The closures of test configurations and algebraic singularity types

Given a K\"ahler manifold $X$ with an ample line bundle $L$, we consider the metric space of $L^1$ geodesic rays associated to the first Chern class $c_1(L)$. We characterize rays that can be approximated by ample test configurations. At the same time, we also characterize the closure of algebraic singularity types among all singularity types of quasi-plurisubharmonic functions, pointing out the very close relationship between these two seemingly unrelated problems. By Bonavero's holomorphic Morse inequalities, the arithmetic and non-pluripolar volumes of algebraic singularity types coincide. We show that in general the arithmetic volume dominates the non-pluripolar one, and equality holds exactly on the closure of algebraic singularity types. Analogously, we give an estimate for the Monge--Amp\`ere energy of a general $L^1$ ray in terms of the arithmetic volumes along its Legendre transform. Equality holds exactly for rays approximable by test configurations. Various other cohomological and potential theoretic characterizations are given in both settings. As applications, we give a concrete formula for the non-Archimedean Monge--Amp\`ere energy in terms of asymptotic expansion, and show the continuity of the projection map from $L^1$ rays to non-Archimedean rays. We also show that the closure of ample test configurations and filtrations gives the same set of rays.


Introduction
We fix notation and terminology for the entire paper. We consider X a compact Kähler manifold of dimension n with an ample line bundle L. We pick a positive smooth Hermitian metric h on L and let ω := i 2π Θ(h) > 0 be the background Kähler form of X, where Θ(h) denotes the curvature form of the Hermitian metric h.
For u ∈ PSH(X, ω), by H 0 (X, L k ⊗ I(ku)) ⊆ H 0 (X, L k ) we denote the space of sections s satisfying the L 2 integrability condition X h k (s, s)e −ku ω n < ∞. We also denote h 0 (X, L k ⊗ I(ku)) := dim C H 0 (X, L k ⊗ I(ku)) .
A major theme in Kähler geometry is to relate algebraic objects to analytic ones. In this work we address two such problems. First, we give cohomological and potential theoretic characterizations for L 1 geodesic rays in the space of Kähler metrics that lie in the closure of test configurations. Second, we characterize the closure of algebraic singularity types in S, with respect to the complete pseudometric d S . Potentials with algebraic singularity types are among the nicest ones one could hope for in practice (see Definition 1.3).
According to our work, it is most natural to treat both of these seemingly different problems at the same time, with our final answers also paralleling each other on many different levels. We now present one facet of this, with slight abuse of precision.
Given ϕ ∈ PSH(X, ω) with algebraic singularity type [ϕ] ∈ S, the arithmetic and nonpluripolar volumes coincide, according to the singular holomorphic Morse inequalities of Bonavero [Bon98] (see Theorem 2.26): lim k→∞ 1 k n h 0 (X, L k ⊗ I(kϕ)) = 1 n! X ω n ϕ . (1.1) All volumes in this work, in particular the one on the right hand side above, are interpreted in the non-pluripolar sense of Guedj-Zeriahi [GZ07], [BEGZ10] (see (2.1) below). Both the left and right hand sides only depend on the singularity type [ϕ] ([WN19, Theorem 1.1]). In Theorem 1.4 we show that in (1.1) the limit on the left exists for all ϕ ∈ PSH(X, ω) and dominates the right hand side in general. Moreover, [ϕ] ∈ S lies in the d S -closure of algebraic singularity types if and only if Bonavero's identity (1.1) holds for ϕ.
Paralleling the above, in [DL20] the authors introduced a metric d c 1 on the space of L 1 geodesic rays R 1 (recalled in Section 2.1), making (R 1 , d c 1 ) a complete geodesic metric space. As is well known, to any ample test configuration one can associate a geodesic ray, a construction going back to Phong-Sturm [PS07]. We show that a geodesic ray {r t } t ∈ R 1 is in the d c 1 -completion of the space of rays induced by test configurations if and only if r τ ∈ PSH(X, ω) satisfies Bonavero's identity (1.1) for all τ ∈ R, wherer τ := inf t>0 (r t − tτ ) is the Legendre transform of the ray {r t } t . In particular, the ray {r t } t ∈ R 1 is approximable by test configurations if and only if the singularity types [r τ ] ∈ S are approximable by algebraic singularity types! This parallels previous characterizations of approximable rays in the non-Archimedean context from [BBJ21].
We refer to Theorem 1.1 and Theorem 1.4 for additional cohomological and potential theoretic characterizations complementing each other in both settings.
In addition, in Theorem 1.2 we express the non-Archimedean Monge-Ampère energy of a ray as the first order term in the expansion of the non-Archimedean Donaldson's L-functional, a result paralleling [BB10,Theorem A].
The closure of the rays induced by test configurations. Let d 1 be the metric on the space of smooth Kähler potentials H ω := {u ∈ C ∞ (X) : ω + dd c u > 0} associated with the L 1 Finsler metric [Dar15]: where V = X ω n is the total volume. By E 1 we denote the d 1 -completion of H ω , that can be identified with a finite energy space studied by Guedj-Zeriahi [GZ07]. Then (E 1 , d 1 ) is a complete geodesic metric space. Its geodesics are limits of solutions to a degenerate complex Monge-Ampère equation [Dar15,Theorem 2]. By R 1 we denote the space of d 1 -geodesic rays {r t } t≥0 in E 1 emanating from r 0 = 0 ∈ H ω . As shown in [DL20, Theorem 1.3, Theorem 1.4], R 1 has a a natural chordal metric d c 1 (see (2.3)), compatible with d 1 , making (R 1 , d c 1 ) a complete geodesic metric space. Of special importance is the subspace T ⊆ R 1 , composed of the rays induced by ample test configurations [PS07], [PS10], [CT08]. Similarly, one can consider the bigger subspace F ⊆ R 1 , the space of rays induced by filtrations [RWN14]. In this work it is advantageous to think of ample test configurations as special kind of filtrations on the ring of sections of (X, L), and we refer to Section 2.5 for details on this. Understanding the closures T and F is one of the main problems we take up in this work.

