{"id":"b44dda6f-b84d-48f3-bd53-3c8bed18ea36","arxiv_id":"2502.06537","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For lci subvarieties of a smooth variety, spectral Hirzebruch-Milnor classes vanish between bounds set by the minimal exponent, yielding homological criteria for higher singularities.","lead":"Defines new spectral Hirzebruch-Milnor classes for local complete intersection varieties and proves that their coefficients vanish exactly in the ranges forced by the minimal exponent. When the singular locus is projective, these classes detect higher Du Bois and higher rational singularities.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The converses of Theorems 4.21 and 4.24 hinge on [10, Lemma 2.6], an unpublished preprint result; if isomorphism (25) fails for non-isolated singular loci, the pullback identification and hence the threshold converses collapse.","rationale":"The reader's CONDITIONAL verdict identifies [10, Lemma 2.6] as the weakest assumption, and my reading agrees. The central claim is the sharp threshold vanishing with converse; the converse needs non-vanishing of the first non-zero spectral coefficient. The only path to that non-vanishing is showing the first non-zero graded de Rham piece is a pullback π*(F), then using projectivity of Sing(X) and the Todd class isomorphism to push it to a non-zero class. Equation (25) is exactly that pullback identification, and it is imported from an unreviewed preprint. Everything else in the paper—the definitions via deformation to the normal cone, the spectral Hirzebruch transformations, and the reduction to [7, Thm 4.3]—appears coherent and is consistent with the known hypersurface case. Thus the appropriate verdict remains CONDITIONAL, pending a full proof or published version of [10, Lemma 2.6]. I do not see grounds to reject the paper, nor to accept it as is.","tokens_in":32388,"tokens_out":11056,"duration_ms":88963,"concrete_test":"Independently re-derive isomorphism (25) directly from the V-filtration and the definition of Sp_X, without invoking [10, Lemma 2.6], under exactly the hypotheses used in Theorem 4.7 (no isolatedness assumption on Sing(X)). In particular, check both injectivity and surjectivity of (F^σ M_{r−1+λ})[z_1,...,z_r] → F^σ M_{λ+Z} when σ is the lowest non-zero Hodge index and the direct-sum equality F^σ M_{λ+Z} = ⊕_{ℓ≥0} F^σ M_{r−1+λ+ℓ} holds. If a counterexample exists with non-isolated singular locus, or if the proof requires an extra hypothesis not stated in §4.4, then Corollary 4.10 and the converse parts of Theorems 4.21 and 4.24 lose their main input.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive step is the microlocal gluing lemma [10, Lemma 2.6] imported in §4.4 to turn the natural map (24) into isomorphism (25): π*(F^σ M_{r−1+λ}) ≅ F^σ M_{λ+Z}. This is what makes the lowest non-zero graded de Rham piece of ϕ_{h,μ}(Q~Y[dim~Y−1]) a pullback sheaf π*(F) on Σ = π^{−1}(Sing X), as stated in Corollary 4.10. The converse directions of Theorems 4.21 and 4.24 both use this pullback form: from F ≠ 0 on projective Sing(X) they conclude td∗(π∗F) ≠ 0 and hence a non-zero spectral coefficient. If (25) fails—for example if F^σ M_{λ+Z} contains monodromic summands below r−1+λ at the same Hodge level, so the multiplication map is not an isomorphism—the first non-zero Gr^F DR need not be a pullback, and the non-vanishing conclusion does not follow. The lemma is quoted from an unpublished preprint and no proof is supplied here. The peer-reviewed input [7, Thm 4.3] is not the weak point; it is the unpublished gluing statement. A secondary unresolved step is the Todd-duality non-vanishing in Theorem 4.24 via [15, Ex. 18.3.19], but the main risk is the unproved isomorphism (25).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines (spectral) Hirzebruch-Milnor type homology characteristic classes for any local complete intersection subvariety X of a smooth complex algebraic variety Y, using the deformation to the normal cone, Verdier-Saito specialization, and the spectral Hirzebruch class transformation. The main results (Theorems 4.21 and 4.24, stated also as Theorems 1.1 and 1.4) establish vanishing of the spectral classes outside an interval determined by the minimal exponent ~α(X), with converses under the assumptions that Sing(X) is projective and lct(X)>r−1. As applications, the authors obtain class-level characterizations of k-Du Bois and k-rational singularities for lci varieties (Theorems 4.29 and 4.30) and new proofs of formulas for the Hodge spectrum of isolated complete intersection singularities (Corollaries 4.23 and 4.25).","tokens_in":32647,"tokens_out":5279,"duration_ms":54998,"significance":"This is a substantial and well-motivated contribution. It