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Microlocal Bernstein--Sato polynomials on singular ambient varieties

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Pith's one-line read Microlocal b-functions detect rational singularities on singular spaces

desk verdict New microlocal b-function for singular ambient varieties, with one direction of the main theorem depending on unpublished work read the letter →

arxiv 2607.06376 v1 pith:HF3NMXBE submitted 2026-07-07 math.AG

Bradley Dirks , Michael Perlman This is my paper · ORCID
classification math.AG MSC 14F1014B0514B1514C30
keywords Bernstein-Satopolynomialmicrolocalb-functionminimalexponentrationalsingularitiesintersectioncohomologyD-modulemixedHodgemodulessingularvarietieslocal
topics The Hodge Conjecture
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

The paper extends the theory of microlocal Bernstein–Sato polynomials from smooth ambient varieties to singular ones. When the ambient variety X is smooth, the microlocal b-function eb_f(s) equals the reduced b-function b_f(s)/(s+1), and its largest root—the minimal exponent—characterizes rational singularities of the divisor D = X ∩ V(f). On singular ambient varieties, the structure sheaf no longer carries a D-module structure, so the authors replace it with the intersection cohomology D-module IC_X and define eb_{(X,f)}(s) via partial microlocalization of the graph embedding of IC_X along f. The key discovery is that on singular ambient varieties, eb_{(X,f)}(s) can strictly divide the reduced b-function b_{(X,f)}(s)/(s+1), so the microlocal polynomial carries genuinely new information. The paper defines the minimal exponent eα(X,f) as the negative of the largest root of eb_{(X,f)}(s) and proves that when X has rational singularities, D has rational singularities if and only if eα(X,f) > 1, generalizing the classical theorem of Saito for smooth X. Additional results include a characterization of purity of local cohomology via absence of integer roots, a Thom–Sebastiani formula for the minimal exponent, a linear combination formula relating b-functions of ideals to microlocal b-functions, and effective algorithms with Macaulay2 implementations for complete intersections with rational singularities.

What carries the argument

intersection cohomology D-module IC_X

What would settle it

If one could exhibit a pair (X, f) where X has rational singularities and eα(X,f) > 1 but D = X ∩ V(f) does not have rational singularities, or conversely D has rational singularities but eα(X,f) ≤ 1, the main theorem would fail.

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Extended reading notes

Core claim

The central object is the microlocal Bernstein–Sato polynomial eb_{(X,f)}(s), defined by applying Saito's partial microlocalization to the graph embedding of the intersection cohomology Hodge module IC_X^H along f. The core discovery is that this polynomial strictly refines the reduced b-function on singular ambient varieties—it can divide b_{(X,f)}(s)/(s+1) properly—and that its largest root, the minimal exponent eα(X,f), provides a sharp criterion for rational singularities: if X has rational singularities, then D = X ∩ V(f) has rational singularities if and only if eα(X,f) > 1. The paper also shows that eb_{(X,f)}(s) having no integer roots characterizes purity of the local cohomology H^1

Load-bearing premise

The converse directions in the filtration comparison theorems require that gr^F(IC_X^H) has no f-torsion, a condition automatic when X is smooth but not guaranteed for singular X. While the main rational singularities theorem circumvents this via a separate argument, the general theory of comparing Hodge and pole-order filtrations depends on this torsion condition without a complete geometric characterization of when it holds.

Editorial extensions

If this is right

  • The minimal exponent eα(X,f) becomes a computable obstruction to rational singularities of divisors on singular ambient varieties, enabling singularity detection beyond the log canonical threshold.
  • The Thom–Sebastiani formula for eα allows singularity analysis of products of singular pairs, which is relevant for degeneration arguments and moduli problems.
  • The Macaulay2 algorithms for complete intersections with rational singularities make these invariants accessible for explicit computation and experimentation.
  • The characterization of purity via integer roots of eb_{(X,f)}(s) connects the vanishing cycle theory of IC_X to Hodge-theoretic properties of local cohomology in new ways.
  • The strict refinement eb_{(X,f)}(s) | b_{(X,f)}(s)/(s+1) reveals that singular ambient geometry introduces conormal obstructions invisible in the smooth setting, opening questions about when equality holds.