1.2)
Another quantity that is linear along a ray {r t } t is the supremum of potentials (Lemma 3.2). For simplicity, we will often assume that sup X r t = 0 for t ≥ 0, and such rays will be called sup-normalized. Note that all our results hold in an appropriate form without normalization, even when these are not specifically mentioned. First, in Theorem 3.7, we develop ideas from [RWN14] further, and show a precise formula for the radial Monge-Ampère energy of sup-normalized rays {r t } t ∈ R 1 : wherer τ ∈ PSH(X, ω) for τ < 0 is the Legendre transform of the ray: We attempt to approximate the non-pluripolar volumes in the integrand of (1.3) using arithmetic volumes (in the spirit of Bonavero's identity (1.1)). In our first main result we show that this fails in general, and it works exactly for rays in the d c 1 -closure of T : Theorem 1.1 (Theorem 4.7, Corollary 5.6). For {r t } t ∈ R 1 with sup X r t = 0 we have (1.4) Moreover, equality holds in (1.4) if and only if the following equivalent conditions hold: A ray satisfying condition (iii) is called a maximal geodesic ray in the work [BBJ21], giving non-Archimedean characterizations of T recalled below. Here we do not use this terminology, to avoid potential confusion with other notions of maximality.
In condition (ii) the operator P [u] I is the I-envelope of the singularity type [u] ∈ S, used explicitly and implicitly in [KS20] and [Cao14]  where I(u) is simply the multiplier ideal sheaf of a quasi-psh function u on X.
As part of showing (1.4), we will argue that the limit on the left hand side exists. We think of (i) as the cohomological characterization of the closure of T . On the other hand, we think of (ii) as the potential theoretic characterization.
The equivalence between (iii) and (iv) indicates that uniform notions of K-stability with respect to test configurations and filtrations are very likely the same (c.f. [CC18, Question 1.12]). To fully confirm this, one needs to show that elements of F can be approximated by T while also preserving the slope of the K-energy functional, as predicted by [Li20, Conjecture 1.5]. Of course, due to the examples of [ACGT08], one might still expect that relative K-stability with respect to F and T are different notions.
It is not hard to see that for many of the rays constructed in [Dar17] there is strict inequality in (1.4), implying T R 1 . This strict containment was noticed in [BBJ21, Remark 5.9] using non-Archimedean methods. However, as a result of condition (i) above and our Theorem 3.7 (iv) (that allows for flexible construction of L 1 rays using test curves) the containment T R 1 is seen to be nowhere d c 1 -dense (c.f. [CC18, Question 1.10]). Finally, let us put our Theorem 1.1 in historical context, and discuss the possible connection with the general version of the Yau-Tian-Donaldson conjecture, seeking to characterize existence of constant scalar curvature Kähler (cscK) metrics cohomolgous to ω in terms of algebro-geometric properties of the bundle (X, L). Despite the difficulties arising due to infinite dimensionality, and the underlying fourth order PDE, by now we have a comprehensive understanding on the analytic side (see [BDL17], [CC21a], [CC21b], [CC18], [DL20]), allowing to characterize existence of cscK metrics in terms of uniform geodesic stability along C 1,1 -rays of the space of Kähler metrics, yielding the essentially optimal version of what was conjectured by Donaldson [Don99].
Similarly, with the development of the non-Archimedean toolbox, we have a very good understanding of the algebraic side as well (see [BBJ21], [BHJ17], [BJ18], [BHJ19], [Der16]), allowing not only to embed test configurations into R 1 (along with their invariants), but to also keeping track of algebraic invariants using non-Archimedean metrics, an intermediate notion lying between the algebraic and analytic data.
The remaining step in the variational program for the Yau-Tian-Donaldson conjecture is to understand what L 1 rays are approximable by ample test configurations, while also preserving the slope of the radial K-energy in the limit. This is the connection point with our characterization theorem above, though here we completely ignored the behavior of the K-energy in the approximation process.
During the final stages of writing up our work we learned of the preprint of C. Li [Li20], who proved that L 1 rays with bounded radial K-energy are in T . Though not a characterization of T , this result is more closely lined up with the variational program, and it is an intriguing prospect to examine the relationship between our results and the ones in [Li20].
Non-Archimedean interpretation. The non-Archimedean approach to singularities in pluripotential theory developed in [BFJ08], [BBJ21] will play a crucial role in our discussion (especially in the form of valuative criteria), and we mention here how Theorem 1.1 can be interpreted in this context.
In this approach T can be identified with H NA , the space of Fubini-Study metrics on the Berkovich analytification (X NA , L NA ) with respect to the trivial valuation on C. On the other hand, the closure T can naturally be identified with the space of finite energy metrics on (X NA , L NA ), leading to a characterization of T = E 1,NA in the non-Archimedean context [BBJ21], in addition to the ones given in Theorem 1.1.
Given an arbitrary ray {r t } t ∈ R 1 , in [BBJ21] the authors introduce a natural projection satisfying r t ≤ Π(r) t and one can think of {Π(r) t } t as the closest ray to {r t } t that is approximable by test configurations (see Section 3.2 for more details). Using Π, one can conveniently introduce the non-Archimedean Monge-Ampère energy as follows: The original definition is given by means of the non-Archimedean Monge-Ampère measures introduced in [CL06], [CLD12] that only depend on the non-Archimedean data r NA (see [BJ18] and references therein). Here we show that Π is d c 1 -continuous, and give the following expansion interpretation for I NA : Theorem 1.2 (Theorem 3.18, Corollary 4.9). The map Π : R 1 → T is d c 1 -continuous. Moreover, for any sup-normalized {r t } t ∈ R 1 we have (1.6) The integral in (1.6) can be interpreted as L NA k {r t }, the non-Archimedean analogue of Donaldson's L-functional (see (4.1) and Proposition 4.4). Theorem 1.2 says that the leading order term in the expansion of L NA k {r t } is given by the non-Archimedean Monge-Ampère energy. This is the non-Archimedean analogue of [BF14,Theorem 3.5], where based on [Don05] and [BB10], it is proved that Donaldson's L-functional from [Don05] admits an expansion whose leading order term is given by the usual Monge-Ampère energy of E 1 .
Similar flavour results in the non-Archimedean setting were obtained in [BE21, Theorem A] and [BGG+20, Theorem A] under different assumptions on the ground field and for continuous metrics. It would be interesting to see if one could extend their results to finite energy metrics in case of trivially valued base fields of characteristic 0, using our Theorem 1.2.
The closure of the space of algebraic singularity types. Finally we discuss approximation in the space of singularity types S. We start with precisely defining algebraic/analytic singularity types. Definition 1.3. We say that [ψ] is an algebraic singularity type (notation: [ψ] ∈ Z ⊆ S), if there exists c ∈ Q + and around every point of X there exists a Zariski open set U and f j ∈ O(U) algebraic, such that ψ| U − c 2 log k j=1 |f j | 2 is smooth. We say that [ψ] is an analytic singularity type (notation: [ψ] ∈ A ⊆ S), if there exists c ∈ R + and around every point of X there exists an open set U (with respect to the analytic topology) and f j ∈ O(U), such that ψ| U − c 2 log k j=1 |f j | 2 is locally bounded.
Many different conventions are in place regarding the definition of analytic/algebraic singularity types in the literature (see [Dem12,Definition 1.10], [MM07, Definition 2.3.9] or [RWN17, (4)]). Out of all possible definitions, our choice of Z is the smallest family one can consider, and A is perhaps the biggest. As we will show below, for purposes of approximation, using A or Z does not make a difference.
Since many of the important invariants involved only depend on the singularity type of the potentials, in [DDNL21]  We refer to Section 2.1 for a detailed discussion on the d S metric, as well as the paper [DDNL21]. We only mention the following double inequality of [DDNL21, Lemma 3.4 and Proposition 3.5], giving intuition about what d S -convergence means: where C > 1 only depends on dim X. The pseudometric d S is slightly degenerate, however d S ([φ], [ψ]) = 0 implies that the singularities of φ, ψ are essentially indistinguishable (for example all Lelong numbers, multiplier ideal sheaves, and mixed complex Monge-Ampère masses need to agree), so in many ways this is a blessing in disguise.
In our last main result we prove the inequality between arithmetic and non-pluripolar volumes for general ω-psh functions, complementing (1. We also give a potential theoretic characterization for elements of Z in terms of the coincidence locus of P [·] I (defined in (1.5)) and its analytic counterpart P [·]: Theorem 1.4 (Theorem 5.5). For u ∈ PSH(X, ω) we have (1.7) Assume that X ω n u > 0. Then equaility holds in (1.7) if and only if one the following equivalent conditions hold: It is part of showing (1.7) that the limit on the left hand side exists. The equality part of (1.7) can be interpreted as singular version of the Riemann-Roch theorem. There are many known examples of potentials u ∈ PSH(X, ω) for which the inequality (1.7) is strict. One can even construct potentials u that have zero Lelong numbers but don't have full mass, i.e., u ∈ E [GZ07]. In particular, Z S. What is more, by taking convex combinations of this u with a potential of Z (and checking failure of condition (i) above), one can see that the containment Z S is nowhere d S -dense.
That the equivalences of Theorem 1.4 are only proved in the presence of positive mass is perhaps not surprising, in light of [DDNL21, Theorem 1.1, Section 4.3], where it was shown that d S is complete only in the presence of such condition. Still, it remains to be seen if this condition is essential in Theorem 1.4.
With different motivation, Rashkovskii studied the approximability of local isolated psh singularties using isolated analytic singularities in [Ras13]. It is an interesting prospect to find the local analog of the d S metric, and to relate our findings to the ones in [Ras13].
As we will see, in all of our main theorems one can allow an additional twisting Hermitian line bundle (T, h T ) as well (see Theorem 4.7, Theorem 4.8, Theorem 5.5 and Corollary 4.9).
Organization. In Section 2 we recall previous results and adapt them to our context. In Section 3 we extend the Ross-Witt Nyström correspondence to finite energy L 1 geodesic rays. In Section 4 we prove Theorem 1.1 and Theorem 1.2. In Section 5 we prove Theorem 1.4.

