{"id":"5a4da236-ee9d-4822-8d1c-db692c917b35","arxiv_id":"2607.25230","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The local analytic Bertini theorem holds: multiplier ideal sheaves of psh functions on polydisc products restrict to fibers outside a pluripolar set.","lead":"This math paper proves a general version of the analytic Bertini theorem: for a plurisubharmonic function on a product of polydiscs, its multiplier ideal sheaf restricts correctly to all fibers outside a tiny (pluripolar) set. This settles a conjecture by Boucksom and opens a path for studying how singularities deform in families.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified — the proof is internally coherent; the main theorem follows conditional on the cited fine-potential-theory and Bergman-kernel results.","rationale":"The reader's ACCEPT verdict is warranted. My read found no internal inconsistency or missing step in the central argument. The single most load-bearing part is Theorem 3.2, whose proof uses fine pluripotential theory in a way that I cannot independently verify without the cited external sources. This is exactly the reader's weakest_assumption. I agree that the cited theorems are the most fragile dependency, but I do not see a concrete route by which they would fail in the given setup. The applications to Corollary 3.5 and to the proof of Theorem 1.1 are logically sound once Theorem 3.2 and Proposition 4.2 are accepted. The acknowledgment of AI-assisted proof generation raises a process concern, not a mathematical one, and does not change the technical evaluation. No verdict adjustment is needed.","tokens_in":11885,"tokens_out":46635,"duration_ms":447389,"concrete_test":"Verify the exact statements of [EKF25, Theorem 5.4.2(iv)] and [EKF25, Theorem 5.4.3(d)] against the construction in Theorem 3.2: confirm that (a) a decreasing limit of F-psh functions on an F-open set is F-psh or identically −∞ without an extra local-boundedness-below assumption, and (b) the −∞ set of a not-identically-−∞ F-psh function is pluripolar. Also check [BGY23, Theorem 1.4] permits the weight ψ to be −∞ on pluripolar sets and applies to the diagonal evaluation of the differential operator D_α used in Proposition 4.2. If any of these statements carry extra hypotheses violated here, Theorem 3.2 would need repair.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After a careful read, I cannot locate a genuine internal gap. The most delicate part is the pluripolarity upgrade in Theorem 3.2. The Step 1 assertion that U_η is continuous on a neighborhood of ∂B is justified because F_η is holomorphic on a neighborhood of the compact set \\(\\overline{2B}\\), and its zero set is closed; if it does not meet ∂B, it stays at positive distance from it. The Step 2 construction of χ as a decreasing limit of psh functions is a standard use of fine pluripotential theory, and Step 3's divergence estimate is correct. The reduction in Section 4.4 from stalkwise equality to the zero sets of jet maps is also exact. The only substantive risk is external: the proof leans on [EKF25, Thm 5.4.2(iv)], [EKF25, Thm 5.4.3(d)], and [BGY23, Thm 1.4]. If those theorems carry hidden hypotheses (for instance, a local-boundedness-below condition on the approximating sequence in the decreasing-limit theorem, or an admissibility condition on the weight in the Bergman kernel theorem), then Theorem 3.2 or Proposition 4.2 could fail. This is a verification risk, not a discovered flaw.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves the local analytic Bertini theorem (Theorem 1.1): for a plurisubharmonic function Φ on the product polydisc X = Δ^m_z × Δ^n_η, there is a pluripolar set P ⊂ Δ^n_η such that for every η outside P the multiplier ideal sheaf I_X(Φ) restricts to the multiplier ideal sheaf of the slice Φ|_{X_η}. This confirms Boucksom's conjecture in the local setting. The proof introduces a new pluripolar zero-set criterion (Theorem 3.2) based on fine pluripotential theory, a jet-bundle rank detector (Section 4), and derives the restriction theorem by reducing stalkwise equality to the vanishing of certain jet maps. Corollary 1.2 extends the result to arbitrary holomorphic maps between σ-compact complex manifolds. The