Reading between the lines

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

  • The condition that gr^F(IC_X^H) has no f-torsion—automatic when X is smooth but not in general—appears to be the key technical hypothesis distinguishing where the full filtration comparison theory works from where it breaks down. A complete geometric characterization of this torsion-freeness condition would clarify the boundary of the theory.
  • The question of whether eb_{(X,f)}(s) = b_{(X,f)}(s)/(s+1) for all f characterizes rational homology manifolds (Question 3.8) suggests a deeper connection between the microlocal b-function and Poincaré duality on singular spaces.
  • The higher microlocal b-functions eb_{(X,f,p)}(s) mentioned in Remark 5.4, defined using higher Hodge pieces F^{c+p}(IC_X^H), could potentially yield criteria for higher Du Bois and higher rational singularities on singular ambient varieties, extending the scope beyond what the current single invariant captures.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 7 minor

Summary. The paper introduces the microlocal Bernstein–Sato polynomial eb_{(X,f)}(s) for a function f on a possibly singular ambient variety X, extending Saito's classical theory. The key construction replaces O_X with the intersection cohomology D-module IC_X and applies partial microlocalization to the graph embedding. The authors prove a division relation (Proposition 3.5) showing eb_{(X,f)}(s) divides the reduced b-function, establish a purity criterion for local cohomology (Theorem 4.1), prove a Thom–Sebastiani formula (Theorem E), and provide a linear combination formula for b-functions of ideals (Theorem F). The central application is Theorem A (Theorem 5.6): when X has rational singularities, D = X ∩ V(f) has rational singularities if and only if the minimal exponent eα(X,f) > 1. Effective algorithms for complete intersections with rational singularities are implemented in Macaulay2.

Significance. This paper makes a substantial contribution to the D-module and singularity theory literature. The definition of eb_{(X,f)}(s) is a natural and well-motivated construction, not circular: it is defined directly from IC_X and the V-filtration via a minimal polynomial, and the minimal exponent eα(X,f) is defined as the negative of the largest root. The observation that eb_{(X,f)}(s) can strictly divide b_{(X,f)}(s)/(s+1) on singular ambient varieties (Example 3.7) is a genuinely new phenomenon. The Thom–Sebastiani formula (Theorem E) and the linear combination formula (Theorem F) are clean generalizations of known results. The Macaulay2 implementations and explicit examples throughout are a significant strength, providing verifiable computations. The purity theorem (Theorem 4.1) and the HRH comparison (Corollary 4.7) are well-executed.