The metric space of L 1 geodesic rays and singularity types
The L 1 metric on H ω and its completion. We recall the basics of the L 1 metric structure of H ω , introduced in [Dar15]. For a survey we refer to [Dar19, Chapter 3], and perhaps [DR17, Section 4] is a convenient quick summary. For historical context, we refer to [Rub20].
The d 1 metric on H ω is simply the path length pseudometric associated with the following L 1 Finsler metric: where V = X ω n is the total volume. One then shows that d 1 is non-degenerate, making (H ω , d 1 ) a bona fide metric space [Dar15, Theorem 1]. When trying to find the d 1 -completion of H ω , one encounters the space E 1 ⊆ PSH(X, ω) that is defined in the following manner. One first defines the space of full mass potentials E ⊆ PSH(X, ω). Potentials in this space are characterized by the property X ω n u = X ω n . Here ω n u is the following limit of measures where ω n max(u,−k) can be made sense of using Bedford-Taylor theory, since max(u, −k) is bounded [BT76]. For a general ω-psh potential u we have X ω n u ∈ [0, X ω n ], with all values taken up. For more on this we refer to the original papers [GZ07] and [BEGZ10] (for a minimalist survey see [Dar19, Chapter 2]).
Then, E 1 ⊆ E is the class of full mass potentials satisfying X |u| ω n u < ∞. By [Dar15, Theorem 2], one can extend the metric d 1 to E 1 . In addition, (E 1 , d 1 ) is a complete geodesic metric space whose geodesics are decreasing limits of C 1,1 -solutions to a degenerate complex Monge-Ampère equation ( [Che00], [CTW18], [Dar15]). Unfortunately, such limits are not the only d 1 -geodesics connecting points of E 1 (see the comments after [Dar15,Theorem 4.17]). However, when talking about d 1 -geodesics, we will only consider this distinguished class of length minimizing segments.
We recall that the definition of the Monge-Ampère energy I : E 1 → R (sometimes denoted Aubin-Yau, or Aubin-Mabuchi energy): (2.2) Using the Monge-Ampère energy one can give the following potential theoretic description of d 1 [Dar15, Corollary 4.14]: where P (u, v) ∈ E 1 is the following rooftop envelope: To understand d 1 -convergence from a purely analytical point of view, the following double estimate is often very useful [Dar15, Theorem 3]: where C is a constant only dependent on n = dim X.
The complete metric space of L 1 rays. Building on the previous paragraph, we recall the basics of the L 1 metric structure of R 1 , the space of d 1 -geodesic rays in E 1 emanating from 0 ∈ H ω , explored in detail in [DL20].
To fix notation, a d 1 -geodesic ray [0, ∞) ∋ t → u t ∈ E 1 with u 0 = 0 will simply be denoted {u t } t ∈ R 1 . The chordal metric d c 1 on R 1 is introduced in the following manner: Proposition 5.1], the limit on the right hand side exists, and one can show that d c 1 is non-degenerate, satisfies the triangle inequality, moreover (R 1 , d c 1 ) is a complete geodesic metric space [DL20, Theorem 1.3, Theorem 1.4]. The subspace R ∞ ⊆ R 1 is the space of bounded geodesic rays {u t } t , satisfying the property u t ∈ L ∞ ∩ E 1 , t ≥ 0. Such rays allow for an important approximation property [DL20, Theorem 1.4] that will be used in this work, as well as its proof: The pseudo-metric space of singularity types. We recall the basics of the pseudometric structure on S, the space of singularity types, first explored in [DDNL21]. First one needs to construct a map from S to R ∞ ⊆ R 1 , using ideas going back to [Dar17]. Starting with [u] ∈ S, one constructs d 1 -geodesic segments [0, l] ∋ t → s(u) l t ∈ E 1 ∩ L ∞ connecting s(u) l 0 = 0 and s(u) l l = max(u, −l). Moreover, using the maximum principle one can show that {s(u) l t } l≥t is an l-increasing sequence converging to r[u] t ∈ E 1 ∩ L ∞ , yielding a geodesic ray Due to non-degeneracy of d c 1 , one immediately sees that is the envelope of the singularity type [χ], first considered in [RWN14] in the Kähler context: Lastly, we mention the following double inequality that often comes handy when discussing d S -convergence in practice [DDNL21, Lemma 3.4 and Proposition 3.5]:

Exponents and filtrations of a family of Hermitian metrics
In this section we relate the log-slope of the volume of a one dimensional family of Hermitian metrics with the associated filtration. In many ways we simply tailor the arguments of [Ber17] to our needs, and for more thorough treatment of related results we refer to [BE21, Part 1]. Let V be a finite dimensional complex vector space of dimension N. By Herm(V ) we denote the set of positive Hermitian inner products on V . Throughout this section, H s ∈ Herm(V ) (s ≥ 0) will denote a continuous family of Hermitian inner products, simply referred to as s → H s .
We denote by V * the dual vector space of V . Recall that given any H ∈ Herm(V ), it naturally induces a dual inner product H * ∈ Herm(V * ).
Definition 2.2. Let I ⊆ R be an interval. We say that a family H s ∈ Herm(V ) (s ∈ I) is negative if its trivial complexification z → H Re z is a Griffiths negative vector bundle on We do not assume that s → H s is positive or negative for the moment.
Hence λ H takes up only a finite number of values. If ∞ is not one of them, then λ H is the exponent of the non-Archimedean pseudometric e λ H , motivating our terminology.
The above properties of the exponent λ H also allow to introduce the associated filtration of s → H s : Notice that F H λ defines an increasing right-continuous filtration of V by linear subspaces. This filtration is bounded from above (in the sense that , one can diagonalize U 1 with respect to U 0 to find eigenvalues e λ 1 , . . . , e λ N counting multiplicity. Then one can introduce the following metric: (2.6) This metric, along with its L p -counterparts, was studied extensively in [DLR20], where it was shown that d V 1 quantizes d 1 in the appropriate context. In particular, (in the appropriate diagonalizing basis) the curve [0, 1] ∋ t → U t := diag(e tλ 1 , . . . , e tλ N ) ∈ Herm(V ) provides a d V 1 -geodesic joining U 0 and U 1 ([DLR20, Theorem 1.1], [BE21, Theorem 3.7]). There are other d V 1 -geodesics joining U 0 , U 1 , but we will only consider the above type of length minimizing curves.
We emphasize the following formula, pointing out that the dualization map U → U * between Herm(V ) and Herm(V * ) is an isometry: This can be verified by picking an appropriate diagonalizing basis of V .
In studying the growth of the volume of the unit ball with respect to H s as s → ∞, we start with the following lemma that one can justify simply by diagonalizing: where the integral on the right is interpreted in the Stieltjes sense. Moreover, dim F H λ 0 = dim λ∈R F H λ = 0 in the middle sum, by convention.
By det H we mean the determinant of a matrix representative of the sesquilinear form H ∈ Herm(V ) with respect to a fixed basis, making det H s /det H 0 in (2.8) well-defined. Note that our convention is different from that in [BE21] by a square.
Using Hadamard's inequality, for s → H s only satisfying λ H < ∞, one can show that in general the left hand side is dominated by the right hand side in (2.8).
As we will see, equality holds in (2.8) when s → H s is only positive, satisfying a mild decay condition. Before we prove this, we will construct a geodesic ray s →H s asymptotic to any s → H s , closely following [Ber17, Proposition 2.2].
Recall the following comparison principle that will be used multiple times in the argument Note that s → log det W s is linear and also log det W s ≤ log det U s . Varying the endpoints a, b we obtain that s → log det H s is concave, whenever s → H s is positive. As a result, the limit on the left of (iv) exists.

