Tamás Darvas

Eleonora Di Nezza

Chinh H Lu

The metric geometry of singularity types

HAL is

Introduction and main results

Let (X, ω) be a Kähler manifold with a smooth closed (1, 1)-form θ. An integrable real-valued potential u belongs to PSH(X, θ) if locally u is the sum of a smooth function and a plurisubharmonic function, moreover θ + i∂ ∂u ≥ 0 in the sense of currents. The set PSH(X, θ) has plenty of members when {θ} ∈ H 2 (X, C) is big, an assumption we will make throughout the paper.

Two potentials u, v ∈ PSH(X, θ) have the same singularity type if and only if there exists C ∈ R such that u -C ≤ v ≤ u + C. This is easily seen to yield an equivalence relation, whose equivalence classes [w], w ∈ PSH(X, θ) give rise to the space of singularity types S(X, θ). This latter space plays an important role in transcendental algebraic geometry, as its elements represent the building blocks of multiplier ideal sheaves, log-canonical thresholds, etc., bridging the gap between the algebraic and the analytic viewpoint on the subject. We refer to the survey [START_REF] Demailly | On the cohomology of pseudoeffective line bundles[END_REF] and references therein for insight into this ever expanding circle of ideas.

The space PSH(X, θ) has a natural complete metric space structure given by the L 1 metric. However the L 1 metric does not naturally descend to S(X, θ) making the study of variation of singularity type quite awkward and cumbersome. Indeed, reviewing the literature, "convergence of singularity types" is only discussed in an ad-hoc manner, under stringent conditions on the potentials involved.

On the other hand, "approximating" an arbitrary singularity type [u] with one that is much nicer goes back to the beginnings of the subject. Perhaps the most popular of these approximation procedures is the one that uses Bergman kernels, as first advocated in this context by Demailly [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF]. Here, using Ohsawa-Takegoshi type theorems one obtains a (mostly decreasing) sequence [u j,B ] that in favorable circumstances approaches [u] in the sense that multiplier ideal sheaves, log-canonical thresholds, vanishing theorems, intersection numbers etc. can be recovered in the limit (see for example [START_REF] Boucksom | On the volume of a line bundle[END_REF][START_REF] Boucksom | Divisorial Zariski decompositions on compact complex manifolds[END_REF][START_REF] Demailly | Numerical characterization of the Kähler cone of a compact Kähler manifold[END_REF][START_REF] Cao | Numerical dimension and a Kawamata-Viehweg-Nadel-type vanishing theorem on compact Kähler manifolds[END_REF][START_REF] Demailly | On the cohomology of pseudoeffective line bundles[END_REF] and references therein). Still, no metric topology seems to be known that could quantify the effectiveness or failure of the "convergence" [u j,B ] → [u] (or that of other approximating sequences, for example the transcendental Bergman kernels suggested in [START_REF] Berman | From Monge-Ampère equations to envelopes and geodesic rays in the zero temperature limit[END_REF]). In this work we propose an alternative remedy to this.

We introduce a natural (pseudo)metric d S on S(X, θ) and point out that it fits well with some already existing approaches in the literature. The precise definition of d S uses the language of geodesic rays from [START_REF] Darvas | L 1 metric geometry of big cohomology classes[END_REF][START_REF] Darvas | Geodesic stability, the space of rays, and uniform convexity in Mabuchi geometry[END_REF] and is delayed until Section 3, however for the sake of a gentle Applications to multiplier ideal sheaves. For [v] ∈ S(X, θ) we denote by J [v] the multiplier ideal sheaf associated to the singularity type [v]. Recall that J [v] is the sheaf of germs of holomorphic functions f such that |f | 2 e -v is locally integrable on X. Providing a positive answer to the Demailly strong openness conjecture [START_REF] Demailly | Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds[END_REF], Guan-Zhou have shown that for any u j , u psh such that u j u a.e. we have that J [u j ] = J [u] for j ≥ j 0 [START_REF] Guan | A proof of Demailly's strong openness conjecture[END_REF][START_REF] Guan | Effectiveness of Demailly's strong openness conjecture and related problems[END_REF], with a partial result obtained earlier by Berndtsson [START_REF] Berndtsson | The openness conjecture and complex Brunn-Minkowski inequalities[END_REF] (see also [START_REF] Demailly | On the cohomology of pseudoeffective line bundles[END_REF][START_REF] Hiep | The weighted log canonical threshold[END_REF][START_REF] Lempert | Modules of square integrable holomorphic germs, Analysis meets geometry[END_REF] for related results). Below we extend the scope of this theorem to the global context, providing a result that uses d S -convergence and avoids the condition u j ≤ u: Theorem 1.3. Let [u], [u j ] ∈ S(X, θ), j ≥ 0, such that d S ([u j ], [u]) → 0. Then there exists j 0 ≥ 0 such that J [u] ⊆ J [u j ] for all j ≥ j 0 .

The proof of this theorem involves an application of Theorem 1.2 and the local Guan-Zhou result for increasing sequences [START_REF] Guan | A proof of Demailly's strong openness conjecture[END_REF][START_REF] Guan | Effectiveness of Demailly's strong openness conjecture and related problems[END_REF]. Lastly, since u j ≤ u trivially gives J [u j ] ⊆ J [u], together with the d S -convergence criteria of Lemma 4.1, our theorem contains the global version of the Guan-Zhou result for increasing sequences of θ-psh potentials.

Motivated by a possible local analog of Theorem 1.3 it would be interesting to see if a local version of the d S metric exists on the space of singularity types of local psh potentials.

Note that equality in the inclusion J [u] ⊆ J [u j ] of Theorem 1.3 can not be expected in general. Indeed, d S ([λu], [u]) → 0 as λ 1 for any u ≤ 0, however if u has log type singularity at some x ∈ X, but is locally bounded on X \ {x}, then

J [u] J [λu] = O X , λ ∈ (0, 1).
Applications to variation of complex Monge-Ampère equations. Finally, we turn to the application that motivated our introduction of the d S -topology.

In a series of works [DDL2, DDL3, DDL4] the authors studied solutions to equations of complex Monge-Ampère type with prescribed singularity. In a nutshell, one starts with a potential φ ∈ PSH(X, θ) and a density 0 ≤ f ∈ L p (X), p > 1, and is looking for a solution ψ ∈ PSH(X, θ) such that θ n ψ = f ω n and [ψ] = [φ]. By [START_REF] Witt-Nyström | Monotonicity of non-pluripolar Monge-Ampère measures[END_REF] the condition X θ n φ = X f ω n > 0 is necessary for the solvability of this equation. Beyond this normalization condition, as it turns out, the necessary and sufficient condition for the well posedness is that [φ] satisfies [φ] = [P [φ]], where

P [φ] := sup{v ∈ PSH(X, θ) : [v] ≤ [u], v ≤ 0}.
Singularity types [φ] satisfying the above condition are of model type, and they appear in many natural contexts, as described in [START_REF] Darvas | Monotonicity of non-pluripolar products and complex Monge-Ampere equations with prescribed singularity[END_REF].

One might ask the question, what happens if one considers a family of such equations, where the prescribed singularity type [φ j ] converges to some fixed singularity type [φ]. In our next result we obtain that in such a case, the solutions ψ j converge to ψ in capacity as expected, further evidencing the practicality of the d S -topology: Theorem 1.4. Given δ > 0 and p > 1 suppose that:

• [φ j ], [φ] ∈ S δ (X, θ), j ≥ 0 satisfy [φ j ] = [P [φ j ]], [φ] = [P [φ]] and d S ([φ j ], [φ]) → 0. • f j , f ≥ 0 are such that f L p , f j L p , p > 1, are uniformly bounded and f j → L 1 f . • ψ j , ψ ∈ PSH(X, θ), j ≥ 0 satisfy sup X ψ j = 0, sup X ψ = 0 and θ n ψ j = f j ω n [ψ j ] = [φ j ] , θ n ψ = f ω n [ψ] = [φ].
Then ψ j converges to ψ in capacity, in particular ψ j -ψ L 1 → 0.

Organization. In Section 2 we recall several results in relative pluripotential theory developed recently by the authors. The metric d S along with its basic properties are introduced in Section 3. Theorem 1.1 is proved in Section 4 where an example is also given showing that the positive mass condition is necessary. Theorem 1.2 is proved in Section 5, Theorem 1.3 is proved in Section 6, and Theorem 1.4 is proved in Section 7.

Acknowledgments. The first named author has been partially supported by NSF grants DMS-1610202 and DMS-1846942(CAREER). This work was finished while the authors participated in the "Research in Paris" program of Institut Henri Poincaré, and we would like to thank the institute for the hospitality and support. We thank the referees for the useful remarks/comments/clarifications which improved the presentation of the paper.

Preliminaries

In this section we recall terminology and relevant results from the literature with focus on the works [DDL1, DDL2, DDL3, DDL4], as well as [START_REF] Darvas | Geodesic stability, the space of rays, and uniform convexity in Mabuchi geometry[END_REF]. We also point out some differences and extend the scope of some results whenever necessary.

Model potentials and relative full mass classes

Let (X, ω) be a compact Kähler manifold of dimension n and fix θ a smooth closed (1, 1)-form whose cohomology class is big. Our notation is taken from [DDL2, DDL3, DDL4] and we refer to these works for further details.

A function u : X → R ∪ {-∞} is called quasi-plurisubharmonic (quasi-psh) if locally u = ρ + ϕ, where ρ is smooth and ϕ is a plurisubharmonic (psh) function. We say that u is θ-plurisubharmonic (θ-psh) if it is quasi-psh and θ u := θ+i∂ ∂u ≥ 0 in the weak sense of currents on X. We let PSH(X, θ) denote the space of all θ-psh functions on X which are not identically -∞. The class {θ} is big if there exists ψ ∈ PSH(X, θ) satisfying θ + i∂ ∂ψ ≥ εω for some ε > 0. By the fundamental approximation theorem of Demailly [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF], if {θ} is big there are plenty of θ-psh functions.

