here, t = log(|z| 2 + ε 2 ). Next, set ψ := χ + (χχ )F, so that

ψ = 1 s 2 + F 1 s - 1 s 2 , ψ = 2 s 3 + F - 1 s + 2 s 2 - 2 s 3 , ψ = 6 s 4 + F 1 s - 3 s 2 + 6 s 3 - 6 s 4 .
From the computations in § 5.3, the Ricci curvature of ω ε is bounded from below uniformly if and only if we have C > 0 such that (5.1)

Fψ 2 + (1 -F)ψ 2 -Fψψ -(1 -F)ψψ ≥ -Cψ 3 .
Case 1. When F stays bounded away from 0 (i.e. very close to the origin). Then we have the asymptotics ψ ≈ F s , ψ ≈ -F s , ψ ≈ F s . The LHS of (5.1) is equivalent to 2F 3 s 2 hence the inequality is satisfied for large C.

Case 2. When F get close to zero (relatively far away from the origin). Brute force computations show that the asymptotics of the LHS are -2 s 6 -

F s 3 + 2F 3 s 2 .
Since ψ 3 ≈ 1 s 6 + F 3 s 3 , one has F s 3 F 3 s 3 . If we work near |z| 2 = ε 2 (log ε) α with α ∈ ( 1 2 , 3), then one can check that F s 3 max{ 1 s 6 , F 3 s 2 }. This shows that (5.1) is violated.

Lower bounds on the

ω ε = √ -1 |z| 2 + ε 2 ∑ i,j χ δ ij + (χ -χ ) • zi z j |z| 2 + ε 2 dz i ∧ d zj
where we write χ in place of χ • log(|z| 2 + ε 2 ) and similarly for χ . In view of the expression above, it is convenient to introduce the matrix

A(z) := zi z j |z| 2 + ε 2 ij
It is semipositive hermitian with rank one and its non-zero eigenvalue coincides with its trace, i.e. |z| 2 |z| 2 +ε 2 . With these notations, the Kähler form ω ε is associated to the hermitian matrix H(ω ε ) := 1 |z| 2 + ε 2 χ I + (χχ )A(z) .

Its eigenvalues are χ

|z| 2 +ε 2 with multiplicity n -1 and

1 |z| 2 + ε 2 χ + (χ -χ ) • |z| 2 |z| 2 + ε 2 = 1 |z| 2 + ε 2 χ + (χ -χ ) • ε 2 |z| 2 + ε 2 with multiplicity 1.
A useful computation is that of dd c log f , where f is positive non-decreasing and convex and as usual, we omit the composition with log(|z| 2 + ε 2 ). Then

dd c log f = dd c f f - d f ∧ d c f f 2 = √ -1 |z| 2 + ε 2 ∑ i,j f f δ ij + f -f f - f 2 f 2 • zi z j |z| 2 + ε 2 dz i ∧ d zj so that H(dd c log f ) = 1 (|z| 2 + ε 2 ) • f 2 [ f • f ] • I + [ f ( f -f ) -f 2 ] • A(z) .
Computation of the Ricci curvature. -The determinant of H(ω ε ) is given by det and we can decompose the hermitian matrix H(Ric ω ε ) associated to Ric ω ε as

H(ω ε ) = 1 (|z| 2 + ε 2 ) n • χ n-1 • χ + (χ -χ ) • ε 2 |z| 2 + ε 2
H(Ric ω ε ) = 1 |z| 2 + ε 2 (H 1 + H 2 + H 3 )
where H 1 = nI -nA(z) has eigenvalues λ 1 = n with multiplicity (n -1)

µ 1 = nε 2 |z| 2 +ε 2
with multiplicity 1 and

H 2 = -(n -1)(|z| 2 + ε 2 )H(dd c log χ ) = -(n -1) χ 2 [χ • χ ] • I + χ (χ -χ ) -χ 2 • A(z) .
has eigenvalues

   λ 2 = -(n -1) χ χ
with multiplicity (n -1)

µ 2 = -(n -1) χ χ -χ χ 2 -ε 2 |z| 2 +ε 2 χ χ -χ χ 2
with multiplicity 1

As for H 3 , it is convenient to set ψ(t) := χ (t) + (χ (t)χ (t)) • ε 2 e -t = χ (t)ε 2 (χ (t)e -t ) so that