Proof. First we interpret the condition λ H
s is the geodesic connecting H 0 and H t and H t s = H s for s > t. By the comparison principle, we get that H t s is t-decreasing for any s ≥ 0 (in fact s → H t s is positive for any t, but this will not be needed). Due to (2.9) we can take the decreasing t-limit to obtainH It is immediate that s → H s is a d V 1 -geodesic ray satisfying (i) and (ii) . Recall that s → log det H s is concave (due to positivity) and of course s → log detH s is linear (since s →H s is a geodesic). Using this, due to the construction of s →H s , one immediately sees that lim s→∞ s −1 (log det H s − log detH s ) = 0, proving (iv).
Since H s ≥H s , comparing with (2.6) we arrive at Because of this, by Lemma 2.5 below, for any ǫ > 0 there exists s 0 such that e −ǫsH s ≤ H s ≤ e ǫsH s , for s ≥ s 0 . This is immediately seen to imply (iii) .
Proof. We fix a basis (e 1 , . . . , e dim V ) that is orthonormal with respect to U 1 and orthogonal with respect to U 2 , with eigenvalues e λ 1 , . . . , e λ dim V . Then by definition, Proof. As discussed below the statement of Lemma 2.4, the limit on the left hand side of (2.10) exists and is finite. In fact, for the ray s →H s constructed in Lemma 2.4 we have that Since λ H = λH implies F H λ = FH λ , the conclusion follows from Lemma 2.3.

Quantization of the Monge-Ampère energy
Recall that we have a positive Hermitian line bundle (L, h) inducing a background Kähler metric ω = i 2π Θ(h) > 0 with class [ω] ∈ c 1 (L). For any k ≥ 1, the metric h induces a Hermitian metric h k on L k .
For the rest of the paper we also fix a holomorphic (twisting) line bundle T on X together with a smooth Hermitian metric h T . By slight abuse of notation, we also denote the induced For each k ≥ 1, define the Hilbert map H k : E 1 → Herm(H 0 (X, T ⊗ L k )) as follows: Define the quantum Monge-Ampère energy I k : Herm(H 0 (X, T ⊗ L k )) → R by the formula . (2.12) The expression I k (U) − I k (V ) is nothing but Donaldson's original L-functional from [Don05]. As we will see, the I k quantizes the usual Monge-Ampère energy I, motivating our notation. Now we define L k : (2.13) Remark 2.7. When (T, h T ) is trivial and ϕ is equal to P (φ) for some continuous function φ on X, the functional L k (ϕ) is defined and studied in [BB10]. Note that our L k (ϕ) corresponds to h 0 (X, L k )L k (X, φ/2) in their paper, with the extra 1/2 due to the difference in conventions.
With the focus of this section on quantization, we define a scaled version of this metric: This convention coincides with the one used in [DLR20]. Let S = {0 < Re z < 1} ⊂ C be the unit strip and π : S × X → X be the natural projection. We say that (0, 1) ∋ t → ϕ t ∈ E 1 is a subgeodesic if its complexification Φ satisfies π * ω + dd c Φ ≥ 0 on S × X in the sense of currents. Let us recall the following version of Berndtsson's convexity theorem [Ber09, Theorem 1.2]. (2.14) The proof follows line by line from that of [DLR20, Corollary 2.13, Lemma 2.14].
Lemma 2.9. For ϕ 0 , ϕ 1 ∈ E 1 we have where C depends only on X.
Proof. Notice that Take a basis (e 1 , . . . , e N k ) of H 0 (X, T ⊗ L k ) which is orthonormal with respect to H k (ϕ 0 ) and is orthogonal with respect to H k (ϕ 1 ). Let λ j := log H k (ϕ 1 )(e j , e j ) for j = 1, . . . , N k . Then By the Riemann-Roch theorem, N k is dominated by V k n , and the result follows.
This result is the twisted version of [DLR20, Theorem 1.2(ii)]. We reproduce the proof for convenience of the reader.
By [BK07] we can find ϕ j 0 , ϕ j 1 ∈ H ω , sequences decreasing to ϕ 0 , ϕ 1 , respectively. We may assume without loss of generality that ϕ 1 0 , ϕ 1 1 ≤ 0. By our assumption, for any j ≥ 1 we have, lim Hence it is enough to show that for any ǫ > 0, we can find j 0 > 0 such that for any j ≥ j 0 . By symmetry, we only prove the former. We fix some real number δ > 1 for now.
. Hence, by comparison of the tangent vectors at ℓ ′ 1 , we conclude for all t ∈ [0, 1]. By (2.14), for any f ∈ H 0 (X, T ⊗ L k ), we have By [DLR20, Lemma 4.5], the left hand side is finite. Now we find a basis e 1 , . . . , e N of H 0 (X, T ⊗ L k ) that is orthonormal with respect to H k (ℓ ′ 1 ) and such that the quadratic form is orthogonal with eigenvalues λ 1 , . . . , λ N . Then, using (2.15) and (2.16), we get Letting k → ∞, it follows from the classical Bergman kernel expansion that (see the elementary calculations following [DLR20, where the last equality follows from [DLR20, Lemma 4.5]. Letting δ ց 1, we find finishing the proof of the claim, and the argument. Next we quantize the Monge-Ampère energy (see (2.2)) on the space E 1 , extending the corresponding result for smooth metrics [Don05], continuous metrics [BB10, Theorem A], and the case of K X -twisting [BF14, Theorem 3.5]: Theorem 2.11. For any ϕ ∈ E 1 , we have lim k→∞ n! k n L k (ϕ) = I(ϕ) . (2.17) Proof. Assume that this result is true for ϕ ∈ H ω . For a general ϕ ∈ E 1 , take a decreasing sequence ϕ j ∈ H ω that converges to ϕ. Then by Lemma 2.9 and Lemma 2.10, for any j ≥ 1, By our assumption, lim k→∞ n! k n L k (ϕ j ) = I(ϕ j ). This implies that Letting j → ∞, we conclude. It remains to prove (2.17) when ϕ ∈ H ω . When (T, h T ) is trivial, this was carried out in [Don05]. Indeed, it suffices to observe that where B k (tϕ) denotes the k-th T -twisted Bergman kernel at tϕ ∈ PSH(X, ω

