Jumping numbers of analytic multiplier ideals (with an appendix by S\'ebastien Boucksom)

Dano Kim; Hoseob Seo

REVIEW 3 major objections 4 minor 39 references

Jumping numbers of analytic multiplier ideals (with an appendix by S\'ebastien Boucksom)

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For toric singularities on the unit bidisk, jumping numbers have cluster points exactly when the Newton convex body fails to meet one of its coordinate asymptotes, and then all cluster points are explicit rational numbers.

desk verdict A genuinely new complete characterization of cluster points of jumping numbers for toric psh functions in dimension 2, with repairable gaps in the proof and a valuable appendix by Boucksom. read the letter →

arxiv 1908.02474 v2 pith:PNOYIFQ4 submitted 2019-08-07 math.AG math.CV

Dano Kim , Hoseob Seo This is my paper · ORCID

classification math.AGmath.CV MSC 14B0514F1832U0532U25

keywords jumpingnumbersmultiplieridealsplurisubharmonicfunctionstoricpshNewtonconvexbodyclusterpointsasymptoticperiodicity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends the theory of jumping numbers of multiplier ideals from algebraic singularities to general plurisubharmonic (psh) functions. Many algebraic properties survive, but two central ones do not: jumping numbers need not be discrete and need not be periodic. For toric psh functions on the unit bidisk, the authors fully characterize when jumping numbers accumulate: clustering happens exactly when one of the coordinate asymptotes of the Newton convex body does not meet the body. In that case the cluster points are precisely the rational numbers $k/m$ formed from the asymptote's distance to the origin, and this yields uncountably many new examples with clustering behavior.

What carries the argument

The load-bearing object is the Newton convex body $P(\phi)$ of a toric psh function, together with the criterion (Proposition 2.7) that the multiplier ideal $J(\phi)$ is monomial and $z_1^{a_1}\cdots z_n^{a_n}\in J(\phi)$ if and only if $(a_1+1,\dots,a_n+1)$ lies in the interior of $P(\phi)$. This turns the analytic question of when multiplier ideals jump into the convex-geometric question of which lattice points cross the boundary $\partial P(c\phi)$ as $c$ varies. In dimension 2, the asymptotic behavior of the boundary of $P(\phi)$ near its two coordinate asymptotes then determines all cluster points.

What would settle it

Take a toric psh function on $D^2$ whose Newton convex body $P$ has a horizontal asymptote $y=y_0>0$ disjoint from $P$ and a vertical asymptote at $x=0$ meeting $P$; Theorem 5.7 predicts exactly the cluster points $\{k/y_0:k\in\mathbb{Z}_{>0}\}$. Compute $\mathrm{Jump}(\phi)_0$ directly as the closure of $C(\phi)$ from lattice points on $\partial P(c\phi)$; any cluster point outside that predicted set, or any missing predicted point, would falsify the characterization.

Watch

Extended reading notes

Core claim

For a toric psh function $\phi$ on $D^2$ with Newton convex body $P(\phi)$, write $x_0=\inf \mathrm{pr}_1(P(\phi))$ and $y_0=\inf \mathrm{pr}_2(P(\phi))$. Theorem 5.7 states that $\mathrm{Jump}(\phi)_0$ has a cluster point if and only if either $x_0>0$ and the vertical line $\{(x_0,t):t\in\mathbb{R}\}$ misses $P(\phi)$, or $y_0>0$ and the horizontal line $\{(t,y_0):t\in\mathbb{R}\}$ misses $P(\phi)$. In that case the set of cluster points is exactly $\{k/m:k\in\mathbb{Z}_{>0},\, m\in S\}$, where $S\subseteq\{x_0,y_0\}$ contains $x_0$ exactly in the first case and $y_0$ exactly in the second. The paper also proves that for toric psh functions on $D^n$, if $c$ is a jumping number then all integer multiples $mc$ are jumping numbers, and that periodicity of jumping numbers fails in general; the appendix proves that asymptotic multiplier ideals of a graded system of ideals equal the analytic multiplier ideals of the associated psh function.