extends the hypersurface theory of Hirzebruch-Milnor classes and their spectral refinements to arbitrary lci varieties, giving global homological refinements of the Hodge spectrum that detect higher Du Bois and rational singularities. The definitions are natural, the machinery (mixed Hodge modules, V-filtrations, specialization) is used carefully, and the exposition is clear. If the main results are accepted, they provide a powerful new tool for studying singularities via characteristic classes, with the minimal exponent threshold behavior encoded directly in the spectral classes. The paper also gives clean new proofs of the isolated spectrum formulas, which are of independent interest.","major_comments":[{"comment":"The identification of the lowest non-zero graded de Rham piece with a pullback sheaf π^*(F) relies crucially on [10, Lemma 2.6], quoted from an unpublished preprint by one of the authors. This lemma is used in Theorem 4.7 and Corollary 4.10, and the converse directions of Theorems 4.21 and 4.24 both depend on the pullback form of this lowest piece. Since the lemma is not proven here, the main converses are not established independently. The authors should either supply a full proof of (25) within the paper or explicitly state it as an assumption and ensure the preprint is publicly available and verifiable for referees and readers.","section":"§4.4, Eq. (25)"},{"comment":"In the converse part of the proof, the authors assert that from td_*([π^*(F)]) ≠ 0 one obtains td_*([D_coh(π^*(F))]) ≠ 0 by citing [15, Example 18.3.19]. This is not a routine consequence of the cited example; duality under the Todd class transformation is subtle for arbitrary coherent sheaves on singular varieties. The authors should spell out the precise statement they are using and give a justification, or provide a direct argument showing that the Todd class of the Grothendieck dual is non-zero in this specific geometric setting. Without this, the converse for the lower-bound vanishing is incomplete.","section":"§4.7, proof of Theorem 4.24"}],"minor_comments":[{"comment":"The phrase \"monodromic module\" should also mention \"mixed Hodge module\" for consistency with the rest of the section.","section":"§4.4, line before Eq. (24)"},{"comment":"In the long display after the sentence \"Finally, this gives the vanishings\", the case λ=1 lists p≤q but the preceding condition for λ=1 was p≤q; this is consistent, so no change needed, but the typography could be clarified.","section":"§4.6, proof of Theorem 4.21"},{"comment":"Reference [10] is an arXiv preprint; the authors should indicate its current status (submitted, under review) or make the relevant lemma available in a published or easily accessible form.","section":"References"},{"comment":"The abstract contains a typographical space error in \"local complete intersection s\"; this should be corrected.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper is well-written and the main results appear plausible, but the heavy dependence on an unreviewed co-author preprint for the central gluing lemma is a real obstacle for a journal publication. I would advise the editor to require the authors to either include a complete proof of [10, Lemma 2.6] or to provide a detailed and verifiable statement, possibly by expanding the paper with a proof of this key ingredient. The secondary point about the Todd class of the dual under [15, Ex. 18.3.19] should also be addressed; it is currently presented too casually for a loaded step."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real advance and deserves a serious referee. The authors define spectral Hirzebruch–Milnor classes for arbitrary lci subvarieties and prove that their vanishing behavior is controlled by the minimal exponent, with converses when the singular locus is projective. That gives class-level characterizations of higher Du Bois and higher rational singularities. The hypersurface cases in [22,23,25] and the complete-intersection classes in [21] do not imply these lci statements. That is the main thing to know.\n\nWhat the paper does well: the machinery is handled carefully. The deformation to the normal cone, Verdier–Saito specialization, and V-filtration formalism are written out in detail. Theorem 4.7, which expresses the minimal exponent as the lowest non-vanishing graded de Rham piece of the vanishing cycle module, is a genuine contribution. The gluing argument in Lemma 4.8 for the sheaves Gr^F_p Gr^λ_V(B_{X,Y}) is a useful technical step. The applications to k-Du Bois and k-rational singularities are mostly formal once the spectral vanishings are in place, but they fill a natural gap.\n\nThe soft spot is real, and it is the one the stress test flags. The isomorphism (25), which identifies the lowest non-vanishing graded de Rham piece as a pullback π*(F), is imported from [10, Lemma 2.6], an unpublished preprint by one of the authors. The converses of Theorems 4.21 and 4.24 rely on having that pullback form and then on the non-vanishing of td*(π*F) under projectivity. If the lemma fails, or needs extra hypotheses for non-isolated singular loci, the non-vanishing conclusions do not follow. This is not a manufactured worry; the paper itself points to [10] for exactly that step. The peer-reviewed input [7, Thm 4.3] is not the weak point. There is a smaller gap in Theorem 4.24 where non-vanishing after Grothendieck duality is justified by [15, Ex. 18.3.19]; that is probably fine, but it is quick.\n\nOne thing I would add to the reader's take: the dependence on [10] is concentrated, not pervasive. The main vanishing directions, without converses, are proved from Theorem 4.7 and hence rest on the peer-reviewed [7]. If the unpublished lemma breaks, much of the paper survives; the converses are the fragile part.\n\nWho this is for: people working on characteristic classes of singular varieties, Hodge spectrum, and higher Du Bois/rational singularities. I would send it to a knowledgeable referee rather than desk reject. The right request is: make the proof of [10, Lemma 2.6] available or point to a published version, and expand the Todd-duality step. With those, the conditional becomes a solid accept.","headline":"New lci Hirzebruch–Milnor classes with a real theorem: minimal exponent controls spectral class vanishing, but the key gluing lemma is quoted from an unreviewed preprint.","tokens_in":33255,"tokens_out":2019,"would_cite":true,"duration_ms":18422,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J17","32S35","32S50","14C17"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that spectral Hirzebruch-Milnor classes of local complete intersections vanish exactly outside a window set by the minimal exponent, and uses that to detect higher Du Bois and rational singularities.","keywords":["minimal exponent","Hirzebruch-Milnor classes","local complete intersections","higher Du Bois singularities","higher rational singularities","Hodge spectrum","V-filtration","characteristic classes"],"falsifier":"Take a codimension $r$ local complete intersection $X\\subset Y$ with non-isolated, projective singular locus and $\\operatorname{lct}(X)>r-1$, such as a cone over a smooth projective complete intersection, and compute the spectral Hirzebruch-Milnor class via the deformation-to-normal-cone construction; any non-zero coefficient outside the window $[\\tilde{\\alpha}(X)-r+1,\\ \\dim Y+r-\\tilde{\\alpha}(X)]$ would falsify Theorems 4.21 and 4.24. Alternatively, directly test the gluing lemma by computing the lowest Hodge piece of $\\phi_h$ for two different regular sequences defining the same ideal and comparing the resulting pullback sheaves; a mismatch would break the converse arguments.","tokens_in":32138,"feed_emoji":"🕳️","tokens_out":9040,"duration_ms":74529,"temperature":0.7,"pith_summary":"The paper introduces Hirzebruch-Milnor type characteristic classes for local complete intersection subvarieties of smooth complex algebraic varieties, using the deformation to the normal cone and the associated specialization functor. Its central claim is that the coefficients of these classes, indexed by a spectral parameter $t$, vanish outside an explicit range controlled by the minimal exponent $\tilde{\\alpha}(X)$: above $\\dim Y+r-\\tilde{\\alpha}(X)$ and below $\\tilde{\\alpha}(X)-r+1$, with converses when the singular locus is projective and the log canonical threshold satisfies $\\operatorname{lct}(X)>r-1$. Because the minimal exponent encodes when a local complete intersection has $k$-Du Bois or $k$-rational singularities, these vanishings turn the classes into homological tests for those higher singularity notions. For isolated singularities, the results recover and refine known formulas for the Hodge spectrum.","feed_headline":"Spectral classes spot singularity depth in complete intersections","feed_subtitle":"Their coefficients vanish exactly outside a window set by the minimal exponent, revealing higher Du Bois and rational singularities.","key_machinery":"The central object is the vanishing-cycle mixed Hodge module $\\phi_h Q^H_{\\tilde Y}$ attached to the deformation to the normal cone $h:\\tilde Y\\to \\mathbb{C}$, whose special fiber is the normal bundle $C_XY$ of $X$ in $Y$. The spectral Hirzebruch-Milnor class is obtained by applying the spectral Hirzebruch class transformation (the Todd class of a spectral motivic Chern class) to this module and pulling back along the zero section of the normal bundle. A gluing lemma, quoted