An algebraic notion of singularity type
Detecting singularities using algebraic tools. In this section, let (X, ω) be a compact Kähler manifold of dimension n. Let θ be a smooth real (1, 1)-form on X representing a pseudo-effective cohomology class. Given u ∈ PSH(X, θ), as pointed out in the literature (see for example [BFJ08], [Kim15]), one can not characterize the singularity type [u] using "mainstream" algebraic data, like multiplier ideal sheaves I(cu), c > 0 or Lelong numbers. Instead, one can introduce an algebraic notion that is coarser than equivalence up to singularity types, considered in [KS20, Section 2.1]: Definition 2.12. Let ϕ, ψ ∈ PSH(X, θ). We put ϕ I ψ in case I(aϕ) ⊆ I(aψ) for all a > 0. Then I is a preorder, with equivalence relation ϕ ≃ I ψ characterized by I(aϕ) = I(aψ) for all a > 0. The corresponding classes are called I-singularity types, and are denoted by [χ] I , where χ ∈ PSH(X, θ) is a representative of the class.
In the above statement ν(ϕ, y) is the Lelong number of ϕ • π at y, in local coordinates defined by Given a prime divisor Z of Y , the generic Lelong number of ϕ along Z is defined as: Due to Siu's semicontinuity theorem, for a set S ⊆ Z of measure zero, we have that ν(ϕ, z) = ν(ϕ, Z) for z ∈ Z \ S, motivating the terminology.
Since we work with smooth models Y , for a coherent ideal J ⊆ O X one can talk about ν(J , y) (ν(J , Z)) as the minimum vanishing order of f j • π at y (along Z) for a finite set of generators {f j } j of J y (J z for some z ∈ Z). Moreover, one can see that ν(J , y) := ν(J , E y ), where E y is the exceptional divisor of p y : Bl {y} Y → Y , the blowing up of Y at y.
Remark 2.15. That (i) implies (ii) in Theorem 2.13 is seen to follow from (2.18). The reverse direction now follows from the local result [BFJ08, Theorem A]. More broadly, the reverse direction is the consequence of the valuative criteria for integrability (see [Bou17,Theorem 10.12] and its proof).
As we saw in the above argument, the class of potentials χ satisfying χ I ϕ are stable under taking max, hence we can introduce the notion of an envelope with respect to Isingularity: (2.19) The above envelope should be compared with the well-known envelope with respect to singularity type (going back to [RWN14] and [RS05] in the local case): We refer the reader to [RWN14], [DDNL18b], [DDNL21] for basic properties of P [u]. We begin to discuss the parallel between the above notions: (ii) By Choquet's lemma we can take ψ j ∈ PSH(X, θ) (j ≥ 0), such that ψ j ≤ 0, ψ j ∼ I ϕ and that ψ j increases to P [ϕ] I a.e.. It follows from Guan-Zhou's strong openness theorem [GZ15] that ϕ ∼ I P [ϕ] I . Example 2.19. Following [BBJ21, Example 6.10], we give an example showing that not all model potentials are I-model. Consider X = P 1 and let ω be the Fubini-Study form on X. Let K ⊆ P 1 be a Cantor set. Then K carries an atom-free probability measure, whose potential v has zero Lelong numbers. Then the pull-back of v to any proper modification of X has zero Lelong numbers as well [ Lemma 2.21. Suppose that {ϕ j } j ∈ PSH(X, θ) and ϕ ∈ PSH(X, θ) are model potentials.
(i) If ϕ j ց ϕ and ϕ j are I-model, then ϕ is I-model as well.
Proof. First we prove (i). Note that P [ϕ] I ≃ I ϕ I ϕ j for any j ≥ 1. Hence by Proposition 2.18, We deal with (ii). Since X θ n ϕ j ց X θ n ϕ > 0 [DDNL21, Proposition 4.8], by [DDNL21, Lemma 4.3] there exists α j ց 0 and v j : where in the first inequality we have used P [ψ] I ∼ I ψ, Theorem 2.13, and additivity of Lelong numbers. Since {ϕ j } j is decreasing, so is {P [ϕ j ] I } j , hence w := lim j P [ϕ j ] I ≥ P [ϕ] I exists. Since α j → 0 and sup X P [v j ] I = 0, comparison with the above gives w = P [ϕ] I . Dealing with (iii) is similar. Since X θ n ϕ j ր X θ n ϕ > 0 [DDNL18b, Theorem 2.3], by [DDNL21, Lemma 4.3] there exists α j ց 0 and v j : where in the first inequality we have used that P [ψ] I ∼ I ψ, Theorem 2.13, and additivity of Lelong numbers. Since {ϕ j } j is increasing, so is {P [ϕ j ] I } j , hence w := lim j P [ϕ j ] I ≤ P [ϕ] I exists. Since α j → 0 and sup X P [v j ] I = 0, comparison with the above yields w = P [ϕ] I .
Remark 2.22. The condition ϕ ≃ I ψ is strictly stronger than requiring ϕ and ψ have the same Lelong number everywhere on X (See [Kim15, Example 2.5]). As we will see in the next section, in terms of valuations, ϕ I ψ means exactly that the induced non-Archimedean functions on the space of divisorial valuations X div Q satisfy ϕ NA ≤ ψ NA . In particular, ϕ NA = ψ NA is equivalent to ϕ ≃ I ψ. See [BFJ08] for further details.
Algebraic approximation of I-model potentials. For the remained of this subsection, we return to the context of an ample line bundle L → X, with hermitian metric h, whose first Chern form is equal to the Kähler form ω. Let us recall the following well known result, originated from [DPS01, Theorem 2.2.1]: Theorem 2.23. Let u ∈ PSH(X, ω). Let u k ∈ PSH(X, ω) be the partial Bergman kernel of V u k := H 0 (X, L 2 k +k 0 ⊗ I(2 k u)): (2.20) where H 2 k (u) is the Hilbert map of L 2 k with twisting T = L k 0 (see Section 2.3). Then u k has algebraic singularity type (See Definition 1.3), and for some k 0 = k 0 (X, L, ω) the following hold: (iii) for all m > 0 and k > m one can find δ k,m > 1 such that I(mδ k,m u k ) ⊆ I(mu) and δ k,m ց 1 as k → ∞.

Sketch of proof. (i) follows from
Step 1 in the proof of [GZ05, Following terminology of [Cao14], approximations {u j } j of u ∈ PSH(X, ω) of the type (2.20), satisfying all three conditions in Theorem 2.23 will be referred to as quasi-equisingular approximations of u.
We arrive at the following result, characterizing the difference between model and Imodel potentials in terms of d S -approximability via quasi-equisingular sequences:

Theorem 2.24. Let ϕ ∈ PSH(X, ω) be a model potential (P [ϕ] = ϕ) with X ω n ϕ > 0. Then ϕ is I-model (P [ϕ] I = ϕ) if and only if [ϕ] is the d S -limit of a quasi-equisingular approximation
Remark 2.25. Due to this theorem and Proposition 2.20, the class of analytic singularity types A are d S -approximable by algebraic singularity types of Z (in the presence of positive mass), already proving the (easy) equivalences between (iv) and (v) in Theorem 1.4.
Proof. Let ϕ ∈ PSH(X, ω) be an I-model potential with X ω n ϕ > 0. Let ϕ k be the corresponding quasi-equisingular approximation of ϕ. By [DDNL18b, Theorem 1.1] the sum n j=0 X ω n−j ∧ ω j ϕ k is decreasing in k, hence converges. By [DDNL21, Lemma 3.4], (2.21) By Lemma 2.20 and Lemma 2.21(i), both ϕ and ϕ ′ are I-model, so it suffices to show that I(mϕ) = I(mϕ ′ ) for any m > 0. Since ϕ ≤ ϕ ′ , the non-trivial inclusion is I(mϕ) ⊇ I(mϕ ′ ). To prove this, by the last statement of the above theorem, we notice that By the strong openness theorem [GZ15], we can let k → ∞ to arrive at I(mϕ) ⊇ I(mϕ ′ ), as desired. .6] now gives that X ω n ϕ ′ ց X ω n ϕ . Since ϕ = P [ϕ], lim j ϕ ′ j ≥ ϕ, and X ω n ϕ > 0, by [DDNL18b, Theorem 3.12] we obtain that lim j ϕ ′ j = ϕ. Finally, Lemma 2.21(i) implies that ϕ is I-model. Before we proceed further, we recall that the conventions set at the beginning of Section 2 guarantee that the leading order Riemann-Roch expansion takes the form h 0 (X, T ⊗ L k ) = V n! k n + O(k n−1 ), where T is an arbitrary line bundle. We recall the following result of Bonavero: Theorem 2.26. Assume that ϕ ∈ PSH(X, ω) has algebraic singularity type ([ϕ] ∈ Z). Then This is indeed a special case of the singular holomorphic Morse inequalities proved by Bonavero [Bon98], surveyed in [MM07, Theorem 2.3.18].
Remark 2.27. Note that our convention for the multiplier ideal sheaves is different from that of Bonavero's. In fact, Bonavero's definition of I(ϕ/2) corresponds to our I(ϕ). But the volume of ϕ/2 in the sense of Bonavero is exactly the same as X ω n ϕ in our sense, hence the holomorphic Morse inequalities take exactly the same form, despite the difference in conventions.
We additionally note that Bonavero proved the above result for potentials with analytic singularity type, however his definition of this notion is less general than ours in Definition 1.3, this being the reason for our more conservative statement above.
where in the first estimate we used that (ω+ǫω+dd c ψ ǫ ) n is supported on {ψ ǫ = 0} [DDNL18b, Theorem 3.8], and in the last equality we used that e bψ ǫ is bounded and quasi-continuous, converging to e bϕ in capacity. Similarly, (ω + dd c ϕ) n is supported on {ϕ = 0}, hence letting b → ∞ we arrive at our claim Let δ > 0 be arbitrary, and take ǫ = p/q ∈ Q + such that 1 n! X (ω + ǫω + dd c ψ ǫ ) n < δ. By the positive mass case of this theorem, For a general k (possibly not divisible by q) write k = dq + r with d ∈ Z ≥0 , r = 0, . . . , q − 1.
for q large enough. Thus, replacing T with T ⊗L r as the twisting line bundle, we are reduced to the case r = 0, dealt with in (2.23). Letting δ → 0, the proof is finished.