Given u, v ∈ PSH(X, θ), we say that • u is more singular than v, i.e., u v, if there exists C ∈ R such that u ≤ v + C;

• u has the same singularity as v, i.e., u v, if u v and v u.

The classes [u] ∈ S(X, θ) of this latter equivalence relation are called singularity types. When θ is non-Kähler, all elements of PSH(X, θ) are quite singular, and we distinguish the potential with the smallest singularity type in the following manner:

V θ := sup{u ∈ PSH(X, θ) such that u ≤ 0}.
A function u ∈ PSH(X, θ) is said to have minimal singularity if it has the same singularity type as

V θ , i.e., [u] = [V θ ].
Given θ 1 , ..., θ n smooth closed (1, 1)-forms and ϕ j ∈ PSH(X, θ j ), j = 1, ...n, following Bedford-Taylor [START_REF] Bedford | The Dirichlet problem for a complex Monge-Ampère equation[END_REF][START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF] in the local setting, it has been shown in [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF] that the sequence of positive measures

1 j {ϕ j >V θ j -k} θ 1 max(ϕ 1 ,V θ 1 -k) ∧ . . . ∧ θ n max(ϕn,V θ n -k) (1)
has total mass (uniformly) bounded from above and is non-decreasing in k ∈ R, hence converges weakly to the so called non-pluripolar product

θ 1 ϕ 1 ∧ . . . ∧ θ n ϕn .
The resulting positive measure does not charge pluripolar sets. In the particular case when ϕ 1 = ϕ 2 = . . . = ϕ n = ϕ and θ 1 = ... = θ n = θ we will call θ n ϕ the non-pluripolar Monge-Ampère measure of ϕ, which generalizes the usual notion of volume form in case θ ϕ is a smooth Kähler form.

An important property of the non-pluripolar product is that it is local with respect to the plurifine topology (see [START_REF] Bedford | Fine topology, Silov boundary, and (i∂ ∂) n[END_REF]Corollary 4.3],[BEGZ10, Section 1.2]). For convenience we record the following version for later use.

Lemma 2.1. Fix closed smooth big (1, 1)-forms θ 1 , ..., θ n . Assume that ϕ j , ψ j , j = 1, ..., n are θ j -psh functions such that ϕ j = ψ j on U an open set in the plurifine topology. Then

1 U θ 1 ϕ 1 ∧ ... ∧ θ n ϕn = 1 U θ 1 ψ 1 ∧ ... ∧ θ n ψn .
Lemma 2.1 will be referred to as the plurifine locality. For practice we note that sets of the form {u < v}, where u, v are quasi-psh functions, are open in the plurifine topology.

As a consequence of Bedford-Taylor theory, the measures in (1) all have total mass less than X θ n V θ , in particular, after letting k → ∞ we notice that X θ n ϕ ≤ X θ n V θ . In fact it was proved in [WN19, Theorem 1.2] that for any u, v ∈ PSH(X, θ) the following monotonicity result holds for the masses:

v u =⇒ X θ n v ≤ X θ n u .
This result was extended in [START_REF] Darvas | Monotonicity of non-pluripolar products and complex Monge-Ampere equations with prescribed singularity[END_REF] for non-pluripolar products building on the following fundamental convergence property.

Theorem 2.2. Let θ j , j ∈ {1, . . . , n} be smooth closed (1, 1)-forms on X whose cohomology classes are big. Suppose that for all j ∈ {1, . . . , n} we have u j , u k j ∈ PSH(X, θ j ) such that u k j → u j in capacity as k → ∞. If χ k ≥ 0 is a sequence of uniformly bounded quasi-continuous functions which converges in capacity to a quasi-continuous function χ ≥ 0, then

lim inf k→+∞ X χ k θ 1 u k 1 ∧ . . . ∧ θ n u k n ≥ X χθ 1 u 1 ∧ . . . ∧ θ n un . (2)
If additionally,

X θ 1 u 1 ∧ . . . ∧ θ n un ≥ lim sup k→∞ X θ 1 u k 1 ∧ . . . ∧ θ n u k n , (3)
then θ 1 u k 1 ∧ . . . ∧ θ n u k n weakly converges to θ 1 u 1 ∧ . . . ∧ θ n un .
Note that this result is slightly more general than [DDL2, Theorem 2.3] but the proof is the same. Shadowing Bedford-Taylor theory [BT82, BT87], the above convergence and monotonicity results opened the door to the development of relative finite energy pluripotential theory, whose terminology we now partially recall from [DDL2, Sections 2-3].

The relative full mass classes E(X, θ, φ). Fixing φ ∈ PSH(X, θ) one can consider only θ-psh functions that are more singular than φ. Such potentials form the set PSH(X, θ, φ). Since the map [u] → X θ n u is monotone increasing, but not strictly increasing, it is natural to consider the set of φ-relative full mass potentials:

E(X, θ, φ) := u ∈ PSH(X, θ, φ) such that X θ n u = X θ n φ .
Naturally, when v ∈ PSH(X, θ, φ) we only have X θ n v ≤ X θ n φ . As pointed out in [DDL2, DDL4], when studying the potential theory of the above space, the following well known envelope constructions are of great help:

P θ (ψ, χ), P θ [ψ](χ), P θ [ψ] ∈ PSH(X, θ).
In the context of Kähler geometry these were introduced by Ross and Witt Nyström [RWN14], using slightly different notation. Given any f : X → [ -∞, +∞] the starting point is the envelope P θ (f ) := usc(sup{v ∈ PSH(X, θ), v ≤ f }). Then, for ψ, χ ∈ PSH(X, θ) we can introduce the "rooftop envelope" P θ (ψ, χ) := P θ (min(ψ, χ)). This allows us to further introduce

P θ [ψ](χ) := usc lim C→+∞ P θ (ψ + C, χ) .
It is easy to see that P θ [ψ](χ) depends on the singularity type [ψ]. When χ = V θ , we will simply write P [ψ] := P θ [ψ] := P θ [ψ](V θ ) and call this potential the envelope of the singularity type

[ψ]. It follows from [DDL2, Theorem 3.8], [Ber18], [GLZ17] that θ n P [ψ] ≤ 1 {P [ψ]=0} θ n . Also, by [DDL3, Proposition 2.3 and Remark 2.5] we have that X θ n P [ψ] = X θ n ψ . Using such envelopes, in [DDL2, Theorem 1.3] we characterized membership in E(X, θ, φ): Theorem 2.3. Suppose φ ∈ PSH(X, θ) and X θ n φ > 0. Then u ∈ E(X, θ, φ) if and only if u ∈ PSH(X, θ, φ) and P [u] = P [φ].
For further results about the connection of envelopes and relative full mass classes we refer to [DDL2, Section 3].

The ceiling operator and model potentials. We consider the ceiling operator C : PSH(X, θ) → PSH(X, θ) defined by

C(u) := usc(sup F u ),
where

F u := v ∈ PSH(X, θ) : [u] ≤ [v], v ≤ 0, X θ k v ∧ θ n-k V θ = X θ k u ∧ θ n-k V θ , k ∈ {0, ..., n} . (4)
As it turns out, there is no reason to take the upper semi-continuous regularization in the definition above, as C(u) is a candidate in its defining family F u . This is confirmed by the next lemma.

Lemma 2.4. Assume that u ∈ PSH(X, θ) and u ≤ 0. Then

C(u) = lim ε→0 + P [(1 -ε)u + εV θ ] ∈ F u . (5) In particular, if φ, ψ ∈ PSH(X, θ) with [φ] ≤ [ψ] then C(φ) ≤ C(ψ), i.e., C is monotone increasing. Proof. We have that lim ε→0 + ((1 -ε)u + εV θ ) = u and that u ε := P [(1 -ε)u + εV θ ] ≥ P [u] ≥ u is decreasing as ε → 0+.
We denote by u 0 its limit. Next, we claim that u 0 ∈ F u . First, observe that since u 0 ≥ u, by [DDL2, Theorem 1.1] we have that, fixing k ∈ {1, 2, ..., n},

where in the first inequality we used [DDL2, Theorem 1.1], in the first equality we used [DDL2, Proposition 2.1, Theorem 2.2] and the definition of P [(1 -ε)u + εV θ ], and in the second equality we used the multilinearity of the non-pluripolar product. Letting ε → 0 we prove the claim, hence u 0 ≤ C(u).

On the other hand, as (1 -ε)u + εV θ satisfies the non-collapsing condition

X θ n (1-ε)u+εV θ ≥ ε n X θ n V θ > 0, it follows from [DDL2, Remark 2.5] and [DDL2, Theorem 3.12] that P [(1 -ε)u + εV θ ] is the maximal element of F (1-ε)u+εV θ .
Due to multilinearity of non-pluripolar products, it follows from the above definition of

F u that (1 -ε)F u + εV θ ⊂ F (1-ε)u+εV θ . (6)
This implies that (1

-ε)C(u) + εV θ ≤ P [(1 -ε)u + εV θ ]. Hence letting ε → 0 + we obtain C(u) ≤ u 0 ,
proving the first statement. The last statement follows from (5) together with the fact that if

[φ] ≤ [ψ] then P [(1 -ε)φ + εV θ ] ≤ P [(1 -ε)ψ + εV θ ].
In this work, we say that a potential φ ∈ PSH(X, θ) is a model potential if φ = C(φ), i.e., if φ is a fixed point of C. Similarly, the corresponding singularity types [φ] are called model type singularities. We note that this definition is seemingly different from the one in [DDL2, [START_REF] Darvas | L 1 metric geometry of big cohomology classes[END_REF][START_REF] Darvas | Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity[END_REF], where we said that φ is model in case φ = P [φ]! Thankfully, this inconsistency will cause little to no disruption: in the important particular case of non-vanishing mass, i.e. X θ n φ > 0, [DDL2, Remark 2.5, Theorem 3.12] gives that P [φ] = C(φ), hence these two definitions are indeed the same. We predict that this is the case in general as well:

Conjecture 2.5. For any φ ∈ PSH(X, θ) we have that P [φ] = C(φ).

