{"id":"b9f854a2-1f41-480b-9562-bd8f22982875","arxiv_id":"2506.13220","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A generalized Lefschetz (1,1)-theorem holds for singular projective varieties after replacing the operational Chow group by an inverse limit over resolutions, the extended operational Chow group.","lead":"The paper proves a version of the Lefschetz (1,1)-theorem for singular projective varieties: every degree (1,1) Hodge class can be realized by an algebraic cycle, provided one uses an enlarged 'extended operational Chow group'. This gives a natural cycle-class statement for isolated singularities and for normal surfaces with rational singularities.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection to the central claim; Theorem 3.4(4) survives scrutiny, though Corollary 2.5's cohomological exactness is more terse than its role warrants.","rationale":"The central theorem is a conditional statement under dim X_sing=0, and every step of its proof checks out once the omitted weight argument in Corollary 2.5 is supplied. The reader's conditional verdict is reasonable because of the two peripheral issues, but those do not constitute an objection to the central claim. I therefore leave the verdict unchanged.","tokens_in":11180,"tokens_out":44977,"duration_ms":505355,"concrete_test":"As a check on the least explicit step, take X the projective cone over a smooth elliptic curve (an isolated non-rational surface singularity) with resolution tilde X a P^1-bundle and exceptional divisor E the elliptic curve. Compute the long exact sequence H^1(E)->H^2(X)->H^2(tilde X)->H^2(E) and verify: (i) the image of H^1(E) lies in W_1H^2(X); (ii) the induced map Gr^W_2H^2(X)->H^2(tilde X) is an isomorphism; (iii) H^2_Hdg(E)=0, so the row 0->H^2_Hdg(X)->H^2_Hdg(tilde X)->H^2_Hdg(E) is exact. Passing this would settle the exactness on which Theorem 3.4(4) rests.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the proof of Theorem 3.4(4), I do not find a load-bearing flaw in the central argument. The proof that the extended cycle class map is surjective for isolated singularities depends on the exact diagram (2.4): injectivity of A^1(X) into A^1(tilde X) and of H^2_Hdg(X) into H^2_Hdg(tilde X), plus the identification of A^1_ext(phi) with the inverse image of H^2_Hdg(X) under cl^1. For p=1 with dim X_sing=0, the omitted verification in Corollary 2.5 is supplied by the standard weight argument: the boundary map H^1(E)->H^2(X) from the resolution long exact sequence has image in W_1H^2(X), so Gr^W_2 is unaffected; hence H^2_Hdg(X) injects and the kernel of i^* equals the image of phi^*. The inverse limit step is also sound because Res(X) is a filtered preorder (any morphism over X is the identity on the dense smooth locus), so all transition maps A^1_ext(phi)->A^1_ext(phi') being isomorphisms implies the projections from the limit are isomorphisms. The peripheral defects noted by the reader (Lemma 3.1 missing the dim X_sing<p hypothesis; Example 4.3's doubtful rational-singularity assertion) do not affect the central theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces, for an irreducible projective variety X and an integer p>0, an extended operational Chow group A^p_ext(X), defined as an inverse limit over all resolutions of the subgroups A^p_ext(phi) of A^p(tilde X_phi) consisting of cycle classes whose restriction to the exceptional locus is cohomologically trivial. It proves that, under the hypothesis dim(X_sing)<p, the ordinary operational Chow group A^p(X) embeds into A^p_ext(X) and the Bloch-Gillet-Soule cycle class map extends to A^p_ext(X). The main theorem (Theorem 3.4(4)) states that when p=1 and X has at worst isolated singularities, the extended cycle class map cl^1: A^1_ext(X) -> H^2_Hdg(X) = Gr^W_2 H^2(X,Q) ∩ H^{1,1} is surjective, giving a Lefschetz (1,1)-theorem for isolated singularities. The paper also proves that for normal projective surfaces with rational singularities the ordinary BGS map is surjective and equals the extended one, gives an