Filtrations, flag ideals and the non-Archimedean formalism
Filtrations of the ring of sections. Let us recall the basics of filtrations in the context of canonical Kähler metrics, going back to work of Székelyhidi [Szé15]. We refer to [BHJ17, Section 1, Section 5] and [BJ18, Section 3] for a much more detailed description. In the sequel, we will focus on the point of view advocated by Ross-Witt Nyström [RWN14]. For r > 0 we will consider R(X, L r ) := ∞ k=0 H 0 (X, L kr ) the graded ring associated to (X, L r ). When dealing with filtrations, we always assume that r is big enough so that R(X, L r ) is generated in degree 1.
A filtration {F λ k } λ∈R,k∈N , r of R(X, L r ) is a collection of decreasing left-continuous for any k, k ′ ∈ N, λ, λ ′ ∈ R) and linearly bounded (there existsλ > 0 big enough such that F −λk k = H 0 (X, L kr ) and Fλ k k = {0}, for k ≥ 0.) Let (C, · ) be the trivially normed complex line. A non-Archimedean graded norm on R(X, L r ) is a norm on R(X, L r ) considered as a (C, · )-algebra satisfying exponential boundedness (there existsλ > 0, such that for any k ∈ N and any non-zero s ∈ H 0 (X, L kr ), e −λk ≤ s k ≤ eλ k ) and sub-multiplicativity ( s · s ′ k+k ′ ≤ s k s ′ k ′ for any s ∈ H 0 (X, L kr ) and s ′ ∈ H 0 (X, L k ′ r )).
It is elementary to verify that there is a bijection between filtrations {F λ k }, r and non-Archimedean graded norms { · k } k∈N on R(X, L r ) given by Due to this, we will use the terms filtrations and non-Archimedean norms interchangeably.

Filtrations induced by test configurations and flag ideals.
A filtration {F λ k }, r is a Z-filtration if the jumping numbers/points of discontinuity of λ → F λ k are integers for all k ≥ 0.
Due to the fact that R(X, L r ) is generated in degree 1, we have a surjective map (2.24) Naturally, · 1 induces a non-Archimedean norm on (H 0 (X, L r )) ⊗k , as well as on any quotient (H 0 (X, L r )) ⊗k /W , where W ⊆ (H 0 (X, L r )) ⊗k is a subspace.
As a result, given a filtration {F λ k }, r , it is possible to define a non-Archimedean graded norm · T k on each H 0 (X, L kr ) only using · 1 and the maps (2.24). We say that {F λ k }, r is induced by an (ample) test configuration if it is a Z-filtration, and the map (2.24) induces an isometry between the graded non-Archimedean norms · T k and · k for any k ≥ 0.
This of course is not the usual definition of (ample) test configurations. However, as pointed out in [BHJ17, Proposition 2.15], this construction is in a one-to-one correspondence with the usual one going back to Tian [Tia97] and Donaldson [Don01].
Flag ideals yield an important (and in many ways exhaustive) class of filtrations induced by test configurations, going back to Odaka [Oda13]. A flag ideal a is a C * -invariant coherent ideal of O X×C , cosupported in X × {0}. Such an ideal is always of the form where a j is an increasing sequence of coherent ideals of O X and τ is the variable in C. As a convention, we write a j = O X , when j ≥ d and a j = 0, when j < 0.
If for some r > 0, the sheaves L r ⊗ a i are globally generated for every i ≥ 0, then we associate a filtration to a in the following way. First we define F λ 0 := H 0 (X, L r ⊗ a ⌊−λ⌋ ). As {F λ 0 } λ is decreasing and left-continuous, it induces a non-Archimedean norm · 1 on H 0 (X, L r ), which further introduces a non-Archimedean graded norm · on R(X, L r ) via the surjections (2.24).
By construction, the underlying Z-filtration is clearly induced by a test configuration, with the jumping numbers of {F λ 0 } λ being exactly the integers j such that a j a j+1 . As pointed out by Odaka [Oda13], essentially all test configurations arise via this construction.

The non-Archimedean formalism.
Here we recall some of the formalism developed in [BHJ17; BHJ19; BBJ21], and later tailor some of their results to our context. By X div Q we denote the set of rational divisorial valuations on X, i.e., valuations v : C(X) → Q of the form v = c ord D , with D being a prime divisor on some smooth variety Y , mapping to X via a projective modification, and c ∈ Q + . By convention, we also take the trivial valuation v triv to be part of X div Q . To any v ∈ X div Q one associates σ(v) ∈ (X × C) div Q , the Gauss extension of v. The construction is described in detail in [BHJ17, Section 4.1].
The Gauss extension is defined as σ(v)( j f j τ j ) := min j (v(f j ) + j), where f j ∈ C(X) and τ is the coordinate of C. It can be immediately verified that σ(v) thus defined is a valuation, moreover as shown in [BHJ17, Lemma 4.5, Theorem 4.6], σ(v) is also divisorial on X × C.
Remark 2.29. In [BBJ21, Section 3] the authors define X div Q as divisorial valuations on normal models of X. However, due to Hironaka's theorem, one can always take log-resolutions of normal models, hence no information is lost if one considers only prime divisors of smooth models, as we do in this work.
The non-Archimedean data of potentials, rays and flag ideals. In the non-Archimedean approach to canonical Kähler metrics one converts both analytic and algebraic data into non-Archimedean data, i.e., various functions on X div Q . We describe how this is carried out with ω-psh functions, geodesic rays and flag ideals.
Given u ∈ PSH(X, ω), one defines u NA : X div Q → R by (2.25) Recall that ν(u, D) is the generic Lelong number of u along D (see Section 2.4). In accordance with the literature, sometimes we will also write V (u) := cν(u, D). Before proceeding, we note the following result, which corresponds to Theorem 2.13 in the non-Archimedean dictionary. Indeed, for any projective modification π ′ : Y → X with Y smooth, the Lelong number for any u ∈ PSH(X, ω) at y ∈ Y is the same as the generic Lelong number along the exceptional divisor of the blowup Bl {y} Y .
Proposition 2.30. Suppose that u, w ∈ PSH(X, ω). Then the following are equivalent: Given a ray {r t } t ∈ R 1 , it is known that t → sup X r t is linear, in particular, there exists l ∈ R such that Φ(s, x) = r − log |s| + l log |s| ∈ PSH(X × D, π * ω), where D is the unit disk and π : X × D → X is the usual projection.
We define r NA : X div Q → R using the Gauss extension in the following manner: is to be interpreted as a suitable multiple of the generic Lelong number along the center divisor E ′ of σ(v). It can be seen that this definition does not depend on l ∈ R, nor on the choice of smooth model hosting E ′ .
Lastly, by the same construction u NA can be defined for sublinear subgeodesic rays {u t } t (as defined in Section 3.1 below).
Given a flag ideal a = d−1 j=0 τ j a j + τ d O X , such that L m ⊗ a j are globally generated for all j, we define the corresponding function non-Archimedean function ϕ NA a : X div Q → R as follows: where v ∈ X div Q , and v(a j ) is the valuation of a j given by v(a j ) := inf{ v(a) : a ∈ a j }.
Approximation by flag ideals. Given a ray {r t } t ∈ R 1 , in [BBJ21] the authors define an approximation scheme by flag ideals a m such that ϕ NA a m ց r NA . We describe the main point of this procedure, as it will be important in the sequel.
For simplicity, let us assume that {r t } t ∈ R 1 satisfies sup X r t ≤ 0 for t ≥ 0. The general case, can easily be reduced to this case, but one needs to slightly extend the definition of flag ideals (to allow for fractional ideals). We have Φ(s, x) = r − log |s| ∈ PSH(X × D, π * ω), and we simply define [BBJ21, Section 5.3]: a m := I(2 m Φ) .
As pointed out in [BBJ21, Lemma 5.6] L 2 m +m 0 ⊗a m j is globally generated for some m 0 > 0 and all m, j. In addition, by the proof of [BBJ21, Theorem 6.2], the subbaditivity of multiplier 3 The structure of R 1 and approximability