To finish this paragraph we list and prove a number of basic properties of the ceiling operator: Proposition 2.6. Suppose v ∈ PSH(X, θ). The following hold:

(i) if X θ n v > 0 then C(v) = P [v]. (ii) lim ε→0 C((1 -ε)v + εV θ ) = lim ε→0 P [(1 -ε)v + εV θ ] = C(v). (iii) C(C(v)) = C(v). (iv) C(P [v]) = C(v).
Proof. The first property is a consequence of [DDL2, Theorem 2.5, Theorem 3.12]. The statement in (ii) follows from Lemma 2.4 together with (i). To prove (iii) we note that

F C(v) ⊂ F v implies that C(C(v)) ≤ C(v) (via Lemma 2.4). Since v ≤ C(v), the other inequality in (iii) follows from monotonicity of C. Note that [DDL2, Remark 2.5] implies that P [v] ∈ F v , hence v ≤ P [v] ≤ C(v).
Applying C to these inequalities together with (iii) gives (iv).

The Monge-Ampère energy I and the finite energy class E 1 (X, θ). As further evidenced by the next lemma, potentials with model type singularity play a distinguished role in the theory (see [DDL4, Lemma 2.2] for a more precise result):

Lemma 2.7. Let φ ∈ PSH(X, θ) and φ = P [φ]. Then, for all u ∈ PSH(X, θ, φ) we have sup X (uφ) = sup X u, and the set

F := {u ∈ PSH(X, θ, φ) : sup X (u -φ) = 0}
is relatively compact in the L 1 -topology of potentials.

Proof. Let u ∈ PSH(X, θ, φ). Since φ ≤ 0 it follows that sup X (u -φ) ≥ sup X u. For the other direction we notice that u -sup X u ≤ φ, hence sup X (u -φ) ≤ sup X u. Relative compactness of F then follows from [GZ17, Proposition 8.5].

We define the Monge-Ampère energy of any θ-psh function u with minimal singularities as

I(u) := 1 n + 1 n k=0 X (u -V θ )θ k u ∧ θ n-k V θ .
We then define the Monge-Ampère energy for arbitrary u ∈ PSH(X, θ) as

I(u) := inf{I(v) : v ∈ PSH(X, θ
), v has minimal singularities, and u ≤ v}.

We let E 1 (X, θ) denote the set of all u ∈ PSH(X, θ) such that I(u) is finite. Since θ will be fixed throughout the paper we will occasionally denote this space simply as E 1 . In the next theorem we collect basic properties of the Monge-Ampère energy:

Theorem 2.8. Suppose u, v ∈ E 1 (X, θ). The following hold:

(i) I(u) -I(v) = 1 n+1 n k=0 X (u -v)θ k u ∧ θ n-k v . (ii) If u ≤ V θ then, X (u -V θ )θ n u ≤ I(u) ≤ 1 n+1 X (u -V θ )θ n u .
(iii) I is non-decreasing and concave along affine curves. Additionally, the following estimates hold:

X (u -v)θ n u ≤ I(u) -I(v) ≤ X (u -v)θ n v .
In particular, we observe that u ≤ v implies I(u) ≤ I(v). We refer to [DDL3, Theorem 2.1 and Proposition 2.2] for a proof.

Lemma 2.9. Let u, v ∈ E 1 (X, θ) be such that u ≥ v and I(u) = I(v). Then u = v.

Proof. Since I(u) = I(v), and u ≥ v, Theorem 2.8(i) implies that X (u -v)θ n v = 0. We then have θ n v (u > v) = 0, hence the domination principle [DDL1, Proposition 2.4] gives u = v.

We recall that, given u, v ∈ E 1 (X, θ), it has been shown in [DDL1, Theorem 2.10] that P (u, v) belongs to E 1 (X, θ) as well. As in [DDL3, Section 3] we define:

d 1 (u, v) = I(u) + I(v) -2I(P (u, v)). (7)
By [DDL3, Theorems 1.1] the space (E 1 (X, θ), d 1 ) is a complete geodesic metric space whose geodesic segments arise as d 1 -limits to solutions to a degenerate complex Monge-Ampère equation (they are sometimes referred to as finite energy geodesics). Also, the Monge-Ampère energy is linear along these geodesics. In the following we adapt some results of [START_REF] Berman | Convexity of the extended K-energy and the long time behaviour of the Calabi flow[END_REF] and [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF] to the big setting.

Proposition 2.10. Suppose [0, 1] t → u t , v t ∈ E 1 are finite energy geodesics. Then the maps t → I(u t ), t → I(v t ) are linear and the map t → I(P (u t , v t )) is concave. Consequently, the map

t → d 1 (u t , v t ) is convex.
Proof. The first statement is a consequence of [DDL1, Theorem 3.12]. The rest follows using the same arguments as in the Kähler case [BDL15, Proposition 5.1], relying on the analogous results from the big case in [DDL1, Theorems 2.10, 3.12 and Proposition 3.2].

Next we point out that the d 1 -geodesics are "endpoint stable":

Proposition 2.11. Let [0, 1] t → u j t ∈ E 1 be a sequence of finite energy geodesic segments such that d 1 (u j 0 , u 0 ), d 1 (u j 1 , u 1 ) → 0. Then d 1 (u j t , u t ) → 0 for all t, where [0, 1] t → u t ∈ E 1 is the finite energy geodesic segment connecting u 0 , u 1 .

Proof. Exactly the same as [BDL15, Proposition 4.3].

Proposition 2.12. Given u, v ∈ E 1 , we have |I(u) -

I(v)| ≤ d 1 (u, v).
Proof. We observe that by the monotonicity of the energy (Theorem 2.8(i)) and since P (u, v) ≤ min(u, v) we have I(P (u, v)) ≤ min(I(u), I(v)). The result then follows directly from the definition of d 1 .

Lastly, we point out the analog of [Dar15, Remark 5.6]:

Proposition 2.13. Given u, v ∈ E 1 , there exists C = C(n) > 1 such that d 1 (u, v) ≤ d 1 (u, max(u, v)) + d 1 (max(u, v), v) ≤ Cd 1 (u, v).
Proof. The first inequality follows from the triangle inequality. The second inequality follows from [DDL3, Theorem 3.7 and eq. ( 6)].

The chordal geometry of geodesic rays

By R(X, θ) we denote the space of finite energy geodesic rays emanating from V θ :

R(X, θ) := {[0, ∞) t → u t ∈ E 1 (X, θ) s.t. u 0 = V θ and t → u t is a d 1 geodesic ray}.
As shorthand convention we will use the notation {u t } t ∈ R(X, θ) when referring to rays.

According to the constructions in [START_REF] Darvas | L 1 metric geometry of big cohomology classes[END_REF] the above space has plenty of elements. In [DL18, Section 4] the first and last author carried out a detailed analysis of the space of L p geodesic rays in the Kähler case. For similar flavour results in the non-Archimedean context we refer to [START_REF] Berman | A variational approach to the Yau-Tian-Donaldson conjecture[END_REF] and references therein.

Given that our main focus here is on the space of singularity types, in the present paper we only focus on the basic analysis of the space of L 1 rays in the more general case of big cohomology classes.

In case of the Euclidean topology of R n , the space of (unit speed) geodesic rays emanating from the origin, is just the collection of half lines emanating from 0, that can be identified with the unit sphere. Inspired by this simple analogy and the chordal metric structure on the sphere, we introduce the chordal L 1 geometry on R(X, θ):

d c 1 ({u t } t , {v t } t ) := lim t→∞ d 1 (u t , v t ) t . (8)
Note that by Proposition 2.10 it follows that t → d 1 (u t , v t ) is convex, hence the map t → d 1 (u t , v t )/t is increasing, implying that the limit in (8) is well defined. We also note the following theorem:

Theorem 2.14. The space (R(X, θ), d c 1 ) is a complete metric space.

One expects that actually (R(X, θ), d c 1 ) is a geodesic metric space, as in the Kähler case [DL18, Theorem 1.4].

Proof. The triangle inequality is inherited from the d 1 -geometry of E 1 (X, θ). To show that d c 1 is non-degenerate, suppose d c 1 ({u t } t , {v t } t ) = 0 and fix t 0 > 0. By Proposition 2.10 we have:

d 1 (u t 0 , v t 0 ) t 0 ≤ lim t→∞ d 1 (u t , v t ) t = 0. (9)
Since d 1 is non-degenerate, we obtain that u t 0 = v t 0 and since t 0 > 0 was arbitrary, we obtain that the geodesics t → u t and t → v t are the same. Now we argue that (R(X, θ), d c 1 ) is complete. Let {t → u k t } k be a d c 1 -Cauchy sequence. For fixed t 0 > 0 by (9) we deduce that {u k t 0 } k is a d 1 -Cauchy sequence in (E 1 (X, θ), d 1 ). By the completeness of the latter space there exists u t 0 ∈ E 1 (X, θ) such that d 1 (u k t 0 , u t 0 ) → 0 as k → +∞. As a result we obtain a "limit curve" [0, ∞) t → u t ∈ E 1 (X, θ). By Proposition 2.11 we obtain that t → u t is in fact a finite energy geodesic ray emanating from u 0 = V θ .

To finish the proof, we have to argue that d c 1 ({u k t }, {u t }) → 0. Fix ε > 0. By (9) there exists j ε > 0 such that d 1 ({u k t }, {u l t }) < εt for all t > 0 and k, l > j ε . Letting k → ∞, we obtain that d 1 ({u t }, {u l t }) < εt, hence d c 1 ({u t } t , {u l t } t ) < ε for all l > j ε , finishing the proof.

