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Analytic approximation of plurisubharmonic singularities

We study several classes of isolated singularities of plurisubharmonic functions that can be approximated by analytic singularities with control over their residual Monge--Amp\`ere masses. They are characterized in terms of Green functions for Demailly's approximations, relative types, and valuations. Furthermore, the classes are shown to appear when studying graded families of ideals of analytic functions and the corresponding asymptotic multiplier ideals.


Introduction
Let u be a plurisubharmonic function on a complex manifold X of dimension n > 1. We are interested in asymptotic behavior of u near its singularity point x (where u takes the −∞ value). Loosely speaking, this asymptotic behavior is what we call the singularity of u at x; more precisely, a singularity is an equivalence set with respect to the relation u ∼ v ⇔ u = v + O(1) near x.
Already in dimension 1, the asymptotic behavior of a subharmonic function can be quite complicated, however everything becomes easy when restricting to the case of functions that are harmonic in a punctured neighborhood of x, namely, u(z) = ν log |z − x| + O(1) near x, where ν = 1 2π ∆u({x}). In several variables, this corresponds to consideration of plurisubharmonic functions u that are maximal outside x; when u is locally bounded outside x, this means that it satisfies the homogeneous Monge-Ampère equation in a punctured neighborhood of x. Nevertheless, even in this class the variety in plurisubharmonic singularities is enormous. For instance, their collection contains those of functions u = a log |F | + O(1) for holomorphic mappings F from a neighborhood of x to C N , N ≥ 1. To some extent, we can consider such analytic singularities as "simple" objects and try to use them for approximation of arbitrary plurisubharmonic singularities. Note that although the passage to maximal singularities can change the asymptotic, it keeps its standard characteristics such as Lelong number, residual Monge-Ampère mass, integrability index, etc. In this sense, we lose little when replacing a plurisubharmonic function by the corresponding swept out maximal function.
There are some indications that an analytic approximation would work. First, it is the classical Lelong-Bremermann theorem on uniform approximation of continuous plurisubharmonic functions by functions of the form max i a i log |f i |. Second, the celebrated Approximation Theorem due to Demailly [3] states that for any plurisubharmonic function u there exists a sequence of functions D k u with analytic singularities, converging to u pointwise and in L 1 loc ; moreover, these functions keep track on the singularity of u -for example, their Lelong numbers converge to the Lelong number of u. On the other hand, assuming x to be an isolated singularity point of u (such functions are called weights), it is not clear if the residual Monge-Ampère masses (dd c D k u) n ({x}) of D k u converge to that of u, even if u is maximal outside x. This is definitely not so (for any analytic approximation!) if u has zero Lelong number at x and positive mass there, because then D k u are locally bounded near x and hence of zero residual mass; existence of such a function u is however a well known open problem. Since the Monge-Ampère mass is a crucially important characteristic of plurisubharmonic singularity (for instance, (dd c log |F |) n ({x}) is the multiplicity of the mapping F at x), this uncertainty is rather regretful.
It turns out that the convergence of the masses of D k u is inseparably linked with good analytic approximability of u. In the present note, we study a few classes of plurisubharmonic singularities that are well-suited for analytic approximations. To start with, we introduce a metric ρ on the collection of isolated maximal singularities, based on the notion of relative type [17] σ(u, ϕ) = lim inf z→x u(z)/ϕ(z), and we say that a singularity ψ is asymptotically analytic if it belongs to the ρ-closure of the set of analytic singularities. Asymptotical analyticity of a weight ψ means actually that for any ǫ > 0 one can find an analytic weight ψ ǫ such that (1 + ǫ)ψ ǫ + O(1) ≤ ψ ≤ (1 − ǫ)ψ ǫ + O (1). Examples of such singularities are exponentially Hölder continuous functions and those depending only on the absolute values of the variables. On the other hand, we have no examples of maximal singularities that are not asymptotically analytic.