H 3 = -(|z| 2 + ε 2 )H(dd c log ψ) = - 1 ψ 2 [ψ • ψ ] • I + ψ(ψ -ψ ) -ψ 2 • A(z) .
has eigenvalues

   λ 3 = ψ ψ
with multiplicity (n -1)

µ 3 = -ψ ψ -ψ ψ 2 -ε 2 |z| 2 +ε 2 ψ -ψ ψ -ψ ψ 2
with multiplicity 1 Therefore, the eigenvalues λ := ∑ 3 i=1 λ i and µ := ∑ 3 i=1 µ i with respective multiplicity (n -1) and 1 of

H 1 + H 2 + H 3 are λ = n -(n -1) χ χ + ψ ψ and µ = -(n -1) χ χ - χ χ 2 + ψ ψ - ψ ψ 2 + ε 2 |z| 2 + ε 2 n + (n -1) χ -χ χ - χ χ 2 + ψ -ψ ψ - ψ ψ

1. 2 .

 2 Diameter control by the modulus of continuity. -1.2.1. Dini-Campanato spaces. -Campanato-Morrey spaces provide a useful integral interpretation of Hölder continuity by means of uniform estimates on mean oscillations of the function.

  where D depends on A, B, C and K. Upper bounds on the diameter had been previously obtained under Ricci lower bound hypotheses by Pȃun [Pȃu01, Theorem 2], Tosatti [Tos09, Theorem 3.1], Zhang [RZ11, Theorem 2.1] and Fu-Guo-Song [FGS20] (see Corollary 2.7). Remark 2.4. -Let us formulate a couple of comments on the result above.

  consequences. -2.3.1. Infinite diameter or oscillation. -This first example exhibits a family of Kähler metrics with unbounded diameters, yet whose Ricci curvature is bounded below.

3. 1 .

 1 The special case of the projective space. -To explain the main idea in a simpler setting, we first treat the case of the projective space X = CP n . In this case H 1,1 (X, R) = R{ω FS } is one-dimensional and the Kähler cone is the half-line R + * {ω FS } generated by the Fubini-Study Kähler class. A rescaling argument thus reduces Theorem B to the following: Theorem 3.1. -Let ω = ω FS + dd c ϕ be a Kähler form cohomologous to the Fubini-Study form ω FS on CP n . Set f ω = ω n /ω n FS and fix A > 0 and p > 2n. There exists C(A, p)

  Finally, we have thanks to (3.14)

4. 3 .

 3 Application to singular Kähler-Einstein metrics. -4.3.1. Intrinsic distance. -We assume here that V ⊂ P N (C) is a closed irreducible analytic subvariety. We let Γ(V) denote the set of continuous arcs γ : [0, 1] → V that are piecewise C 1 -smooth, and set (γ) := 1 0 γ (t) ω FS dt. Definition 4.5. -For (x, y) ∈ V, we set

α t 2

 2 (log(-t)) 1-α . Since -∞ dt |t|(log(-t)) 1+α 2 dt = +∞, we infer that diam(B * , d α ) = +∞. Next, we have -∞ χ α (t)(χ α (t)) n-1 |t| n h(log |t|) n dt = -∞ h(log |t|) n (log |t|) nα dt |t| log n |t| = +∞ h(s) n ds

  Ricci curvature. -We analyze whether the Ricci curvature of smooth Kähler metrics approximating the radial examples is bounded below. Computation of the metric. -We set ω ε := dd c χ • log(|z| 2 + ε 2 ) and observe that (5.2)

  and therefore, the Ricci curvature of ω ε is given byRic ω ε = n dd c log(|z| 2 + ε 2 ) =:(I) -(n -1)dd c log χ =:(II) -dd c log χ + (χχ ) • ε 2 |z| 2 + ε 2 =:(III)

2

  and we have Ric ω ε ≥ -Cω ε for some constant C > 0 if and only ifλ ≥ -Cχ and µ ≥ -C(χ + (χχ ) • ε 2 |z| 2 + ε 2 ).

  Special case when ε = 0. -In order to show that the Ricci curvature of ω ε cannot be uniformly bounded from below, it is enough (though not sufficient in general) to show that the Ricci curvature of ω is unbounded from below. The computations become simpler since one has then λ = n -(n -1) χ χ + χ (3) has to disprove one of the following inequalities λ ≥ -Cχ or µ ≥ -Cχ .

-

  In honor of the breakthrough result of S.Kołodziej[START_REF] Kołodziej | The complex Monge-Ampère operator[END_REF], we introduce the following notion.