paper is a direct continuation of the author's global Bertini theorem [Xia22].","tokens_in":12208,"tokens_out":20336,"duration_ms":179404,"significance":"If correct, Theorem 1.1 is a substantial result: it settles Boucksom's local restriction conjecture in full generality and, through Corollary 1.2, gives a pluripolar restriction theorem for multiplier ideals under arbitrary holomorphic maps. The proof introduces a genuinely new ingredient, the pluripolar upgrade via plurifine potential theory (Theorem 3.2 and Corollary 3.5), which is likely to be useful beyond this application. I checked the main architecture carefully: the reduction from stalkwise equality to jet ranks, the construction of the rank detector from weighted Bergman kernels, and the zero-set argument are coherent and internally consistent. The paper is honest about its reliance on several deep external results, especially [EKF25] and [BGY23].","major_comments":[],"minor_comments":[{"comment":"The proof invokes '[Xia, Proposition 1.2.6]' for the standard fact that a plurisubharmonic function which is ≤0 almost everywhere is ≤0 everywhere. This is a self-citation to unpublished lecture notes. Please replace it by a standard textbook reference or give the short argument (e.g., using the mean-value inequality), so the proof is more self-contained.","section":"§3, Lemma 3.1"},{"comment":"The application of [EKF25, Theorem 5.4.2(iv)] and [EKF25, Theorem 5.4.3(d)] is the load-bearing point of the pluripolar upgrade. Since these results are less widely known than the rest of the paper, it would improve clarity to state explicitly the hypotheses of the theorems being used and to verify them for the specific objects χ, U_k, and int_F K. I do not see a gap; this is a presentation request.","section":"§3, Theorem 3.2, Step 2"},{"comment":"The notation 'im2' in the definition of q_Θ, and similarly in the subsequent display for q_η(s), is unclear. If it is intended to denote i^{m^2}, please typeset it unambiguously; as printed it is easy to misread.","section":"§3, Theorem 3.2, Step 2"},{"comment":"Reference [BGY23] is cited as an arXiv preprint. If a journal-published version now exists, it would be helpful to update the citation. Also, the paper cites [EKFW11] and [EMW06] in §3; both are listed in the bibliography, but the reference [Xia] used in Lemma 3.1 is lecture notes and should be marked as such.","section":"References"}],"recommendation":"accept","confidential_remarks":"I concur with the reader's positive assessment. The proof is internally coherent and the central claims are clearly supported. The main residual risk is the correctness and precise applicability of the deep external fine-pluripotential-theory and Bergman-kernel theorems cited from [EKF25] and [BGY23]; I could not verify those independently, but the manuscript's reliance on them is explicit and standard. I see no grounds for rejection or for requiring further substantive revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper proves Boucksom's conjecture in the local setting: for a psh function on a product polydisc, the multiplier ideal restricts correctly off a pluripolar set of base points. That's a genuine advance — prior work had global projective fibrations (Xia's own) and null exceptional sets (Fujino–Matsumura, Meng–Zhou). The new machinery is the pluripolar zero-set criterion (Theorem 3.2), the pluripolar Fubini lemma, and the jet-bundle rank detector; these are real ingredients, not repackaged old results.\n\nThe proof is coherent and I couldn't find an internal gap. The strategy — represent the stalkwise equality as a statement about ranks of jet maps, use the log-psh of weighted Bergman kernels to get psh detector functions, then upgrade the Lebesgue-null set to pluripolar via fine potential theory — works. The stress-test pass confirms Step 2 of Theorem 3.2 is sound; the gluing via regularized maximum is legitimate. The countability arguments in Corollary 3.5 are careful. The only soft spot is the dependence on external theorems, [EKF25, Thm 5.4.2(iv), 5.4.3(d)] and [BGY23, Thm 1.4]. If any of those have unstated hypotheses, the pluripolar upgrade would fail. That's a verification risk, not a discovered flaw; a referee with fine potential theory expertise should check them. Minor: the proof of Lemma 3.1 cites [Xia, Prop 1.2.6], which is an unpublished lecture note, but that's a standard fact and not circular.