The radial Monge-Ampère energy of R(X, θ). For {u t } t ∈ R(X, θ) it is natural to introduce the radial Monge-Ampère energy I{•} : R(X, θ) → R by the formula I{u t } = lim t I(ut) t

= I(u 1 ), where in the last equality we have used the linearity of I along geodesic rays Theorem 2.10. By the d 1 -Lipschitz property of I on E 1 (X, θ) (Proposition 2.12) and the fact that the map t → d 1 (u t , v t )/t is increasing it follows that:

I{u t } -I{v t } ≤ d c 1 ({u t } t , {v t } t ), {u t } t , {v t } t ∈ R(X, θ). (10)
The metric decomposition inequality of R(X, θ). The analog of the Pythagorean formula holds for the space (R(X, θ), d c 1 ) (see [Xia19, Example 3.2] for the argument in the Kähler case that translates easily to our context as well). However this formula does not descend to S(X, θ). Out of this reason, we derive the radial analog of Proposition 2.13 instead. Though perhaps not as "flashy" as the Pythagorean formula, this "decomposition inequality" has a number of similar consequences, and it also descends to S(X, θ) as well (see Section 3 below).

First we define the maximum of two geodesic rays {u t } t , {v t } t ∈ R(X, θ). This is simply the smallest ray {h t } t ∈ R(X, θ) that lies above both {u t } t , {v t } t . It is elementary to see that such a ray does exist. Indeed, h t = usc(lim l→∞ w l t ), where [0, l] t → w l t ∈ E 1 (X, θ) is the finite energy geodesic segment joining w l 0 = V θ and w l l = max(u l , v l ). Since {max(u t , v t )} t is a subgeodesic ray, by the comparison principle [DDL1, Proposition 3.3] it can be seen that each sequence {w l t } l is increasing, proving that {h t } t is indeed a geodesic ray. By construction this ray has to be the smallest ray lying above {u t } t , {v t } t , hence it makes sense to introduce the notation: max R (u t , v t ) := h t , t ≥ 0.

Next we show that the rays u t , v t , max R (u t , v t ) satisfy a "metric decomposition inequality": Proposition 2.15. There exists C > 1 such that, for all {u t } t , {v t } t ∈ R(X, θ),

d c 1 ({u t } t , {v t } t ) ≤ d c 1 ({u t } t , {max R (u t , v t )} t )+d c 1 ({max R (u t , v t )} t , {v t } t ) ≤ Cd c 1 ({u t } t , {v t } t ). ( 11
)
Proof. The first estimate follows from the triangle inequality. Since max R (u t , v t ) ≥ u t , from (7) we have that

d c 1 ({u t } t , {max R (u t , v t )}) = lim t→∞ I(max R (u t , v t )) -I(u t ) t = I(max R (u 1 , v 1 )) -I(u 1 ), (12)
where the last identity follows from the linearity of I along geodesic rays (Theorem 2.10). By construction of the ray t → max R (u t , v t ) it follows that

I(max R (u 1 , v 1 )) = lim l→∞ I(max(u l , v l )) l ,
and this last limit exists as l → I(max(u l , v l )) is convex thanks to [DDL1, Theorem 3.8]. Consequently, we can build on (12) in the following manner:

d c 1 ({u t } t , {max R (u t , v t )} t ) = lim l→∞ I(max(u l , v l )) l -I(u 1 ) = lim l→∞ I(max(u l , v l )) -I(u l ) l = lim l→∞ d 1 (max(u l , v l ), u l ) l ≤ C lim l→∞ d 1 (u l , v l ) l = Cd c 1 ({u t } t , {v t } t ),
where in the last inequality we have used Proposition 2.13. Using symmetry of {u t } t , {v t } t , the proof is finished.

The metric geometry of singularity types

The aim of this section is to show that S(X, θ) embeds naturally in R(X, θ), endowing the former space with a natural pseudo-metric structure. Given ψ ∈ PSH(X, θ) with ψ ≤ 0, generalizing the methods of [Dar13, Section 4], it is possible to define a geodesic ray {r[ψ] t } t ∈ R(X, θ) whose potentials have minimal singularities. The specific construction is as follows. Let [0, l] t → r(ψ) l t ∈ E 1 (X, ω) be the geodesic segment with minimal singularity type joining r(ψ) l 0 = V θ and r(ψ) l l = max(ψ, V θ -l). Using the comparison principle ([DDL1, Proposition 3.2]) numerous times, it can be shown that for any fixed t > 0 the family {r(ψ) l t } l≥0 is increasing as l → ∞, and its limit equals the geodesic ray with minimal singularity type t → r[ψ] t . Along the way we also obtain the lower bound max(ψ, V θ -t) ≤ r[ψ] t for all t ∈ [0, ∞).

Since ψ ≤ ψ implies that r[ψ] t ≤ r[ψ ] t and r[ψ] t = r[ψ + C] t , C ∈ R, we obtain that the construction of the ray only depends on the singularity type, giving us a map:

r[•] : S(X, θ) → R(X, θ). ( 13
)
The basic idea will be to pull back the metric geometry of R(X, θ) recalled in the previous section to S(X, θ) via this map. Before we do this we carry out some preliminary analysis. Since [0, ∞) t → max(ψ, V θ -t) is a subgeodesic ray with minimal singularities we have that t → I(max(ψ, V θ -t)) is convex by [DDL1, Theorem 3.8] and non-increasing by Theorem 2.8. Via our embedding in (13), we can introduce the Monge-Ampère energy of singularity types

I S [ψ] := I{r[ψ] t }.
Theorem 3.1. For ψ ∈ PSH(X, θ) we have

I S [ψ] = - X θ n V θ + 1 n + 1 n j=0 X θ j V θ ∧ θ n-j ψ . ( 14
)
The proof of the above theorem is analogous to [Dar13, Theorem 2.5] that deals with the case when θ is Kähler. Nevertheless, given its central role in this work, we are going to give the details for the reader's convenience.

Proof. We can assume w.l.o.g. that ψ ≤ V θ ≤ 0. Setting ψ t := max(ψ, V θ -t), by [DDL1, Lemma 3.15] we have that

I S [ψ] = lim t→+∞ I(r[ψ] t ) t = lim t→+∞ I(ψ t ) t = 1 n + 1 n k=0 lim t→+∞ X ψ t -V θ t θ k ψt ∧ θ n-k V θ .
Note that X θ n ψt = X θ n V θ since ψ t has minimal singularity. By Lemma 2.1, we have

X ψ t -V θ t θ ψt = {ψ>V θ -t} ψ t -V θ t θ n ψt - {ψ≤V θ -t} θ n ψt = {ψ>V θ -t} ψ -V θ t θ n ψ - X θ n ψt + {ψ>V θ -t} θ n ψt = {ψ>V θ -t} ψ -V θ t θ n ψ - X θ n V θ + {ψ>V θ -t} θ n ψ .
The functions 1 {ψ>V θ -t}

ψ-V θ t
are uniformly bounded in [-1, 0] and they converge pointwise to 0 outside a pluripolar set on which the measure θ n ψ vanishes. Hence

lim t→+∞ X ψ t -V θ t θ n ψt = - X θ n V θ + X θ n ψ .
Since for any j = 1, . . . , n,

X θ n-j ψt ∧ θ j V θ = X θ n V θ , the exact same arguments give lim t→+∞ X ψ t -V θ t θ n-j ψt ∧ θ j V θ = - X θ n V θ + X θ n-j ψ ∧ θ j V θ , j = 1, . . . , n.
This gives the conclusion.

Finally, we list and prove the properties of the map r[•] that will be most important to us, finding a link with the ceiling operator C in the process:

Proposition 3.2. Suppose [ψ], [χ] ∈ S(X, θ) such that ψ, χ ≤ 0. Then the following hold: (i) r[ψ] ∞ := lim t→∞ r[ψ] t = C(ψ). (ii) r[ψ] t = r[C(ψ)] t . In particular, r[C(ψ)] ∞ = C[ψ]. (iii) r[ψ] t = r[χ] t if and only if C(ψ) = C(χ). (iv) P [C(ψ)] = C(ψ).
In particular, part (i) of this proposition proves that the image of the ceiling operator C is exactly the collection of θ-psh functions that can arise as time limits of geodesic rays of the type {r

[ψ] t } t . Proof. First, we claim that given u, v ∈ PSH(X, θ) such that v ∈ F u then r[u] = r[v], and in particular r[u] ∞ = r[v] ∞ . Indeed, by (14) we know that lim t→+∞ I(r[u]t) t = lim t→+∞ I(r[v]t) t
. The linearity of the energy I (Theorem 2.10) then insures that

I(r[v] ) = I(r[u] ) for any ≥ 0. Since r[u] ≤ r[v] it follows from Lemma 2.9 that r[u] = r[v] , ∀ ≥ 0. Now, set u := r[ψ] ∞ and we claim that r[u] t = r[ψ] t . Since u ≥ ψ, we get that r[u] t ≥ r[ψ] t . For the other direction, we note that r[ψ] t ≥ u and r[ψ] t ≥ V θ -t, thus r[ψ] t ≥ max(u, V θ -t). The inequality r[u] t ≤ r[ψ] t then
follows by the construction of {r[u] t } t together with the comparison principle.

As a consequence of this second claim we have that I(r[u] t ) = I(r[ψ] t ). From ( 14) and [DDL2, Proposition 3.1] we get

X θ k u ∧ θ n-k V θ = X θ k ψ ∧ θ n-k V θ , ∀k = 0, • • • , n.
This means that u ≤ C(ψ). Moreover, our claims give r

[ψ] t = r[u] t = r[C(ψ)] t for any t, hence u = r[u] ∞ = r[C(ψ)] ∞ ≥ C(ψ). Thus u = C(ψ) addressing (i). The statement in (ii) is just a consequence of the second claim above. If r[ψ] = r[χ], then by (i) we have that C(ψ) = r[ψ] ∞ = r[χ] ∞ = C(χ). Conversely if C(ψ) = C(χ), then by (ii) we have r[ψ] = r[χ]. This proves (iii). Lastly, since inf t∈[0,∞) r[ψ] t = r[ψ] ∞ = C(ψ)
, by [DDL1, Lemma 3.17] we have that P [C(ψ)] = C(ψ), establishing (iv).