Definition 1.5. -A

  positive Radon measure µ on X satisfies Condition (K) if there exists dV X a smooth volume form, f ≥ 0 a Lebesgue-measurable function, w : R + → [1, +∞) a convex increasing weight s.t. µ = f dV X with X w • f dV X < +∞, where

	w(t) = t(log t) n (h • log • log(t + 3)) n and	+∞ dt h(t)	< +∞.

  • Condition (K) is satisfied if and only if α > 1/n; • X f α | log f α | n log(log( f α + 3)) p dV < +∞ iff p < n(1 + α) -1; • 0 t -1 m v α (t)dt = +∞ for all α > 0; • diam(B * , d ω α ) < +∞ if and only if α > 1;

Next, as we explained above, one can write π * ω KE n = f ω n X where f ∈ L p (X) for some p > 1. If q is the conjugate exponent of p, and α := β/q, we have by Hölder inequality X e αρ x π * ω n KE ≤ ( X e βρ x ω n X ) 1/q ( X f p ω n X ) 1/p and the lemma follows.

-It is known that the local potentials ϕ KE are often continuous near V sing (see [START_REF] Guedj | Continuity of singular kähler-einstein potentials[END_REF]). However examples from Section 5 show that an extra information on the modulus of continuity is required, in order to get control on the diameter of (V reg , ω KE ).

When ω is a Kähler form on X, it follows from [Koł08, DDG + 14] that the unique (normalized) solution to (4.6) is Hölder continuous. The Kähler-Einstein potentials ϕ KE are also Hölder continuous if the singularities V sing are quotient. If V sing consists of isolated and globally smoothable ordinary double points, it has been shown by Hein-Sun [START_REF] Hein | Calabi-Yau manifolds with isolated conical singularities[END_REF] that ω KE is asymptotic to the Stenzel metric dd c |z| 2 n-1 n , whose potential is also Hölder continuous (see [START_REF] Chiu | Higher regularity for singular Kähler-Einstein metrics[END_REF] for some recent generalization).

For all these reasons we conjecture the following result.

Conjecture 4.10. -The Kähler-Einstein potentials ϕ KE are Hölder continuous on V.

Here, Hölder continuous is meant with respect to the euclidean distance induced by local embeddings V → C N . A straightforward application of Theorem 4.4 (or simply its weaker version mentioned a few lines above it) allows one to establish the finiteness of diam(V reg , ω KE ) assuming Conjecture 4.10.

Theorem 4.11. -If Conjecture 4.10 is correct, then diam(V reg , ω KE ) < +∞.

More generally, we conjecture that any solution of (4.5) is Hölder continuous. Note that if ϕ solves (4.5), then ω ϕ := ω V + dd c ϕ induces a Kähler metric on V reg . Similarly to Theorem 4.11, the latter conjecture would imply that the diameter of (V reg , ω ϕ ) is finite.

Proof. -Let π : X → V be a log resolution of V and let ω X be a Kähler form on X such that ω X ≥ π * ω V . Let T := π * ω KE = π * ω V + dd c (π * ϕ). The closed positive current T is a Kähler form outside of the exceptional locus E of π. On X 2 , we have d ω X ≥ π * d ω V . Moreover, under Conjecture 4.10, the function ϕ is Hölder continuous with respect to d ω V , hence π * ϕ is Hölder continuous with respect to d ω X . The result now follows from Theorem 4.4 applied to the current T.

Remark 4.12. -Conjecture 4.10 is very strong and we do not need its full force to derive finiteness of the diameter of (V reg , ω KE ). Indeed, the proof above shows that it would be enough to prove that for a log resolution π : X → V of V, the function π * ϕ has a modulus of continuity m ϕ satisfying m ϕ (r) (log log(-r)) -1-δ for some δ > 0.

Examples

Radial

Examples. -We assume here that the quasi-psh functions ϕ under consideration are smooth in X \ {p}. We choose a local chart biholomorphic to the unit ball B of C n , with p corresponding to the origin. We further assume that ϕ has a radial singularity at p, i.e. it is invariant under the group U(n, C) near p, and so is ω. The singularity type of ϕ only depends on its local behavior near p.

We thus consider the local situation of a psh function v = χ • L, where χ : R -→ R -is a smooth strictly convex increasing function and L : z ∈ B → log |z| 2 ∈ R -. A standard computation shows that