\n\nThe AI acknowledgment is honest; it doesn't affect the math, but it does mean independent checking is even more worthwhile, not less. The paper is dense but well-organized.\n\nThis is a serious within-field contribution. I'd send it to peer review without hesitation, with a referee who knows fine potential theory and Bergman kernels. I'd cite it if I worked on multiplier ideals; it will be used in deformation theory of psh singularities.","headline":"The local analytic Bertini theorem is proven in full generality via a genuinely new pluripolarity criterion; the proof is intricate, honest, and rests on credible external results — worth engaging seriously.","tokens_in":12678,"tokens_out":2517,"would_cite":true,"duration_ms":24334,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32U05","32A36","14F18"],"pacs":[],"model":"deepseek-v4-flash","headline":"A local analytic Bertini theorem holds: for any psh function on a product of polydiscs, multiplier ideals restrict to the fiber correctly outside a pluripolar set of parameters.","keywords":["analytic Bertini theorem","multiplier ideal sheaves","plurisubharmonic functions","pluripolar sets","fine pluripotential theory","weighted Bergman kernels","jet bundles","Boucksom conjecture"],"falsifier":"Construct an explicit plurisubharmonic function Φ on a product polydisc where the set of parameters η for which I_X(Φ)·O_{X_η} ≠ I_{X_η}(Φ|_{X_η}) is not pluripolar. Since the paper proves this set is always pluripolar, finding any example with a non-pluripolar exceptional set would refute the theorem. Alternatively, check whether the cited fine-pluripotential theorems (e.g., [EKF25, Theorem 5.4.2]) apply verbatim to the decreasing limit of regularized psh functions constructed in Step 2 of Theorem 3.2; a missing hypothesis there would also invalidate the proof.","tokens_in":11782,"feed_emoji":"📐","tokens_out":2011,"duration_ms":20544,"temperature":0.7,"pith_summary":"The paper proves that the multiplier ideal sheaf of a plurisubharmonic function on a product polydisc, when restricted to a fiber, agrees with the multiplier ideal sheaf of the restricted function for every parameter outside a pluripolar set. This was previously known only for almost every fiber by Fubini, and the upgrade to pluripolar exceptional sets was an open conjecture of Boucksom. The result is local, so it extends to arbitrary holomorphic maps between complex manifolds, giving a clean statement about the behavior of psh singularities in families. A sympathetic reader would care because it shows that psh singularities deform in a highly controlled way, with the only exceptions forming a thin (pluripolar) set of parameter values.","feed_headline":"Multiplier ideals restrict correctly off a pluripolar set","feed_subtitle":"The local analytic Bertini theorem holds for all psh functions, extending the global case to arbitrary families.","key_machinery":"The argument combines three tools: (1) a jet-bundle rank detector that compares the fiberwise multiplier ideal to the restricted ambient multiplier ideal by looking at jets of functions; (2) the log-plurisubharmonicity of fiberwise weighted Bergman kernels (a theorem of Bao–Guan–Yuan extending Berndtsson) to show these rank detectors are psh; and (3) a new pluripolar zero-set criterion (Theorem 3.2) that upgrades a Lebesgue null set of forbidden parameters to a pluripolar set, using fine pluripotential theory. The jet construction converts the equality of ideal sheaves into a statement about vanishing loci of holomorphic maps into Hilbert spaces of jets, and the criterion then shows the bad","core_discovery":"The central claim is Theorem 1.1: Let X = Δ^m_z × Δ^n_η and Φ a plurisubharmonic function on X. For each η, let X_η be the fiber over η and Φ_η the restriction. Then there exists a pluripolar set P ⊆ Δ^n_η such that for all η ∉ P, the image of the ambient multiplier ideal sheaf I_X(Φ) in the fiber equals the multiplier ideal sheaf I_{X_η}(Φ_η). The inclusion one way follows from Ohsawa–Takegoshi; the innovation is proving the reverse inclusion holds outside a pluripolar set, not merely a measure-zero set. The paper also derives a corollary for any holomorphic map between complex manifolds, with fibers that are smooth and where the same multiplier ideal restriction formula holds outside a loc","pith_inferences":["One plausible extension: the same jet-Bergman machinery could show that the jumping locus of multiplier ideals along a family has not just pluripolarity but a more precise structure, e.g., being a countable union of analytic sets in many cases, though the paper only establishes pluripolarity.","The proof's reliance on the log-plurisubharmonicity of weighted Bergman kernels suggests that similar restriction formulas might hold for other canonical objects defined by L^2 methods, such as Nadel–Ohsawa sheaves or the multiplier ideals of singular Hermitian metrics on vector bundles.","The technique of upgrading conull sets to pluripolar sets via fine pluripotential theory may apply to other problems in analytic geometry where one only knows a property holds almost everywhere, effectively making the exceptional set 'small' in the pluripotential sense.","A testable consequence: if one takes a family of algebraic varieties degenerating over a polydisc and a psh weight coming from a section of a line bundle, the jumping locus of the multiplier ideal should be pluripolar in the base, a statement that could be checked by explicit computations in examples."],"forward_implications":["Boucksom's conjecture on the restriction of multiplier ideals is confirmed in full local generality, removing the compactness assumptions of earlier global results.","The result extends to arbitrary holomorphic maps between σ-compact complex manifolds, giving a pluripolar exceptional set for the fiberwise restriction formula.","The new pluripolar zero-set criterion (Theorem 3.2 and Corollary 3.5) provides a general mechanism to upgrade measure-zero exceptional sets to pluripolar ones in other problems involving holomorphic families of zero sets.","The deformation theory of psh singularities, which the author indicates as the next step, can now build on a stable local statement for how multiplier ideals vary in families."],"fun_headline_variants":["Multiplier ideals restrict correctly off pluripolar sets","Local Bertini theorem proved for all psh functions","Pluripolar sets resolve multiplier ideal restriction","Reverse inclusion holds off a pluripolar set","Bertini local: ideals restrict off pluripolar"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof depends on externally cited theorems in fine pluripotential theory (El Kadiri–Fuglede) and the log-plurisubharmonicity of fiberwise weighted Bergman kernels; if these theorems carry unstated hypotheses that fail in the specific construction used here, the upgrade to pluripolarity would break down and the main theorem would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Multiplier ideals restrict correctly off pluripolar sets","Local Bertini theorem proved for all psh functions","Pluripolar sets resolve multiplier ideal restriction","Reverse inclusion holds off a pluripolar set","Bertini local: ideals restrict off pluripolar"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00085,"raw_usage":{"total_tokens":3436,"prompt_tokens":549,"completion_tokens":2887,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":293,"completion_tokens_details":{"reasoning_tokens":2817}},"tokens_in":293,"tokens_out":2887,"duration_ms":18338,"temperature":1.0,"reasoning_tokens":2817,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T03:01:56.201229+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an explicit plurisubharmonic function Φ on a product polydisc where the set of parameters η for which I_X(Φ)·O_{X_η} ≠ I_{X_η}(Φ|_{X_η}) is not pluripolar. Since the paper proves this set is always pluripolar, finding any example with a non-pluripolar exceptional set would refute the theorem. Alternatively, check whether the cited fine-pluripotential theorems (e.g., [EKF25, Theorem 5.4.2]) apply verbatim to the decreasing limit of regularized psh functions constructed in Step 2 of Theorem 3.2; a missing hypothesis there would also invalidate the proof.","supporting_citations":[],"review_version":1}