Load-bearing premise

The central argument assumes that a toric psh function's multiplier ideal is completely encoded by the integer lattice points inside its Newton convex body, via the criterion that $z^A$ belongs exactly when $A+\mathbf{1}$ lies in the interior; if that encoding failed for even one toric psh function, the lattice-point boundary analysis behind the cluster-point theorem would collapse.

Editorial extensions

If this is right

If a toric psh function in dimension 2 has one cluster point, then it has infinitely many cluster points, and they form an unbounded set.
Every cluster point of jumping numbers is itself a jumping number, so the failure of discreteness occurs at values that still belong to the jumping set.
For the earlier psh example whose jumping numbers accumulate at 1, the full set of cluster points is precisely the positive integers.
Periodicity of jumping numbers, which holds in the algebraic case, can fail for general psh functions; an explicit example with no period is presented.
The asymptotic multiplier ideal of a graded system of ideals equals the analytic multiplier ideal of the associated psh function, so algebraic and analytic constructions can be used interchangeably in this setting.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the asymptote-disjoint condition is open in the space of Newton convex bodies, cluster-point behavior should be generic among toric psh singularities in dimension 2, not a rare pathology.
The same asymptote criterion can be tested for mixed jumping numbers $\mathrm{Jump}(\phi;\psi)_0$ by Minkowski addition of the two Newton bodies, giving an explicit recipe for clustering in additive families.
In higher dimensions the natural extension is that cluster points are governed by coordinate hyperplane asymptotes of $P(\phi)$; the dimension-2 proof suggests the cluster set will be generated by reciprocals of the missing asymptote distances, possibly with new phenomena such as cluster points of cluster points.
The equality between asymptotic and analytic multiplier ideals suggests one can engineer psh singularities with prescribed cluster sets by choosing Newton convex bodies with the desired asymptote structure, which may inform the open algebraic question of whether cluster points occur for multiplier ideals on singular varieties.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this paper is worth a serious look. The main result, Theorem 5.7, gives a complete convex-geometric criterion for when a toric psh function on the unit polydisk in C^2 has cluster points of jumping numbers, and it identifies all cluster points as the positive rationals k/x0 and/or k/y0. That is a real advance: previously only two sporadic examples (Guan–Li and ELSV) were known, and this paper shows clustering is a frequent phenomenon and answers Question 1.1 affirmatively in this toric setting. The appendix by Boucksom, proving that analytic multiplier ideals of Siu psh functions equal asymptotic multiplier ideals, is an independent and cleanly proved result that will be cited on its own.

The paper does several things well. The reduction to Newton convex bodies is natural, and Proposition 3.3, linking jumping numbers to lattice points on the boundary of P(cϕ), is a useful tool. Corollary 5.11 sharpening the Guan–Li example, and the explicit computations in Examples 3.6 and 3.12, strengthen the overall picture. The writing is clear and the authors are honest about what depends on external theorems like Guenancia–Rashkovskii and the openness conjecture.

That said, the proof of the central theorem, as printed, has some genuine gaps. In the proof of Theorem 5.7, condition (2) says A_k+1 ∈ intP((c−ε_k)ϕ) but A_k+1 ∉ intP((c−ε_{k+1})ϕ). Since ε_k > ε_{k+1}, the second set is larger, so that containment pattern is impossible. The later text and the intended argument use “not in intP(cϕ)”, so this looks like a simple typo, but it has to be fixed. More substantively, the proof asserts without argument that a subsequence exists with one coordinate constant and the other strictly increasing. That extraction is not a formal consequence of the preceding line; it needs the bounded-coordinate argument the stress-test note sketches. This is the load-bearing step in the necessity direction, so it cannot be left as is. Theorem 5.3 has a smaller version of the same issue: it only proves mC(ϕ) ⊂ C(ϕ), then jumps to all jumping numbers without explicitly using the closure statement of Proposition 3.3 for c ∈ Jump(ϕ) \ C(ϕ). All these look repairable, and I do not see a reason to doubt the main theorem, but as printed the proof is incomplete.