Finally, as previously advertised, we consider the L 1 (pseudo)metric structure of S(X, θ), by pulling back the chordal metric structure from R(X, θ):

d S ([ψ], [χ]) := d c 1 ({r[ψ] t } t , {r[χ] t } t ).
Our main result about (S(X, θ), d S ) in this subsection characterizes the singularity types that are at zero distance apart with respect to the d S pseudo-metric:

Theorem 3.3. (S(X, θ), d S
) is a pseudo-metric space. More precisely, the following are equivalent:

(i) d S ([ψ], [χ]) = 0. (ii) r[ψ] = r[χ]. (iii) C(ψ) = C(χ).
Proof. Theorem 2.14 gives the equivalence (i) ⇔ (ii). Proposition 3.2(iii) gives that (ii) ⇔ (iii).

In case [u] ≤ [v], using (14), the expression for d S ([u], [v]) is especially simple :

Lemma 3.4. If [u], [v] ∈ S(X, θ) is such that [u] ≤ [v] then d S ([u], [v]) = 1 n + 1 n j=0 X θ j V θ ∧ θ n-j v - X θ j V θ ∧ θ n-j u .
Observe that, it is a direct consequence of the above that if

[u] ≤ [v] ≤ [w] then d S ([u], [w]) ≥ d S ([v], [w]). When [u] ≤ [v]
, then a similar simple expression for d S ([u], [v]) may not be available, however one can find a useful expression that totally governs the behavior of d S ([u], [v]), as shown in Proposition 3.5 below.

It is clear that for [u], [v] ∈ S(X, θ) it makes sense to define [max(u, v)] ∈ S(X, θ), which does not depend on the choice of representatives u, v ∈ PSH(X, θ). We start by arguing that max Next we notice that (15) and Proposition 2.15 allow to establish the following decomposition inequality for the d S pseudo-metric:

R (r[u] t , r[v] t ) = r[max(u, v)] t , t ≥ 0, ( 15
)
where {max R (r[u] t , r[v] t )} t is
Proof. We can assume that d S ([u j ], [u j+1 ]) ≤ C -2j
, where C is the constant of Proposition 3.5. Let

v j := usc sup k≥j u k ∈ PSH(X, θ). (17)
Lemma 4.1 implies that lim l d S ([v l j ], [v j ]) = 0, where v l j = sup k∈{j,...,j+l} u k . To finish the argument, we show that lim

l d S ([u j ], [v l j ]) = d S ([u j ], [v j ]) → 0 as j → ∞.
Using the triangle inequality and Proposition 3.5 we get

d S ([u j ], [v l j ]) = d S ([u j ], [max(u j , v l-1 j+1 )]) ≤ Cd S ([u j ], [v l-1 j+1 ]) ≤ C(d S ([u j ], [u j+1 ]) + d S ([u j+1 ], [v l-1 j+1 )]).
After iterating the above inequality l times and observing that d S ([u j+l ], [v 0 j+l ]) = 0, we conclude that

d S ([u j ], [v l j ]) ≤ k∈{j,...,j+l-1} C k+1-j d S ([u k ], [u k+1 ]) = k∈{j,...,j+l-1} C k+1-j 1 C 2k ≤ k≥j C k+1-j 1 C 2k = k≥j 1 C k+j-1 ≤ 1 C j-1 C C -1 . ( 18
)
4.1 Completeness of S δ (X, θ)

Under the assumption of non-collapsing mass, we will show below that decreasing d S -Cauchy sequences do indeed converge, implying completeness of S δ (X, θ), via Proposition 4.2. For this we will need the following important lemma:

Lemma 4.3. Assume that u, v ∈ PSH(X, θ), u ≤ v, X θ n u > 0 and b > 1 is such that b n X θ n u > (b n -1) X θ n v . (19)
Then

P (bu + (1 -b)v) ∈ PSH(X, θ).
Proof.

If P (bu + (1 -b)C(v)) ∈ PSH(X, θ) then so does P (bu + (1 -b)v) since v ≤ C(v) and
(1 -b) < 0. Therefore, after possible replacing v with C(v), we can assume that

v = C(v). Since X θ n v ≥ X θ n u > 0, we have that P [v] = C(v) ([ DDL2
, Remark 2.5, Theorem 3.12]). For j ∈ N we set u j := max(u, v -j) and ϕ j := P (bu j +(1-b)v). Observe that ϕ j is a decreasing sequence of θ-psh functions, whose singularity type is equal to [v]. The proof is finished if we can show that lim j ϕ j ≡ -∞ is a θ-psh function. Assume by contradiction that sup X ϕ j → -∞. It follows from Lemma 4.4 below that

θ n ϕ j ≤ b n 1 {ϕ j =bu j +(1-b)v} θ n u j . (20)
Fix j > k > 0. We note that u j = u on {u > v -k} and, since u j has singularity type equal to [v], we have by [WN19, Theorem 1.2] and the plurifine locality,

{u≤v-k} θ n u j = X θ n u j - {u>v-k} θ n u j = X θ n v - {u>v-k} θ n u .
Proof. Let ψ k := max(ψ, V θ -k) and ϕ k := max(ϕ, V θ -k). It follows from a result of Demailly (see [Bl1, Theorem 2.2.10]) that

θ n max(ϕ k ,ψ k ) ≥ 1 {ψ k ≤ϕ k } θ n ϕ k + 1 {ϕ k <ψ k } θ n ψ k
holds in the ample locus of {θ} where all the functions above are locally bounded. As the nonpluripolar products are extended trivially over X, we see that the above inequality holds over X in the sense of measures. Multiplying with 1 {ϕ>V θ -k}∩{ψ>V θ -k} , and using plurifine locality (Lemma 2.1) we arrive at

1 {ϕ>V θ -k}∩{ψ>V θ -k} θ n max(ϕ,ψ) ≥ 1 {ϕ>V θ -k}∩{ψ>V θ -k}∩{ψ≤ϕ} θ n ϕ + 1 {ϕ>V θ -k}∩{ψ>V θ -k}∩{ϕ<ψ} θ n ψ .
Letting k → ∞, (23) follows.

Next we prove that along a decreasing sequence of fixed points of C the total masses converge.

Proposition 4.6. Let u j , u ∈ PSH(X, θ) be such that sup X u j = sup X u = 0, and u j converges in capacity to u. Then lim sup

j→+∞ {u j =0} θ n u j ≤ {u=0} θ n u .
If additionally

u j = C(u j ) then lim j→+∞ X θ n u j = lim j→+∞ {u j =0} θ n u j = {u=0} θ n u = X θ n u .
Proof. For each C > 0 and each v ∈ PSH(X, θ) we set v C := max(v, V θ -C). Since u j ≤ V θ , we have that {u j = 0} ⊂ {V θ = 0}. For each β > 0, using plurifine locality (Lemma 2.1) we can write lim sup

j→+∞ {u j =0} θ n u j = lim sup j→+∞ {u j =0} θ n u C j ≤ lim sup j→+∞ {V θ =0} e βu j θ n u C j ≤ {V θ =0} e βu θ n u C ,
where the last inequality follows from [GZ17, Theorem 4.26]. Indeed, since e βu j is a sequence of bounded quasi-continuous functions converging in capacity to e βu and u C j converges in capacity to u C all having minimal singularities V θ -C ≤ u C j ≤ 0, by [GZ17, Theorem 4.26] we have that e βu j θ n u C j converges weakly to e βu θ n u C in Ω, the ample locus of {θ}. Since {V θ = 0} is a compact subset of Ω, the last inequality follows.

Letting β → +∞ and noting that e βu decreases to 1 {u=0} we arrive at lim sup

j→+∞ {u j =0} θ n u j ≤ {u=0} θ n u C = {u=0} θ n u ,
where we used again Lemma 2.1. This finishes the proof of the first part. Assume now that u j = C(u j ). In case X θ n u j > 0 then we have C(u j ) = P [u j ], and by [DDL2, Theorem 3.8] we know that θ n u j is supported on the contact set {u j = 0}. In case X θ n u j = 0 this same fact is trivially true.

Observe that the inequality lim inf j X θ n u j ≥ X θ n u follows from [DDL2, Theorem 2.3]. From this and the previous step we obtain

X θ n u ≤ lim inf j X θ n u j ≤ lim sup j X θ n u j = lim sup j {u j =0} θ n u j ≤ {u=0} θ n u ≤ X θ n u . (24)
The conclusion follows.

For the rest of this subsection assume that X is equal to the complex surface described in [DPS94, Example 1.7]: X := P(E) where E is a rank 2 vector bundle over an elliptic curve Γ = C/Z + τ Z, Im τ > 0. By [START_REF] Demailly | Compact complex manifolds with numerically effective tangent bundles[END_REF] the line bundle N := O E (1) is nef and the only positive current in c 1 (N ) is the current of integration along a curve C, i.e. c 1 (N ) = {[C]}.

Naturally we have a projection map π : P(E) → Γ. Since the elliptic curve E is projective, so is X = P(E) [Har77, Section II.7], allowing to find an ample line bundle L → X.

Since L is ample there exists a Hermitian metric on L whose curvature form η is Kähler. We also fix a Hermitian metric on N whose curvature form is denoted by θ. By possibly replacing L with a high tensor power, we can assume that η + θ and ω := 2η + θ are Kähler forms on X.

Due to existence of sections for high powers of L, up to replacing L with a high tensor power, there exists u ∈ PSH(X, η), sup X u = 0, such that η u := η + i∂ ∂u = [D] for some smooth curve D ⊂ X. In particular, X η u ∧α = 0 for any Kähler form α, since we are working with non-pluripolar products throughout.

Set

φ := C ω (u) ∈ PSH(X, ω) ≤ 0. Since η u ∧ (η + θ) = η 2 u = 0, using multilinearity we can write, (ω + i∂ ∂u) 2 = (η + θ + η u ) 2 = (η + θ) 2 .
It then follows from [DDL4, Theorem 3.3] that u -φ is bounded.