from an unpublished preprint, identifies the lowest non-vanishing graded de Rham piece of the vanishing-cycle module with a pullback sheaf $\\pi^*(F)$ on the normal bundle over the singular locus; this identification is what converts non-vanishing of a coherent sheaf on a projective singular locus into non-vanishing of the corresponding characteristic class coefficient.","core_discovery":"The central discovery is that the spectral Hirzebruch-Milnor class $M^{sp}_{t*}(X\\subset Y)$ of a codimension $r$ local complete intersection $X$ in a smooth variety $Y$ is a sharp homological shadow of the minimal exponent $\\tilde{\\alpha}(X)$. Under the hypothesis $\\operatorname{lct}(X)>r-1$, the paper proves that the $t^{\\alpha}$-coefficient vanishes for all $\\alpha>\\dim Y+r-\\tilde{\\alpha}(X)$ and for all $\\alpha<\\tilde{\\alpha}(X)-r+1$; if $\\operatorname{Sing}(X)$ is projective, the converses hold, so the vanishing range characterizes the minimal-exponent bound. The proof first establishes a new formula for $\\tilde{\\alpha}(X)$ as $r-1+\\min\\{p+\\lambda\\}$, where the minimum is taken over the lowest non-vanishing graded de Rham piece of the vanishing-cycle mixed Hodge module of the deformation to the normal cone. As applications, the paper shows that the $y$-coefficients of the (unipotent and non-unipotent) Hirzebruch-Milnor classes vanish in specified ranges exactly when $X$ has $k$-Du Bois or $k$-rational singularities, with converses under the same projectivity and log canonical threshold assumptions.","pith_inferences":["The proof's reliance on Hodge-theoretic duality suggests that the spectral classes satisfy a $t\\leftrightarrow t^{-1}$ symmetry with dimension twist, extending the duality formula known for Hodge spectra of isolated singularities; making this symmetry explicit could give a direct bridge between the two vanishing theorems.","Because $\\operatorname{lct}(X)>r-1$ is automatic for Du Bois lci singularities, the converses apply to the entire Du Bois regime, so the class-level tests are complete precisely where the minimal exponent is governed by Hodge-theoretic vanishing.","The new minimal-exponent formula in Theorem 4.7 does not itself involve characteristic classes and could be used independently to compute $\\tilde{\\alpha}(X)$ from local $V$-filtration data in examples with non-isolated singular locus.","The gluing lemma that identifies the lowest Hodge piece with a pullback sheaf may be the natural tool to extend the results to analytic singularities or to other homology theories admitting a Todd-class Riemann-Roch, although the paper notes that compactness alone does not replace projectivity."],"forward_implications":["For isolated singularities, the Hodge spectrum's smallest non-zero exponent is $\\tilde{\\alpha}(X)-r+1$ and its largest is $\\dim Y-\\tilde{\\alpha}(X)$, refining earlier spectrum formulas.","A local complete intersection with projective singular locus and $\\operatorname{lct}(X)>r-1$ is $k$-Du Bois if and only if the non-unipotent coefficients vanish for all $p\\geq \\dim Y-k$ and the unipotent coefficients vanish for all $p\\geq \\dim Y+1-k$.","Under the same hypotheses, $X$ is $k$-rational if and only if $[M_{y*}(X\\subset Y)]_p=0$ for all $p\\geq \\dim Y-k$, with the additional unipotent vanishing $[M^{\\{1\\}}_{y*}(X\\subset Y)]_{k+1}=0$.","Smooth local complete intersections have identically zero Hirzebruch-Milnor classes, while singular ones have a non-zero coefficient of the fundamental class of the singular locus, giving a simple singularity detector.","The lci results extend the hypersurface spectral vanishing theorems from codimension one to arbitrary codimension, with the same sharp threshold behavior in terms of the minimal exponent."],"supporting_citations":[{"why":"Supplies the V-filtration characterization of the minimal exponent for local complete intersections, which is the starting point for the new minimal-exponent formula.","marker":"[7]"},{"why":"Provides the microlocalization and gluing lemma that identifies the lowest non-vanishing graded de Rham piece of the vanishing-cycle module with a pullback sheaf, driving Theorem 4.7 and the converses.","marker":"[10]"},{"why":"Introduced the spectral Hirzebruch class transformation and spectral Hirzebruch-Milnor classes for hypersurfaces, the construction generalized here to lci.","marker":"[22]"},{"why":"Proved hypersurface vanishing theorems for higher Du Bois and rational singularities in the normalized setting that this paper extends to local complete intersections.","marker":"[23]"},{"why":"Proved the un-normalized hypersurface spectral vanishing theorems that motivate and are recovered by the lci results.","marker":"[25]"},{"why":"Defined the Hodge