example where they differ, and constructs a further enlargement A^p_wide(X) whose cycle class map is surjective in all codimensions conditional on the Hodge conjecture for smooth projective varieties.","tokens_in":11443,"tokens_out":25389,"duration_ms":314091,"significance":"If the main theorem is correct, it gives a natural and original formulation of the Lefschetz (1,1)-theorem for singular varieties: for isolated singularities, every rational (1,1) class in the weight-graded cohomology becomes algebraic in the extended operational Chow group, despite the fact that the ordinary BGS map need not be surjective. The construction is well motivated, the use of the classical Lefschetz theorem on smooth resolutions is a legitimate external input rather than a circular dependence, and the paper contains explicit examples illustrating the difference between A^1(X) and A^1_ext(X). However, several technical points need correction before the results can be regarded as fully established: the hypotheses of Lemma 3.1 and Definition 3.2 are not stated precisely, the proof of the key cohomological exactness in Corollary 2.5 is too terse for its central role, and the rational-singularity assertion in Example 4.3 appears to be wrong.","major_comments":[{"comment":"Lemma 3.1 is stated for an arbitrary projective variety X with no hypothesis on dim(X_sing), but its proof invokes Corollary 2.5, which is proved only under the hypothesis dim(X_sing)<p. Since Definition 3.2 uses Lemma 3.1 to form the inverse system defining A^p_ext(X), the construction is not justified as written. Either restrict Lemma 3.1 and Definition 3.2 to dim(X_sing)<p (the hypothesis later used in Theorem 3.4), or prove the lemma directly from functoriality of cycle classes: for a morphism psi: tilde X_{phi'} -> tilde X_phi of resolutions, one has i'^* psi^* = (psi|_{E'})^* i^*, so psi^* automatically preserves the condition i^* cl^p(alpha)=0. This is a load-bearing point because the whole inverse limit construction depends on Lemma 3.1.","section":"3.1, Lemma 3.1 and Definition 3.2"},{"comment":"The bottom exact sequence 0 -> H^{2p}_Hdg(X) -> H^{2p}_Hdg(tilde X) -> H^{2p}_Hdg(E) is asserted to follow from H^{2p}(X_sing,Q)=0 together with Corollary 2.4 and functoriality, but that vanishing alone does not give the required exactness on Hodge classes. One needs to pass to Gr^W_{2p} in the long exact sequence associated to the resolution and use the weight bound on the boundary map H^{2p-1}(E) -> H^{2p}(X), whose image lies in W_{2p-1}H^{2p}(X); this yields the injectivity of H^{2p}_Hdg(X) -> H^{2p}_Hdg(tilde X) and exactness at the middle term. This argument should be written out, since Corollary 2.5 is used in the proofs of Lemma 3.1, Theorem 3.4(4), and Theorem 4.1.","section":"2.2, Corollary 2.5"},{"comment":"The claim that after contracting C' the resulting surface X has isolated rational singularities is not supported and appears to be false. The exceptional curve C' is a singular cubic, hence has arithmetic genus p_a(C')=1 and h^1(O_{C'})=1. For a normal surface singularity obtained by contracting such a curve, the fundamental cycle has h^1(O) = 1, so R^1 f_* O_tilde is nonzero and the singularity is not rational. Consequently the application of Corollary 4.2 in this example is unjustified. Since this example is presented as an application of the main theorem to Totaro's examples, it should be corrected or replaced by an example whose singularity type is correctly verified.","section":"4.1, Example 4.3"}],"minor_comments":[{"comment":"The proof says 'Theorem 3.4 implies A^1(X)=A^1_ext(X)', but the equality is the content of the second assertion of Theorem 4.1, not of Theorem 3.4; the citation should be corrected.","section":"4.1, Corollary 4.2 proof"},{"comment":"The heading 'Lefshetz (1,1) and other properties' contains a typo; it should read 'Lefschetz'.","section":"3.2, title"},{"comment":"The containment A^p(X) subset A^p_ext(X) and the extension of the cycle class map are asserted only under dim(X_sing)<p, but the