Who is this for? Anyone working on pluripotential theory, multiplier ideals, or singularities of psh functions will get something from it. The gaps are annoying but not fatal. I would recommend that the editors send it to a serious referee who can ask for a repaired proof. My own default would be to accept it after that repair.

I would bring it to a reading group and would cite the Boucksom appendix in my own work.

Referee Report

3 major / 4 minor

Summary. The paper extends the study of jumping numbers of multiplier ideals from the algebraic setting (Ein–Lazarsfeld–Smith–Varolin) to general plurisubharmonic functions. After establishing basic properties via the Guan–Zhou openness theorem, the authors analyze v-equivalence of psh functions, construct Siu psh functions from graded systems of ideals, and prove several examples. The main result, Theorem 5.7, gives a complete characterization, in terms of the Newton convex body, of when a toric psh function on the unit polydisk in C^2 has cluster points of jumping numbers at the origin, and identifies the set of such cluster points. An appendix by Boucksom proves that analytic multiplier ideals of Siu psh functions coincide with the asymptotic multiplier ideals of the underlying graded system.

Significance. If the proof gaps are repaired, Theorem 5.7 would be a substantial contribution: it completely describes the cluster-point phenomenon for toric psh functions in dimension two, generalizing the two known examples of Guan–Li and Ein–Lazarsfeld–Smith–Varolin and producing uncountably many new examples. The convex-geometric formulation is elegant, and the appendix by Boucksom is of independent interest, as it settles a natural comparison between analytic and algebraic asymptotic multiplier ideals. The paper is clearly written and the main statement is plausible and consistent with the worked examples.

major comments (3)

[§5.2, proof of Theorem 5.7] The sequence conditions (1)–(2) in the proof are internally inconsistent. Since ε_{k+1} < ε_k, the set intP((c−ε_{k+1})ϕ) is a superset of intP((c−ε_k)ϕ), so no lattice point A_k+1 can satisfy both A_k+1 ∈ intP((c−ε_k)ϕ) and A_k+1 ∉ intP((c−ε_{k+1})ϕ). The same inconsistency appears in the sufficiency direction with B_j. The intended condition presumably involves intP(cϕ) or intP((c+ε_{k+1})ϕ); as written, the existence of the sequence is impossible, so the proof of the cluster-point characterization is not valid.
[§5.2, proof of Theorem 5.7] The subsequence extraction is unjustified and, as stated, false. From 'exactly one coordinate of A_k increases at each step' it does not follow that there is a subsequence with one coordinate constant and the other strictly increasing; for example, the sequence (0,0),(0,1),(1,1),(1,2),(2,2),(2,3),... has both coordinates unbounded along every subsequence. The proof needs an additional argument showing that because A_k+1 ∉ intP(cϕ), at least one coordinate is bounded along a subsequence; this is exactly the step that converts a cluster point into a vertical or horizontal asymptote, so the main theorem is not established as written.
[§5.1, proof of Theorem 5.3] The proof only treats the case c ∈ C(ϕ). For a jumping number c that lies in the closure of C(ϕ) but not in C(ϕ), Proposition 3.3 gives c ∈ closure(C(ϕ)), but the argument showing mC(ϕ) ⊂ C(ϕ) does not directly imply mc ∈ Jump(ϕ)_0. One needs an approximation argument using the fact that Jump(ϕ)_0 is closed, or an alternative argument; as printed, the theorem is not proved for cluster-point jumping numbers.

minor comments (4)

