Convexity of multiplicities of filtrations on local rings

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.


Introduction
Throughout the article, (R, m) denotes an n-dimensional, analytically irreducible, Noetherian, local domain.
The Hilbert-Samuel multiplicity of an m-primary ideal a ⊂ R is a fundamental invariant of the singularities of a and satisfies various convexity properties such as Teissier's Minkowski inequality [Tei78].In this article, we consider m-filtrations of R, which generalize the filtrations of R given by the powers of a single ideal, and prove various convexity properties for such filtrations.The results have applications to the study of K-stability, volumes of valuations, and problems in commutative algebra.
An m-filtration is a collection a • = (a λ ) λ∈R >0 of m-primary ideals of R that is decreasing, graded, and left continuous.The latter three conditions mean a λ ⊂ a µ when λ > µ, a λ • a µ ⊂ a λ+µ , and a λ = a λ−ǫ when 0 < ǫ ≪ 1, respectively.The key examples of m-filtrations are the following: (1) A trivial example is given by taking powers (b ⌈λ⌉ ) λ∈R >0 of a fixed m-primary ideal b ⊂ R.
(3) If (b λ ) λ∈Z >0 is a decreasing, graded sequence of m-primary ideals,1 then (b ⌈λ⌉ ) λ∈R >0 is an m-filtration.where the existence of the above limit and the equality were proven in increasing generality by [ELS03,Mus02,LM09,Cut13,Cut14].This invariant is the local counterpart of the volume of a graded linear series of a line bundle and has been studied both in the context of commutative algebra [CSS19,Cut21] and recently in work of C. Li and others on the normalized volume of a valuation [Li18,LLX20].
In this article, we prove two properties of the multiplicity of m-filtrations.The first is a convexity result, which has applications to volumes of valuations and C. Li's normalized volume function.The second is a generalization of a classical theorem of Rees [Ree61] from the setting of ideals to filtrations and results in a characterization of when the Minkowski inequality for filtrations is an equality.
We call (a •,t ) t∈ [0,1] the geodesic between a •,0 and a •,1 , since it is the local analogue of the geodesic between two filtrations of the section ring of a polarized variety [BLXZ21,Reb23].This definition is also related to a construction in [XZ21].
In [BLXZ21], it was shown that several non-Archimedean functionals from the theory of Kstability are strictly convex along geodesics in the global setting.In a similar spirit, we prove a convexity result in the local setting for the multiplicity along geodesics.
The term smooth in Theorem 1.1.1means E(t) extends to a C ∞ function on (−ǫ, 1 + ǫ) for some ǫ > 0. The symbol • in Theorem 1.1.3denotes the saturation of an m-filtration, which is defined in Section 3.1.This notion is an analogue of the integral closure of an ideal in the setting of filtrations and discussed further in Section 1.2 below.
The proof of Theorem 1.1 is inspired by a related argument in the global setting [BLXZ21] and relies on constructing a measure on R 2 that encodes the multiplicities of the filtrations along the geodesic.In the special case when a •,0 is the m-filtration of a valuation minimizing the normalized volume function over a klt singularity, the proof of [XZ21] can be used to show Theorem 1.1(2).Theorem 1.1 removes these strong restrictions and is proven without the theory of K-stability for valuations introduced in [XZ21, Section 3.1].
This invariant is a local analogue of the volume of a line bundle and also plays a role in the study of K-stability of Fano varieties and Fano cone singularities.Theorem 1.1 can be applied to show that the volume of a valuation is strictly log convex on simplices of quasi-monomial valuations in the valuation space of (R, m).This gives an affirmative answer to [LLX20,Question 6.23].We note that the volume was previously shown to be Lipschitz continuous on such a simplex [BFJ14,Corollary D].
In the above theorem, v α denotes the quasi-monomial valuation of Frac(R) with weights α on D 1 , . . ., D r .See Section 2.2.3 for a detailed definition.
1.1.2.Applications to normalized volume.In [Li18], Chi Li defined the normalized volume of a valuation over a klt singularity and proposed the problem of studying its minimizer.The notion plays an important role in the study of K-stability of Fano varieties and in the study of klt singularities.The invariant has been extensively studied in the recent years; see [LLX20] and [Zhu23] for surveys on this topic.
The fundamental problem in the study of the normalized volume function is the Stable Degeneration Conjecture proposed by Li [Li18,Conjecture 7.1] and Li and Xu [LX18, Conjecture 1.2].The conjecture predicts that there exists a valuation minimizing the normalized volume function and that the minimizer is unique up to scaling, quasi-monomial, has finitely generated associated graded ring, and induces a degeneration of the klt singularity to a K-semistable Fano cone singularity.These five statements were proven in [Blu18, LX18, Xu20, XZ21, XZ22].In particular, the Stable Degeneration Conjecture is now a theorem.
Using Theorem 1.1, we revisit Xu and Zhuang's theorem stating that the minimizer of the normalized volume function is unique up to scaling [XZ21].(The uniqueness was also previously proven in [LX18] under the assumption that the minimizer has finitely generated associated ring.The latter was recently shown in [XZ22].)In particular, we give a proof of the uniqueness result independent of the theory of K-semistability for valuations developed in [XZ21].
Corollary 1.3 (Uniqueness of minimizer).If x ∈ (X, D) is a klt singularity defined over an algebraically closed field of characteristic 0, then any minimizer of vol X,D,x (see Definition 2.8) is unique up to scaling.
The new proof of Corollary 1.3 takes the following direct approach.Fix two valuations v 0 and v 1 that minimize vol X,D,x and consider the geodesic (a •,t ) t∈[0,1] between a • (v 0 ) and a • (v 1 ).Using Theorem 1.1(2), a characterization of the infimum of the normalized volume function in terms of normalized multiplicities [Liu18, Theorem 27], and an inequality of log canonical thresholds [XZ21, Theorem 3.11], we show that e(a •,t ) is linear.Theorem 1.1(3) then implies cv 0 = v 1 for some c > 0.
1.2.Rees's theorem.A theorem of Rees [Ree61] states that if a ⊂ b are two m-primary ideals, then the following statements are equivalent: (1) e(a) = e(b).
( To remedy this issue, we introduce the saturation a • of an m-filtration a • in Section 3. (The definition may be viewed as a local analogue of a construction studied by Boucksom and Jonsson in [BJ21]; see Section 6).The saturation is defined using divisorial valuations, analogous to the valuative definition of the integral closure of an ideal.Using this notion, we prove a version of Rees's theorem for filtrations.
The above result can be explained as follows.The multiplicity of an m-filtration is determined by its valuative properties, not by the integral properties of its Rees algebra.These two properties coincide for ideals, but do not always coincide for filtrations by Example 3.7.
and the equality holds if and only if a The inequality statement of the above corollary is not new and is due to Mustat ¸ǎ [Mus02], Kaveh and Khovanskii [KK14], and Cutkosky [Cut15] in increasing levels of generality.In the equality statement, the forward implication follows easily from [Cut21, Theorem 10.3] and the definition of the saturation, while the reverse implication relies on Theorem 1.4.This paper is organized as follows: In Section 2, we recall definitions and basic facts concerning filtrations, valuations, and multiplicities.In Section 3, we introduce the saturation of a filtration and prove Theorem 1.4.In Section 4, we define the geodesic between two filtrations and prove Theorem 1.1.In Section 5, we deduce Corollaries 1.2, 1.3, and 1.5 as consequences of results in the previous two sections.In Section 6, we discuss relations between the results in this paper and global results in the K-stability literature.The appendix of the paper is devoted to an alternate proof of a special case of Theorem 1.1 using the theory of Okounkov bodies.
Acknowledgement.We would like to thank Mattias Jonsson, Linquan Ma, Sam Payne, Longke Tang, Chenyang Xu, and Ziquan Zhuang for helpful discussions.LQ would like to thank his advisor Chenyang Xu for constant support, encouragement and numerous inspiring conversations.We are also grateful to the referees whose comments and corrections improved the quality of this paper.
HB is partially supported by NSF grant DMS-2200690.YL is partially supported by NSF Grant DMS-2148266.