abstract and Theorem 3.4(1) do not state this hypothesis explicitly enough in the abstract; the abstract should mention the isolated-singularity or dim(X_sing)<p condition for the containment claim.","section":"Abstract and Theorem 3.4(1)"},{"comment":"In the sentence 'Since p-q is non-torsion ... This implies cl_E(L|_E)=0', the vanishing is not a consequence of non-torsion in Pic^0(C'); rather, it holds because the rational cycle class of a degree-zero line bundle on a curve is zero. The non-torsion statement is used to show L is not pulled back from X, and this distinction should be made explicit.","section":"4.2, Theorem 4.4 proof"},{"comment":"The condition on morphisms in Res_wide(X) refers to 'homologically trivial cycles' without specifying the coefficient group; since the paper works with rational coefficients throughout, the definition should say CH^p_hom(-) tensor Q or an equivalent formulation.","section":"5, Definition 5.1"}],"recommendation":"major_revision","confidential_remarks":"The central theorem appears sound and the paper is within scope for a journal in algebraic geometry. The stress-test concern about Corollary 2.5 is a real presentational gap rather than a fatal flaw; the reader's condition is fair. I see no circularity and no misattribution of prior work. The Example 4.3 error is peripheral to the main theorem but should be fixed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — this paper has a real new idea: an extended operational Chow group A^p_ext(X) defined as an inverse limit over all resolutions of cycles whose class vanishes on the exceptional divisor. The payoff is a clean Lefschetz (1,1) theorem for projective varieties with isolated singularities: the extended cycle class map from A^1_ext(X) to Gr^W_2 H^2 ∩ H^{1,1} is surjective. That's a genuine advance over Arapura's result, which only captures the F^1 part of cohomology. The rational surface case, Theorem 4.1/Corollary 4.2, is also a nice application, and the functoriality argument for p=1 is mostly convincing. I checked the central surjectivity proof; the exactness of the resolution sequence under the isolated-singularity hypothesis supplies what's needed, and the inverse limit step is justified because resolutions form a filtered preorder. So the main theorem appears sound.\n\nThere are two soft spots, neither touching the main result. First, Lemma 3.1 is stated without the hypothesis dim(X_sing) < p, but its proof invokes Corollary 2.5, which carries exactly that hypothesis. As written, the lemma is false in general, and the definition of A^p_ext only makes sense as a group in that range. The fix is simple: state the hypothesis in the lemma and in Definition 3.2. Second, Example 4.3 claims that contracting a singular cubic gives a rational surface singularity because the normalization has genus 0. That inference is wrong — an exceptional curve of arithmetic genus 1 normally gives an elliptic singularity. The example's conclusion may still be true, but the stated justification is not.\n\nWho should read this: Hodge theory and algebraic cycles people, especially those thinking about Lefschetz-type statements for singular varieties. The paper is a legitimate contribution that deserves a serious referee. I'd send it to peer review with a request to fix the two issues above; neither should sink the paper if corrected.","headline":"A genuine Lefschetz (1,1) theorem for isolated singularities via a new extended operational Chow group; main proof is sound, but two peripheral errors need correction.","tokens_in":11977,"tokens_out":6596,"would_cite":true,"duration_ms":77794,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C15","14C30","32S35","32G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a Lefschetz (1,1)-theorem for singular projective varieties: for an irreducible projective variety with at worst isolated singularities, every Hodge (1,1)-class is the cycle class of an element of the extended operational…","keywords":["Lefschetz (1,1)-theorem","extended operational Chow