The extended Ross-Witt Nyström correspondence
The results of this subsection hold for an arbitrary Kähler manifold (X, ω). The goal is to give a duality between the finite energy geodesic rays of R 1 ω and certain maximal test curves, reminiscent of [RWN14] and [DDNL18a], but to also give a formula the Monge-Ampère slope of L 1 rays in terms of their Legendre transforms. To do this we consider a wider context and generalize the discussion going back to [RWN14], revisited in [DDNL18a].
Due to t-convexity, we obtain some immediate properties of sublinear subgeodesic rays: Lemma 3.1. Suppose that {u t } t is a sublinear subgeodesic ray. Then the set {u t > −∞} is the same for any t > 0. In particular, for any x ∈ X the curve t → u t (x) is either finite and convex on (0, ∞), or equal to −∞ on this interval.
A psh geodesic ray is a sublinear subgeodesic ray that additionally satisfies the following maximality property: for any 0 < a < b, the subgeodesic (0, 1) ∋ t → v a,b t := u a(1−t)+bt ∈ PSH(X, ω) can be recovered in the following manner: We note the following properties of the map v → sup X v along rays: Lemma 3.2. For any psh geodesic ray {u t } t , the map t → sup X u t is linear. For sublinear subgeodesics, the map t → sup X u t is only convex.
The statement for subgeodesics is a consequence of t-convexity. To argue the statement for psh geodesic rays, one can simply use [Dar17, Theorem 1] together with approximation by bounded geodesics, and the continuity of u → sup X u in the weak L 1 -topology of PSH(X, ω).
Making small tweaks to [RWN14,Definition 5.1], we are ready to give the definition of test curves: is concave, decreasing and usc for any x ∈ X.
Note that this definition is more general than the one in [RWN14] (where the authors only considered potentials with small unbounded locus), even more general than the one in [DDNL18a] (where the authors considered only bounded test curves). Moreover, condition (ii) allows for the introduction of the following constant: (3.2) Remark 3.4. We adopt the following notational convention: psh test curves will always be parametrized by τ , whereas rays will be parametrized by t. Hence {ψ t } t will always refer to some kind of ray, whereas {φ τ } τ will refer to some type of test curve. As we prove below, rays and test curves are dual to each other, so one should think of the parameters t and τ to be dual to each other as well.
Definition 3.5. A psh test curve {ψ τ } τ can have the following properties: (iv) We say (ψ τ ) is bounded if ψ τ = 0 for all τ small enough. In this case, one can introduce the following constant, complementing (3.2): In the above definition, we followed the convention P [−∞] = P [−∞] I = −∞. Note that bounded test curves are clearly of finite energy.
We recall the Legendre transform, that will help establish the duality between various types of maximal test curves and geodesic rays. Given a convex function f : [0, +∞) → R, its Legendre transform is defined aŝ (3.5) The (inverse) Legendre transform of a decreasing concave function g : We point out that there is a sign difference in our choice of Legendre transform compared to the convex analysis literature, however this choice will be more suitable for us. As it is well known, for every τ ∈ R we have thatĝ(τ ) ≥ g(τ ) with equality if and only if g is additionally τ -usc. Similarly,f (t) ≤ f (t) for all t ≥ 0 with equality if and only if f is t-lsc. In general,ĝ is the τ -usc envelope of g, andf is the t-lsc envelope of f . We will refer to these facts commonly as the involution property of the Legendre transform.
Proof. Let S = {Re s > 0}. In the proof, usc(·) will denote the usc regularization on S × X. We consider the usual complexification of the inverse Legendre transform: where usc u is the usc regularization of u on S × X. Let π : S × X → X be the natural projection. It will be enough to show that usc u = u.
We introduce E = {u < usc u} ⊆ S × X. As both u and usc u are R-invariant in the imaginary direction of S, it follows that E is also R-invariant, i.e., there exists B ⊆ (0, ∞)×X such that E = B × R.
As E has Monge-Ampère capacity zero, it follows that E has Lebesgue measure zero. By Fubini's theorem B ⊆ (0, ∞) × X has Lebesgue measure zero as well. For z ∈ X, we introduce the slices: By Fubini's theorem again, we have that B z has Lebesgue measure 0 for all z ∈ X \ F , where F ⊆ X is some set of Lebesgue measure 0. By slightly increasing F , but not its zero Lebesgue measure(!), we can additionally assume that u t (z) > −∞ for all t > 0 and z ∈ X \ F (indeed, at least one potential ψ τ is not identically equal to −∞).
Let z ∈ X \ F . We argue that B z is in fact empty. By our assumptions on F , both maps t → u t (z) and t → (usc u)(t, z) are locally bounded and convex (hence continuous) on (0, ∞). As they agree on the dense set (0, ∞) \ B z , it follows that they have to be the same, hence B z = ∅. This allows to conclude that (3.7) By duality of the Legendre transform ψ τ (x) = inf t>0 [u t (x) − tτ ] for all x ∈ X and τ ∈ R (here is where the τ -usc property of τ → ψ τ is used). From this and (3.7) it follows that ψ τ = χ τ a.e. on X, for all τ ∈ R. Since both ψ τ and χ τ are ω-psh (the former by definition, the latter by Kiselman's minimum principle [Dem12,Theorem I.7.5]), it follows that in fact ψ τ ≡ χ τ for all τ ∈ R.
Given a sublinear subgeodesic ray {φ t } t (psh test curve {ψ τ } τ ), we can associate its (inverse) Legendre transform at x ∈ X aŝ (3.8) Our main theorem describes a duality between various types of rays and maximal test curves, extending various particular cases from [RWN14], [DDNL18a]: (i) psh test curves and sublinear subgeodesic rays, (ii) maximal psh test curves and psh geodesic rays, (iii) [RWN14], [DDNL18a] maximal bounded test curves and bounded geodesic rays. In this case, we additionally have that (iv) maximal finite energy test curves and finite energy geodesic rays. In this case, we additionally have that Recall that the functional I is defined in (1.2).
We prove (ii). From [Dar17, Propisition 5.1] (that only uses the maximum principle (3.1)) we obtain that for any psh geodesic ray {u t } t , the curve {û τ } τ is a maximal psh test curve.
Let {ψ τ } τ be a maximal psh test curve. We will show that the sublinear subgeodesic {ψ t } t is a psh geodesic ray. By elementary properties of the Legendre transform we can assume that τ + ψ = 0, in particular {ψ t } t is t-decreasing.
By the maximality of {ψ τ } τ , we obtain that ψ τ =φ τ . An application of the Legendre transform now gives thatψ t = φ t , a contradiction. Hence {ψ t } t is a psh geodesic ray.
The duality of (iii) is simply [DDNL18a, Theorem 1.3], closely following [RWN14]. We deal with (iv). As before, we may assume that τ + ψ = 0. As a preliminary result, in Proposition 3.8 below we prove (3.9) for bounded maximal test curves.
By Proposition 3.8 below The right hand side is bounded from below, since {ψ τ } τ is a finite energy test curve. Since X ω n ψ k τ ց X ω n ψτ , we can take the limit on both the left and right hand side, to arrive at (3.9), also implying that {ψ t } t is a finite energy geodesic ray.
Conversely, assume that {φ t } t is a finite energy geodesic ray, with decreasing approximating sequence of bounded rays {φ k t } t , as detailed above. For similar reasons we have Since I{φ k t } ց I{φ t }, the monotone convergence theorem gives that (3.9) holds for {φ τ } τ , finishing the proof.
As promised, to complete the argument of Theorem 3.7, we prove the next proposition, whose argument can be extracted from [RWN14, Section 6] with additional references to [DDNL18b]. We recall the precise details here as the results of [RWN14] were proved in the context of potentials with small unbounded locus.
Proposition 3.8. Suppose that {ψ τ } τ is a bounded maximal test curve with τ + ψ = 0. Then Proof. Without loss of generality we assume that V = 1. For N ∈ Z + , M ∈ Z and t > 0, we introduce the following:ψ , since it is a maximum of a finite number of ω-psh potentials (here we also used that {ψ τ } τ is a bounded test curve). Moreover, we now argue that Indeed, for elementary reasons: , and using τ -concavity we notice that Moreover, on U t we havě Fixing N, let M be the biggest integer to the left of 2 N τ − ψ . Then repeated application of (3.11) yields We now notice that we have Riemann sums on both the left and right of the above inequality. Using Lemma 3.9 below, it is possible to let N → ∞ and obtain (3.10), as desired.
By working harder, using [DDNL21, Theorem B], one can show that τ → X ω n ψτ 1/n is concave, however we will not need this in the sequel.