major comments (2)
  1. Theorem 5.6 (Theorem A), the paper's central application, relies on the injectivity of f on IC_X / O^GR_X for the converse direction (D rational ⟹ eα > 1). This injectivity is attributed to [CDO26b], listed as 'in preparation.' The alternative argument sketched in Remark 5.7 does not visibly close the gap: Corollary 4.7 gives HRH(D) ≥ 0 ⟺ p(φ_{f,1}(IC^H_X)) ≥ 2 − d_X, which does not by itself imply φ_{f,1} = 0 (equivalently eα > 1) without additional argument. The forward direction (eα > 1 ⟹ D rational) is established without [CDO26b], so only half of the 'if and only if' is fully verified from available references. This is a verifiability concern, not evidence of incorrectness — the result is consistent with Saito's smooth-case theorem and all computed examples. The authors should either make [CDO26b] available for refereeing, provide a self-contained proof of the injectivity claim, or,
  2. Theorem B (stated in the Introduction) asserts converses under the hypothesis that gr^F(IC^H_X) has no f-torsion. The more precise version, Theorem 5.3, states converses under the weaker hypothesis that (M,F) is f-saturated up to level k (Definition 5.2). The relationship between these two conditions should be clarified more explicitly in the Introduction's statement of Theorem B, so that the reader understands the precise scope of the converse claims. Currently, the Introduction states the converses hold 'if gr^F(IC^H_X) has no f-torsion,' which is sufficient but not the weakest hypothesis used in the body.
minor comments (7)
  1. The notation eα(X,f) for the minimal exponent is introduced in (5.6) but used earlier in the Introduction and Theorem A without a forward reference to (5.6). A cross-reference would help the reader.
  2. In the proof of Theorem 5.6, the inclusion IC_X / O^GR_X ↪ H^c_X(O_Y) / F^0 H^c_X(O_Y) is stated to hold 'by strictness of the Hodge filtration.' A brief justification or reference for this inclusion would be helpful, as the strictness argument is not spelled out.
  3. Example 3.7(3): the statement 'it is possible that eb_{(X,f)}(s) = b̃_{(X,f)}(s), even if eb_{(X,h)}(s) ≠ b̃_{(X,h)}(s) for some h' is slightly confusing because in this example eb = 1 and b̃ = 1, so they are equal, but the point about dependence on the pair (X,f) could be stated more clearly.
  4. Question 3.8 asks whether eb_{(X,f)}(s) = b̃_{(X,f)}(s) for all f when X is a rational homology manifold. It would be useful to mention whether the examples in the paper (e.g., Example 4.5, Example 4.9(2)) provide positive evidence for this question.
  5. In Section 7.2, the bound d' in Proposition 7.12(3) is noted to be non-sharp in Remark 7.13. It would be helpful to state in the algorithm (Algorithm 7.14) that d' is an upper bound, to avoid confusion for users of the implementation.
  6. The reference [CDO26b] is cited as 'In preparation (2026).' If this reference is load-bearing for Theorem A, its status should be clarified before publication.
  7. Minor typo: in the abstract, 'Bernstein–Sato' uses an en-dash, while in the title it uses a double hyphen. Consistency would be preferable.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for a careful reading and for recognizing the paper's contributions. The referee raises two major comments: (1) a verifiability concern regarding the converse direction of Theorem A (Theorem 5.6), which relies on [CDO26b] (in preparation) for an injectivity claim, and (2) a request to clarify the relationship between the hypothesis stated in the Introduction's Theorem B and the weaker hypothesis used in the body (Theorem 5.3). We address both below.

read point-by-point responses

  1. Referee: Theorem 5.6 (Theorem A), the paper's central application, relies on the injectivity of f on IC_X / O^GR_X for the converse direction (D rational ⟹ eα > 1). This injectivity is attributed to [CDO26b], listed as 'in preparation.' The alternative argument sketched in Remark 5.7 does not visibly close the gap. The authors should either make [CDO26b] available for refereeing, provide a self-contained proof of the injectivity claim, or restate Theorem A as a one-directional result.

    Authors: We thank the referee for identifying this verifiability concern, which is legitimate and well-articulated. We address it on both fronts. First, regarding the alternative argument in Remark 5.7: the referee is correct that Corollary 4.7 alone does not close the gap. The argument in Remark 5.7 requires the additional input that D having Du Bois singularities (which follows from eα(X,f) ≥ 1 via [Dir25, Cor. 1.5]) combined with X having rational singularities implies HRH(D) ≥ 0. This implication is a consequence of the fact that rational singularities imply HRH ≥ 0 (by [DOR25, Corollary E]) and that Du Bois singularities on D, together with X rational, give the needed comparison — but this last step itself uses properties of the Hodge filtration on local cohomology that are part of the theory developed in [CDO26b]. So we agree that Remark 5.7 does not provide a fully self-contained alternative. Second, and more fundamentally, the injectivity of f on IC_X / O^GR_X is indeed currently established only in [CDO26b], which is listed as 'in preparation.' We acknowledge that this creates a verifiability gap for the converse direction (D rational ⟹ eα > 1) of Theorem 5.6. The forward direction (eα > 1 ⟹ D rational) is fully self-contained, as the referee notes. To resolve this concern, we will take the following action in the revised manuscript: we will restate Theorem A (Theorem 5.6) to clearly separate the two directions. The forward implication will be stated as a theorem with a complete proof. The converse will be stated as a theorem whose proof depends on [CDO26b], with a clear flag that this reference is not yet available for verification. We will also revise the abstract and introduction to accurately reflect this status. If the referee or editor requires the converse to be降 revision: yes

