{"id":"daf12d2a-e027-4869-ab0b-629f0c07639c","arxiv_id":"2607.06376","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The microlocal b-function on singular ambient varieties strictly refines the reduced b-function and its minimal exponent detects rational singularities of divisors.","lead":"This paper defines microlocal Bernstein–Sato polynomials for functions on singular varieties, showing they differ from the reduced b-function unlike in the smooth case. It generalizes Saito's minimal exponent to detect rational singularities of divisors and provides Macaulay2 algorithms for complete intersections.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The 'only if' direction of Theorem 5.6 (D rational ⟹ eα>1) relies on the unpublished [CDO26b] for injectivity of f on IC_X/O^GR_X, and the alternative proof sketched in Remark 5.7 does not visibly close this gap.","rationale":"The reader correctly identified the [CDO26b] dependency in the rationale ('The main caveat is reliance on unpublished work [CDO26b] for a key injectivity step in Theorem 5.6') and also identified the f-torsion condition for Theorem B as the weakest assumption. These are related but distinct concerns: the f-torsion condition limits the converse directions in Theorem B (filtration comparison), while the [CDO26b] dependency specifically affects the 'only if' direction of the headline Theorem 5.6 (rational singularities characterization). My concern is slightly more pointed than the reader's in two respects: (1) the alternative proof in Remark 5.7 does not visibly fill the gap, so there is no known independent route to the 'only if' direction; (2) the 'only if' direction is part of the 'if and only if' in the paper's central application, so the headline theorem is only half-established without [CDO26b]. That said, the forward direction is fully proved, the structural theory (division relation, purity, Thom-Sebastiani, linear combination formula) is independent of [CDO26b], and the algorithms are implemented and tested. The concern is about verifiability of one direction of one theorem, not about internal inconsistency or mathematical error. The verdict of ACCEPT with correctness_risk 'unknown' is appropriate — the unknown risk is precisely the [CDO26b] dependency. I do not think the verdict needs to change, but the concern should be understood as somewhat more load-bearing than 'a key injectivity step' suggests: it is the enabling ingredient for the converse direction of the paper's main theorem.","tokens_in":39302,"tokens_out":7692,"duration_ms":231784,"concrete_test":"Verify the injectivity of f on IC_X/O^GR_X directly in a specific complete-intersection-with-rational-singularities example using the Macaulay2 framework of Section 7. For instance, take X=V(x_1^2+...+x_6^2, x_1x_2+x_3x_4+x_5x_6)⊆A^6 (Example 5.8(2)) with f=x_1−x_2 (where D is NOT rational and eα=1). Compute IC_X/O^GR_X as a D-module quotient and check whether multiplication by f is injective. If it fails in this example, the [CDO26b] claim as used in Theorem 5.6 would be in question. If it holds, this provides computational evidence (though not proof) supporting the dependency.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 5.6 uses Corollary 5.5 in both directions. The forward direction (eα>1 ⟹ D rational) is established without [CDO26b]: it uses only the forward implication of Corollary 5.5, which requires no injectivity hypothesis. However, the converse direction (D rational ⟹ eα>1) requires the converse of Corollary 5.5, which needs injectivity of f on IC_X/O^GR_X. This injectivity is deduced from a stronger claim — injectivity of f on H^c_X(O_Y)/F^0 H^c_X(O_Y) — attributed to [CDO26b], listed as 'in preparation.' The inclusion IC_X/O^GR_X ↪ H^c_X(O_Y)/F^0 H^c_X(O_Y) transfers the injectivity via strictness, but the source result is unavailable for verification. Remark 5.7 sketches an alternative proof via Corollary 4.7, but the logical step is incomplete: Corollary 4.7 gives HRH(D)≥0 ⟺ p(φ_{f,1}(IC^H_X)) ≥ 2−d_X, which does not imply φ_{f,1}=0 (equivalently eα>1) without additional argument. Thus, without [CDO26b], only half of the 'if and only if' in the headline theorem is fully proved. This is a verifiability concern, not evidence of incorrectness — the result is consistent with Saito's smooth-case theorem and with all computed examples. But it is load-bearing: the characterization of rational singularities is the paper's central application.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","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.","tokens_in":39645,"tokens_out":1418,"duration_ms":210994,"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":[{"comment":"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,","section":null},{"comment":"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.","section":null}],"minor_comments":[{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"Minor typo: in the abstract, 'Bernstein–Sato' uses an en-dash, while in the title it uses a double hyphen. Consistency would be preferable.","section":null}],"recommendation":"major_revision","confidential_remarks":"The reliance on [CDO26b] for the converse direction of Theorem 5.6 is the main concern. If [CDO26b] can be made available and verified, or if a self-contained proof of the injectivity claim can be supplied, the recommendation would likely drop to minor_revision. The paper is otherwise strong, with careful proofs and valuable computational contributions. The authors may also consider splitting Theorem A into a fully proved statement (forward direction) and a conditional converse, pending [CDO26b], if the reference cannot be finalized."