spectrum for arbitrary subvarieties via the deformation to the normal cone, which the paper globalizes to characteristic classes and uses for the isolated-singularity corollaries.","marker":"[9]"},{"why":"Introduced Verdier specialization, the functor used to build the deformation-to-normal-cone setup and the vanishing-cycle module.","marker":"[39]"},{"why":"Supplies the mixed Hodge module framework and the specialization functor underlying the definitions of the classes.","marker":"[30]"},{"why":"Defines the Hirzebruch class transformation used to turn mixed Hodge modules into characteristic classes.","marker":"[2]"}],"fun_headline_variants":["Spectral classes pin singularity depth via minimal exponent","Minimal exponent controls vanishing of spectral Milnor classes","Higher singularities read from spectral class vanishing ranges","Complete intersections: spectral classes expose singularity type"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central equivalence rests on a gluing lemma borrowed from an unreviewed preprint that identifies the first non-vanishing Hodge piece of the vanishing-cycle module with a coherent sheaf pulled back from the singular locus; if that lemma fails for non-isolated singularities, the vanishing theorems and their converses collapse.","fun_headline_variants_meta":{"raw":{"variants":["Spectral classes pin singularity depth via minimal exponent","Minimal exponent controls vanishing of spectral Milnor classes","Higher singularities read from spectral class vanishing ranges","Complete intersections: spectral classes expose singularity type"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000363,"raw_usage":{"total_tokens":1933,"prompt_tokens":894,"completion_tokens":1039,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":980}},"tokens_in":510,"tokens_out":1039,"duration_ms":9173,"temperature":1.0,"reasoning_tokens":980,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T15:07:59.891483+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a codimension $r$ local complete intersection $X\\subset Y$ with non-isolated, projective singular locus and $\\operatorname{lct}(X)>r-1$, such as a cone over a smooth projective complete intersection, and compute the spectral Hirzebruch-Milnor class via the deformation-to-normal-cone construction; any non-zero coefficient outside the window $[\\tilde{\\alpha}(X)-r+1,\\ \\dim Y+r-\\tilde{\\alpha}(X)]$ would falsify Theorems 4.21 and 4.24. Alternatively, directly test the gluing lemma by computing the lowest Hodge piece of $\\phi_h$ for two different regular sequences defining the same ideal and comparing the resulting pullback sheaves; a mismatch would break the converse arguments.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the V-filtration characterization of the minimal exponent for local complete intersections, which is the starting point for the new minimal-exponent formula."},{"cited_title":"Some applications of microlocalization for local complete intersection subvarieties","cited_arxiv_id":"2310.15277","evidence_quote":"Provides the microlocalization and gluing lemma that identifies the lowest non-vanishing graded de Rham piece of the vanishing-cycle module with a pullback sheaf, driving Theorem 4.7 and the converses."},{"cited_title":"Maxim, M","cited_arxiv_id":null,"evidence_quote":"Introduced the spectral Hirzebruch class transformation and spectral Hirzebruch-Milnor classes for hypersurfaces, the construction generalized here to lci."},{"cited_title":"Maxim, M","cited_arxiv_id":null,"evidence_quote":"Proved hypersurface vanishing theorems for higher Du Bois and rational singularities in the normalized setting that this paper extends to local complete intersections."},{"cited_title":"Dimca, P","cited_arxiv_id":null,"evidence_quote":"Defined the Hodge spectrum for arbitrary subvarieties via the deformation to the normal cone, which the paper globalizes to characteristic classes and uses for the isolated-singularity corollaries."},{"cited_title":"V erdier, Sp´ecialisation de faisceaux et monodromie mod ´er´ee, Analyse et Topologie sur les Espaces singuliers (II-III), Ast´ erisque 101–102 (1983), 332–364","cited_arxiv_id":null,"evidence_quote":"Introduced Verdier specialization, the functor used to build the deformation-to-normal-cone setup and the vanishing-cycle module."},{"cited_title":"Saito, Mixed Hodge Modules , Publ","cited_arxiv_id":null,"evidence_quote":"Supplies the mixed Hodge module framework and the specialization functor underlying the definitions of the classes."},{"cited_title":"Brasselet, J","cited_arxiv_id":null,"evidence_quote":"Defines the Hirzebruch class transformation used to turn mixed Hodge modules into characteristic classes."}],"review_version":1}