Preliminaries
Throughout this section, (R, m, κ) denotes an n-dimensional, analytically irreducible, 2 Noetherian, local domain.We set X := Spec(R) and write x ∈ X for the closed point corresponding to m.
By convention, we set a 0 := R. The definition is a local analogue of a filtration of the section ring of a polarized variety in [BHJ17].
2 Recall that (R, m) is analytically irreducible if its completion R is a domain.and (2) v(f + g) ≥ min{v(f ), v(g)}.
By convention, we set v(0) := ∞.We say v is centered at We write m v for the maximal ideal of R v and κ v := R v /m v .
2.2.1.Real valuations.When Γ = R with the usual order, we say that v is a real valuation.We denote by Val R,m the set of real valuations centered at m. 3 In geometric settings, we will instead denote the set by Val X,x .
For v ∈ Val R,m and λ ∈ R >0 , we define the valuation ideal where the existence of the limit and second equality is [JM12, Proposition 2.3].

Divisorial valuations. A valuation
We write DivVal R,m ⊂ Val R,m for the set of such valuations.Divisorial valuations appear geometrically.If µ : Y → X is a proper birational morphism with Y normal and E ⊂ Y a prime divisor, then there is an induced valuation ord E : Frac(R) × → Z.If µ(E) = x and c ∈ R >0 , then c•ord E ∈ DivVal R,m .When R is excellent, all divisorial valuations are of this form; see, for example, [CS22, Lemma 6.5].

2.2.3.
Quasi-monomial valuations.In the following construction, we always assume R contains a field.Let µ : Y := Spec(S) → X = Spec(R) be a birational morphism with R → S finite type and η ∈ Y a not necessarily closed point such that O Y,η is regular and µ(η) = x.Given a regular system of parameters y 1 , . . ., y r of O Y,η and α A valuation of the above form is called quasi-monomial.
Let D = D 1 + • • • + D r be a reduced divisor on Y such that y i = 0 locally defines D i and µ(D i ) = x for each i.We call η ∈ (Y, D) a log smooth birational model of X.We write QM η (Y, E) ⊂ Val X,x for the set of quasi-monomial valuations that can be described at η with respect to y 1 , . . ., y r and note that QM η (Y, D) ≃ R r ≥0 \ 0.

2.2.4.
Izumi's inequality.The order function R \ 0 → N is defined by The following version of Izumi's inequality compares ord m to a fixed quasi-monomial valuation.
Lemma 2.4.Let v ∈ Val R,m .If (i) v is divisorial or (ii) R contains a field and v is quasimonomial, then there exists a constant c > 0 such that If (i) holds, the the existence of c follows from Izumi's theorem for divisorial valuations as phrased in [RS14, Remark 1.6].4If (ii) holds, then there exists a log smooth birational model η ∈ (Y, D) of x ∈ X and α ∈ R r such that v = v α .Choose γ ∈ Z r >0 such that α i ≤ γ i for each i = 1, . . ., r and consider the valuation w := v γ .We claim that w ∈ DivVal R,m .Assuming the claim, then (i) implies that there exists c > 0 such that w(f To verify that w ∈ DivVal R,m , note that w : where w is the valuation that sends β∈Z r To see that the last equality holds, note that the complete local ring R is a domain by assumption and thus equidimensional.Therefore R is universally catenary by [Sta, Tag 0AW6] and so the dimension formula [Sta,Tag 02II] gives the last equality.Therefore, w is divisorial as desired.
2.3.Intersection numbers.The theory of intersection numbers of line bundles on a proper scheme over an algebraically closed fields was developed in [Kle66].We will use a more general framework developed in [Kle05, Appendix B].
2.3.1.Definition.Let Z be a proper scheme over an Artinian ring Λ.For line bundles L 1 , . . ., L r on Z, the function This is well defined, since each prime divisor D i is proper over Spec(κ).
Since the intersection product in 2.3.1 is symmetric, F 1 • . . .• F n is independent of the order of F 1 , . . ., F n−1 .To deduce the full result, we rely on intersection theory [Ful98].
A subtle issue is that the results in [Ful98, §1-18] are stated for schemes of finite type over a field and, hence, do not immediately apply to Y .Fortunately, the Chow group of a scheme of finite type over a regular base scheme can be defined and the results of [Ful98,§2] extend to this setting by [Ful98, §20.1] (see also [Sta,Chapter 02P3] for the results in an even more general setting).
Proof.By Cohen's Structure Theorem, there exists a surjective map A ։ R, where A is a regular local ring.Since the composition Y → X := Spec(R) → Spec(A) is finite type, the framework of [Ful98, §20.1] applies.Using intersection theory on Y and its subschemes, we compute where D i and Yκ denote the degree maps induced by the proper morphisms D i → Spec(κ) and Y κ → Spec(κ) [Ful98, Definition 1.4] and M i := O Y (F i ).The first equality holds by [Ful98, Example 18.3.6]with the fact that D i is a proper scheme over κ, the second by the equality and the last by the definition of the first Chern class.Since and the latter is independent of the order of the F i by [Ful98, Corollary 2.4.2], the proposition holds. 2.4.Multiplicity.