},"author_rebuttal":{"model":"glm-5.2","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"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."},{"response":"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_made":"yes","referee_comment":"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."}],"tokens_in":39187,"tokens_out":1114,"duration_ms":204826,"standing_objections":["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."]},"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper defines a microlocal Bernstein–Sato polynomial eb_{(X,f)}(s) for functions on singular ambient varieties, replacing O_X with IC_X in Saito's partial microlocalization. The definition is clean and the phenomenon is genuinely new — eb can strictly divide the reduced b-function b/(s+1), which has no smooth analogue (Example 3.7). The structural results are substantial: a division relation (Proposition 3.5), a purity theorem for H^1_f(IC_X) (Theorem 4.1), a Thom–Sebastiani formula (Theorem E), and a linear combination formula for ideals (Theorem F). The Macaulay2 algorithms for complete intersections with rational singularities are concretely useful and the examples are illuminating throughout. The proofs are carefully executed, particularly the kernel analysis in Proposition 3.5 and the monodromy filtration argument in Theorem 4.1. The f-torsion condition for converses in Theorem B is a genuine limitation but is clearly stated, not hidden, and is automatic in the smooth case. Now the soft spot. The stress-test concern about Theorem 5.6 (Theorem A) is real and lands. The forward direction (eα > 1 implies D rational) goes through without issue. The converse (D rational implies eα > 1) needs injectivity of f on IC_X/O^GR_X, which the authors deduce from a stronger injectivity statement attributed to [CDO26b], listed as 'in preparation.' Remark 5.7 sketches an alternative via Corollary 4.7, but the logical step from HRH(D) ≥ 0 to φ_{f,1} = 0 is not filled in. So without [CDO26b], only half the iff is fully proved. This is a verifiability gap, not evidence of incorrectness — the result is consistent with Saito's smooth-case theorem and all computed examples agree. But it is load-bearing: the rational singularities characterization is the paper's headline application. For researchers in D-modules and singularity theory working on singular ambient varieties. The algorithms make it practically useful beyond the theory. Deserves a serious referee. The [CDO26b] dependency should be flagged: either the paper should wait for that preprint to be available, or the authors should complete the alternative argument in Remark 5.7.","headline":"New microlocal b-function for singular ambient varieties, with one direction of the main theorem depending on unpublished work","tokens_in":40415,"tokens_out":573,"would_cite":true,"duration_ms":105306,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F10","14B05","14B15","14C30"],"pacs":[],"model":"glm-5.2","headline":"Microlocal b-functions detect rational singularities on singular spaces","keywords":["Bernstein-Sato polynomial","microlocal b-function","minimal exponent","rational singularities","intersection cohomology D-module","mixed Hodge modules","singular varieties","local cohomology"],"falsifier":"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.","tokens_in":39413,"feed_emoji":"🔍","tokens_out":1211,"duration_ms":157245,"temperature":0.7,"pith_summary":"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.","feed_headline":"Microlocal b-functions detect rational singularities on singular spaces","feed_subtitle":"A new invariant strictly refines Bernstein–Sato theory when the ambient variety is singular, with a sharp criterion for rational singularity","key_machinery":"intersection cohomology D-module IC_X","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Microlocal b-functions strictly refine Bernstein-Sato theory on singular spaces","Minimal exponents from microlocal b-functions detect rational singularities","A sharp criterion for rational singularities via microlocal Bernstein-Sato","Microlocal b-functions yield a sharp test for rational singularities","Microlocal Bernstein-Sato polynomials strictly refine b-functions"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Microlocal b-functions strictly refine Bernstein-Sato theory on singular spaces","Minimal exponents from microlocal b-functions detect rational singularities","A sharp criterion for rational singularities via microlocal Bernstein-Sato","Microlocal b-functions yield a sharp test for rational singularities","Microlocal Bernstein-Sato polynomials strictly refine b-functions"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1601,"prompt_tokens":513,"completion_tokens":1088,"prompt_tokens_details":null},"tokens_in":513,"tokens_out":1088,"duration_ms":44407,"temperature":1.0,"reasoning_tokens":910,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T07:47:20.306593+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"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.","supporting_citations":[],"review_version":1}