We now set φ t := P ω (tφ), for t ∈ [1, 2]. Clearly, 2u + V θ is ω-psh and it is smaller than 2φ, hence φ t ∈ PSH(X, ω) and it is a model potential thanks to Lemma 4.10.

We estimate the mixed mass of φ t for each

t ∈ [1, 2). Recall that {θ} is nef, hence {(2 -t)η + θ} is Kähler if t ∈ [1, 2). Let ϕ t be a smooth negative ((2 -t)η + θ)-psh function. Then φ t is less singular than tu + ϕ t ∈ PSH(X, ω), hence X (ω + i∂ ∂φ t ) ∧ ω ≥ X (2η + θ + i∂ ∂(tu + ϕ t )) ∧ ω ≥ X ((2 -t)η + θ) ∧ (η + (η + θ)) ≥ {θ}.{η} = C η > 0,
where in the third inequality above we used the fact that {η + θ} is a Kähler class. Also, φ 2 = P ω (2φ) is more singular than 2u. The potential P ω (2φ) -2u is then bounded from above and it satisfies:

θ + i∂ ∂(P ω (2φ) -2u) = θ + 2η + i∂ ∂P ω (2φ) ≥ 0 on X \ D, since η + i∂ ∂u = 0 in X \ D. Therefore φ 2 -2u extends over X as a θ-psh function. Thus φ 2 -2u = V θ + C 1 , for some constant C 1 . Since θ + i∂∂V θ = [C]
is a current of integration, the following holds for the mixed mass of φ 2 :

X (ω + i∂ ∂φ 2 ) ∧ ω = X (2η u + θ V θ ) ∧ (2η + θ) = 0.
} t ∈ R(X, θ) such that d c 1 ({r[φ s ] t } t , {r t } t ) → 0.
The construction of the geodesic ray {r t } is explicit: for each t > 0, r t is the limit in (E 1 , d 1 ) of r[φ s ] t as s → 2. In this case since {r[φ s ] t } t is s-decreasing we have that r[φ s ] t decreases to r t as s → 2. If S(X, θ) is indeed complete, then r t = r[ψ] t for some [ψ] ∈ S(X, θ). But then we must have that r

[φ s ] t ≥ r t = r[ψ] t , t ≥ 0. Letting t → ∞, since C(φ s ) = φ s (Lemma 4.10), Proposition 3.2(i) implies that φ s ≥ C(ψ) ≥ ψ for any s ∈ [1, 2). In particular φ s ≥ φ 2 ≥ C(ψ) ≥ ψ. As a result, Lemma 3.4 gives that d S ([φ s ], [φ 2 ]) ≤ d S ([φ s ], [ ψ
]) → 0 as s → 2, a contradiction with our above findings, hence S(X, θ) is incomplete.

The volume diamond inequality

Lemma 5.1. Assume that u, v, w ∈ PSH(X, θ) are such that X θ n u + X θ n v > X θ n w and max(u, v) ≤ w. Then P (u, v) ∈ PSH(X, θ).

Proof. We can assume without loss of generality that u, v, w ≤ 0. Replacing w with P [εV θ +(1-ε)w] for small enough ε > 0 we can also assume that X θ n w > 0 and w = C(w). For j ≥ 0 we set u j := max(u, w -j), v j := max(v, w -j), h j := P (u j , v j ). Observe that u j , v j , h j have the same singularity type as w. We fix s > 0 big enough, such that for all j > s, we have {u>w-s}

θ n u j + {v>w-s} θ n v j = {u>w-s} θ n u + {v>w-s} θ n v > X θ n w ,
where in the equality above we used Lemma 2.1. It follows from [DDL2, Lemma 3.7] and the above estimate that for j > s,

{h j ≤w-s} θ n h j ≤ {u j ≤w-s} θ n u j + {v j ≤w-s} θ n v j = 2 X θ n w - {u>w-s} θ n u - {v>w-s} θ n v < X θ n w ,
where in the identity above we used the fact that {u j ≤ w -s} = {u ≤ w -s}. Since u j , v j decrease to u, v respectively, it follows that h j P (u, v). We now rule out the possibility that P (u, v) ≡ -∞. Indeed, suppose sup X h j decreases to -∞. From Lemma 2.7 we obtain that sup X h j = sup X (h j -w) -∞. But then, for j large enough the set {h j ≤ w -s} coincides with X, contradicting our last integral estimate, since each h j has the same singularity type as w.

Plainly speaking, by the next lemma, the fixed point set of the map ψ → P [ψ] is stable under the operation (ψ, φ) → P (ψ, φ). Lemma 5.2. Suppose u 0 , u 1 ∈ PSH(X, θ) are such that P (u 0 , u 1 ) ∈ PSH(X, θ), and P [u 0 ] = u 0 and P [u 1 ] = u 1 . Then P [P (u 0 , u 1 )] = P (u 0 , u 1 ).

Proof. As P (u 0 , u 1 ) ≤ min(u 0 , u 1 ) ≤ 0 and P [P (u 0 , u 1 )] ≤ min(P [u 0 ], P [u 1 ]), it follows that P (u 0 , u 1 ) ≤ P [P (u 0 , u 1 )] ≤ P (P [u 0 ], P [u 1 ]) = P (u 0 , u 1 ).

This shows that all the inequalities above are in fact equalities.

Proposition 5.3. Let φ, ψ ∈ PSH(X, θ) be such that φ = P [φ], ψ = P [ψ], and P (φ, ψ) ∈ PSH(X, θ). If u ∈ E(X, θ, φ), v ∈ E(X, θ, ψ) and X θ n P (φ,ψ) > 0 then P (u, v) ∈ E(X, θ, P (φ, ψ)). Proof. We can assume that u ≤ φ and v ≤ ψ.

Step 1. We first prove that P (u, ψ) ∈ E(X, θ, P (φ, ψ)). By assumption we have Comparing the total mass via [START_REF] Witt-Nyström | Monotonicity of non-pluripolar Monge-Ampère measures[END_REF] and letting t → +∞ we see that u b ∈ E(X, θ, φ). The previous argument then gives P (u b , ψ) ∈ PSH(X, θ). On the other hand we also have

u ≥ b -1 u b + (1 -b -1 )φ,
that e -εψ j → e -εψ in and sup j e -εψ j L s (X,ω n ) < +∞. This is possible as we explain below. For x = -εψ j , y = -εψ we have that x, y ≥ 0 and an elementary argument gives |e x -e y | s ≤ e s(x+y) |x -y| s .

Thus, after applying Hölder's inequality twice, we obtain X |e -εψ j -e -εψ | s ω n ≤ ε s X e -sε(ψ j +ψ) |ψ j -ψ| s ≤ ε s X e -2sε(ψ j +ψ) ω n .

The convergence statement for e -εψ j then follows because ψ j converges to ψ in any L t , t > 1, while, since sup X ψ j = sup X ψ = 0, Skoda's uniform integrability theorem ([Zer01],[GZ17, Theorem 2.50]) ensures that both e -4sεψ j and e -4sεψ are uniformly bounded in L 1 for ε > 0 small enough. Now set h j := e -εψ j f j , h := e -εψ f . We have X |h j -h|ω n ≤ X e -εψ j |f j -f |ω n + X |e -εψ j -e -εψ |f ω n .

Applying Hölder's inequality with exponents r and s we conclude that h j -h L 1 → 0. Up to extracting again we can assume that h j , h ≤ g where g ∈ L 1 (X, ω n ) is constructed exactly as the function g at the beginning of the proof. From (37) we have w k j -C ≤ χ k j := P (ψ j , ψ j+1 , . . . , ψ j+k ) ≤ w k j , giving that χ k j is a θ-psh function. Observe then that the Monge-Ampère equation for ψ j rewrites as θ n ψ j = e εψ j h j ω n . Thus, Lemma 7.2 below gives where χ j := lim k χ k j . Also w j -C ≤ χ j ≤ w j . Now we argue that the increasing limit χ := lim j χ j = ψ. Indeed, we can apply [DDL2, Theorem 2.3] and the dominated convergence theorem to conclude that θ n χ ≤ e εχ h ω n = e ε(χ-ψ) f ω n .

and

  Lemma 5.1 gives P (u, ψ) = P (u, P (φ, ψ)) ∈ PSH(X, θ). Fixing b > 1, it follows from Lemma 4.3 that u b := P θ (bu -(b -1)φ) ∈ PSH(X, θ). For 1 < b < t we have φ ≥ u b ≥ bt -1 u t + (1 -bt -1 )φ.

≤ e εχ k j sup l≥j h l ω n .≤

 n From the first statement of [DDL2, Theorem 2.3] we haveθ n χ j ≤ lim inf k e εχ j sup l≥j h l ω n ,

  hence χ ∈ E(X, θ, φ). Recall that we also have ψ ∈ E(X, θ, φ). By the comparison principle, [DDL2, Corollary 3.16], and (38), we have{χ<ψ} θ n ψ ≤ {χ<ψ} θ n χ ≤ {χ<ψ} e ε(χ-ψ) f ω n = {χ<ψ} e ε(χ-ψ) θ n ψ ≤ {χ<ψ} θ n ψ .

  the smallest ray that lies above max(r[u] t , r[v] t ) that was constructed in Section 2.2. But {r[max(u, v)] t } t has this "extremal" property as well. Indeed by construction we have max(r[u]

t , r[v] t ) ≤ r[max(u, v)] t , and any ray {w t } t that satisfies max(r[u] t , r[v] t ) ≤ w t has to also satisfy r[max(u, v)] t ≤ w t .

  Hence φ s φ 2 but due to Lemma 3.6 we have that d S ([φ s ], [φ 2 ]) → 0 as s → 2. Due to Lemma 3.4, we have that {[φ s ]} s∈[1,2) does form a d S -Cauchy sequence. By Theorem 2.14 there exists {r t

Proposition 3.5. There exists C > 1 such that, for all [u], [v] ∈ S(X, θ),

we note the following corollary of (14), somewhat reminiscent of [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]Proposition 4.9]: Lemma 3.6. Suppose u j , u ∈ PSH(X, θ) are such that either

, k ∈ {0, . . . , n}.