We obtain several characterization of such singularities. Some of them use the notion of generalized Green functions with respect to plurisubharmonic singularities [22]: given a maximal singularity ϕ at a point x in a bounded hyperconvex domain D, the corresponding Green function G ϕ is the upper envelope of negative plurisubharmonic functions u in D such that σ(u, ϕ) ≥ 1. Since G ϕ = ϕ + O(1), it depicts the same singularity as ϕ. We will use the Green functions as a kind of uniformization for families of the singularities. First, we test the Green functions for the Demailly approximations D k ψ of a maximal weight ψ. In spite of the convergence of D k ψ to ψ, there is no direct reason for their Green functions to converge to G ψ . Moreover, this again faces the aforementioned problem on zero Lelong number and nonzero Monge-Ampère mass. We prove that a subsequence of the Green functions of D k ψ -namely, G D m! ψ -decreases to the function G ψ = inf k G D k φ ≥ G ψ ; the proof is based on the Subadditivity Theorem for multiplier ideals [6]. We show that G ψ can be viewed as an "analytic greenification" of ψ, that is, the upper envelope of all negative plurisubharmonic functions v in D such that σ(v, ϕ) ≥ σ(ψ, ϕ) for all analytic weights ϕ. It is proved to coincide with G ψ if and only if ψ can be approximated from above by analytic singularities whose residual Monge-Ampère masses converge to the mass of ψ; we call such singularities inf analytic, and they form our other main class of singularities that are close to analytic. Any asymptotical analytic singularity is inf analytic, and a stronger convergence describes asymptotical analyticity: G D k ψ /G ψ → 1 uniformly on D \ {x}; it implies, in particular, (dd c D k ψ) n ({x}) → (dd c ψ) n ({x}) and exp G ψ ∈ C(D).
Another characteristic property of asymptotically analytic weights makes use of the notion of asymptotic multiplier ideals [7]. We show first that, given a graded family a • of primary ideals a k ⊂ O x , their Green functions G a k [19] converge, after a rescaling, to a weight G a• , while the Green functions G j k of the corresponding asymptotic multiplier ideals j k converge (also after rescaling) to a weight G j• ≥ G a• . The multiplicities e(a • ) and e(j • ) (in the sense of [14]) are just the Monge-Ampère masses of G a• and G j• , respectively, and they coincide if and only if the limit functions are equal.
The equality is then proved for the case of families generated by "good" (in particular, asymptotically analytic) singularities. Specifically, we consider a k defined by the conditions σ(log |f |, ψ) ≥ k. The upper regularization of sup k h k , where h k = k −1 G a k , is shown to coincide with G ψ if and only if ψ can be approximated from below by analytic singularities whose residual Monge-Ampère masses converge to the mass of ψ, and for such a singularity, h k → G ψ in L n (D). The weight ψ is asymptotically analytic if and only if h k /G ψ → 1 uniformly on D \ {x}.
The lower envelope inf k H k of the functions H k = k −1 G j k equals G ψ when ψ has analytic approximations from both below and above, and, if this is the case, H k converge to G ψ in L n (D) as well. When ψ is asymptotically analytic, we have again Previously, the equality e(a • ) = e(j • ) was known in few situations -for instance, when a k are determined by Abhyankar valuations [8] (which can be interpreted as a very special case of the weight ψ; in return, a stronger result is proved there), or when they are monomial [14]. A new feature of our approach is that we show what the "limits" of a • and j • -or, more precisely, the scaled limits of their logarithmic images log |a • | and log |j • | -are: they are the collections of plurisubharmonic functions u satisfying σ(u, G a• ) ≥ 1 and σ(u, G j• ) ≥ 1, respectively.
Another portion of results concerns types relative to asymptotically analytic and inf analytic weights. While the residual Monge-Ampère masses of Demailly's approximations characterize inf analytic weights, asymptotical analyticity can be completely described in terms of relative types. Namely, σ(D k u, ψ) → σ(u, ψ) for every u if and only if ψ is asymptotically analytic. Since the Lelong number is the type relative to the weight ψ(z) = log |z − x|, this gives a stronger property of Demailly's approximations than that on convergence of their Lelong numbers.
In addition, the type of u relative to an inf analytic weight ψ turns out to coincide with the infimum of weighted divisorial valuations arising as generic Lelong numbers of pullbacks µ * u of u along exceptional primes of proper modifications µ :X → X over x. As a consequence, it can be represented as the lower envelope of the disk functionals where γ are analytic maps from the unit disk to X such that γ(0) = x. When both u and ψ have analytic singularities, this was proved (by algebraic methods) in [12]; note that in commutative algebra, the object corresponding to the relative type is known as Samuel's asymptotic function. For tame weights, a slightly smaller class of plurisubharmonic weights than asymptotically analytic ones, these representations were essentially proved in [1]; we believe however that our methods are more elementary.
The obtained characterizations of asymptotically/inf analytic weights show limits of using Demailly's approximations in describing plurisubharmonic singularitiesunless all maximal singularities are such, which is doubtful (although this remains an open question).
We conclude with two results that perhaps are true not only for asymptotic analytic singularities, however we do not know how to prove them in the general case; our proofs just use the corresponding statements from algebraic geometry and the good approximability of asymptotical analytic singularities. The first result extends an inequality due to Teissier from [9] to the plurisubharmonic situation, giving thus τ where τ w is the residual Monge-Ampère mass of w at x. The second statement concerns the multiplier ideal of the function max{u, v} and is based on a result from [15].