group","Bloch-Gillet-Soule cycle class map","Hodge classes","isolated singularities","mixed Hodge structure","resolution of singularities","rational surface singularities"],"falsifier":"A concrete test would be to take a projective variety $X$ with two isolated singular points whose resolutions have non-isomorphic $\\mathrm{Pic}^0$ on the exceptional components and compute the inverse limit $A^1_{\\mathrm{ext}}(X)$ explicitly: the theorem asserts this limit is isomorphic to the extended group of any single resolution and that the induced map to $H^2_{\\mathrm{Hdg}}(X)$ is surjective. Exhibiting a class in $H^2_{\\mathrm{Hdg}}(X)$ not in the image, or showing the inverse limit is strictly smaller than a single-resolution extended group, would refute the central claim.","tokens_in":10940,"feed_emoji":"📐","tokens_out":15876,"duration_ms":159435,"temperature":0.7,"pith_summary":"This paper establishes Lefschetz (1,1)-theorems for singular projective varieties. It proves that on a normal projective surface with at worst rational singularities, the Bloch-Gillet-Soule cycle class map from the operational Chow group $A^1(X)$ to the space of Hodge (1,1)-classes is surjective. For varieties where that map fails, it introduces the extended operational Chow group $A^1_{\\mathrm{ext}}(X)$, an inverse limit over all resolutions of classes whose cycle class vanishes on the exceptional divisor, and shows this larger group contains $A^1(X)$ and still carries a natural cycle class map. The main theorem is that when $X$ has at worst isolated singularities, the extended map $\\mathrm{cl}^1: A^1_{\\mathrm{ext}}(X) \\to H^2_{\\mathrm{Hdg}}(X)$ is surjective, so every Hodge (1,1)-class of such a variety is algebraic in the extended sense.","feed_headline":"For isolated singularities, every Hodge (1,1)-class is algebraic","feed_subtitle":"Extended operational Chow group recovers the classical Lefschetz theorem for isolated singularities.","key_machinery":"The central object is the extended operational Chow group $A^p_{\\mathrm{ext}}(X)$ (Definition 3.2), defined as the inverse limit over the cofiltered category of resolutions $\\phi: \\widetilde X_\\phi \\to X$ of the subgroups $A^p_{\\mathrm{ext}}(\\phi) = \\{\\alpha \\in A^p(\\widetilde X_\\phi) : i^*_\\phi(\\mathrm{cl}^p(\\alpha)) = 0\\}$, where $i_\\phi: E_\\phi \\to \\widetilde X_\\phi$ is the inclusion of the exceptional locus. The argument is carried by Corollary 2.5's exact sequences, valid when $\\dim(X_{\\mathrm{sing}}) < p$: $0 \\to A^p(X) \\to A^p(\\widetilde X) \\to A^p(E)$ and $0 \\to H^{2p}_{\\mathrm{Hdg}}(X) \\to H^{2p}_{\\mathrm{Hdg}}(\\widetilde X) \\to H^{2p}_{\\mathrm{Hdg}}(E)$. These sequences let the cohomology of $X$ be seen through a single resolution and make $A^p(X)$ sit inside each $A^p_{\\mathrm{ext}}(\\phi)$. For $p=1$, the additional fact that $\\mathrm{Pic}^0$ is a birational invariant makes the inverse limit collapse to any one resolution, so surjectivity of the extended map follows from the classical Lefschetz (1,1)-theorem on the smooth resolution.","core_discovery":"The paper's central discovery is Theorem 3.4(4): if $X$ is an irreducible projective variety over $\\mathbb{C}$ with at worst isolated singularities, then the extended cycle class map $\\mathrm{cl}^1: A^1_{\\mathrm{ext}}(X) \\to H^2_{\\mathrm{Hdg}}(X) = \\mathrm{Gr}^W_2 H^2(X,\\mathbb{Q}) \\cap H^{1,1}$ is surjective. Here $A^1_{\\mathrm{ext}}(X)$ is the inverse limit over all resolutions $\\phi: \\widetilde X_\\phi \\to X$ of the subgroup $A^1_{\\mathrm{ext}}(\\phi)$ of $A^1(\\widetilde X_\\phi)$ consisting of classes whose cycle class restricts to zero on the exceptional divisor $E_\\phi$. The same theorem also shows that $A^p(X)$ embeds into $A^p_{\\mathrm{ext}}(X)$, that the BGS map extends to the extended group, and that for $p=1$ the extended group is functorial under morphisms preserving smooth loci. Along the way, the paper proves that the usual BGS map is already surjective for normal projective surfaces with rational singularities, and gives an