It follows from (10) that

). By the next lemma it turns out that a similar statement holds for the individual components of the sum in (14) as well:

Lemma 3.7. There exists C > 1 such that for all k ∈ {0, . . . , n} and [u], [v] ∈ S(X, θ) we have

Proof.

, k ∈ {0, . . . , n}. As a result of this and ( 14) we can write that

For general [u], [v] ∈ S(X, θ), using this last inequality and Proposition 3.5 we can conclude:

Discussion of completeness

In this section we prove the completeness of the spaces (S δ (X, θ), d S ), for δ > 0. Recall from the introduction that S δ (X, θ) = {[u] ∈ S(X, θ) s.t. X θ n u ≥ δ}. On the other hand, we will also show that (S(X, θ), d S ) is not complete.

First we show that "increasing" sequences always have a d S -limit inside S(X, θ):

Lemma 4.1. Let u j ∈ PSH(X, θ) such that {u j } j is increasing and u j ≤ 0. Then d S ([u j ], [u]) → 0, where u j u a.e. on X.

, the proof is finished after an application of Lemma 3.6. Proposition 4.2. Suppose that [u j ] ∈ S(X, θ) is a d S -Cauchy sequence with u j ≤ 0. Then there exists a decreasing sequence

Since {u j ≤ v -k} = {u ≤ v -k}, from the above and (20) we obtain

we have sup X (ϕ j -v) = sup X ϕ j → -∞ by Lemma 2.7. From this we see that {ϕ j ≤ v -bk} = X for j large enough, k being fixed. Thus, letting j → +∞ in (21), and then k → +∞ gives

. Consequently, ϕ j decreases to a θ-psh function, finishing the proof.

Lemma 4.4. Assume that b ≥ 1, and u, v, P θ (bu + (1 -b)v) ∈ PSH(X, θ). Then the measure

Proof. Up to rescaling, we can assume that θ ≤ ω, and hence PSH(X, θ) ⊂ PSH(X, ω). Let u j ∈ C ∞ (X) ∩ PSH(X, ω) be such that u j u. This is possible thanks to [START_REF] Blocki | On regularization of plurisubharmonic functions on manifolds[END_REF], [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF]. Set

In the integrand we take minimum with 1 (one can simply take any positive constant) so that the functions min(bu j + (1 -b)v -ψ j , 1) are uniformly bounded, with values in the interval [0, 1]. These functions are additionally quasi-continuous, and (since u j and ψ j are ω-psh functions decreasing to u and ψ respectively) they converge in capacity to min(bu + (1 -b)v -ψ, 1), which is quasicontinuous and bounded on X. It follows from Theorem 2.2 that after letting j → +∞ in the above equality we obtain

This implies that θ n ψ vanishes in the set {ψ

Note that ϕ ≤ u. By Lemma 4.5 below we then have

Moreover, by the above we know that

Lemma 4.5. Let ϕ, ψ ∈ PSH(X, θ). Then

In particular, if ϕ ≤ ψ then 1 {ϕ=ψ} θ n ϕ ≤ 1 {ϕ=ψ} θ n ψ .

Corollary 4.7. Assume that u j ∈ PSH(X, θ) and u j = C(u j ). If u j u, then u = C(u). If u j converges in capacity to u and X θ n u > 0 then u = C(u).

Proof. If u j u then u ≤ C(u) ≤ C(u j ) = u j , hence u = C(u). Assume now that u j converges in capacity to u and X θ n u > 0. It follows from (24) that {u=0} θ n u = X θ n u . Hence θ n u is supported on {u = 0} = {u = P [u] = 0}, where we used that u ≤ P [u] ≤ 0. In particular, u ≥ P [u] a.e. with respect to θ n u . Since X θ n P [u] = X θ n u > 0 we can use the domination principle, [DDL2, Proposition 3.11], and Proposition 2.6(i) to conclude that u = P [u] = C(u).

In the presence of non-vanishing mass, we can also show the convergence of the mixed masses of decreasing model potentials:

Letting j → +∞ the result follows.

Finally, we summarize the above findings in our main theorem:

Theorem 4.9. Fix δ > 0. The pseudo metric space (S δ (X, θ), d S ) is complete.

Incompleteness of S(X, θ)

In this short section we show that (S(X, ω), d S ) is in general not complete. In particular, completeness fails even in the Kähler case. We start with the following general lemma:

Proof. Fix 1 < s < t and assume that P (th) ∈ PSH(X, ω). Let v ≤ 0 be a ω-psh function more singular than P (sh). Then s -1 v is more singular than h. Since s -1 v is ω-psh and h = C(h) = P [h], it follows that s -1 v ≤ h, hence v ≤ sh. Now, since v is ω-psh it follows that v ≤ P (sh). Hence P [P (sh)] = P (sh). Since s t P (th) ≤ P (sh) and s ∈ (1, t), the mass of P (sh) is positive. It follows that P (sh) is a model potential (Proposition 2.6).

Lastly, P (th) is the decreasing limit of the model potentials P (su), s ∈ (1, t). It then follows from Corollary 4.7 that P (th) is also a model potential, i.e. C(P (th)) = P (th). therefore P (u, ψ) ≥ b -1 (u b , ψ) + (1 -b -1 )P (φ, ψ). Comparing the total mass via [START_REF] Witt-Nyström | Monotonicity of non-pluripolar Monge-Ampère measures[END_REF] and letting b → +∞ we arrive at X θ n P (u,ψ) ≥ X θ n P (φ,ψ) , hence the conclusion.

Step 2.

We prove that P (u, v) ∈ PSH(X, θ). It follows from [START_REF] Witt-Nyström | Monotonicity of non-pluripolar Monge-Ampère measures[END_REF], the assumption v ∈ E(X, θ, ψ), and the first step that

Since max(P (u, ψ), v) ≤ ψ, Lemma 5.1 can be applied giving P (u, v) = P (P (u, ψ), v) ∈ PSH(X, θ).

Step 3. We conclude the proof. Fixing b > 1, it follows from Lemma 4.3 that

Comparing the total mass via [START_REF] Witt-Nyström | Monotonicity of non-pluripolar Monge-Ampère measures[END_REF] and letting t → +∞ we see that v b ∈ E(X, θ, ψ). By the second step we have that P (u, v b ) ∈ PSH(X, θ). On the other hand we also have

Comparing the total mass via [START_REF] Witt-Nyström | Monotonicity of non-pluripolar Monge-Ampère measures[END_REF] and letting b → +∞ we arrive at X θ P (u,v) ≥ X θ n P (u,ψ) . Combining this and the first step we arrive at the conclusion.

Finally, we prove the main result of this section:

Proof. It is enough to check (25) only in the case when u = P [φ] and v = P [ψ] for some φ, ψ ∈ PSH(X, θ). Indeed, we first note that, for each t > 0, max(P (u + t, 0), P (v + t, 0)) and max(u, v) have the same singularity type. Since max(P (u + t, 0), P (v + t, 0)) increases a.e. to max(P [u],

, while in the zero mass case, the equality follows from [START_REF] Witt-Nyström | Monotonicity of non-pluripolar Monge-Ampère measures[END_REF]. For the rest of the argument we assume that u = P [φ] and v = P [ψ], and for convenience we introduce w := max(u, v) ≤ 0. For t > 0 we set u t := max(u, w -t) and v t := max(v, w -t). Observe that, by [DDL2, Theorem 3.8], θ n u is supported on {u = 0} = {u t = 0} which is contained in {u > w -t}, for t > 0. From this and plurifine locality, Lemma 2.1 we have

By the same argument applied for v t we have

Integrating over X and noting that, by [START_REF] Witt-Nyström | Monotonicity of non-pluripolar Monge-Ampère measures[END_REF],

Building on (26), ( 27), an application of Lemma 3.7] gives

In particular, θ n P (ut,vt) is supported on the union of the disjoint sets A t := {u ≤ w -t} ∪ {v ≤ w -t} and {P (u t , v t ) = 0}. From here, since P (u t , v t ) has the same singularity type as w, an integration allows to conclude that:

where in the inequality we have used (29). Now, using the above inequality, (28), and Proposition 4.6 we let t → +∞ to get

finishing the proof.

Remark 5.5. If dim X = 1 then we have actually equality in (25). Indeed, since max(u,v)+P (u,v) 2 ≤ u+v 2 , an application of the main result of [START_REF] Witt-Nyström | Monotonicity of non-pluripolar Monge-Ampère measures[END_REF], yields the equality in (25). On the other hand, equality can not hold in general. Consider X = CP 2 with ω := ω F S the Fubini Study metric and we view (z 1 , z 2 ) ∈ C 2 as a chart of CP 2 . Let ρ be the local potential of ω F S . Set

Then w ≤ P (u, v), hence P (u, v) is a ω F S -psh function and X ω 2 u = X ω 2 v = X ω 2 P (u,v) = 0. On the other side, max(u, v) is bounded, hence X ω 2 max(u,v) = X ω 2 > 0.