Preliminaries
For basics on plurisubharmonic functions and the complex Monge-Ampère operator, we refer the reader to [10]. In this section, we recall some notions of particular importance for us and set the corresponding notation.

Plurisubharmonic singularities
Let X be a complex manifold of dimension n, and let PSH x be the collection of germs of plurisubharmonic functions at x ∈ X. We will say that a plurisubharmonic function u is singular at x if u is not bounded (below) in any neighborhood of x.
The equivalence class cl(u) of u ∈ PSH x with respect to the relation "u ∼ v if u(z) = v(z) + O(1)" will be called the plurisubharmonic singularity of u; in [22], a closely related object was introduced under the name "standard singularity".
Germs with isolated singularities at x (i.e., that are locally bounded outside x) will be occasionally called weights and their collection will be denoted by W x . The Monge-Ampère operator (dd c ) n is well defined on such functions; the residual mass A weight ϕ ∈ W x is called maximal if it is a maximal plurisubharmonic function on a punctured neighborhood of x (satisfies (dd c ϕ) n = 0 outside x). A basic example is ϕ = log |F | for an equidimensional holomorphic mapping F with isolated zero at x, and in this case τ ϕ equals the multiplicity of F . The collection of all nonconstant maximal weights will be denoted by MW x , and cl(MW x ) is the collection of all maximal singularities.

Relative types and Lelong numbers
For any function u ∈ PSH x , we denote its type relative to a weight ϕ ∈ MW x [17] by It is a finite nonnegative number, and maximality of ϕ implies the bound Evidently, σ(u, ϕ) = σ(v, ψ) for any u ∈ cl(u), ψ ∈ cl(ϕ), so the relative type is a function on singularities.
If f is an analytic function and F is an equidimensional holomorphic mapping, then, as follows from [12], σ(log |f |, log |F |) equals Samuel's asymptotic function [20] ν for the ideal I generated by the components of the mapping F .
When ϕ(z) = log |ζ(z) − ζ(x)|, where ζ are local coordinates near x, σ(u, ϕ) is the Lelong number ν(u, x) of u at x. Since only local behavior of plurisubharmonic functions is considered here, we will omit the indication on coordinate functions and use the denotation |z − x|.
Given an irreducible analytic variety M ⊂ X, the value

Green and Green-like functions
We will use the following extremal function introduced by V. Zahariuta [22], see also [23]. Let D ⊂ X be a bounded hyperconvex domain; P SH − (D) will mean the collection of all negative plurisubharmonic functions on D.
The function is plurisubharmonic in D, maximal in D \{x}, G ϕ ∈ cl(ϕ), and G ϕ = 0 on ∂D; moreover, it is a unique function with these properties. Furthermore, if ϕ is continuous near x, then G ϕ is continuous on D. (The continuity of φ ∈ W x is understood here as continuity of exp φ.) We will refer to this function as the Green (or Green-Zahariuta) function with singularity ϕ.
If ϕ(z) = log |z − x|, then G ϕ is the standard pluricomplex Green function G x,D of D with pole at x.
Any function u ∈ PSH x generates various maximal functions that can be viewed as its greenifications. First, let P be an arbitrary collection of maximal weights; consider the family M P u, is called the type-greenification of u with respect to the collection P [17]. For φ ∈ W x , the function h P φ ∈ MW x , equals 0 on ∂D and satisfies σ(h P φ , ϕ) = σ(φ, ϕ) for all ϕ ∈ P ; it can also be represented as the best plurisubharmonic minorant of the family , when the only available information on u is the values σ(u, ϕ) for ϕ ∈ P .
Alternatively, we can consider the class F φ,D of negative plurisubharmonic functions v in D such that v(z) ≤ φ(z) + O(1) near x, then the regularization of its upper envelope, is a plurisubharmonic function in D, the complete greenification of φ at x in D [17]. It is maximal on D \ {x} and equals zero on ∂D. If ϕ ∈ MW x , then g ϕ = G ϕ , the Green-Zahariuta function for ϕ. When ϕ is not maximal, one need not have the equality ϕ = g ϕ + O(1); nevertheless, the relative types and the residual Monge-Ampère mass remain the same [17, Prop. 5.5, 5.6]: σ(ϕ, ψ) = σ(g ϕ , ψ) for every ψ ∈ MW x and τ ϕ = τ gϕ . (2.7) The greenifications are related by

Multiplier ideals and Demailly's approximation
Given u ∈ PSH(X), let J (u) denote the multiplier ideal for u, that is, the ideal of functions f ∈ O(X) such that |f |e −u ∈ L 2 loc . A fundamental property of special importance for us is the Subadditivity Theorem [6, Thm. 2.6] We refer to [4], [11] for detailed information on multiplier ideals and their applications to analysis and algebraic geometry. The notion was used by Demailly for constructing approximations of plurisubharmonic functions by ones with analytic singularities. For any D ⋐ X, the restriction of J (u) to D is generated as O D -module by any basis of the Hilbert space Demailly's Approximation Theorem [3] says that any function u ∈ PSH x can be approximated, in a neighborhood D of x, by the functions where {f k,i } i is an orthonormal basis for H(ku), in a way that (2.10) Since there exist finitely many functions all the functions D k u have analytic singularities. As a consequence of (2.10), they converge to u pointwise and in L 1 loc . In addition, the Lelong numbers ν(D k u, x) of the functions D k at x tend to the Lelong number ν(u, x).

Asymptotically analytic singularities
We will say that u ∈ PSH x has analytic singularity if c log |F | ∈ cl(u) for some c > 0 and a holomorphic mapping F of a neighborhood of x to C N .
If a weight φ ∈ W x has analytic singularity, then the mapping F has isolated zero at x. As is known, the integral closure of the ideal generated by the components F j of F has precisely n generators ξ k = a k,j F j (generic linear combinations of F j ), k = 1, . . . , n, so log |F | = log |ξ| + O(1). Therefore, φ is equivalent to a maximal weight. The collection of all weights with analytic singularities at x will be denoted by AW x .
We are going to deal with singularities that are "close" to analytic ones, in the sense of the following metric.
is a metric on maximal singularities cl(MW x ).
We will say that ψ ∈ MW x has asymptotically analytic singularity if it belongs to the closure (in the sense of the metric ρ) of AW x . The collection of all such weights will be denoted by AAW x . Proposition 3.2 A weight ψ ∈ MW x has asymptotically analytic singularity if and only if for every ǫ > 0 there exists a weight ψ ǫ ∈ AW x such that Conversely, assuming the asymptotical analyticity, we can find weights ψ ǫ ∈ AW x such that Relations (3.2) do not use the maximality of ψ and can be thus used as a definition of asymptotical analyticity for non-maximal weights (and even for arbitrary plurisubharmonic functions).
Unlike analytic weights, an asymptotically analytic weight need not be equivalent to a maximal one. Take, for example, ψ = log |z| − | log |z|| 1/2 in the unit ball, then (1 + ǫ) log |z| − C ǫ ≤ ψ ≤ log |z|, while g ψ = log |z|. On the other hand, there is always a weight g ψ ∈ AAW x such that lim ψ/g ψ = 1, which follows from Proposition 3.3 Let ψ ∈ W x be a weight with asymptotically analytic singularity and let D be a hyperconvex neighborhood of the point x. Then there exists the limit where g ψ is the complete greenification of ψ (2.6) in D. As a consequence, for any u ∈ PSH x one can set Proof. Given ǫ ∈ (0, 1), choose a weight ψ ǫ ∈ AW x and a constant C ǫ > 0 such that is the Green function for ψ ǫ in D, and thus Remark. Without the asymptotic analyticity assumption the result need not be true (even in dimension 1, just take ψ ∈ W 0 such that lim sup z→0 ψ(z)/ log |z| = ∞).
Example 3.4 According to [1], a continuous weight ϕ ∈ W x is called tame if there exists a constant C > 0 such that for every t > C and every analytic germ f from the multiplier ideal J (tϕ) of tϕ at x (that is, the function f e −tϕ is L 2 -integrable near x), one has log |f | ≤ (t − C)ϕ + O(1). For maximal weights ϕ, the latter can be written as σ(log |f |, ϕ) ≥ t − C. Demailly's approximations D k ϕ (2.9) of a tame weight ϕ satisfy, by [1, Lemma 5.9], near x; moreover, conditions (3.3) characterize tame weights. Therefore, all tame weights have asymptotically analytic singularity.
Example 3.6 Let x = 0 ∈ X ⊂ C n . Any multicircular weight ϕ ∈ MW 0 (depending on |z 1 |, . . . , |z n | only) has asymptotically analytic singularity, which can be shown as follows. First of all, such a function ϕ is equivalent to its indicator at 0, that is the Green function for ϕ in the unit polydisk D n [13], so we assume ϕ = G ϕ . This implies ϕ(A c z) = c ϕ(z) for every c > 0, where Then the functionφ(t) := ϕ(e t 1 , . . . , e tn ) is convex and positive homogeneous in R n − , equal to zero on ∂R n − , and thus represents as ϕ(t) = sup{ a, t : a ∈ Γ}, t ∈ R n − , where Γ ⊂ R n + . Therefore, for any ǫ > 0 there exist positive integers m and N and monomials Take any w ∈ D n with ϕ(w) = t < −1 and let z = A c w with c = 1/|2t|. Then ϕ(z) = −1/2, A 1 w = A |2t| z, and Since ϕ(w) = t, this implies for all w with ϕ(w) < −1.
Remark. We have no example of ϕ ∈ MW x \ AAW x .

Convergence of Green functions
In what follows, we fix a bounded hyperconvex domain D ⊂ X containing x. Given φ ∈ MW x , let G φ denote the Green function of D for the singularity φ. Since any analytic weight is equivalent to a maximal weight, the Green functions G D k φ for Demailly's approximations D k φ (2.9) of φ are well defined, too.

Green functions of Demailly's approximations
By Demailly's approximation theorem, the functions D k φ converge to φ in L 1 loc for any φ ∈ W x . We do not know if this implies convergence of the Green functions G D k φ to G φ . This is certainly not so if φ has zero Lelong number (because then G D k φ ≡ 0), however existence of such a maximal weight φ is equivalent to a famous open problem on plurisubharmonic weights with zero Lelong number and positive residual Monge-Ampère mass.
We can show that there is a subsequence of the Green functions decreasing to the lower envelope G φ of the functions G D k φ (and thus G φ is plurisubharmonic).
Proof. Let J (kφ) be the multiplier ideals for the functions kφ, k ∈ Z + . By the Subadditivity Theorem (2.8), J ((mk)φ) ⊆ J (kφ) m for all m, k ∈ Z + . Since the multiplier ideals J (kφ) generate Demailly's approximations D k φ, we have D mk φ ≤ D k φ + C(k, m) and so, which implies that the sequence G D m! φ is decreasing to some function G φ , plurisubharmonic in D and maximal in D \ {x}. Moreover, (4.1) yields The function G φ has a nice interpretation in terms of greenifications. Namely, let h A φ denote the type-greenification (2.5) with respect to the family AW x : As a consequence, there exists a sequence ϕ j ∈ AW x such that G ϕ j decrease to h A φ .
Proof. As was mentioned in Section 2.3, the function h A φ is the best plurisubharmonic minorant of the family {G ψ : ψ ∈ AW x , σ(φ, ψ) ≥ 1}. Since all the functions G D k φ belong to that family, h A φ ≤ G D k φ and thus h A φ ≤ G φ . On the other hand, the Finally, the sequence ϕ j = D j! φ provides a sequence of Green functions decreasing to h A φ . Proposition 4.2 means that the singularity of G φ is the best upper bound on the singularity of φ when the latter can be tested on all analytic weights. This gives us one more motivation on the problem if G φ coincides with G φ .
It turns out that the relation G φ = G φ is completely controlled by the behavior of the Monge-Ampère masses of the functions D k φ.
Proof. (i) Since the Monge-Ampère operator is continuous with respect to decreasing sequences, the function ψ is maximal on D \ {x} and τ ϕ j → τ ψ . Assuming τ ψ = τ ϕ , we get the two functions, ψ ≥ ϕ, maximal on D \ {x}, equal to 0 on ∂D, and with the same residual Monge-Ampère mass at x. By Lemma 4.5 below, ψ = ϕ.
(ii) Similar proof, the relation τ ϕ j → τ η * in this case being due to Bedford-Taylor's result for increasing sequences of bounded plurisubharmonic functions, applied to the functions ψ j = max{ϕ j , −1}, since τ ϕ j = (dd c ψ j ) n (D). (Alternatively, one can refer to the convergence results for increasing sequences in more general classes of plurisubharmonic functions [2], [21].)

Lemma 4.5 [17, Lemma 6.3] Let D be a bounded hyperconvex domain and let
If φ is a tame weight, then (3.3) implies and, therefore, uniform convergence of G D k φ /G φ to 1. We will prove that the convergence holds true for any weight φ ∈ AAW x as well. In particular, this will give G φ = G φ for such weights.
Theorem 4.6 Given ψ ∈ MW x , let G ψ and G D k ψ be the Green functions of a bounded hyperconvex domain D with the singularities ψ and D k ψ, respectively. Then if and only if ψ ∈ AAW x . For such a weight ψ, we get thus τ D k ψ → τ ψ and G ψ = G ψ .

Corollary 4.7
The Green function G φ of a bounded hyperconvex domain D with singularity φ ∈ AAW x is continuous on D.
So, asymptotical analyticity is rather a strong property with regard to behavior of the singularities of D k φ. In order to get the largest class of weights φ such that G φ = G φ , we will say that a weight φ ∈ MW x has inf analytic singularity if there exists a sequence of weights φ j ∈ AW x such that φ ≤ φ j + O(1) and τ φ j → τ φ . The class of these weights will be denoted by IAW x ; evidently, IAW x ⊇ AAW x .
In view of Proposition 4.2, this can be restated as follows.

Asymptotic multiplier ideals
Another nice property of asymptotically analytic weights comes from consideration of asymptotic multiplier ideals, the notion introduced in [7]. Recall that a family In some cases the ideals j • are not much bigger than a • in the sense e(a • ) = e(j • ) (for instance, when a k are defined by Abhyankar valuations [8] or when they are monomial [14]). Let us apply a machinery of Green functions in order to deal with such families of ideals. Given a bounded hyperconvex neighborhood D of x, let G a k denote the Green function of D with singularity along a k [19], i.e. with the singularity ϕ k = log |F k |, where F k are holomorphic mappings whose component generate and a k , and let h k = k −1 G a k . Similarly, we denote H k = k −1 G j k ; as follows from (4.8), (4.10) Since the family a • is graded, we have a m k ⊆ a km and thus h k ≤ h km , hence we can argue as in the proof of Proposition 4.1. In doing so, we derive that the sequence h m! is increasing to the function h a• := sup k h k . Let us denote its upper regularization (h a• ) * by G a• .
In the same manner, relation (4.7) implies H k ≥ H km , so the sequence H m! is decreasing to the function G j• := inf k H k ∈ MW x . By (4.10), To show that the monotone convergence of h m! and H m! implies a convergence of all h k and H k , we need the following Proof. This is a particular case of [16,Prop. 3.4]. Proof. (i) Note first that, since the Samuel multiplicity e(I) of a primary ideal I generated by f 1 , . . . , f m , equals the residual Monge-Ampère masse of the function log |f | [5, Lemma 2.1], we have e(a k ) = k n τ h k and e(j k ) = k n τ H k . Since h m! increase to h a• and H m! decrease to G j• , we have τ h m! → τ a• and τ H m! → τ j• , so (4.6) and (4.9) give us τ h k → τ a• and τ H k → τ j• . In addition, h k ≤ G a• and H k ≥ G j• and all these functions are maximal on D \ {x}, equal zero on ∂D. Therefore, the statement follows from Lemma 4.10.
We can use results from Section 4.1 to find conditions for the equality G a• = G j• .
In view of Theorem 4.8, this gives us the following Let us now specify , it is a graded family of ideals. To provide they are different from {0} and are primary, we assume that the weight φ has finite Lojasiewicz exponent, i.e.
lim sup We will see that, for such a choice of a k , the relation G a• = G j• = G φ is true for asymptotically analytic and even more general singularities φ.
Since σ(h k , φ) ≥ 1, we have h k ≤ G φ and thus We will say that a weight φ ∈ MW x has sup-analytic singularity if φ ≥ φ j + O(1) for some weights φ j ∈ AW x with τ φ j → τ φ ; the collection of such weights is denoted by SAW x . Again, SAW x ⊇ AAW x . Note also that any weight with sup-analytic singularity has finite Lojasiewicz exponent. Proof. Let φ ∈ SAW x . In view of (4.12), we need to show only that G φ ≤ G a• .
Denote a (j) The other direction is evident.
Theorem 4.14 Let the functions G a• and G j• be defined as above for a weight φ ∈ IAW x ∩ SAW x . Then G a• = G j• = G φ and e(a • ) = e(j • ) = τ φ .
Proof. Proposition 4.13 implies G φ = G a• . Then Proposition 4.12 becomes applicable and, in view of Proposition 4.11(ii), completes the proof.
Remark. Since every Abhyankar valuation is generated by a relative type [1], Theorem 4.16 extends the mentioned result from [8] to arbitrary asymptotically analytic weights (note however that for Abhyankar valuations a stronger result is proved there).
Since AAW x ⊆ IAW x ∩ SAW x , the statement of Theorem 4.14 holds true for all asymptotically analytic singularities. It is not surprising that for such weights one can claim even more.
We start with a characterization of AAW x in terms of the functions h k .
Proof. Let φ have asymptotically analytic singularity, then for any ǫ ∈ (0, 1) one can find a holomorphic mapping F ǫ and a positive integer k such that so all the components of F ǫ belong to a k and thus h k ≥ (1 − ǫ) −1 G φ , which gives (4.13). Conversely, the inequality G φ /h k > (1 − ǫ) on D \ {x} implies (4.14).

Theorem 4.16
If the functions h k , H k , G a• , and G j• are defined as above for a weight φ ∈ AAW x , then h k /H k → 1 uniformly on D \ {x} and G a• = G j• = G φ .
Observe that our approach gives a more precise meaning to the fact that the families a • and j • are "close". Since all the ideals a k and j k are integrally closed, they are in a one-to-one correspondence with their Green functions G a k and G j k in the sense that, for example, f ∈ a k ⇔ log |f | ≤ G a k + O(1). These functions have the scaled limits G a• = lim and for "good" weights φ, the limit objects equal {u ∈ PSH x : σ(u, φ) ≥ 1}.

Relative types
Let us notice the following simple properties of types relative to asymptotically analytic weights.
Proof. This is a direct consequence of Theorem 4.6.
The corresponding property of types relative to inf analytic singularities reads as follows.
Proof. Let us denote ϕ j = G D j! φ . By Proposition 4.2 and Corollary 4.9, the functions ϕ j decrease to G φ ; therefore, σ(u, ϕ j ) is a decreasing sequence for any u ∈ PSH x . We can set then σ(u) = lim j→∞ σ(u, ϕ j ).
More advanced special properties of types relative to asymptotically/inf analytic weights are given below.

Types of Demailly's approximations
As stated in Demailly's Approximation Theorem, the Lelong numbers of D k u converge to that of u for every u ∈ PSH x . Here we extend this to types of D k u relative to asymptotical analytic weights.
We first relate the ϕ-types of the functions u k to the ϕ-types of u for exponentially Hölder continuous weights ϕ.
Lemma 5.4 Let ϕ ∈ MW x satisfy (3.4) and let u ∈ PSH x . Then the types σ(D k u, ϕ) of its Demailly approximations D k u satisfy the relations Proof. The first inequality in (2.10) implies the relation σ(D k u, ϕ) ≤ σ(u, ϕ).
This implies the convergence σ(D k u, ψ) → σ(u, ψ) for ψ ∈ AAW x ; moreover, this is one more characteristic property of asymptotically analytic weights.

Representation by divisorial valuations
Let µ be a proper modification over a neighborhood of a point x ∈ X. For any function f ∈ O x , the generic multiplicity of µ * f over an irreducible component of the exceptional divisor µ −1 (x) is a divisorial, or Rees, valuation. One can extend this notion to plurisubharmonic functions by replacing the multiplicity of µ * f with the generic Lelong number of µ * u over the component. Here we will represent types relative to inf analytic weights as envelopes of such valuations. Let us take ϕ = log |F | ∈ MW x . By the Hironaka desingularization theorem, there exists a log resolution for the mapping F , i.e., a proper holomorphic mapping µ of a manifoldX to a neighborhood U of x, that is an isomorphism between X\µ −1 (x) and U \{x}, such that µ −1 (x) is a normal crossing divisor with components E 1 , . . . , E N , and in local coordinates centered at a generic point p of a nonempty intersection E I = ∩ i∈I E i , where I ⊂ {1, . . . , N}, We need the following elementary Lemma 5. 6 Let v(t) be a negative convex function on R k − , k > 1, increasing in each t i . Then Proof of Lemma 5.6. Denote the left hand side of (5.4) by A and From the convexity of v it follows that for any point t * ∈ R k − the ratio and so on, the last inequality being Summing them up, we get Since the reverse inequality is evident, the lemma is proved.
From Lemma 5.6 applied to the function where ν E i (µ * u) is the generic Lelong number of µ * u along E i (2.4). We denote this value by R j (u) and call it the divisorial valuation of u ∈ PSH x along E i :

Now (5.3) gives us the following result.
Theorem 5.7 For any weight ϕ = log |F | ∈ AW x there exist finitely many divisorial valuations R j and positive integers m j such that Remark. For the case u = log |f | this follows also from [12, Thm. 4.1.6]. Conversely, one can deduce Theorem 5.7 from that result by applying Theorem 5.5.
Theorem 5.8 If ψ ∈ MW x , then ψ ∈ IAW x if and only if there exist denumerably many divisorial valuations R j and positive numbers s j such that Proof. Let ψ ∈ IAW x . By Theorem 5.7, for each j the type σ(u, D j! ψ) is the lower envelope of finitely many weighted divisorial valuations s j R j (u). Then the theorem follows from Proposition 5.3. Conversely, (5.5) means that G ψ is the best plurisubharmonic minorant of the function inf j G ψ j , where the weights ψ j represent the valuations s j R j in the sense σ(u, ψ j ) = s j R j (u), see the proof of Proposition 5.3. All ψ j are tame (and thus asymptotically analytic), and since the best plurisubharmonic minorant of the minimum of finitely many tame weights is tame as well, this means that there exists a sequence of weights ϕ j with analytic singularities such that the Green functions G ϕ j decrease to G ψ . Therefore, ψ ∈ IAW x .
Remarks. 1. For asymptotically analytic weights, the arguments were briefly sketched in [18]. For tame weights, the representation follows from what was proved (by a different method) in [1].
2. Relative types have the obvious property σ( α j u j , ϕ) ≥ α j σ(u j , ϕ) and hence are concave functionals on PSH x , while for the divisorial valuations R j there is always an equality. From this point of view, relation (5.5) is similar to the representation of concave functions as lower envelope of linear ones, which holds on linear topological spaces. This can be put into the general picture of tropical analysis on plurisubharmonic singularities [17], [18].

Analytic disks
Another (although related) representation for the types can be given by means of analytic disks. To do so, we use arguments from [12]. Given u ∈ PSH x and ϕ ∈ MW x , denote σ * (u, ϕ) = inf γ∈Ax lim inf ζ→0 γ * u(ζ) γ * ϕ(ζ) , (5.6) where A x is the collection of all analytic maps γ : D → X such that γ(0) = x, and γ * u is the pullback of u by γ.
Proof. For the case of analytic weight ψ = log |F | we will use the idea from the proof of [12,Prop. 5.4]. By Theorem 5.7, σ(u, log |F |) = min j m −1 j R j (u), where m j and R j (u) are generic Lelong numbers of log |µ * F | and µ * u, respectively, along the exceptional primes E j of a log resolution µ for the mapping F . Take a germ of an analytic curveγ j ⊂X passing transversally through a pointx j ∈ E j and such that the generic Lelong number of µ * u along E i equals the Lelong number of the restriction of µ * u toγ j atx j (which is possible bye Siu's theorem). Then for the curve γ j = µ * γ j we get For arbitrary ψ ∈ IAW x , we refer to Theorems 5.8.

Other results
In this section we present some results that perhaps are not specific to functions with asymptotically analytic singularities, but we do not know how one can prove them without such an assumption.
If I and J are two primary ideals in the ring O x , then e(I · J) 1/n ≤ e(I) 1/n + e(J) 1/n , where e(I) denotes the Samuel multiplicity of the ideal I.
If an ideal I is generated by analytic functions f 1 , . . . , f m , then e(I) = τ log |f | , see [5]. Let us for any u ∈ W x denoteτ u = τ 1/n u . Theorem 6.1 If the complete greenifications (2.6) of u, w ∈ W x have asymptotically analytic singularities, thenτ u+v ≤τ u +τ v .

Musţatǎ's summation formula
Given an ideal I = F 1 , . . . , F r ⊂ O x and γ > 0, we denote by J (I γ ) the multiplier ideal of the plurisubharmonic function function γ log |F |.
In [15] it was shown that J ((I + J) γ ) ⊆ α+β=γ J (I α )J (J β ) (6.2) (as is easy to see, the sum contains finitely many distinct terms). Its plurisubharmonic counterpart will be about the multiplier ideal of the function max{u, v}.