Multiplicity of an ideal.
The multiplicity of an m-primary ideal a is e(a) := lim The following intersection formula for multiplicities commonly appears in the literature when x ∈ X is a closed point on a quasi-projective variety [Laz04,pp. 92].In the generality stated below, it follows from [Ram73].
Proposition 2.6.Let a ⊂ R be an m-primary ideal and where E = a i D i and L i := L| E i .Indeed, the first equality is [Kle05, Lemma B.12] and the second equality holds, since O Y,ξ i is a DVR and O E,ξ i = O Y,ξ /(π a i ), where π is a uniformizer of the DVR.
To finish the proof, notice that D i is defined over κ, since there is a natural inclusion which completes the proof.

Multiplicity of a filtration. Following [ELS03]
, the multiplicity of a graded sequence of m-primary By [ELS03, Mus02, LM09, Cut13, Cut14] in increasing generality, the above limit exists and e(a Proof.By Lemma 2.4, there exists c > 0 such that a 2.5.Normalized volume.Assume R is the local ring of a closed point on a algebraic variety defined over an algebraically closed field of characteristic 0. We say x ∈ (X, ∆) is a klt singularity if ∆ is an R-divisor on X such that K X + ∆ is R-Cartier, and (X, ∆) is klt as defined in [KM98].
The following invariant was first introduced in [Li18] and plays an important role in the study of K-semistable Fano varieties and Fano cone singularities.
Definition 2.8.For a klt singularity x ∈ (X, ∆), the normalized volume function vol (X,∆),x : where A X,∆ (v) is the log discrepancy of v as defined in [JM12,BdFFU15].The local volume of a klt singularity x ∈ (X, ∆) is defined as The above infimum is indeed a minimum by [Blu18,Xu20].
3.1.Saturation.Let a ⊂ R be an m-primary ideal.Recall that the integral closure a of a can be characterized valuatively by See, for example, [HS06, Theorem 6.8.3] and [Laz04, Example 9.6.8].We define the saturation of an m-filtration in a similar manner.
Definition 3.1.The saturation a • of an m-filtration a • is defined by Remark 3.2.Corollary 3.19 shows that it is equivalent to define the saturation using all positive volume valuations, rather than only divisorial valuations.
Remark 3.3.The saturation is a local analogue of the maximal norm of a multiplicative norm of the the section ring of a polarized variety, which was defined and studied in [BJ21].See Section 6.
Remark 3.4.Definition 3.1 differs from the definition of saturation used in [Mus02, Section 2] for monomial ideals, which coincides with the ideals in Lemma 3.6 defined using the integral closure of the Rees algebra.
Example 3.7.In general, the ideals a m and a ′ m appearing above do not coincide.For example, let R := k[x] (x) and m = (x).Consider the m-filtration a • defined by a λ := (x ⌈λ+1⌉ ).Using that ord m (a λ ) = ⌈λ + 1⌉ and ord m (a • ) = 1, we compute The following lemma states basic properties of the saturation.
for each m ∈ Z >0 , where the first inequality follows from the definition of a m and the second from (1).Sending m → ∞ gives v(a We say two m-filtrations a • and b • are equivalent if a • = b • .The following proposition gives a characterization of when two filtrations are equivalent after possible scaling.Proposition 3.9.Let a • and b • be m-filtrations and c ∈ R >0 .The following statements are equivalent: (1) . Therefore, (1) implies (2) follows from Lemma 3.8(2), while (2) implies (1) follows from the definition of the saturation.
3.2.Saturation and completion.Let ( R, m) denote the m-adic completion of (R, m) and write ϕ : R → R for the natural morphism.(Note that R is a domain by the assumption throughout the paper that R is analytically irreducible.)For an m-filtration a • , we set a • R := (a λ R) λ>0 , which is an m-filtration.
The proposition shows that completion and saturation commute, as is the case with the integral closure of an ideal [HS06, Proposition 1.6.2].As a consequence of the proposition, many results regarding saturations reduce to the case when R is a complete local domain.
Proof.By [HS06, Theorem 9.3.5],there is a bijective map ϕ * : DivVal R, m → DivVal R,m that sends v to the valuation v defined by composition Note that v(a R) = v(a) for any ideal a ⊂ R. Fix λ ∈ R >0 .Using the previous observation twice and the definition of the saturation, we compute v( Thus, a λ R ⊂ b λ .To prove the reverse inclusion, note that cR = b λ for some ideal c ⊂ R, since b λ is m-primary.As before, we compute Proof.Since R/a λ and R/b λ are isomorphic as Artinian rings, ℓ(R/a λ ) = ℓ( R/b λ ) for each λ > 0. Therefore, the equality of multiplicities holds.