As a consequence of (25) we show that every d S -convergent sequence in S δ (X, θ) has a subsequence that can be sandwiched between an increasing and a decreasing d S -convergent sequence: Theorem 5.6. Let δ > 0, and suppose that [u j ], [u] ∈ S δ (X, θ) satisfies d S ([u j ], [u]) → 0 and u j = P [u j ], u = P [u]. Then there exist a subsequence u j k and decreasing/increasing sequences

As we will see below, for the appropriate subsequence u j k , the potentials v j k and w j k can be chosen as follows:

Proof. We will pass to subsequences multiple times during the proof. Thus we can assume without loss of generality that d S ([u j ], [u j+1 ]) ≤ 1 C 2j , where C > 1 is the constant from Proposition 3.5. To start, we introduce the following decreasing sequence:

Trivially, u k ≤ v j for all k ≥ j, and it follows from Proposition 4.2 and its proof (see (18)) that d S ([u], [v j ]) → 0. Now we construct the sequence w j After possibly taking another subsequence, we can assume that d

Fix j 0 > 0 large enough so that 2 -j 0 +3 < δ. We claim that, for all j > j 0 , k ≥ 0, we have P (u j , ..., u j+k ) ∈ PSH(X, θ), and

We argue inductively assuming the claim for k -1, as the case k = 0 is obvious. From ( 31),(32) we have

where the last inequality follows from the choice of j 0 . Since P (u j , u j+1 , . . . u j+k-1 ), u j+k ≤ v j+k-1 , it then follows from Lemma 5.1 that P (P (u j , ..., u j+k-1 ), u j+k ) = P (u j , ..., u j+k ) ∈ PSH(X, θ). We next apply Theorem 5.4 to obtain X θ n u j+k + X θ n P (u j ,u j+1 ,...,u j+k-1 ) ≤ X θ n max(P (u j ,...,u j+k-1 ),u j+k ) + X θ n P (u j ,u j+1 ,...,u j+k )

where in the second inequality we used [DDL2, Theorem 1.1]. The claim follows from (33) and the above inequality. Set w k j := P (u j , . . . , u j+k ). It follows from Lemma 5.2 that w k j = P [w k j ], hence sup X w k j = 0. Therefore, the decreasing limit lim k w k j is a θ-psh function (it is not identically -∞). Proposition 4.6 now gives that X θ n w k j → X θ n w j . Putting this together with (32) we obtain that 0

Moreover from Corollary 4.7 together with Proposition 2.6(i) we know that C(w j ) = P [w j ] = w j . Let w ∈ PSH(X, θ) be the increasing limit of {w j } j , and v be the decreasing limit of C(v j ) = P By monotonicity of {w j } j and {v j } j we see that w j ≤ w ≤ P [v j ], j ≥ 0. From here and the above integral estimate we have that X θ n w = X θ n u and w ≤ u, hence u = w since they are both model potentials.

Finally, according to Lemma 4.1, lim j d S ([w j ], [u]) = lim j d S ([w j ], [w]) = 0, finishing the proof.

Semicontinuity of multiplier ideal sheaves

For [u] [u] the multiplier ideal sheaf associated to the singularity type [u]. Recall that J [u] is the sheaf of germs of holomorphic functions f such that |f | 2 e -u is locally integrable on X. We now give a version of a theorem of Guan and Zhou [START_REF] Guan | A proof of Demailly's strong openness conjecture[END_REF][START_REF] Guan | Effectiveness of Demailly's strong openness conjecture and related problems[END_REF] adapted to our context:

Then there exists j 0 ≥ 0 such that J [u] ⊆ J [u j ] for all j ≥ j 0 .

Proof. We first assume that there exists δ > 0 such that u j , u ∈ S δ (X, θ), for all j ≥ 0.

We note that we can assume that P [u j ] = u j and P [u] = u. Indeed, since P [u j ] is the increasing limit of the potentials P (0, u j + c) and [P (0,

By contradiction let us assume that J [u] is not a subsheaf of J [u j ] for big enough j. Then there exists a subsequence of [u j ], again denoted by [u j ], such that

After possibly taking another subsequence, via Theorem 5.6, we can further assume that there exists {w j } j ⊂ PSH(X, θ) increasing such that w j ≤ u j and w j u. Using [Dem15, Theorem 0.8] again, it follows that J [u] = J [w j ] ⊆ J [u j ] for all j greater than some fixed j 0 . But this is a contradiction with our assumption (34).

We now treat the general case. Using [Dem15, Theorem 0.8] we can find ε > 0 small enough such that J [u] = J [u + εV θ ]. By Lemma 6.2 below, d S,(1+ε)θ ([u j + εV θ ], [u + εV θ ]) → 0. Thus, by the first step we have that J [u] = J [u + εV θ ] ⊂ J [u j + εV θ ], for j ≥ j 0 , where j 0 depends on ε. But J [u j + εV θ ] ⊂ J [u j ], hence the conclusion. Lemma 6.2. For ε ∈ [0, 1] there exists C = C(n) > 1 such that for all u, v ∈ PSH(X, θ) we have

Proof. Let us assume that u ≤ v. The general case reduces to this particular situation using Proposition 3.5. Set u ε := u + εV θ , v ε := v + εV θ . Then Lemma 3.4 gives the following:

where 0 ≤ c j = O(ε), j = 0, ..., n are positive constants depending only on n, ε. From (35), (36), and the fact that X θ j u ∧ θ n-j V θ ≤ X θ j v ∧ θ n-j V θ (which follows from [DDL2, Theorem 2.4]) we obtain the desired estimate.

7 Stability of solutions to CMAE with prescribed that solutions to a family of complex Monge-Ampère equations with varying singularity type converge as governed by the d S -topology: Theorem 7.1. Given δ > 0 and p > 1 suppose that:

• f j , f ≥ 0 are such that f L p , f j L p , p > 1, are uniformly bounded and f j → L 1 f . • ψ j , ψ ∈ PSH(X, θ), j ≥ 0 satisfy sup X ψ j = 0, sup X ψ = 0 and

.

Then ψ j converges to ψ in capacity, in particular ψ -ψ j L 1 → 0.

Proof. First we claim that it is enough to show that any subsequence of ψ j contains a subsequence that converges in capacity to ψ. Indeed, suppose that ψ j does not converge to ψ in capacity. Then there exists ε > 0 such that lim sup j Cap ω ({|ψ j -ψ| > ε}) > δ for some δ > 0. In particular, there exists j k → ∞ such that Cap ω ({|ψ j k -ψ| > ε}) > δ for all j k . But then {ψ j k } k would contain a subsequence converging to ψ in capacity, giving a contradiction. We take a subsequence of f j , again denoted by f j , such that f j -f j+1 L 1 ≤ 1 2 j+1 , j ≥ 0. By an elementary argument g := f 0 + j≥0 |f j+1 -f j | ∈ L 1 (ω n ) and f j , f ≤ g for all j ≥ 0. Now let us take a subsequence of φ j , again denoted by φ j , such that there exists w j , v j ∈ PSH(X, θ) increasing/decreasing sequences with w j ≤ φ j ≤ v j such that d S ([w j ], [φ]) → 0 and d S ([v j ], [φ]) → 0. This is possible due to Theorem 5.6. Moreover we recall that v j = usc sup k≥j φ k , and w j arises as the decreasing limit w j := lim k w k j , where w k j := P (φ j , φ j+1 , . . . , φ j+k ) (see (30)). We consider γ j := usc sup k≥j ψ k ≥ ψ j . Observe that sup X γ j = 0, j ≥ 0. For this sequence [DDL2, Lemma 4.27] gives that θ n

In particular, we have that w j -C ≤ γ j ≤ v j for all j ≥ 0. Hence the monotonicity of the sequences w j and v j implies that

Letting j → ∞, we obtain that

Consequently, the conditions of [DDL2, Theorem 2.3] hold for the decreasing sequence {γ j } j , yielding the estimate θ n γ ≥ f ω n . By comparing total masses again, we conclude that in fact θ n γ = f ω n . By uniqueness of solutions in E(X, θ, φ) ([DDL2, Theorem 4.29]), and noting that sup X γ = sup X ψ = 0, we obtain that γ = ψ.

This also shows that ψ j converges in L 1 (and a.e.) to ψ. Indeed since sup X ψ j = 0 we can assume that, up to extracting, ψ j converges to some ψ ∞ in L 1 and a.e.. Then (by construction) γ j also does converge to ψ ∞ . But the limit of γ j is γ = ψ.

We fix r ∈ (1, p). By our assumptions on the f j , f , and the Hölder inequality we obtain that f j → f in L r . Let s > 1 be the conjugate exponent of r, i.e. 1/s + 1/r = 1. Take ε > 0 so small It then follows that all the above inequalities become ψ) = 0. Therefore, all terms in the above are zero. In particular θ n χ (χ < ψ) = 0, and by the domination principle, [DDL2, Proposition 3.11], we have that χ ≥ ψ.

On the other side, by construction of χ j and γ j we have that χ j ≤ γ j , and so χ ≤ γ = ψ finally giving χ = ψ.

To summarize, we proved existence of two monotone sequences χ j , γ j such that χ j ≤ ψ j ≤ γ j with γ j decreasing to ψ and χ j increasing to ψ. In particular χ j and γ j converge in capacity to ψ ([GZ17, Proposition 4.25]). This implies that ψ j converges to ψ in capacity, finishing the proof. Lemma 7.2. Assume that u, v, P (u, v) ∈ PSH(X, θ), and µ is a positive non pluripolar measure, ε > 0, 0 ≤ f, g ∈ L 1 (µ). If θ n u ≤ e εu f µ, θ n v ≤ e εv gµ, then θ n P (u,v) ≤ e εP (u,v) max(f, g)µ.

Proof. By replacing µ with 1 X\P µ, where P := {u = v = -∞}, we can assume that µ(P ) = 0.

Since µ(X) < +∞, the function r → µ({u ≤ v + r}) is monotone increasing. Such functions have at most a countable number of discontinuities, hence for almost every r ≥ 0 we have that µ({u = v + r}) = 0. For such r we set ϕ r := P θ (u, v + r), and note that ϕ r P θ (u, v) as r → 0. It then follows from [DDL2, Lemma 3.7] that we can write θ n ϕr ≤ 1 {ϕr=u} θ n u + 1 {ϕr=v+r} θ n v ≤ 1 {ϕr=u} e εu f µ + 1 {ϕr=v+r} e εv gµ ≤ 1 {ϕr=u} e εϕr max(f, g)µ + 1 {ϕr=v+r} e εϕr max(f, g)µ ≤ e εϕr max(f, g)µ,

where in the last inequality we used the fact that µ({u = v + r}) = 0. Letting r 0, we use [DDL2, Theorem 2.3] to arrive at the conclusion.