  2. Referee: Theorem B (stated in the Introduction) asserts converses under the hypothesis that gr^F(IC^H_X) has no f-torsion. The more precise version, Theorem 5.3, states converses under the weaker hypothesis that (M,F) is f-saturated up to level k (Definition 5.2). The relationship between these two conditions should be clarified more explicitly in the Introduction's statement of Theorem B.

    Authors: The referee is correct that the Introduction's statement of Theorem B uses a stronger hypothesis than necessary. As noted in the manuscript immediately after the statement of Theorem B, the condition that gr^F(IC^H_X) has no f-torsion is sufficient but not necessary for f-saturatedness (Definition 5.2), and Theorem 5.3 uses the weaker hypothesis. We agree that the Introduction should make this relationship more explicit. In the revised manuscript, we will add a sentence to the Introduction's statement of Theorem B (or immediately following it) clarifying that the no-f-torsion condition is a sufficient condition for f-saturatedness, and that Theorem 5.3 establishes the converses under the weaker hypothesis of f-saturatedness up to the relevant level. We will also briefly recall the definition of f-saturatedness in the Introduction so that the reader understands the precise scope of the converse claims without needing to consult Section 5 first. revision: yes

standing simulated objections not resolved
  • The converse direction of Theorem A (D rational ⟹ eα > 1) depends on the injectivity of f on IC_X / O^GR_X, which is established in [CDO26b] (in preparation) and cannot be verified from currently available references. We cannot provide a self-contained proof of this injectivity at present without reproducing substantial material from that work. We will flag this dependency clearly in the revised manuscript.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity found; definitions are genuine constructions and theorems relate independently defined quantities. Self-citations exist but are not load-bearing in a circular sense.

full rationale

The paper defines eb_{(X,f)}(s) as the minimal polynomial of s = -∂_t t on Gr^0_G(eB_f) (Definition 3.2), which is a direct algebraic construction from IC_X and the microlocal V-filtration. The minimal exponent eα(X,f) = min{λ | eb_{(X,f)}(-λ) = 0} (eq. 5.6) is a definition, not a fitted parameter. The main theorems relate this exponent to independently defined geometric properties: rational singularities (Theorem 5.6), purity of local cohomology (Theorem 4.1), and Hodge filtration comparisons (Theorem 5.1). The division relation eb | b̄ (Proposition 3.5) is proved directly via the surjectivity of π: Gr^{-1}_G(B_f) → Gr^{-1}_G(eB_f) and kernel analysis. The Thom-Sebastiani formula (Theorem E) uses the external [MSS20, Thm 1.2]. The linear combination formula (Theorem F) uses a V-filtered isomorphism from [Dir24]. Several self-citations appear ([Dir25] for the non-microlocal b-function definition, [CDO26a] for a proof technique, [CDO26b] for an injectivity lemma, [DOR25]/[DOR26] for HRH theory), but none reduce to the paper's conclusions by construction. The [CDO26b] dependency for one direction of Theorem 5.6 is a verifiability concern (unpublished), not circularity—the injectivity of f on H^c_X(O_Y)/F^0 H^c_X(O_Y) is an independent technical statement not equivalent to eα > 1. The algorithms in Section 7 compute polynomials from explicit algebraic data and verify the theorems on examples, with no fitting-to-prediction loop. Score 2 reflects the presence of multiple self-citations that, while not circular, are not all independently verifiable.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The paper introduces no free parameters — all objects are defined canonically from IC_X and the V-filtration. The axioms are standard results in Hodge module theory, with [MSS20] and [CDO26b] being domain-specific dependencies. The two invented entities (microlocal b-function and minimal exponent on singular ambients) are given explicit constructions with independent verification via smooth recovery and Macaulay2 computations.

assumptions (6)
  • standard math Saito's theory of mixed Hodge modules [Sai90], including the six functor formalism, V-filtration, and nearby/vanishing cycles
    Used throughout as the foundational framework. Section 2.2 recalls the relevant structure.
  • standard math Existence and properties of the V-filtration along t on the graph embedding direct image (Kashiwara–Malgrange)
    Used in Section 2.4 to define the V-filtration on B_f and eB_f, and to relate roots of b-functions to vanishing cycles.
  • domain assumption The filtered Thom–Sebastiani formula of [MSS20, Theorem 1.2]
    Used in the proof of Theorem E (Section 5.3) to decompose Gr^F Gr^V of eB_{f1+f2} into external products.
  • domain assumption For complete intersections with rational singularities, F^0(IC_X^H(-c)) = S·[1/G] (from [Ola23, CDM24])
    Used in Section 7.1 to identify the cyclic generator needed for the algorithms.
  • domain assumption Injectivity of f on IC_X / O^GR_X and on H^c_X(O_Y)/F^0 H^c_X(O_Y) from [CDO26b]
    Used in the proof of Theorem 5.6 to verify the f-saturation condition needed for the converse.
  • standard math Walther's algorithms [Wal02] for computing Bernstein–Sato polynomials and annihilators of rational functions
    Used in Section 7 as the computational backbone for the algorithms.
invented entities (2)
  • Microlocal Bernstein–Sato polynomial eb_{(X,f)}(s) on singular ambient varieties independent evidence
    purpose: Minimal polynomial of s = -∂_t t on Gr^0_G(eB_f), the zeroth graded piece of the G-filtration on the partial microlocalization of the graph embedding of IC_X
    Defined via an explicit construction (Definition 3.2) and verified to recover the classical microlocal b-function when X is smooth. Its properties are derived from the V-filtration theory, not postulated. The Macaulay2 computations provide independent computational verification.
  • Minimal exponent eα(X,f) on singular ambient varieties independent evidence
    purpose: Negative of the largest root of eb_{(X,f)}(s); generalizes Saito's minimal exponent to singular ambients
    Defined directly from eb_{(X,f)}(s). Theorem 5.6 provides a falsifiable characterization: eα(X,f) > 1 iff D has rational singularities (when X has rational singularities), which is independently checkable.

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Pith. "Pith review of Microlocal Bernstein--Sato polynomials on singular ambient varieties." pith.science (2026). https://pith.science/paper/HF3NMXBE

@misc{pith2026260706376,
  author       = {Pith},
  title        = {Pith review of: Microlocal Bernstein--Sato polynomials on singular ambient varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HF3NMXBE}},
  note         = {Machine review of arXiv:2607.06376}
}
abstract

We introduce the microlocal Bernstein--Sato polynomial of a function on a possibly singular ambient variety, extending the theory of Saito. We show that, contrary to the smooth ambient setting, these polynomials are not generally equal to the reduced $b$-functions obtained by removing the trivial root. We define the minimal exponent and use it to study the singularities of the divisor and the Hodge filtration on local cohomology. Our main results include a generalization of Saito's theorem relating the minimal exponent to rational singularities, a characterization of purity of local cohomology, a Thom--Sebastiani formula for the minimal exponent, and a linear combination formula for Bernstein--Sato polynomials of ideals. When the ambient variety is a complete intersection with rational singularities, we provide effective algorithms for these Bernstein--Sato polynomials and implement them in Macaulay2.

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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