Saturation and multiplicity.
In this section we prove Theorem 1.4 and a number of corollaries.The theorem is a consequence of the following two propositions.
The proposition was shown when v is divisorial in [Cut21, Theorem 7.3] using Okounkov bodies.The proof below instead follows the strategy of [LX20, Proposition 2.7] and [XZ21, Lemma 3.9].
Proof.Suppose the statement is false.Then there exists Now, consider the map φ : R → b lm /a lm sending g → f m g.
We claim ker(φ) ⊂ a (l−k)m (v).Indeed, if g ∈ ker φ, then f m g ∈ a lm .Hence, as desired.Using the claim, we deduce where the first inequality is by (3.1) and the second uses that v has positive volume.This contradicts our assumption that e(a • ) = e(b • ).
The proposition and its proof are local analogues of [Szé15, Lemma 22], which concerns the volumes of graded linear series of projective varieties.
Remark 3.14.The assumption that R is complete in Proposition 3.13 implies that R is Nagata [Sta, Lemma 032W] (see [Sta, Definition 033S]).The latter property will be used in the proof of Proposition 3.13 to ensure certain normalization morphisms are proper.
Proof.We claim that there exists a sequence of proper birational morphisms the each X i is normal and the sheafs a l • O Xm and b l • O Xm are line bundles when l ≤ m.Such a sequence can be constructed inductively as follows.Let X i → X i−1 be defined by the composition is finite by the definition of Nagata, and we conclude the composition X i → X i−1 is proper, which completes the proof of the claim.Next, consider a proper birational morphism Y → X with Y normal and factoring as , which are relatively nef over X and satisfy where the sums run through prime divisors E ⊂ Y .Throughout the proof, we will without mention replace Y with higher birational models so it factors through certain X m → X.
Given ǫ > 0, by Lemma 2.6 and (2.1), there exists m 0 > 0 such that where the second equality uses that F m 0 is the pullback of π m 0 * F m 0 and the projection formula we may choose m 1 sufficiently large and divisible by m 0 such that , where the first equality is Proposition 2.5.
For any multiple m 2 of m 1 , and the terms F ′ m 1 and F m 0 are nef over X. Similarly to the previous paragraph, we compute and, hence, we may choose m 2 sufficiently large and divisible by m 1 so that Remark 3.15.When x ∈ X is an isolated singularity on a normal variety, Proposition 3.13 follows easily from the intersection theory for nef b-divisors developed in [BdFF12].Indeed, the assumption v(a The proposition then follows from [BdFF12, Remark 4.17].However, when x is not an isolated singularity, a satisfactory intersection theory for nef b-divisors seems missing from the literature.
We are now ready to prove Theorem 1.4 and a number of its corollaries.
Proof of Theorem 1.4.By Propositions 3.10 and 3.11, we may assume (R, m) is complete.This condition will be needed below to apply Proposition 3.13.
By Propositions 3.12 and 3.13, e(a . By Proposition 3.9, the latter condition holds if and only if a Proof.By Lemma 3.8, a • ⊂ a • and a is saturated.Thus, Theorem 1.4 implies e(a • ) = e( a • ).
Corollary 3.17.Let a • and b • be m-filtrations.The following statements are equivalent: (1) Proof of Corollary 3.17.Assume (1) holds.Observe that e(a where the first and third equality follow from Corollary 3.16.Thus, it remains to show that e(b where the inequality uses that a m + b m ⊂ a m + b m = a m + a m = a m by the assumption that (1) holds.Therefore, (2) holds.
Conversely, assume (2) holds.Applying Theorem 1.4 to both (a The following result was proven when R is regular in [Mus02, Theorem 1.7.2]. Corollary 3.18.An m-filtration a • is linearly bounded if and only if e(a • ) > 0.
Proof.If a • is linearly bounded, then there exists c > 0 such that a λ ⊂ m ⌈cλ⌉ for all λ > 0. Thus, e(a • ) ≥ c n e(m) > 0 as desired.
3.4.Saturation and finite-volume valuations.Using results from the previous section, we show that the saturation can be defined using positive volume valuations, rather than only divisorial valuations.
Proposition 3.19.If a • is an m-filtration and λ ∈ R >0 , then R,m } The proposition will be deduced from the following lemma.
Lemma 3.20.If {v i } i∈I is a collection of valuations in Val + R,m and {c i } i∈I of non-negative real numbers, then the m-filtration a • defined by Suppose the statement is false.Then there exists some λ ∈ R >0 such that a λ a λ .Thus, there exist f ∈ a λ and i ∈ I such that v i (f ) where the second equality uses that
This definition is a local analog of the geodesic between two filtrations of the section ring of a polarized variety [BLXZ21,Reb23].See Section sec:global for details.
>0 , let v α : Frac(R) × → R be the monomial valuation with weight α 1 and α 2 with respect to x and y, that is This computation unfortunately does not generalize to the case of quasi-monomial valuations.See [LXZ22, Section 4] for a study of this failure in the global setting.
To prove Theorem 1.1, we define a measure on R 2 that encodes the multiplicities along the geodesic.The argument may be viewed as a local analogue of the construction in [BLXZ21, Section 3.1] for the geodesic between two filtrations of the section ring of a polarized variety.The latter global construction is motivated by [BC11] and [BHJ17, Thmeorem 4.3], which constructs a measure on R associated to a single filtration of the section ring of polarized variety.
Before proceeding with the proof, fix integers C > 0 and D > 0 such that Notice that H m is non-decreasing and left continuous in each variable.Using the sequence (H m ) m≥1 , we define a sequence of measures on R 2 .
Proposition 4.3.The distributional derivative µ m := − ∂ 2 Hm ∂x∂y is a discrete measure on R 2 and has support contained in Before proving the proposition, we prove the following lemma allowing us to reduce to the case of complete local rings.(1) ℓ(R/a a,0 ∩ a b,1 ) = ℓ( R/b a,0 ∩ b b,1 ) for each a, b ≥ 0.
Proof.For m-primary ideals c ⊂ R and d ⊂ R, Therefore, (a a,0 ∩ a b,1 ) R = b a,0 ∩ b b,1 and a a,t R = b a,t .Statements (1) and (2) follow from these equalities.
Proof of Proposition 4.3.By Lemma 4.4, it suffices to prove the statement when (R, m) is complete.Since R contains a field by assumption and is complete, there is an inclusion κ ֒→ R such that the composition κ ֒→ R → κ is the identity.This will be helpful, since any finite-length R-module is naturally a finite-dimensional κ-vector space via restriction of scalars.
Fix an integer N > 0 and consider the finite-dimensional κ-vector space V := R/m N Cm .The m-filtrations a •,0 and a •,1 induce decreasing filtrations Observe that when x < N and y < N , where the first equality uses that m N Cm ⊂ a mx,0 ∩ a my,1 .
To analyze the dimension of F mx 0 V ∩ F my 1 V , we use that any finite-dimensional vector space with two filtrations admits a basis simultaneously diagonalizing both filtrations (see [BE21, Proposition 1.14] and [AZ22, Lemma 3.1]).This means there exists a basis (s 1 , . . ., s ℓ ) for V such that each F λ j V is the span of some subset of the basis elements.Hence, if we set Therefore, the restriction of Since N > 0 was arbitrary, µ m is a discrete measure on R 2 .It remains to analyze the support of µ m .Note that . Using (i), (ii), and (4.1), we see that supp Using (4.1), we show that µ m encodes the colengths of a •,t .
Proposition 4.5.For each t ∈ [0, 1] and a, b ∈ R ≥0 , Proof.By Lemma 4.4, it suffices to prove the result when (R, m) is complete.In this case we continue with the notation from the previous proof, but additionally fix N > max{ a 1−t , b t }.Now, consider the filtration where the sum runs through all µ, ν ∈ R satisfying ma = (1 − t)µ + tν.Now, we compute where the first equality uses that m N Cm ⊆ a ma/(1−t),0 + a mb/t,1 ⊂ a ma,t , the third (4.2), and the fourth (4.1).Therefore, the first formula in the proposition holds.
To verify the second formula, we fix N > max{a, b} and similarly compute where the first equality uses that m N Cm ⊂ a ma,0 ∩ a mb,1 .
4.3.Limit measure.We now construct a limit of the sequence of measures (µ m ) m≥1 that encodes the multiplicity along the geodesic.
Consider the function H : R 2 → R defined by Above, we use the convention that e(R) = 0, which occurs when x ≤ 0 and y ≤ 0.
Proof.We claim that H m converges to H in L 1 loc (R 2 ).Assuming the claim, then H m converges to H as distributions, and, hence, µ m converges to µ := − ∂ 2 H ∂x∂y as distributions as well.Since each µ m is a measure, [Hor03, Theorem 2.1.9]implies µ is a measure and µ m converges to µ weakly as measures.
It remains to prove the above claim.First, observe that H m converges to H pointwise by the definition of the multiplicity as a limit.Next, fix an integer N > 0. For −N ≤ x, y < N , observe that Therefore, the dominated convergence theorem implies that lim m→∞ (−N,N )×(−N,N ) |H m − H| = 0.
Since N > 0 was arbitrary, H m converges to H in L 1 loc (R 2 ) as desired.Proposition 4.7.For each t ∈ [0, 1] and a, b ∈ R ≥0 , (1) a n e(a Proof.Fix ǫ > 0. Since (µ m ) m≥1 converges to µ weakly as measures by Proposition 4.6, Using Proposition 4.5(1) to compute the lim sup and lim inf, we deduce Sending ǫ → 0 completes the proof of (1).The proof of the second is similar, but uses Proposition 4.5.2.
For the reverse implication, assume (2) holds.We claim Thus, the claim holds when cx ≥ y and similar reasoning treats the case when cx < y.Using the claim, we conclude (1) holds.
Proposition 4.9.The following statements hold: (1) The support of µ is contained in Proof.By Propositions 4.3 and 4.6, supp(µ where the second equality is by Proposition 4.7.Similar reasoning shows µ({y = 0}).Thus, (1) holds.Statement (2), follows from the fact that H is homogeneous of degree n, meaning H(cx, cy) = c n H(x, y) for each c > 0 and (x, y) ∈ R 2 .4.4.Measure on the segment.To prove Theorem 1.1, it will be convenient to work with a measure induced by µ on the interval [0, 1].
Proof.For simplicity, we assume t ≤ 1 2 , which ensures g t (z) := (1−t)z+t(1−z) is non-decreasing.(The case when t > 1 2 can be proved similarly, but using left-hand approximations below.)For each m ≥ 1, consider the simple function where and g t,m → g pointwise as m → ∞, the monotone convergence theorem implies (Above we are using that g t (z) −n is defined and continuous on (z, t) where the second equality uses Proposition 4.9.2.Next, notice that A m ⊂ A m+1 and Combining the previous two computations, we conclude where the final equality is Proposition 4.7(1).
To show the latter quantity is non-positive, we compute where the first inequality is by the Cauchy-Schwarz inequality.Therefore, d 2 dt 2 E(t) −1/n ≤ 0 for all t ∈ (0, 1), which proves (2).
To prove (3), note that E(t) is linear if and only if d 2 dt 2 E(t) −1/n = 0 for all t ∈ (0, 1).The latter holds if and only if the inequalities in (4.4) are all equalities.The inequalities are equalities iff (2z −1)g(z, t) −(n+2)/2 and g(z, t) −n/2 are linearly dependent in L 1 ( µ).Equivalently, 2z − 1 and g(z, t) are linearly dependent in L 1 ( µ).The linear dependence holds exactly when µ is supported at a single point.By Lemma 4.10 and Corollary 3.17, the latter condition is equivalent to the existence of c ∈ R such that a •,0 = a c•,1 .

Applications
In this section, we prove Corollaries 1.2, 1.3, and 1.5 using results from previous sections.

Proof of Corollary
where the second inequality is Theorem 1.1(2).Additionally, by Theorem 1.1(3) and Lemma 3.20, the second inequality is strict unless there exists c ∈ R >0 such that a •,0 = a c•,1 .Since a •,0 = a c•,1 if and only if cα 0 = α 1 , the result follows.5.2.Uniqueness of normalized volume minimizer.Throughout this section, we assume R is essentially of finite type over a field of characteristic 0.
To prove Corollary 1.3, we need the following proposition, which will be deduced from [XZ21, Theorem 3.11] where the inequality is [XZ21, Theorem 3.11].It remains to verify the claim.Since c m ⊂ a t,m for each integer m > 0, v(c Using the previous inclusions, we see that We can now compute . By rescaling, we may assume A X,∆ (v i ) = 1 and, hence, vol(v i ) = vol(x, X, D) for i = 0, 1.We seek to show where the first inequality is Proposition 5.1 and the second uses that v i (a where there first inequality holds by [Liu18, Theorem 27], the second by (5.1), and the third by Theorem 1.1(2).Therefore, every inequality above, in particular the third one, is an equality.By Theorem 1.1(3) and Lemma 3.20, there exists c > 0 such that a Remark 5.2.[Liu18, Theorem 27] was originally proved for Q-coefficients, but the proof works for R-coefficients with little change.
Thus, it remains to analyze when the inequality is an equality.If (5.2) is an equality, then Propositions 5.3 and 3.9 imply there exists c > 0 such that a • = b c• .Conversely, assume there exists c > 0 such that a where the first equality uses that v(ambm) m = v(am) m + v(bm) m for all m ∈ N and the second is by Proposition 3.9.Therefore, Proposition 3.9 implies a where the first and third equality hold by Corollary 3.17.

Relation to global results
In this section, we explain the relationship between the local constructions in this paper and certain global constructions in the K-stability literature.
Throughout, let X be an n-dimensional normal projective variety over a field k and L be an ample line bundle on X.The section ring of (X, L) is 6.1.Filtrations and multiplicity.The analogue of an R-filtration of a local ring in the global setting is the following definition [BHJ17, Section 1.1], which plays an important role in the Kstability literature.
A filtration F is linearly bounded if there exists C > 0 such that F Cm R m = 0 for all m > 0.
The data of a filtration F can be encoded as a norm χ : R(X, L) → R ∪ {+∞} by setting for details.Following loc.cit., we write N R for the set of such norms χ : R(X, L) → R ∪ {+∞} that arise from linearly bounded R-filtrations of R(X, L).Definition 6.2.The energy of a linearly bounded filtration F of R(X, L) is where gr λ F R m = F λ R m /F λ+ǫ R m and 0 < ǫ ≪ 1.The limit in the definition exists as a consequence of [BHJ17, Theorem 5.3].
The energy measures the size of a filtration and is an analogue of the multiplicity of an Rfiltration in the global setting.The invariant appears under different names in the K-stability literature such as the energy in [BHJ17] and the S-invariant in [Xu21].For a norm χ ∈ N R of R(X, L), vol(χ) denotes the energy of the associated filtration in [BJ21].6.2.Saturation and Rees's theorem.Rees's theorem for R-filtrations (Theorem 1.4) says that two R-filtrations a • ⊂ b • have equal volume if and only if their saturations are equal.As we will explain, an analogue of this result was previously proven by Boucksom and Jonsson in the global setting.
For a norm χ ∈ N R , Boucksom and Jonsson introduce the notion of its maximal norm χ max [BJ21, Definition 6.16].Similar to the saturation in Definition 3.1, it is defined using divisorial valuations.When char(k) = 0, they prove that two norms χ, χ ′ ∈ N R with χ ≤ χ ′ satisfy vol(χ) = vol(χ ′ ) if and only if χ max = χ ′ max [BJ21, Lemma 3.11 and Theorem 6.22].The proof relies on results from non-Archimedean pluripotential developed in [BJ22].6.3.Geodesics.The notion of a geodesic between two R-filtrations (Definition 4.1) is inspired by the following definition in the global setting.Definition 6.3 ([BLXZ21, Reb23]).For two linearly bounded filtrations F 0 and F 1 of R(X, L) and t ∈ (0, 1), we define a filtration F t of R(X, L) by setting We call (F t ) t∈[0,1] the geodesic between F 0 and F 1 .
This definition was introduced by the first two authors, Xu, and Zhuang in [BLXZ21, Section 3.1.2]to prove uniqueness results for certain optimal destabilizations of Fano varieties that arise from limits of Kähler-Ricci flow [BLXZ21, Section 3].Independently, Reboulet introduced an equivalent definition phrased in the language of norms on the section ring, rather than filtrations, and used it to define geodesics between metrics on a line bundle in non-Archimedean pluripotential theory [Reb23].
A different and possibly more intuitive way to understand the above definition is in terms of a well-chosen basis.By [AZ22, Lemma 3.1], there exists a basis (s 1 , . . ., s Nm ) of R m that is compatible with both F 0 and F 1 .Here, 'compatible' means that F λ j R m is the span of some subset of (s 1 , . . ., s Nm ) for each j ∈ {0, 1} and λ ∈ R. If we set λ i,j := max{λ ∈ R | s i ∈ F λ j R m }, then a computation shows (A similar computation is made in the proof of Proposition 4.3.)Imprecisely, this expression shows that (F t ) t∈[0,1] interpolates between F 0 and F 1 .
As shown in [BLXZ21, Section 3.2] and [Reb22, Section 4.7], various functionals on the space of filtrations are convex along geodesics.These convexity results are proven in [BLXZ21, Section 3.2] using a measure on R 2 (similarly to the proof of Theorem 1.1) and imply the uniqueness of minimizers of the h-functional on the space of valuations on a Fano variety [BLXZ21, Section 3.3].Thus, Theorem 1.1 and Corollary 1.3 can be viewed as local counterparts to these global results.
Following the calculation in [Cut13, Section 4], we see that the value semigroup S contains β ′ and β ′ + e i for some β ′ and 1 ≤ i ≤ n, where e i ∈ Z n is the ith standard basis vector.Hence S generates Z n as a group.
A.2.A volume formula for convex bodies.The following lemma will be useful in proving the convexity of multiplicities.It is a slight generalization of [Izm14, Lemma 4.4] which follows from the same argument, so we omit the proof.A.3.The cutting function induced by a filtration.To proceed, we fix a quasi-monomial valuation w ∈ Val X,x and a good Z n -valued valuation v associated to w given by Lemma A.2. Let S = v(R\{0}) ∪ {0} ⊂ Z n be the value semigroup of v. Let a • be a linearly bounded m-filtration.The goal of this section is to construct a cutting function on the strongly convex cone C(S) satisfying the conditions in Lemma A.3.
The following lemma is an easy generalization of Izumi's inequality to filtrations.For a family version, see Lemma A.10.
Lemma A.4.With notation as above, for any m ∈ R ≥0 , there exists Proof.By Lemma 2.4, there exists r > 0 such that ξ(v(f )) = w(f ) ≤ r•ord m (f ).By assumption, there exists d ∈ Z >0 such that m d ⊂ a 1 .So if we choose β 0 ∈ S such that ξ(β 0 ) ≥ rd(⌈m⌉ + 1), then for v(f ) β 0 , we have ord m (f ) ≥ d(⌈m⌉ + 1).Hence, We define a function m a• : S → R ≥0 associated to a • by for any β ∈ S. Since a • is left continuous, the above supremum is indeed a maximum.Now we choose an element f a•,β ∈ v −1 (β) ∩ a ma • (β) .We will write m(β) and f β respectively, if there is no chance of ambiguity.We first prove some properties of m(•) and construct a function h on S using it, which will play a key role in estimating the multiplicities.
Proposition A.5.Let m = m a• : S → R ≥0 be the function associated to a linearly bounded m-filtration a • as above.Then the following statements hold.
(1) m is superadditive, that is, (2) There exists M 2 > 0 such that for any β ∈ S, we have m(β) ≤ M 2 ξ(β). .As before, we will drop a • from the notation of h(β) if there is no ambiguity.
(4) Choose finitely generated subsemigroups S 1 ⊂ S 2 ⊂ • • • of S such that ∪ ∞ l=1 S l = S and each S l generates Z n as a group.By [Kho92, §3, Proposition 3], there exists γ l ∈ S l such that (C(S l ) + γ l ) ∩ N n ⊂ S l .
For any β ∈ Int(C(S l )), there exists k 0 such that for any k ≥ k 0 , we have ⌊kβ⌋ ∈ (C(S l ) + γ l ) ∩ N n ⊂ S l .Since the above definition does not depend on l, we have a well-defined function h on Int(C(S)) = ∪ ∞ l=1 Int(C(S l )).Clearly h is homogeneous of degree 1.

So we can define
Next we prove the continuity of h.It suffices to prove the continuity of h in Int(C(S l )).We first show that for any β, β 1 ∈ Int(C(S l )), we have h(β + β 1 ) ≥ h(β).
Hence, h is continuous on Int(C(S l )).The concavity of h follows easily from its continuity, homogeneity and (1).
(5) By definition we have for any β ∈ S. Now (A.2) follows easily from the homogeneity and continuity of h.
The following proposition asserts that f β gives a basis for a m /a >m .
Proposition A.6.For any m ∈ R ≥0 , the set {[f β ] | m(β) = m} forms a basis for the -vector space a m /a >m .
Proof.We may assume that m is a jumping number of a • .Let We first show that the set {[f β ]} is linearly independent.Assume to the contrary that there exist c β ∈ , not all zero, such that It remains to prove that for any f ∈ a m \a >m , there exist By condition (1) of Definition A.1, there exists a unique sequence {a β } β∈S such that for any Let β 0 be as defined in Lemma A.4.In particular, we have By the choice of a β , we know that v(g 1 ) ≥ β 0 .Hence g 1 ∈ a >m by the choice of β 0 .
Claim: For any β β 0 with β / ∈ B m , if a β = 0, then m(β) > m.We prove the claim by induction.Let β β 0 and assume that the claim is true for any Hence v −1 (β)∩a m = ∅ and m(β) ≥ m.Since m(β) = m by assumption, we have strict inequality and the claim is proved.
By the claim we know that g 2 ∈ a >m .Thus we conclude that that is, (A.5) holds with c β = a β .
A.4. Multiplicities and Okunkov bodies.In this section we apply the strategy as in [LM09,KK14] to estimate the multiplicities.Let a •,i be two linearly bounded m-filtrations for i = 0, 1. Recall that the geodesic (a •,t ) t∈[0,1] between a •,0 and a •,1 is defined as Then a •,t is also a linearly bounded m-filtration.
From now on, we follow the notation of Theorem A.7.Let v be a good valuation associated to w given by Lemma A.2. Lemma A.8. Let a •,t be as above for t ∈ [0, 1].Let m (t) = m a•,t : S → R ≥0 be the function defined in (A.1).Then we have for any t ∈ [0, 1] and any β ∈ S.
Proof.Fix an arbitrary element β ∈ S. For simplicity, denote by m We first show that m t ≥ (1 − t)m 0 + tm 1 .By assumption, we have w Next, we show that m t ≤ (1−t)m 0 +tm 1 .Choose f ∈ v −1 (β)∩a mt,t .We may write f = i f i as a finite sum such that f i ∈ a µ i ,0 ∩ a ν i ,1 where (1− t)µ i + tν i = m t for every i.Since f i ∈ a µ i ,0 = a µ i (w), we have w(f i ) ≥ µ i .Hence after replacing (µ i , ν i ) by (w(f i ), t −1 (m t − (1 − t)w(f i )) the assumption f i ∈ a µ i ,0 ∩ a ν i ,1 still holds.Furthermore, if we have µ i = µ j for some i = j then we may replace (f i , f j ) by f i + f j .After finitely many steps of replacements and permutations, we obtain a decomposition f = l i=1 f i such that f i ∈ a µ i ,0 ∩ a ν i ,1 where (1 − t)µ i + tν i = m t and µ i = w(f i ) for every i, and which completes the proof.
As an immediate corollary to Lemma A.8, we get the following result.
Corollary A.9. Let h t := h a•,t : Int(C(S)) → R >0 be defined as in The following lemma can be viewed as an Izumi-type estimate in our setting of filtrations.
We will use these sets to estimate the multiplicities of a •,t .The following property for Γ (t) is one of the main ingredients for our proof of Theorem A.7. Proof.Note that the jumping numbers of the m-filtration a •,t form a discrete set as for any µ > λ > 0, ℓ(a λ,t /a µ,t ) < ∞.By Proposition A.6, the quotient ring R/a m,t has a basis {[f ∈ a m,t }.Note that if ξ(v(f ∈ a m,t , we have ξ(β) = ξ(v(f β / ∈ a m,t }, and the proposition follows. We now check that the Γ and Γ (t) are semigroups satisfying the conditions (2.3-5) of [LM09].

and
(4) if we replace M by a proper multiple of it in the definition of Γ, then Γ (t) and Γ contain a set of generators of Z n+1 as a group. Proof.
From now on we assume that Γ and Γ (t) satisfy the conditions of Lemma A.12.As in [LM09, (2.1)], we define ∆ as the closed convex hull of m Γ m /m, and define ∆ (t) as the closed convex hull of m Γ (t) m /m for t ∈ [0, 1].Then ∆ (t) ⊂ ∆.By Lemma A.12, Proposition A.11 and [LM09, Proposition 2.1], we have the following result.Next, we show that the Γ (t) can be characterized by the function h t defined as in Corollary A.9.
The proof is finished.
Next we prove Theorem A.7. (A.6)By Corollary A.9 we have h t = (1 − t)h 0 + th 1 , hence d dt h t = h 1 − h 0 .By Proposition A.5(4), we can differentiate under the integral sign, and hence E(t) is a smooth function.Thus (1) is proved.
To prove (2), we use a slight variant of the formula (A.6).Let h := h 1 − h 0 .Then by homogeneity we have Hence by Cauchy-Schwarz we have which implies (2).
Such sequences have been well studied in the literature [Laz04, Section 2.4.B].Following work of Ein, Lazarsfeld, and Smith, the multiplicity of an m-filtration is vol(a • ) := lim N∋m→∞ ℓ(R/a m ) m n /n! = lim N∋m→∞ e(a m ) m n , ) m∈N a m = m∈N b m .(3) a = b.The symbol • in (2) denotes the algebraic closure in R[t], while in (3) it denotes the integral closure of an ideal.The equivalence between (2) and (3) follows from definitions.It is natural to ask for a generalization of the above result for m-filtrations a • ⊂ b • .In [Cut21], Cutkosky studies whether the two conditions (1) e(a • ) = e(b • ) (2) m∈N a m = m∈N b m are equivalent.While (2) ⇒ (1) holds by [CSS19, Theorem 6.9], (1) ⇒ (2) can fail even in very simple examples (see Example 3.7).That said, (1) ⇒ (2) holds for special classes of m-filtrations [Cut21, Theorem 1.4].

1. 3 .
Minkowski inequality.By work of Teissier [Tei78], Rees and Sharp [RS78], and D. Katz [Kat88] in increasing generality, for two m-primary ideals a and b of R, e(ab) 1/n ≤ e(a) 1/n + e(b) 1/n and the equality holds if and only if there exist c, d ∈ Z >0 such that a c = b d .For two m-filtrations a • and b • , we let a • b • denote the m-filtration (a λ b λ ) λ∈R >0 .Using the saturation of a filtration, we characterize when the Minkowski inequality for filtrations is an equality.Corollary 1.5 (Minkowski Inequality).For m-filtrations a • and b • with positive multiplicity,

Remark 1. 6 (
Relation to work of Cutkosky).Cutkosky proved a version of the equality part of Corollary 1.5 in the special case when a • and b • are bounded filtrations [Cut21, Definition 1.3], which roughly means that their integral closure is induced by a finite collection of divisorial valuations [Cut21, Theorem 1.6].For such filtrations, the integral closure and saturation agree by Lemma 3.20.Thus, Corollary 1.5 may be viewed as a generalization of the latter result.Similarly, Theorem 1.4 may be viewed as a generalization of [Cut21, Theorem 1.4].
Remark 3.5.After the first version of this paper was posted on the arXiv, Cutkosky and Praharaj introduced an operation on R-filtrations that is defined using certain asymptotic Hilbert-Samuel functions (see the definition in [CP22, Theorem 1.3]).As shown by [CP22, Example 7.2], their operation does not always coincide with the saturation.Lemma 3.6 ([Cut21, Lemma 5.6]).If a • be an m-filtration, then the integral closure of Rees(a • by Lemma 2.6 and (2.1), we see that e(b • ) < e(a • ) + (n + 1)ǫ.Therefore, e(b • ) ≤ e(a • ).
the claim holds.By Proposition 3.12 and the claim, e( a • ) < e(a • ).The latter contradicts Theorem 1.4.Proof of Proposition 3.19.Let b • denote the m-filtration defined by b where the second inclusion uses that all divisorial valuations have positive volume.Taking saturations gives a • ⊂ b • ⊂ a • .Since b • and a • are saturated by Lemma 3.20, we conclude a • ⊂ b • ⊂ a • .