[Throughout] There are several typographical errors: 'absense' in the introduction, 'Newon' in Example 3.12, and 'Theorem 5.10' in the proof of Corollary 5.11 should likely be Theorem 5.7 or 5.3.
[§3.1, Proposition 3.3] The proof of the inclusion for non-cluster jumping numbers assumes that a single lattice point A can be chosen independently of ε; this deserves a brief justification using Noetherianity of the stalks.
[§5.2, sufficiency direction] The construction of the sequence (α_j) in the sufficiency direction is only sketched; it would be helpful to explain explicitly why such lattice points can be chosen with the stated nesting property, especially given the inconsistency noted in the first major comment.
[Appendix] The proof of Theorem 2.2 is very terse; in particular, the passage from v(ϕ) = v(a•) to the multiplier ideal equality uses the valuative criteria without recalling the normalization condition on v in the singular case. This is understandable for an appendix by an expert but could be clarified for the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central characterization is built from external convex-geometric and valuative results, with no fitted parameter or self-referential definition.

full rationale

The paper's main theorem (Theorem 5.7) reduces jumping numbers for toric psh functions to lattice points on the Newton convex body using Proposition 2.7, a result of Guenancia and Rashkovskii that is external to the paper. The other load-bearing inputs—Guan-Zhou openness, the valuative descriptions of multiplier ideals in [BFJ] and [BFFU], and the Boucksom appendix proving Theorem 2.2—are independent of the paper's conclusions and are not derived from the cluster-point statement. No parameter is fitted to the jumping numbers it later predicts: the cluster points are computed from P(ϕ) by a convex-geometric boundary analysis. Citations to the authors' own work (K15, K19, KR) appear only for auxiliary notions such as v-equivalence, coefficient-dependence of Siu psh functions, and Cegrell-class ideal behavior; none is load-bearing for Theorem 5.7. The skeptic's concern about an unjustified subsequence extraction in the necessity direction of Theorem 5.7 (and a similar gap in Theorem 5.3) is a possible proof gap, not a circularity: replacing that step by a bounded-coordinate argument would repair the proof, and at no point does the argument assume the target set of cluster points as an input.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on established theorems from pluripotential theory, valuation theory, and the convex geometry of toric singularities. The paper introduces no new entities, no fitted constants, and no ad hoc assumptions specific to the paper. The main external inputs are listed above; the proof of Theorem 2.2 (Boucksom) supplies the bridge between analytic and algebraic multiplier ideals within the paper itself.

assumptions (6)

standard math Guan-Zhou strong openness theorem: for psh functions ψ,ϕ, ⋃_{ε>0} J(ψ+εϕ) = J(ψ).
Invoked in Section 3.1 (Definition 3.1) to guarantee that multiplier ideals are constant on half-open intervals and that jumping numbers are well-defined; also used in Section 2.1 for the v-equivalence characterization.
standard math For a toric psh function ϕ, J(ϕ) is a monomial ideal and z^a ∈ J(ϕ) iff (a+1) ∈ intP(ϕ) (Guenancia, Rashkovskii).
Used throughout, in particular in Proposition 2.7, Proposition 3.3, and the proof of Theorem 5.7. This is the central bridge between analytic multiplier ideals and the convex body P(ϕ).
standard math Valuative characterization of multiplier ideals: f∈J(cϕ) iff there exists ε>0 such that v(f) ≥ (1+ε)v(cϕ) − A(v) for all divisorial valuations v (BFJ).
Used in the appendix (Boucksom's proof of Theorem 2.2).
standard math Asymptotic multiplier ideal membership via normalizing subschemes: f∈J(c·a•) iff for every normalizing subscheme N and every 0<δ≪1, v(f) ≥ c v(a•) − A(v) + δ v(I_N) (BFFU).
Used in the appendix to compare analytic and asymptotic multiplier ideals.
standard math Subadditivity of multiplier ideals: J(ψ+ϕ) ⊆ J(ψ)J(ϕ) (DEL, Theorem 2.6).
Used in Proposition 3.2 to bound the gap between consecutive jumping numbers.
standard math Valuation of a graded system: v(a•)=inf_k v(a_k)/k = lim_k v(a_k)/k (JM, Lemma 2.3).
Used in the appendix to prove v(ϕ)=v(a•).

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Pith. "Pith review of Jumping numbers of analytic multiplier ideals (with an appendix by S\'ebastien Boucksom)." pith.science (2026). https://pith.science/paper/PNOYIFQ4

@misc{pith2026190802474,
  author       = {Pith},
  title        = {Pith review of: Jumping numbers of analytic multiplier ideals (with an appendix by S\'ebastien Boucksom)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PNOYIFQ4}},
  note         = {Machine review of arXiv:1908.02474}
}

read the original abstract

We extend the study of jumping numbers of multiplier ideals due to Ein-Lazarsfeld-Smith-Varolin from the algebraic case to the case of general plurisubharmonic functions. While many properties from Ein-Lazarsfeld-Smith-Varolin are shown to generalize to the plurisubharmonic case, important properties such as periodicity and discreteness do not hold any more. Previously only two particular examples with a cluster point (i.e. failure of discreteness) of jumping numbers were known, due to Guan-Li and to Ein-Lazarsfeld-Smith-Varolin respectively. We generalize them to all toric plurisubharmonic functions in dimension 2 by characterizing precisely when cluster points of jumping numbers exist and by computing all those cluster points. This characterization suggests that clustering of jumping numbers is a rather frequent phenomenon. In particular, we obtain uncountably many new such examples.

Figures

Figures reproduced from arXiv: 1908.02474 by the authors.

**Figure 1.** Two different choices of a sequence of closed rational convex polyhedra for P in Example 2.8 and for the asymptotic multiplier ideals J (ma•) by [JM, Proposition 8.4], respectively. Such difference contradicts Theorem 2.2 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 2.** Theorem 5.7 exactly says that the jumping numbers of [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 2.** All the possible asymptotic behaviors of the boundary of P(ϕ) with respect to the two axes. Proof of Theorem 5.7. Assume that c is a cluster point of jumping numbers of ϕ at 0. In view of Proposition 2.7, we then can find a sequence (ǫk) of positive real numbers and a sequence (Ak) in Z 2 ≥0 such that (1) ǫk > ǫk+1 for every k ≥ 1 and ǫk → 0, (2) Ak + 1 ∈ int P((c − ǫk)ϕ) but Ak + 1 ∈/ int P((c − ǫk+1)ϕ) for every k… view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Boucksom, P

S. Boucksom, P. Eyssidieux, V. Guedj and A. Zeriahi, Monge-Amp\`ere equations in big cohomology classes, Acta Math. 205 (2010), 199 - 262

work page 2010

[2] [2]

Boucksom, T

S. Boucksom, T. de Fernex, C. Favre and S. Urbinati, Valuation spaces and multiplier ideals on singular varieties, Recent advances in algebraic geometry, 29 - 51, London Math. Soc. Lecture Note Ser., 417, Cambridge Univ. Press, Cambridge, 2015

work page 2015

[3] [3]

Boucksom, C

S. Boucksom, C. Favre and M. Jonsson, Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci 44 (2008), no.2, 449-494

work page 2008

[4] [4]

Blickle, K

M. Blickle, K. Schwede, S. Takagi and W. Zhang, Discreteness and rationality of F-jumping numbers on singular varieties, Math. Ann. 347 (2010), no. 4, 917–949

work page 2010

[5] [5]

Budur and M

N. Budur and M. Saito, Multiplier ideals, V-filtration, and spectrum, J. Algebraic Geom. 14 (2005), no. 2, 269–282

work page 2005

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Cegrell, The general definition of the complex Monge-Amp\`ere operator, Ann

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Demailly, Monge-Amp\`ere operators, Lelong numbers and intersection theory, Complex analysis and geometry, 115-193, Univ

J.-P. Demailly, Monge-Amp\`ere operators, Lelong numbers and intersection theory, Complex analysis and geometry, 115-193, Univ. Ser. Math., Plenum, New York, 1993

work page 1993