Rays induced by filtrations and approximability
We fix a filtration ({F λ k }, r) on R(X, L r ). Following [RWN14, Section 7], one can associate a maximal test curve to this filtration in the following manner. The corresponding construction for test configurations is due to Phong-Sturm [PS07], [PS10]. For τ ∈ R, let Since each F τ k k is finite dimensional, one notices thatû k τ ∈ PSH(X, ω) has analytic singularity type. Moreover, by the multiplicativity of the filtration we have that (3.14) As a result, Fekete's lemma implies thatû τ := lim kû k τ = sup kû k τ ∈ PSH(X, ω) exists, and the curve τ →û τ has a number of special properties. We give a very brief sketch of the argument. As elaborated in [RWN14, Section 7], that {û τ } τ is τ -concave and τ -decreasing is a consequence of the multiplicativity of the filtration. Boundedness follows from linear boundedness of the filtration. To make sure that {û τ } τ is τ -usc we takeû τ + Let v ∈ X div Q , and σ(v) ∈ (X × C) div Q be the corresponding Gauss extension (see Section 2.5). Since τ + r ≤ 0, by Lemma 3.14 below we conclude that −r Lemma 3.14. Let Ω be a complex manifold. Let F be a non-empty family of non-positive psh functions on Ω and ψ := usc sup ϕ∈F ϕ . Then for any x ∈ Ω, (3.17) Proof. By Choquet's lemma, we may assume that F consists of only countably many functions ϕ j (j ∈ N) and ψ = usc sup j∈N ϕ j . By upper semicontinuity of Lelong numbers, ν(ψ, x) ≥ lim j∈N ν(max{ϕ 0 , . . . , ϕ j }, x). In addition, by monotonicity of Lelong numbers, ν(ψ, x) = lim j∈N ν(max{ϕ 0 , . . . , ϕ j }, x) = inf j ν(ϕ j , x), where in the last step we used [Bou17, Corollary 2.10].
Given a ray {r t } t ∈ R 1 , we define two associated envelopes, based on the ideas of [BBJ21].
A priori, it is not even clear that {Π(r t )} t is a geodesic ray. On the other hand, following the argument of [BBJ21, Theorem 6.6], for t ≥ 0 we can also consider π(r t ) := sup {r ′′ t } t ∈ R 1 : r ′′ NA = r NA .
As we prove now, these two projections coincide to give a ray, whose maximal test curve can be described concretely: Theorem 3.15. Let {r t } t ∈ R 1 . Then {Π(r) t } t is an approximable geodesic ray. Moreover the following hold: Since {Π(r) t } t is always approximable, we get that Π • Π = Π, i.e., Π is a projection. It is not clear if (i) and (ii) hold for τ = τ + r , though this is not essential to our discussion. Proof. First we observe that no generality is lost if we assume the condition sup X r t ≤ 0 for t ≥ 0, after possibly replacing r t with r t − mt for some m ∈ N big enough.
We note that (i) immediately implies (ii), as P [r τ ] I ≃ Irτ (Proposition 2.18(ii)). On the other hand, due to Theorem 2.13 and (3.15), we have that (ii) implies (iii) as well. Lastly, (i) together with Lemma 3.16 below imply that {Π(r) t } t is approximable.
Hence we only need to argue (i). In fact, from Lemma 3.16 below we know that {π(r t )} t is an approximable ray, so Π(r) t ≤ π(r) t , since r t ≤ π(r) t . To show that Π(r) t ≥ π(r) t , it suffices to show that for any Phong-Sturm ray {w t } t satisfying w t ≥ r t , we have w t ≥ π(r) t .
Lemma 4.1. (i) Let {r t } t be a sublinear subgeodesic ray and r ′ t := r t + tc for some c ∈ R.
Proof. (i) is obvious. Let us argue (ii). One can see that To conclude, one observes that both the first and second lines are positive quantities.
Next we provide an important estimate for the radial Monge-Ampère energy of approximable rays, in terms of L NA k .
Proposition 4.2. Let {r t } t ∈ R 1 be an approximable ray, i.e., {r τ } τ is I-maximal. Then Proof. By Lemma 4.1, we may assume that τ + r = 0. By Lemma 3.9, we have X ω n rτ > 0 for τ < 0. Moreover,r τ is I-model for all τ ∈ R by Theorem 3.17. We can calculate where the first line we used (3.9), in the second line we used the Riemann-Roch theorem together with Theorem 2.28, and in the third line we used Fatou's lemma.
Using the results of Section 2.2, in the next lemma we provide a formula that will be an Next we link the non-Archimedean functional L NA k to the classical functional L k for sufficiently positive subgeodesic rays: Proposition 4.4. Let {r t } t be a sublinear subgeodesic ray and δ > 0 such that r t ∈ E 1 , and ω rt ≥ δω for all t ≥ 0. Then there exists k 0 (δ) > 0 such that t → L k (r t ) is convex, moreover (4.8) As it will be clear from the proof below, in case T = K X and h T is dual to ω n , one can omit the condition ω rt ≥ δω from the assumptions.
Proof. By Lemma 4.1, we may assume that sup X r t = 0 for any t ≥ 0.
By Lemma 4.3 for f ∈ H 0 (X, T ⊗ L k ) we have that λ H k (f ) = lim t→∞ t −1 log H k (r t )(f, f ) = −k sup λ < 0 : f ∈ H 0 (X, T ⊗ L k ⊗ I(kr λ )) . (4.9) In particular, for λ ≥ 0, where I − (kr τ ) := λ<τ I(kr λ ), and F H k λ is the filtration associated to λ H k , defined in (2.5). As H k (r s ) is increasing in s, H k (r s ) * is decreasing in s, hence the exponent λ H * k of H k (r s ) * on H 0 (X, T ⊗ L k ) * is bounded above. Moreover, the family (H k (r t )) t≥0 is positive when k ≥ k 0 (δ) by Theorem 2.8. As a result, t → L k (r t ) is convex (see the comments after Lemma 2.4) and the conditions of Theorem 2.6 are satisfied to imply that for k ≥ k 0 (δ), where in the first line we also used (4.10). As I(kr τ ) ⊆ I − (kr τ ) ⊆ I(kr τ −ǫ ) for any ǫ > 0, and F H k λ can only have finitely many jumping numbers, we get Comparing with (4.1), the proof is finished.
Finally, since T ⊆ F , we obtain that (i) implies (ii) . For the other direction, it is enough to show that elements of F are approximable. However the rays of F are all I-maximal, due to Theorem 3.11, so they are approximable due to Theorem 3.17, proving (i) .
Theorem 4.8. Let {r t } t ∈ R 1 with sup X r t = 0 for any t ≥ 0. Then lim k→∞ n! k n L NA k {r t } exists and can be estimated the following way (4.14) Proof. Consider the approximable ray {Π(r) t } t ∈ R 1 . In the argument (iii) implies (iv) of the previous theorem, we actually showed that the limit on the left hand side of (4.14) exists and is equal to I{Π(r) t }. The inequality now readily follows from the fact that r t ≤ Π(r) t , implying I{r t } ≤ I{Π(r) t }.
For {r t } t ∈ R 1 , it is possible to introduce the non-Archimedean Monge-Ampère energy in the following manner: I NA {r t } := I{Π(r) t } . (4.15) In particular, when {r t } t ∈ T we have I NA {r t } = I{r t }. Comparing with (4.13), we obtain a new interpretation for the non-Archimedean Monge-Ampère energy: This proves the second part of Theorem 1.2.

The closure of algebraic singularity types
We start with the following result about approximable bounded geodesic rays.
Proposition 5.1. Let {r t } t ∈ T ⊆ R 1 be a ray of bounded potentials. Then for all τ ∈ (τ − r , τ + r ) we have X ω n rτ = lim k→∞ n! k n h 0 (X, T ⊗ L k ⊗ I(kr τ )) . (5.1) In particular the limit on the right hand side exists. Adding these estimates, we get lim , contradicting (5.6).
Finally, we state our last main result, collecting many of our previous findings: