A non-Archimedean approach to K-stability, II: divisorial stability and openness

To any projective pair $(X,B)$ equipped with an ample $\mathbb{Q}$-line bundle $L$ (or even any ample numerical class), we attach a new invariant $\beta(\mu)\in\mathbb{R}$, defined on convex combinations $\mu$ of divisorial valuations on $X$, viewed as point masses on the Berkovich analytification of $X$. The construction is based on non-Archimedean pluripotential theory, and extends the Dervan-Legendre invariant for a single valuation--itself specializing to Li and Fujita's valuative invariant in the Fano case, which detects K-stability. Using our $\beta$-invariant, we define divisorial (semi)stability, and show that divisorial semistability implies $(X,B)$ is sublc (i.e. its log discrepancy function is non-negative), and that divisorial stability is an open condition with respect to the polarization $L$. We also show that divisorial stability implies uniform K-stability in the usual sense of (ample) test configurations, and that it is equivalent to uniform K-stability with respect to all norms/filtrations on the section ring of $(X,L)$, as considered by Chi Li.


Introduction
Consider, for the moment, a complex projective manifold X, equipped with an ample Q-line bundle L. The notion of K-stability of the polarized manifold (X, L) was originally phrased in [Don02] (building on [Tia97]) in terms of certain equivariant one-parameter degenerations of (X, L) known as test configurations.The 'uniform' version 1 of the Yau-Tian-Donaldson (YTD) conjecture states that the cohomology class c 1 (L) contains a unique constant scalar curvature Kähler (cscK) metric iff (X, L) is uniformly K-stable (see [BHJ17,Der16]).By [BDL20,CC21], the former condition is known to be equivalent to the coercivity of the Mabuchi K-energy functional, which implies in turn uniform K-stability [BHJ19].
As originally pointed out in [WN12], (ample) test configurations for (X, L) can be understood as filtrations of finite type on the section ring of (X, L); following the work of G. Székelyhidi [Szé15] and the authors' preprints [BoJ18a,BoJ18b], C. Li [Li22] studied K-stability for general filtrations (see Definition 5.7), and proved the remarkable result that uniform K-stability for filtrations implies coercivity of the Mabuchi K-energy functionaland hence the existence of a unique cscK metric.To sum up, for any polarized manifold (X, L) we have uniform K-stability for filtrations ⇒ unique cscK metric ⇒ uniform K-stability, (0.1) and the missing part of the (uniform) YTD conjecture thus consists in proving the (purely algebro-geometric) statement that uniform K-stability for filtrations already follows from its version for test configurations, viewed as filtrations of finite type.
Our primary goal in this paper is to provide a valuative characterization of uniform Kstability for filtrations, as used by Li in [Li22].Using this, we will prove: Main Theorem.Uniform K-stability for filtrations is an open condition on the polarization.
More precisely, the condition of (X, L) being uniformly K-stable for filtrations only depends on the numerical class of L, and is an open condition on the ample cone.This result holds also when X is singular, see Theorem 4.12.In the smooth case, the openness property ties in well with the corresponding result for cscK metrics [LS94].
To explain the proof of the main theorem, let us first consider the Fano case L = −K X .In this situation, the YTD conjecture was first established in full generality in [CDS15], and the missing implication in (0.1) is thus known.It can alternatively be obtained by purely algebro-geometric means, combining techniques from the Minimal Model Program (MMP), which allows us to convert uniform K-stability into uniform Ding-stability [LX14, BBJ15,Fuj19a], together with the non-Archimedean version of Berman's 'thermodynamical formalism', which yields the equivalence between uniform Ding-stability and uniform Kstability for filtrations (see Corollary 5.10).
Crucially, the MMP techniques of [LX14] further show that K-stability of a Fano manifold (or, more generally, any log Fano pair) can be tested using only 'special' test configurations, which correspond to so-called 'dreamy' divisorial valuations [Fuj19a]; this leads to a purely valuative characterization of K-stability [Li17,Fuj19a], which has been instrumental in the recent spectacular progress towards the deeper understanding of K-stability of log Fano varieties and the construction of moduli spaces thereof; see [BLXZ23,LXZ22], to name just a few.
Motivated by this, R. Dervan and E. Legendre initiated in [DL20] the study of a valuative criterion in the case of a general polarization, by providing a purely valuative expression β(v) ∈ R for the Donaldson-Futaki invariant of a special test configuration in terms of the corresponding 'dreamy' divisorial valuation v. Their invariant β(v) in fact makes sense for any divisorial valuation v (see (0.3) below), and gives rise to a notion of (uniform) valuative stability, which was recently proved in [Liu23] to be an open condition with respect to the polarization.
In contrast with the log Fano case, valuative stability is not expected to imply K-stability for a general polarization, and the MMP will likely play a less important role in that case as well.Using our previous works [BoJ22,BoJ21], we will extend the β-invariant to convex combinations of divisorial valuations, viewed as atomic measures on the Berkovich analytification of X, and show that the corresponding stability notion, which we call divisorial stability is equivalent to (uniform) K-stability for filtrations.
Valuative stability.As in the main body of the paper, we consider from now an arbitrary polarized pair (X, B; ω), where X is a normal projective variety over an algebraically closed field k of characteristic 0, B is a (not necessarily effective) Q-Weil divisor on X such that K X,B := K X + B is Q-Cartier, and ω ∈ Amp(X) ⊂ N 1 (X) is a (possibly irrational) ample numerical class, of volume V ω = (ω n ) with n := dim X.
Before describing our notion of divisorial stability, let us first briefly revisit the definitions of [DL20,Liu23] in the present setting.Denote by X div the set of divisorial valuations v : k(X) × → R, of the form v = s ord F where F is a prime divisor on a smooth birational model π : Y → X and s ∈ Q ≥0 (the case s = 0 corresponding to the trivial valuation v triv ).The volume function vol : N 1 (Y ) → R ≥0 is continuous, and positive precisely on the big cone.Set This quantity, which appeared in [MR15,Li17,Fuj19a] under various normalization and notation, coincides with the expected vanishing order S L (v) when ω = c 1 (L) with L ∈ Pic(X) Q (see [BlJ20]).The function ∥ • ∥ ω : X div → R ≥0 so defined is homogeneous with respect to the scaling action of Q >0 , and vanishes precisely on the trivial valuation v triv .
Using the differentiability of the volume function on the big cone [BFJ09,LM09], one checks that ∥v∥ ω is a differentiable function of ω ∈ Amp(X), see §2.5.This allows us to define where A X,B (v) = s A X,B (F ) ∈ Q is the log discrepancy of v = s ord F .Differentiating under the integral sign in (0.2) yields an expression of the invariant β(v) that coincides with the one in [DL20,Liu23] (up to a factor V ω ), see §4.1 for details.The β-invariant defines a Q >0 -homogeneous function β : X div → R. Following [DL20,Liu23], we say that (X, B; ω) is valuatively semistable if β ≥ 0 on X div , and valuatively stable2 if β ≥ ε∥ • ∥ ω for some ε > 0.
Divisorial stability.Our stability notion involves the larger space M div of divisorial measures on X.This is the set of probability measures on X div of the form where r ≥ 1, m i ∈ R >0 , v i ∈ X div , and i m i = 1.If L is an ample Q-line bundle, then any ample test configuration (X , L) for (X, L) defines a divisorial measure, whose support is exactly the set of divisorial valuations associated with the irreducible components of the central fiber of the normalization of X , see [BHJ17].However, not every divisorial measure is of this form.For example, a Dirac mass δ v with v ∈ X div is of this form iff v is dreamy with respect to L (i.e. the corresponding filtration of the section ring is of finite type).
There is an obvious embedding X div → M div given by v → δ v , and we can extend the functional β from X div to M div as follows.The first term in (0.3) is extended by linearity: following [BHJ17,BHJ19] we define the (non-Archimedean) entropy of a measure µ = i m i δ v i as Ent X,B (µ) := i m i A X,B (v i ) = ˆAX,B µ; this does not depend on the class ω.To extend the second term in (0.3), we interpret the invariant (0.2) as the energy of the Dirac mass δ v in the sense of non-Archimedean pluripotential theory [BoJ22].
To explain this, recall first that the space X div admits a natural compactification X an , the Berkovich analytification of X (with respect to the trivial absolute value on k), whose points can be viewed as semivaluations on X, i.e. valuations v : k(Y ) × → R for some subvariety Y of X.We can therefore embed M div into the space M of Radon probability measures on the compact Hausdorff space X an , and define the energy of any such measure µ ∈ M, as follows.Assume first that ω ∈ Amp(X) is rational, i.e. ω = c 1 (L) with L ∈ Pic(X) Q .Any function φ ∈ C 0 (X an ) induces a filtration on the vector space R m := H 0 (X, mL) for m sufficiently divisible, given by where X an can be replaced with the dense subset X div , by continuity.Suitably normalized, the volumes of these filtrations (i.e. the average of their jumping numbers) converge to a number vol L (φ) ∈ R, see [BC11].It further follows from [BoJ22] that vol L (φ) = vol ω (φ) only depends on ω = c 1 (L), and that ω → vol ω (φ) uniquely extends by continuity to the whole ample cone Amp(X) ⊂ N 1 (X).
vol ω (φ) − ˆφ µ for µ ∈ M. The energy is convex, lower semicontinuous in the weak topology of M, and vanishes precisely at the trivial measure µ triv := δ v triv ; it is further homogeneous with respect to the natural scaling action of R >0 .
As mentioned before, the energy ∥δ v ∥ ω of the Dirac mass associated to a divisorial valuation v ∈ X div turns out to coincide with (0.2); in particular, it is finite, and the energy functional therefore restricts to a finite-valued, convex functional ∥ • ∥ ω : M div → R ≥0 .
When ω = c 1 (L) with L ∈ Pic(X) Q and µ ∈ M div is associated to a (semi)ample test configuration (X , L) for (X, L), ∥µ∥ L equals the minimum norm of (X , L) in the sense of Dervan [Der16]; this corresponds to I NA − J NA in the notation4 of [BHJ17], and can be expressed as a certain linear combination of the intersection numbers ( Ln+1 ) and (L X • Ln ), with ( X , L) → P 1 the canonical compactification of (X , L) → A 1 , and L X the pull-back of L. In fact, this fully characterizes the energy functional, as any µ ∈ M div can be written as the weak limit of measures (µ j ) associated to test configurations in such a way that ∥µ j ∥ L → ∥µ∥ L .
Our first main result is as follows: superfix 'NA' to the corresponding functional in Kähler geometry; in later works [BBJ21,BoJ22], this superfix was dropped when no ambiguity can arise.
Theorem A. For any measure µ ∈ M div , the function ω → ∥µ∥ ω is of class C 1 on Amp(X).
The result actually holds for all measures of finite energy, i.e. ∥µ∥ ω < ∞ (a condition that is independent of ω).The proof, which relies on rather sophisticated estimates for Monge-Ampère integrals from [BoJ22], ultimately deriving from the Hodge index theorem, further yields Hölder estimates for the derivative of the energy, a key ingredient in the proof of Theorem B below.
Given ω ∈ Amp(X), θ ∈ N 1 (X), and µ ∈ M div , we write for the directional derivative of the energy with respect to ω.By approximation, this is again fully characterized by the case where ω = c 1 (L) and µ is associated to a (semi)ample test configuration (X , L) for (X, L), for which ∇ θ ∥µ∥ ω can be expressed as a certain linear combination of the intersection numbers ( Ln+1 ) and (θ X • Ln ) (see (2.6), (2.8)).In particular, ∇ ω ∥µ∥ ω = ∥µ∥ ω .Note, however, that Theorem A does not directly follow from the description of ∥µ∥ ω in terms of intersection numbers, which cannot be used anymore for the perturbed energy ∥µ∥ ω+tθ .For any polarized pair (X, B; ω), we may now define the desired extension β : M div → R of (0.3) (with respect to the embedding X div → M div ) by setting We say that (X, B; ω) is divisorially semistable if β ≥ 0 on M div , and divisorially stable if β ≥ ε∥ • ∥ ω for some ε > 0. By considering Dirac masses, it follows immediately that divisorial (semi)stability implies valuative (semi)stability, and we expect the converse to fail in general; see [DL20,Example 2.28].
When ω = c 1 (L) with L ∈ Pic(X) Q and µ is associated to an ample test configuration (X , L), it follows from the description of ∇ K X,B ∥µ∥ ω in terms of intersection numbers mentioned above that β(µ) coincides with the (non-Archimedean) Mabuchi K-energy of (X , L), i.e. its Donaldson-Futaki invariant up to a simple error term that disappears after base change (see [BHJ17]).As a result, divisorial semistability implies K-semistability, and divisorial stability implies uniform K-stability; here we conjecture that the converse does hold: more on this below.
Theorem B. For any polarized pair (X, B; ω), the following holds: Recall that the pair (X, B) is sublc (a short-hand for sub-log canonical ) if its log discrepancy function A X,B : X div → Q is non-negative; the pair (X, B) is then lc in the usual sense iff B is further effective.As an immediate consequence of Theorem B, we get: Corollary C. If a polarized pair (X, B; ω) is divisorially semistable, then (X, B) is necessarily sublc.Furthermore, divisorial stability of (X, B; ω) is an open condition on ω ∈ Amp(X).
Note that the last part of Corollary C implies the Main Theorem above.The first part can be viewed as a version in our context of a celebrated result of Y. Odaka [Oda13] (see also [BHJ17]), to the effect that any polarized pair (X, B; L) with L ∈ Pic(X) Q ample and B effective that is K-semistable is necessarily lc.While the proof of the latter result relies on the MMP through the existence of log canonical blowups, the proof of Theorem B (i) is of an entirely different nature, and rests on an estimate for the energy on certain affine segments in M div .
Part (ii) of Theorem B is a consequence of a refined version of Theorem A, involving Hölder estimates for the directional derivative ω → ∇ θ ∥µ∥ ω of the energy (which actually show that ω → σ div (X, B; ω) is locally Hölder continuous).While delicate, these estimates derive in a rather formal way from the Hodge index theorem; this is studied in [BoJ23], where versions of Theorem A and Theorem B (ii) are shown to hold over an arbitrary valued (possibly Archimedean) field.Theorem B (ii) should also be compared with the work [Liu23] of Yaxiong Liu, who considers a similar threshold of valuative stability, defined using only Dirac masses µ = δ v , v ∈ X div , and shows that this threshold is a continuous function on the ample cone.In fact, given any subset M ⊂ M div , one could consider the threshold defined by restricting the infimum in (0.4) to M , and our proof yields the continuity with respect to ω ∈ Amp(X), which recovers Liu's result.However, we cannot prove that uniform K-stability is an open condition5 in this way, since the subset of M div used to test the K-stability of (X, L) depends on L.
In the log Fano case, the divisorial stability threshold essentially reduces to the δ-invariant defined in [FO18,BlJ20].In fact, for any sublc polarized pair (X, B; ω) such that −K X,B ≡ λω with λ ∈ R, the identity ∇ ω ∥ • ∥ ω = ∥ • ∥ ω mentioned above yields the simpler expression By linearity of the entropy and convexity of the energy, this implies ∥v∥ ω coincides with the usual δ-invariant.As a consequence, divisorial stability, (uniform) valuative stability, and uniform K-stability are all equivalent in the log Fano case (see Corollary 4.11).
K-stability for filtrations.Next we discuss the relation between divisorial stability and the stability notions by C. Li [Li22].We note that Li assumed that X is smooth, as he relied on the preprint [BoJ18b], and specifically continuity of envelopes, which is not yet known in the singular case.Using [BoJ21], we are able to bypass this obstacle.
Consider thus a polarized pair (X, B; ω), and assume that (X, B) is subklt, i.e. its log discrepancy function A X,B : X div → Q is positive outside {v triv }.This function then admits a maximal lsc extension A X,B : X an → [0, +∞], which coincides with the one constructed in [JM12,BdFFU15] on the subspace X val ⊂ X an of valuations on k(X), and is infinite outside it (see Appendix A).We may thus define an lsc extension Ent X,B : M → [0, +∞] of the entropy functional by setting Ent X,B (µ) := ´AX,B µ.This yields in turn an extension β : M 1 → R ∪ {+∞} of the β-functional to the space M 1 ⊂ M of measures µ of finite energy, i.e. ∥µ∥ ω < +∞, which we characterize as the maximal lsc extension of β : M div → R with respect to the natural (strong) topology of M 1 .In particular, the divisorial stability threshold can be computed using all measures in M 1 , i.e.
where the last equality holds by homogeneity with respect to the scaling action of R >0 .
Assume further ω = c 1 (L) with L ∈ Pic(X) Q ample.As in [BoJ21], denote by N R the set of norms χ : R(X, dL) → R ∪ {+∞} on the section ring of (X, dL), with d ≥ 1 sufficiently divisible (depending on χ).Such norms are in 1-1 correspondence with the more commonly used (multiplicative, graded, linearly bounded) filtrations, via the inverse maps where R m := H 0 (X, mL) for m sufficiently divisible.The space N R comes with a translation action (c, χ) → χ + c of R, such that (χ + c)(s) := χ(s) + mc for s ∈ R m .
The Rees construction yields an identification of the subset T Z ⊂ N R of Z-valued norms of finite type χ with the set of ample test configurations for (X, L).Each χ ∈ T Z thus defines a divisorial measure MA(χ) ∈ M div , called the Monge-Ampère measure of χ.By [BoJ21], the space N R is equipped with a natural pseudometric d 1 , with respect to which T Z is dense, and the Monge-Ampère operator admits a unique d 1 -continuous extension MA : N R → M 1 , which is invariant under the translation action of R. We may now define the Mabuchi K-energy and the minimum norm of any χ ∈ N R by M(χ) := β(MA(χ)), ∥χ∥ := ∥ MA(χ)∥ L .
Theorem D. The divisorial stability threshold of any polarized subklt pair (X, B; L) satisfies In particular, (X, B; L) is divisorially semistable (resp.stable) iff M ≥ 0 on N R (resp.M ≥ ε∥ • ∥ for some ε > 0), which respectively correspond to K-semistability and uniform Kstability for filtrations, as considered in [Li22].Note that this notion of K-stability for filtrations a priori differs from the one in [Szé15] and relies on working on the full Berkovich space X an rather than just X div .
In view of (0.5), the main step in the proof of Theorem D consists in showing that the image of MA : N R → M 1 contains the set M div of divisorial measures.This follows from [BoJ21], where it is proved that the Monge-Ampère operator induces a 1-1 map where N div R ⊂ N R denotes the set of divisorial norms, of the form χ = min i {χ v i + c i } for a finite set (v i ) in X div and c i ∈ R. We then have MA(χ) = i m i δ v i for the same set of valuations (v i ), where m i = m i (c) is a certain (non-linear) function of c = (c i ).
As a consequence, the infimum in Theorem D can be computed on the space N div R of divisorial norms; we show that it can be further restricted to the subspace N div Q of rational divisorial norms, whose coefficients c i above can be chosen rational.By [BoJ21], such norms arise from arbitrary (i.e.not necessarily ample) test configuration for (X, L), called models in [Li22], and it follows that divisorial stability is also equivalent to uniform K-stability for models in the sense of C. Li; see §5.4.
A fortiori, divisorial stability implies uniform K-stability, as we already noted above.The two notions are equivalent if a certain entropy regularization conjecture holds, as first formulated in [BoJ18b].A stronger, and more precise conjecture, goes as follows: Conjecture.Let χ ∈ N div Q be a rational divisorial norm, and let (χ d ) be its sequence of canonical approximants, where χ d is generated in degree 1 by the restriction of χ to H 0 (X, dL) for d sufficiently divisible.Then lim d Ent X,B (MA(χ d )) = Ent X,B (MA(χ)).
Granted this conjecture, divisorial stability is the same as uniform K-stability-and the uniform YTD conjecture thus holds for any polarized complex manifold, as discussed in the beginning of the introduction.See also [Li23].
In the Fano case, divisorial stability and valuative stability are equivalent.In the general case, we do not expect this to be true, but one can ask whether it is enough to consider divisorial measures with a fixed bound on the cardinality of their supports.For example, [Don02, Proposition 5.3.1]shows that on a toric surface, it suffices to consider measures supported on a set of cardinality at most two.
Structure of the paper.The article is organized as follows.
• Section 1 recalls some aspects of our previous work [BoJ22,BoJ21] of relevance for the present paper.• Section 2 is devoted to the proof of Theorem A (cf. Theorem 2.15), along with some key estimates that will lead to the proof of Theorem B (ii). • Section 3 studies the entropy functional of a pair, the associated δ-invariant, and its relation to Ding-stability.Theorem 3.6 is the main ingredient in the proof of Theorem B (i). • Section 4 introduces the main concepts of this paper, the β-invariant of a divisorial measure, and the associated notion of divisorial stability.It completes the proof of Theorems A and B. • Section 5 compares divisorial stability and K-stability.Theorem D is proved, and the entropy regularization conjecture is discussed.Along the way, we prove the Main Theorem above.• Finally, Appendix A reviews the properties of the log discrepancy function of a pair, and its extension to the Berkovich space in the subklt case.

Preliminaries
The main purpose of this preliminary section is to recall results from non-Archimedean pluripotential theory, as developed in [BoJ22], which form the building blocks of our approach.
• We say that a function f : Z → R defined on a set Z endowed with an action of R >0 (or a subgroup thereof) is homogeneous if it satisfies the equivariance property f (t • x) = tf (x) for t ∈ R >0 and x ∈ Z. • A net in a set Z is a family (x i ) i∈I of elements of Z indexed by a directed set, i.e. a partially preordered set in which any two elements are dominated by a third one.• If Z is a Hausdorff topological space, and φ : Z → R ∪ {±∞} is any function, then the usc regularization φ ⋆ of φ is the smallest usc function with φ ⋆ ≥ φ.Concretely, φ ⋆ (x) = lim sup y→x φ(y).The lsc regularization is defined by φ ⋆ = −(−φ) ⋆ .• For x, y ∈ R + , x ≲ y means x ≤ C n y for a constant C n > 0 only depending on n, and x ≈ y if x ≲ y and y ≲ x.Here n will be the dimension of a fixed variety X over k.
1.2.Quasi-metric spaces.A quasi-metric on a set Z is a function d : Z × Z → R ≥0 that is symmetric, separates points, and satisfies the quasi-triangle inequality for some constant ε > 0. A quasi-metric space (Z, d) comes with a Hausdorff topology, and even a uniform structure.In particular, Cauchy sequences and completeness make sense for (Z, d).Such uniform structures have a countable basis of entourages, and are thus metrizable, by general theory.
A continuous function f : Z → R on a quasi-metric space is uniformly continuous if, for each ε > 0, there exists δ > 0 such that d(x, y) The following standard result will be used several times in the paper: Lemma 1.1.Let (Z, d) be a quasi-metric space, D ⊂ Z a dense subset, and f : D → R a uniformly continuous function.Then f admits a unique uniformly continuous extension f : Z → R.
Proof.Uniqueness is clear, by density of D. Pick x ∈ Z, and a sequence (x i ) in D such that x i → x.Then εd(x i , x j ) ≤ d(x i , x) + d(x, x j ) tends to 0 as i, j → ∞.By uniform continuity of f , it follows that (f (x i )) is a Cauchy sequence, which thus admits a limit f (x) ∈ R. Using again uniform continuity, it is further easy to check that the limit is independent of the choice of (x i ), and that the extension constructed this way is uniformly continuous.□ 1.3.Positive numerical classes.In the entire paper, we work over an algebraically closed field k of characteristic 0, and X denotes an irreducible projective variety over k, i.e. an integral projective k-scheme (not necessarily normal for now).We set n := dim X.
Denote by N 1 (X) the finite dimensional R-vector space of numerical classes of R-Cartier divisors on X. Ample classes form a nonempty open convex cone Amp(X) ⊂ N 1 (X).We generally denote by θ an element of N 1 (X), and by ω an element of Amp(X).The closure Nef(X) ⊂ N 1 (X) of Amp(X) is the closed convex cone of nef classes.We denote by ≥ the corresponding partial order on N 1 (X), i.e. (1.1) Each ω ∈ Amp(X) induces a norm ∥ • ∥ ω on N 1 (X), defined by We will occasionally use the Thompson metric of the open convex cone Amp(X), defined by δ(ω, ω ′ ) := max{δ ≥ 0 | e −δ ω ≤ ω ′ ≤ e δ ω}.
(1.3) It is locally equivalent to the metric on Amp(X) induced by any norm on N 1 (X) (see [Tho63,Lemma 3]).
The volume of ω ∈ Amp(X) is defined as We define the trace of θ ∈ N 1 (X) with respect to ω ∈ Amp(X) as Note that tr ω (θ) is linear with respect to θ, with The trace computes the logarithmic derivative of the volume, i.e.
computes the mean value S of the scalar curvature of any Kähler form representing ω.
1.4.Berkovich analytification and psh functions.The Berkovich analytification X an of X (with respect to the trivial absolute value on k) is a compact Hausdorff topological space, whose points are semivaluations on X, i.e. valuations v : k(Y ) × → R for a subvariety Y ⊂ X.It contains as a dense subset the space X val of actual valuations on X, endowed with the topology of pointwise convergence as maps k(X) × → R.
The space X an comes with a continuous scaling action R >0 × X an → X an (t, v) → tv, which fixes the trivial valuation v triv ∈ X val , defined by v triv ≡ 0 on k(X) × .
The subspace X div ⊂ X val of divisorial valuations is already dense in X an .Each v ∈ X div is of the form v = t ord F for a prime divisor F on a smooth birational model Y → X and t ∈ Q ≥0 (the case t = 0 corresponding to v triv ).
We denote by C 0 (X) the Banach space of continuous functions φ : X an → R, endowed with the supnorm, and by C 0 (X) ∨ its topological dual, i.e. the space of (signed) Radon measures on X an .It contains the subspace M = M(X) ⊂ C 0 (X) ∨ of Radon probability measures, which is convex and compact for the weak topology.The scaling action of R >0 on X an induces an action (t, µ) → t ⋆ µ on C 0 (X) ∨ preserving M.
The space C 0 (X) contains a dense subspace PL(X) of piecewise linear (PL) functions, see [BoJ22,§2].Among various possible descriptions, each such function is of the form φ D ∈ PL(X) for a vertical Q-Cartier divisor D on some test configuration X → A 1 for X, where φ D (v) = σ(v)(D) for v ∈ X an , with σ : X an → X an denoting Gauss extension.This construction is invariant under pullback to a higher test configuration, and one can thus always assume that X dominates the trivial test configuration X × A 1 .
To each ω ∈ Amp(X), one associates a class PSH(ω) of ω-psh functions φ : and characterized as follows: • a PL function φ, written φ = φ D as above, is ω-psh iff ω X + D is relatively nef on X , with ω X ∈ N 1 (X /A 1 ) the pullback of ω to X ; • each φ ∈ PSH(ω) can be written as the pointwise limit of a decreasing net6 (φ i ) in PL(X) ∩ PSH(ω).
Each φ ∈ PSH(ω) is usc, and hence bounded above.Further, sup φ = φ(v triv ).Define T = T ω : The (Borel) set and hence v ∈ X lin iff v is a valuation of linear growth in the sense of [BKMS15].
In particular, every ω-psh function is finite-valued on X div ⊂ X lin ; it is further determined by its restriction to X div , and we equip the space PSH(ω) with the (Hausdorff) topology of pointwise convergence on X div .The scaling action of R >0 on X an induces an action on the topological space PSH(ω), denoted by (t, φ) → t • φ where (1.12) 1.5.Energy pairing and functions of finite energy.The energy pairing is first defined as a symmetric (n + 1)-linear map on N 1 (X) × PL(X), that takes a tuple (θ i , φ is the pull-back of θ i to the canonical compactification X → P 1 , and the right-hand side is an intersection number computed on X .See [BoJ22, §3.2].The energy pairing (which is simply an extension to numerical classes of [BHJ17, Definition 6.6]) satisfies (θ 0 , 0) (1.16) For any ω ∈ Amp(X), the Monge-Ampère energy 7 of φ ∈ PL(X) with respect to ω is defined as . (1.17) This normalization guarantees the equivariance property The space of ω-psh functions of finite energy is defined as and the strong topology of E 1 (ω) is defined as the coarsest refinement of the subset topology from PSH(ω) (i.e. the topology of pointwise convergence on X div ) in which E ω : E 1 (ω) → R becomes continuous.The vector space ⃗ E 1 generated by E 1 (ω) (interpreted as functions X lin → R) turns out to be independent of ω ∈ Amp(X).It contains PL(X), and we have E 1 (ω) = ⃗ E 1 ∩ PSH(ω) for any ω ∈ Amp(X).See [BoJ22,§7] for details on this and the remainder of §1.5.
7 This should not be confused with the extended Monge-Ampère energy of φ as in [BoJ22,§8].
Since PL(X) is a dense subspace of C 0 (X), we can now associate to any tuple (θ i , φ i ) ∈ N 1 (X) × ⃗ E 1 , i = 0, . . ., n, a mixed Monge-Ampère measure )), and it is positive when θ i is ample and φ i is θ i -psh.
1.6.Measures of finite energy.Fix ω ∈ Amp(X), and set for simplicity The energy8 of a Radon probability measure µ ∈ M is defined as for any φ ∈ E 1 .See [BoJ22, § §9-10] for details on this and on what follows.
The energy functional ∥ • ∥ : M → [0, +∞] is convex; it is also lsc for the weak topology of M, as a simple regularization argument shows that E 1 can be replaced by PL ∩ PSH(ω) in the right-hand side of (1.25).Despite the chosen notation, the energy is not actually a norm; however, it satisfies (1.27) The space M 1 ⊂ M of measures of finite energy is defined as the domain of the energy functional.By definition, any µ ∈ M 1 satisfies E 1 ⊂ L 1 (µ), and the converse holds as well.
In contrast with E 1 = E 1 (ω), the space M 1 turns out to be independent of ω ∈ Amp(X), a property that plays a key role in the present paper.More precisely, for any ω ′ ∈ Amp(X) and s ≥ 1 such that s −1 ω ≤ ω ′ ≤ sω, we have for all µ ∈ M, where C n := 2n 2 + 1. See [BoJ22, Theorem 9.24].
As in the case of E 1 , the strong topology of M 1 is defined as the coarsest refinement of the (weak) subspace topology induced by M such that the energy functional ∥ • ∥ : M 1 → R becomes continuous.The strong topology of M 1 is also independent of ω ∈ Amp(X).
By Example 1.6, the Monge-Ampère operator defines a map MA : E 1 → M 1 .By [BoJ22, Theorem A], this induces a topological embedding with dense image with respect to the strong topology on both sides.Further, a net (φ i ) in E 1 and µ ∈ M 1 satisfy MA(φ i ) → µ strongly in M 1 iff (φ i ) is a maximizing net for µ, i.e.J µ (φ i ) → 0.
The Monge-Ampère operator maps E 1 onto M 1 (and hence induces a homeomorphism E 1 /R ∼ → M 1 ) iff the envelope property (aka continuity of envelopes) holds for ω, which is the case when X is smooth.
In [BoJ22] and [BoJ21], it was respectively shown that the strong topology of M 1 can be defined by a certain quasi-metric I ∨ = I ∨ ω , and also by a (Darvas-type) metric d 1 = d 1,ω , these two (quasi-)metrics being further Hölder equivalent on bounded sets.The metric d 1 on M 1 will not be used in this paper; as to the quasi-metric I ∨ , it will be advantageously replaced by the following equivalent, more natural quasi-metric, directly induced by Aubin's I-functional on E 1 .Theorem 1.8.There exists a unique strongly continuous functional for all φ, ψ ∈ E 1 .For all µ 1 , µ 2 , µ 3 , µ ∈ M 1 and φ ∈ E 1 , we further have: (iii) I is a quasi-metric on M 1 that defines the strong topology, and the quasi-metric space (M 1 , I) is complete; (iv) I(µ, MA(φ)) ≈ J µ (φ) (see (1.26)); in particular, I(µ) := I(µ, µ triv ) ≈ ∥µ∥.Finally, for all φ, ψ ∈ E 1 and µ, ν ∈ M 1 we have Proof of Theorem 1.8.By [BoJ22, §10], Theorem 1.8 is valid for a certain quasi-metric I ∨ on M 1 in place of I, except for (1.34), which is replaced by Next, note that uniqueness is clear, by density of the image of MA, and that (1.34) is equivalent to for all µ, ν ∈ M 1 in the image of MA, by (1.32).Pick µ, µ ′ , ν, ν ′ in the image of MA, and write Using I(φ µ , φ µ ′ ) ≈ I ∨ (µ, µ ′ ) and I(φ ν , φ ν ′ ) ≈ I ∨ (ν, ν ′ ), the 'I ∨ -version' of (1.35) yields a Hölder estimate (1.37) with M := max{∥µ∥, ∥µ ′ ∥, ∥ν∥, ∥ν ′ ∥}.This shows that I is uniformly continuous on a dense subspace of the quasi-metric space M 1 × M 1 , and hence that it admits a unique (uniformly) continuous extension The main purpose of this section is to establish the differentiability of the energy of a measure with respect to the ample class (Theorem 2.15 below, which corresponds to Theorem A in the introduction), along with Hölder estimates for the derivative (Theorem 2.12) that will be the main ingredient for the continuity of the divisorial stability threshold (Theorem B (ii) in the introduction).
As before, X denotes an arbitrary (possibly non normal) projective variety, of dimension n, defined over an algebraically closed field k (whose characteristic can in fact be arbitrary in this section).
Proposition 2.2.For all φ ∈ ⃗ E 1 we have As a special case of [BoJ22, Theorem 3.25], we also have: The twisted energy is closely related to the differential of E ω with respect to ω: given by (2.6) Note that (2.6) is translation invariant as a function of φ (see (2.1) and (1.18)).
2.5.The case of a Dirac mass.In this section, we fix ω ∈ Amp(X) and θ ∈ N 1 (X).For any v ∈ X an , recall that v ∈ X lin ⇔ δ v ∈ M 1 , and that we then write, for simplicity, ∥v∥ ω := ∥δ v ∥ ω ; we similarly set By (1.27) and (2.13), we have In what follows, we provide explicit formulas for these invariants when v ∈ X div is a divisorial valuation, i.e. v = t ord F for a prime divisor F ⊂ Y on a smooth birational model π : Y → X and t ∈ Q >0 .By (2.26), we assume t = 1, to simplify notation.
Proof of Theorem 2.18.By (1.28), the left-hand side of (2.27) is continuous with respect to ω.The same holds for the right-hand side, using ´+∞ 0 vol(ω−λF ) dλ = ´Tω(v) 0 vol(ω−λF ) dλ and the continuity ω → T ω (v) and the volume function.By (1.33), (2.27) holds when ω is rational, and the general case follows.As a consequence, for any t ∈ R small enough we get for λ ̸ = λ(t) := T ω+tθ (v).On the other hand, Theorem 2.15 yields and a simple computation thus shows that (2.28) is equivalent to While this appears to be a simple differentiation under the integral sign, the slight twist here is that f (t, λ) might fail to be differentiable at (t, λ(t)).To circumvent this, note first that t → λ(t) is locally Lipschitz continuous.Indeed, pick C > 0 such that −Cω ≤ θ ≤ Cω.For any t, s ∈ R we then have which proves that t → λ(t) is locally Lipschitz, since it is locally bounded.Next, we may assume θ ∈ Amp(X), by linearity of the desired formula with respect to θ.In that case, t → λ(t) is further strictly increasing, and f (t, λ) is thus C 1 on {(t, λ) | t > 0, λ < λ(0)}.By usual differentiation under the integral sign, we thus have and it remains to see ˆ+∞ as t → 0 + .But |λ(t) − λ(0)| ≤ Ct, and f (t, λ(t)) = 0, which yields the desired estimate by (uniform) continuity of f , thereby finishing the proof.□

Entropy and the δ-invariant
In what follows, (X, B; ω) denotes a polarized pair, i.e.X is a normal projective variety, B is any Q-Weil divisor such that K X,B := K X + B is Q-Cartier, and ω ∈ Amp(X) is an ample numerical class.We introduce the entropy functional of (X, B), defined on the space of divisorial measures, and study the associated δ-invariant.When (X, B) is subklt, we prove that the entropy admits a natural extension to all measures of finite energy, and show that the δ-invariant is a threshold for Ding-stability.

Divisorial measures.
Definition 3.1.We define a divisorial measure on X an as a Radon probability measure with support a finite subset of X div .We denote by M div the set of such measures.
A divisorial measure µ ∈ M div is thus a measure of the form for a finite set (v i ) of divisorial valuations and m i ∈ R ≥0 such that i m i = 1.In other words, M div is the convex hull of X div → M. As X div ⊂ X an is stable under the scaling action of Q >0 , the same holds for M div , i.e.
Lemma 3.3.The space M div of divisorial measures sits as a dense subset of M 1 for the strong topology.
Proof.For any v ∈ X div ⊂ X lin , δ v lies in M 1 , and hence M div ⊂ M 1 , by convexity of ∥ • ∥ ω : M → [0, +∞].Next, pick µ ∈ M 1 , and choose a maximizing sequence (φ j ) in PL ∩ PSH(ω) for µ.Then µ j := MA(φ j ) lies in M div (see Example 3.2), and µ j → µ strongly in M 1 .Thus M div is strongly dense in M 1 .□ 3.2.The entropy functional of a pair.Following [BHJ17], we define the entropy functional Ent X,B : M div → R of the pair (X, B) by setting Ent X,B (µ) := ˆAX,B µ for all µ ∈ M div .Here A X,B : X div → Q is the classical log discrepancy function of the pair (X, B), see Appendix A.
Remark 3.4.When k = C, the above (non-Archimedean) entropy computes the 'slope at infinity' of the usual (relative) entropy functional along certain rays of smooth volume forms (see [BHJ19, Theorem 3.6]); this explains the chosen terminology.
The entropy functional is clearly affine on the convex set M div , and simply given by for µ ∈ M div and t ∈ Q >0 , by homogeneity of A X,B .
Note that (iv) implies in particular that δ(X, B; ω) is continuous with respect to ω ∈ Amp(X) (see [Zha21,Theorem 1.7] for an extension to the big cone).
Next, assume that (X, B) is sublc.Suppose given c ∈ R ≥0 such that A X,B (v) ≥ c∥v∥ for all v ∈ X div , and pick µ ∈ M div .Write µ = i m i δ v i for a finite set (v i ) in X div and by convexity of ∥ • ∥.This proves (iii).Finally, (iv) is a direct consequence of (1.28).□ Remark 3.8.Note that (3.4) fails when (X, B) is not sublc.Indeed, using dual cone complexes as in Appendix A, one can show that the right-hand side of (3.4) is finite for any polarized pair (X, B; ω), whether (X, B) is sublc or not.

3.4.
Extending the entropy functional.We assume in this section that the pair (X, B) is subklt (i.e.A X,B > 0 on X div \ {v triv }).The log discrepancy function A X,B : X div → Q ≥0 then admits a greatest lsc extension A X,B : X an → [0, +∞], which further satisfies Note also that Ent X,B : M → [0, +∞] is lsc in the weak topology, since A X,B is lsc on X an .Its restriction to M 1 is thus a fortiori strongly lsc, but it is not strongly continuous in general.
Theorem 3.10.Any µ ∈ M 1 can be written as the strong limit of a sequence (µ i ) in M div such that Ent X,B (µ i ) → Ent X,B (µ).Equivalently, Ent X,B : M 1 → [0, +∞] is the greatest (strongly) lsc extension of the entropy functional.
When X is smooth and B = 0, the result follows from [Li22, Proposition 6.3] and its proof (itself based in part on the authors' preprint [BoJ18a]).
Corollary 3.11.The δ-invariant of any polarized subklt pair (X, B; ω) satisfies δ(X, B; ω) = inf Ent X,B (µ). (3.11) Proof.By definition of δ := δ(X, B; ω), we have Ent X,B ≥ δ∥ • ∥ on M div .By Theorem 3.10 and the strong continuity of ∥ • ∥, this inequality extends to M 1 ; this proves (3.11), the right-hand equality being a consequence of the homogeneity of Ent X,B and ∥ • ∥ with respect to the scaling action of R >0 on M 1 .□ We will rely several times on the following simple observation: Lemma 3.12.Let f : Z → R ∪ {+∞} be an lsc function on a topological space, and pick a convergent net Proof.Since f is lsc, we have f (x) ≤ lim inf i f (x i ), while the assumption yields lim sup i f (x i ) ≤ f (x).The result follows.□ The next two results are the key ingredients in the proof of Theorem 3.10.
Lemma 3.13.Assume X is smooth.Let X be an snc test configuration for X, and denote by p X : X an → ∆ X the retraction onto the associated dual complex ∆ X → X an (see §A.4). Then: for the topology of uniform convergence; (iii) for any weakly convergent net µ i → µ in M(∆ X ) → M, ´φ µ i → ´φ µ uniformly with respect to φ ∈ PSH(ω).
Proof.(i) and (ii) follow from Theorems A.1 and A.4 of [BoJ22] (the latter being a consequence of the uniform Lipschitz estimates of [BFJ16a, Theorem C]).Now consider a convergent net µ i → µ as in (iii).Since ´φ(µ i − µ) is invariant under translation of φ by a constant, it is enough to show ´φ µ i → ´φ µ uniformly for φ ∈ PSH sup (ω).Denote by K the (compact) closure of PSH sup (ω)| ∆ X in C 0 (∆ X ), equipped with the supnorm metric.
Since A X is lsc and A X • p X ≤ A X (see Theorem A.10), Lemma 3.12 similarly yields Ent X (µ X ) → Ent X (µ), which proves (iii) when B = 0.In the general case, observe that (iii) trivially holds if Ent X,B (µ) = +∞, again by Lemma 3.12.We therefore assume Ent X,B (µ) < +∞, and hence µ(X an \ X fld ) = 0, by (3.10).By Proposition A.4, we have A X,B = A X − ψ B on X val ⊃ X fld , where ψ B (v) := v(B).Write B = B 1 − B 2 with B i ≤ 0, i = 1, 2. Pick an ample line bundle L such that B i + L is semiample, for i = 1, 2, and hence ψ B i ∈ PSH(L) (see [BoJ22, Lemma 6.7]).By Theorem A.6 (ii), each ψ B i satisfies 0 ≤ −ψ B i ≤ CA X,B on X an for some C > 0, and hence is integrable with respect to µ and all µ X .Since A X,B = A X + ψ B 2 − ψ B 1 on X fld , A X,B is integrable with respect to µ and µ X as well, and Ent X,B (µ X ) = Ent X (µ X ) + ´ψB 2 µ X − ´ψB 1 µ X .By (ii) and the case B = 0 of (iii), this converges to Ent X,B (µ) = Ent X (µ) + ´ψB 2 µ − ´ψB 1 µ, and the general case of (iii) follows. □ Proof of Theorem 3.10.Observe first that the result is trivial if Ent X,B (µ) = +∞.By Lemma 3.3, we can indeed pick a sequence (µ i ) in M div converging strongly to µ, and we then have Ent X,B (µ i ) → Ent X,B (µ), by Lemma 3.12.We therefore assume Ent X,B (µ) < +∞, and hence µ(X an \ X fld ) = 0, by (3.10).Since the strong topology of M 1 is defined by the quasi-metric I (see Theorem 1.8), it is enough to show that, for any ε > 0, there exists ν ∈ M div such that To see this, pick a resolution of singularities π : X ′ → X.Since the induced map X ′an → X an is surjective, it follows from general theory that π ⋆ : M(X ′ ) → M(X) is surjective as well (see for instance [SW99, V.5.4]), and we can thus write µ = π ⋆ µ ′ for some measure µ ′ ∈ M(X ′ ).For any snc test configuration X for X ′ , set . By Lemma 3.14, we have lim X µ ′ X = µ ′ weakly in M(X ′ ), and hence lim X µ X = µ weakly in M(X).Next, pick ω ∈ Amp(X), and ω ′ ∈ Amp(X ′ ) such that π ⋆ ω ≤ ω ′ .For any By Lemma 3.12, we infer lim X ∥µ X ∥ ω = ∥µ∥ ω , and hence µ X → µ strongly in M 1 (X) (the strong topology being the coarsest refinement of the weak one in which Replacing µ with µ X , it is thus enough to show (3.12) when µ = π ⋆ µ ′ with µ ′ ∈ M(∆ X ).By density of the set of rational points ∆ X (Q) in the simplicial complex ∆ X , we can now write µ ′ as the weak limit of a sequence of measures µ ′ i ∈ M(∆ X ) with finite support in ∆ X (Q) = ∆ X ∩ X ′div , and hence lying in M div (X ′ ).We claim that µ i := π ⋆ µ ′ i ∈ M div (X) satisfies the desired estimate (3.12) for i large enough.
Proof.By translation invariance of the Ding functional (see (3.14)), we may assume that φ, ψ are normalized by sup φ = sup ψ = 0. We first claim that L X,B (φ) ≥ −C I(φ), where for each v ∈ X div .The right-hand side is easily seen to be bounded below by −C(n, δ) I(φ), and the claim follows.By (3.16), we can further find B = B(n, δ) > 0 such that and the same holds for ψ.Now set M := max{I(φ), I(ψ)}.By (1.35) for any v ∈ X div with ∥v∥ ≤ BM , we have In view of (3.17), this implies . By (2.12), a similar estimate holds for E ω , and (3.15) follows, by (3.13).□ As we next show, the δ-invariant is a threshold for Ding-stability (see also [BBJ21,BL22]).
Theorem 3.16.Set δ := δ(X, B; ω).Then: By strong continuity of the Ding functional (see Proposition 3.15), it is enough to test the conditions in the second halves of (i) and (ii) on the dense subset PL ∩ PSH(ω).

Divisorial stability
As before, (X, B; ω) denotes an arbitrary polarized pair.We introduce the main concepts of this paper, the β-invariant of a divisorial measure and the associated notion of divisorial stability, and prove Theorem B and Corollary C in the introduction.4.1.The β-functional.We now introduce the key functional in this paper.
To simplify notation, we slightly abusively use K X,B to also denote the image of the log canonical class in N 1 (X).By Theorem 2.15, (4.1) can be rewritten for all µ ∈ M div , t ∈ Q >0 and s ∈ R >0 , by (3.2) and (2.13).
(4.5)The main advantage of β over M lies in the fact that both the domain and the entropy term of the former are independent of ω.
Consider next v ∈ X lin , hence δ v ∈ M 1 , and set for simplicity , Theorem 2.18 provides the following explicit description: Lemma 4.2.For any prime divisor F on a smooth birational model π : Y → X, v := ord F satisfies vol(ω − λF ) dλ, the right-hand side of (4.6) coincides (up to the factor V ω ) with the invariant introduced in [DL20] (generalizing the Fano case of [Fuj19a]).
4.2.Divisorial stability.We are now in a position to introduce the main new concept in this paper.
As we next show, divisorial stability also implies the usual notion of K-stability (see §5.5 for a more detailed discussion).
According to an important result of Odaka [Oda13] (see also [BHJ17,Theorem 9.1]), any pair (X, B) with B ≥ 0 such that (X, B; L) is K-semistable for some ample L ∈ Pic(X) Q is necessarily log canonical, i.e.A X,B ≥ 0 on X div .Theorem 3.6 directly yields the following version for divisorial stability.
This proves Theorem B (i) and the first part of Corollary C in the introduction.
Proof.By (2.14) and linearity of ∇ θ ∥ • ∥ ω with respect to θ, we have and the β-functional thus takes the simpler form This directly implies (4.10), and the rest follows from Theorem 3.6.□ Corollary 4.11.For any polarized subklt pair (X, B; ω) such that ω ≡ −K X,B , we have: (i) (X, B; ω) is divisorially semistable iff it is Ding-semistable; (ii) (X, B; ω) is divisorially stable iff it is uniformly Ding-stable.If we further assume that B is effective, i.e. (X, B; ω) is log Fano, then (i) and (ii) are also equivalent to (X, B) being K-semistable and uniformly K-stable, respectively.
Proof.(i) and (ii) are direct consequences of Theorem 3.16 and Proposition 4.10, while the final assertion follows from [Fuj19a,Corollary 6.11].□ We may now state one of the main results of this article.

Comparison to other stability notions
In this final section, we consider a polarized pair (X, B; L), where L is ample Q-line bundle, and provide a detailed comparison of divisorial stability with usual K-stability, and its version for norms/filtrations.5.1.Norms and filtrations.We use [BoJ21] as a reference for what follows.For any d ∈ Z >0 such that dL is an honest line bundle, we set We denote by N R = N R (L) the set of (superadditive, k × -invariant, linearly bounded) norms χ : R(X, dL) → R∪{+∞} for some sufficiently divisible d ∈ Z >0 ; these are in 1-1 correspondence with the more commonly used (multiplicative, graded, linearly bounded) filtrations, via the inverse maps The space N R comes with a natural scaling action (t, χ) → tχ of R >0 , which fixes the trivial norm χ triv ∈ N R , such that χ triv (s) = 0 for s ∈ R m \ {0}; there is also a translation action (c, χ) → χ + c of R, where (χ + c)(s) := χ(s) + mc for s ∈ R m .
The space N R is equipped with a natural pseudometric d 1 , and we endow it with the corresponding (non-Hausdorff) topology (see [BoJ21,§3]).It contains as a dense subset the space T Z ⊂ N R of Z-valued norms of finite, which can be identified with the set of ample test configurations (X , L) for (X, L), thanks to the Rees construction.
The Monge-Ampère operator is the unique continuous map (with respect to the d 1 -topology on N R and the strong topology on M 1 ) that takes the norm χ ∈ T Z corresponding to an ample test configuration (X , L) to the divisorial measure MA(χ) = E m E δ v E , where E ranges over the irreducible components of the central fiber X 0 of the normalization ( X , L) of (X , L), v E ∈ X div is the associated divisorial valuation, and Extending [Der16], we define the minimum norm of χ ∈ N R as the energy of the Monge-Ampère measure MA(χ).It depends continuously on χ, and vanishes iff d 1 (χ, χ triv + c) = 0 for some c ∈ R.
for a finite subset Σ ⊂ X div and c v ∈ R. The notation means χ(s) = min v∈Σ {v(s) + mc v } for s ∈ R m , the corresponding filtration thus being The subset N div R ⊂ N R of divisorial norms is preserved by the scaling action of Q >0 and the translation action of R. A divisorial norm χ ∈ N div R is rational if it is Q-valued, which equivalently means that the c i in (5.2) can be chosen in Q.They form a subset N div Q ⊂ N div R , which is dense in N R .By [BoJ21, Corollary 7.16], the Monge-Ampère operator induces a homeomorphism where the left-hand side is equipped with the quotient topology and the right-hand side with the strong topology.In particular, a divisorial norm χ for any s ∈ R m and d ≥ 1.To any χ ∈ N R we can associate a homogenization χ hom defined by χ hom (s) := lim d d −1 χ(s d ).This is the smallest homogeneous norm such that χ ≤ χ hom .We have d 1 (χ, χ hom ) = 0 and MA(χ hom ) = MA(χ) for any χ ∈ N R , see Corollary 5.2 and Proposition 7.4 in [BoJ21].
To any (not necessarily ample, nor normal) test configuration (X , L) for (X, L) we can associate a norm χ = χ X ,L ∈ N R , see [WN12,BHJ17].
Proof.For any test configuration (X , L) for (X, L), the homogenization χ hom X ,L of χ X ,L lies in N div Q , see [BoJ21,Corollary A.8]. Conversely, consider any norm χ ∈ N div Q .By [BoJ21, Theorem 6.12], χ = IN(φ) for some PL function φ ∈ PL(X).Moreover, such a PL function φ is of the form φ X ,L for a normal test configuration (X , L) for (X, L), see [BoJ22,Theorem 2.31].By [BoJ21, Proposition A.3] χ = χ hom X ,L .□ 5.2.Divisorial stability in terms of divisorial norms.Consider the functional β = β X,B;L : M div → R of the polarized pair (X, B; L); thus for µ ∈ M div .As we just saw, the Monge-Ampère operator induces a homeomorphism MA : N div R /R ∼ → M div ; in line with (4.5), we define the Mabuchi K-energy functional Theorem 5.2.The divisorial stability threshold of any polarized pair (X, B; L) satisfies In particular, (X, B; L) is divisorially semistable iff M(χ) ≥ 0 for all rational divisorial norms χ ∈ N div Q , and it is divisorially stable iff there exists ε > 0 such that M(χ) ≥ ε∥χ∥ for all χ ∈ N div Q .Recall that a divisorial norm χ ∈ N div R satisfies ∥χ∥ = 0 ⇔ χ = χ triv + c for c ∈ R. Lemma 5.3.For any χ ∈ N div R , there exists a sequence Proof.Pick a finite subset Σ ⊂ X div such that χ = min v∈Σ {χ v + c v } with c ∈ R Σ .As recalled above, we then have supp MA(χ) ⊂ Σ. Writing c as the limit of a sequence (c i ) in Using Lemma 5.3, it is further easy to see that the latter infimum can be computed using norms in N div Q , and the result follows.□ Remark 5.4.In [Fuj16], K. Fujita defined a notion of divisorial (semi)stability for a Q-Fano variety X by requiring that η(D) := 1 − V −1 ´∞ 0 vol(L − λD)dλ be (semi)positive for each effective Weil divisor D on X, where L = −K X (see also [Gri20]).By monotonicity of the volume, it suffices to check this when D = i D i is reduced.In this case, one can show that η(D) = M(χ D ) with χ D := min i χ ord D i .This shows that the present notion of divisorial stability implies that of [Fuj16].
By continuity of MA : N R → M 1 , (5.6) is an lsc extension of (5.3); it is in fact characterized as the maximal lsc extension, as follows from the next result.
Lemma 5.6.For any χ ∈ N R , there exists a sequence Proof.By Lemma 5.3, it is enough to produce such a sequence in N div R .By Theorem 3.10, we can find a sequence (µ i ) in M div such that µ i → MA(χ) strongly and β(µ M(χ).
In particular, (X, B; L) is divisorially semistable iff it is K-semistable for filtrations, and it is divisorially stable iff it is uniformly K-stable for filtrations.
Note that Theorem 5.8 and Theorem 4.12 together imply the Main Theorem stated in the beginning of the introduction.
In view of Corollary 4.11, we also get the following result in the log Fano case.
Corollary 5.10.For any polarized subklt pair (X, B; L) with L = −K X,B , we have: (i) (X, B; L) is K-semistable for filtrations iff it is Ding-semistable; (ii) (X, B; L) is uniformly K-stable for filtrations iff it is uniformly Ding-stable.
This means that divisorial stability is equivalent to Li's notion of uniform K-stability for models introduced in [Li22].10Indeed, a (normal) test configuration is called a model in loc.cit.Li assumes X is smooth as his definition relies on continuity of envelopes as in [BoJ18b], but we can bypass this using [BoJ21].We note that Proposition 6.3 in [Li22] can be used to show that uniform K-stability for models is equivalent to uniform K-stability for filtrations, in the setting considered there.5.5.Divisorial stability vs. K-stability.We return to the setting of an arbitrary polarized pair (X, B; L).
Let N hom R ⊂ N R denote the set of homogeneous norms.We have N div R ⊂ N hom R , and Recall that the Rees construction yields an identification of the set T Z of Z-valued norms of finite type with that of ample test configurations for (X, L).We denote by T int Z ⊂ T Z the subset corresponding to normal (i.e.integrally closed) test configurations; it contains the set T hom Z of homogeneous Z-valued norms of finite type, which correspond to test configurations with reduced central fiber.Homogenization χ → χ hom induces a 1-1 map T int Z ∼ → T hom Q (see [BoJ21,Theorem A.11]).
In particular, σ K (X, B; L) ≥ σ div (X, B; L). ∥χ∥ .Since M and ∥ • ∥ are both invariant under homogenization, this is also equal to σ Q .By homogeneity, σ Q is also equal to inf χ∈T Q , ∥χ∥=1 M(χ), since T Q is preserved by the scaling action of Q >0 and the minimum norm ∥χ∥ of any χ ∈ T Q is rational.The proof is complete.□ We conjecture that equality holds in Theorem 5.13: Conjecture 5.15.For any polarized pair (X, B; L) we have σ div (X, B; L) = σ K (X, B; L).
for all v ∈ X div and D ∈ |L| Q .Since the maximal vanishing order satisfies T L (v) = sup{m −1 v(s) | s ∈ H 0 (X, mL) \ {0}, m sufficiently divisible} we get A X,B ≥ α T L on X div , and hence also on X an , since T L is lsc.This proves (i)⇒(ii), and (ii)⇒(iii) is trivial, since X lin = {T L < +∞} is contained in X val .Finally, assume (iii).Pick v ∈ X div \ {v triv }, and denote by Z ⊂ X the closure of the center of v, which is a strict subvariety since v ̸ = v triv .We obtain a trivial semivaluation v Z,triv ∈ X an \ X val , such that lim t→+∞ tv = v Z,triv in X an .As A X,B is lsc on X an , we infer lim inf t→+∞ tA X,B (v) ≥ A X,B (v Z,triv ) = +∞.
Thus A X,B (v) > 0, which proves (iii)⇒(i).□ A.3.Valuations of finite log discrepancy.If (X, B) is a pair, Proposition A.4 shows that the locus {A X,B < ∞} ⊂ X val does not depend on B, and it is a birational invariant.This naturally leads to the following notion.
Definition A.7.For any (not necessarily normal) projective variety X, we define the set X fld ⊂ X val of valuations of finite log discrepancy as the subset {A Y < +∞} of Y val = X val for some (or any) projective birational morphism Y → X with Y smooth.
Clearly X fld is a birational invariant of X.We also note Lemma A.8.The set X fld is a Borel subset of X an for any projective variety X.
Proof.We may assume X is smooth.Applying Theorem A.6 to the klt pair (X, 0) shows that X fld = {A X < +∞} ⊂ X an , which is a Borel set since A X is lsc.□ Remark A.9. Similarly X lin ⊂ X an is a Borel set as X lin = {T ω < +∞} for any ω ∈ Amp(X).We do not know whether X val ⊂ X an is a Borel set when k is uncountable.
A.4. Log discrepancy via snc test configurations.We refer to [BoJ22, Appendix A] for details on the following discussion.For any projective variety X, Gauss extension provides an embedding σ : X an → (X × P 1 ) an onto the set of k × -invariant semivaluations w ∈ (X × P 1 ) an such that w(ϖ) = 1, where ϖ denotes the coordinate on A 1 ⊂ P 1 .If v ∈ X an , then σ(v) is a valuation iff v is.Now assume that X is smooth, and consider an snc test configuration for X, i.e. a test configuration dominating the trivial test configuration, such that X is nonsingular and X 0,red is snc.The canonical compactification X → P 1 provides a log smooth pair ( X , X 0,red ) over X × P 1 .As in §A.1, the dual cone complex ∆ X := ∆(X , X 0,red ) embeds in (X × P 1 ) val .The preimage ∆ X := σ −1 ( ∆ X ) ⊂ X val is compact, and σ div (∆ X ) ⊂ ∆ X is a compact simplicial complex cut out by the equation w(ϖ) = 1.We view ∆ X as a simplicial complex itself, and each simplex has an integral affine structure with respect to which the rational points are the divisorial points; hence ∆ X ∩ X div is dense in ∆ X .