explicit contracted-cubic example where the usual map is not surjective but the extended map is.","pith_inferences":["The extended group $A^1_{\\mathrm{ext}}(X)$ could serve as a singular analogue of the Néron-Severi group: its rational rank should equal the dimension of $H^2_{\\mathrm{Hdg}}(X)$ for isolated singularities, and the quotient $A^1_{\\mathrm{ext}}(X)/A^1(X)$ appears to measure exactly the failure of the ordinary BGS map.","The same inverse-limit-over-resolutions recipe might apply to other cycle theories with cycle class maps to graded pieces of mixed Hodge structures; conditional surjectivity in higher codimension would then follow from the Hodge conjecture for smooth resolutions, along the lines the paper sketches in its final section.","The contraction example in Section 4.2 suggests that the difference between $A^1(X)$ and $A^1_{\\mathrm{ext}}(X)$ is controlled by the abelian part $\\mathrm{Pic}^0$ of the exceptional divisor; one could test this by computing the quotient for varieties whose exceptional divisor has positive-dimensional $\\mathrm{Pic}^0$.","Because Theorem 3.4(3) gives pullbacks for morphisms preserving smooth loci, one could try to organize the extended groups into a presheaf or stack on the category of isolated-singularity varieties, yielding a singular version of the Picard functor; the paper does not do this."],"forward_implications":["On normal projective surfaces with at worst rational singularities, the ordinary BGS cycle class map $\\mathrm{cl}^1: A^1(X) \\to H^2_{\\mathrm{Hdg}}(X)$ is surjective, so no extended group is needed there.","On any irreducible projective variety with at worst isolated singularities, every Hodge (1,1)-class is the extended cycle class of an element of $A^1_{\\mathrm{ext}}(X)$, giving a Lefschetz (1,1)-theorem for isolated singularities.","The extended group contains the usual operational Chow group, and the extended cycle class map restricts to the BGS map, so all previously known algebraic (1,1)-classes remain algebraic in the extended sense.","For $p=1$, morphisms between isolated-singularity varieties that preserve smooth loci induce pullback maps on extended groups, so the extended algebraic classes behave functorially.","If the Hodge conjecture holds for smooth projective varieties, the wider group $A^p_{\\mathrm{wide}}(X)$ makes the extended cycle class map surjective in every codimension $p$, extending the Lefschetz picture beyond $p=1$."],"supporting_citations":[{"why":"Supplies the Bloch-Gillet-Soule cycle class map as a natural transformation to Hodge cohomology, the map whose surjectivity the paper studies.","marker":"[2]"},{"why":"Provides the definition and basic properties of operational Chow groups $A^p(X)$.","marker":"[3]"},{"why":"Gives the description of $R^0CH^p(Y)$ as $A^p(Y)$ and the exact sequences used to derive Corollaries 2.4 and 2.5.","marker":"[6]"},{"why":"Contributes the contracted-cubic example where the usual first Chern class and BGS maps are not surjective, which motivates the extended group.","marker":"[8]"},{"why":"Gives the earlier motivic Lefschetz (1,1)-theorem for singular varieties whose target is strictly smaller than $H^2_{\\mathrm{Hdg}}(X)$, setting up the comparison.","marker":"[1]"}],"fun_headline_variants":["Extended Chow group proves Lefschetz for isolated singularities","Hodge (1,1) classes algebraic for isolated singularities","Lefschetz theorem extended to singular varieties","New Chow group recovers Lefschetz for singular spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the singular locus has dimension less than $p$, and for $p=1$ that the singularities are isolated; this is what makes the exact sequences $0 \\to A^p(X) \\to A^p(\\widetilde X) \\to A^p(E)$ and the corresponding cohomology sequences valid, and without it none of the extended group's cycle-class control goes through.","fun_headline_variants_meta":{"raw":{"variants":["Extended Chow group proves Lefschetz for isolated singularities","Hodge (1,1) classes algebraic for isolated singularities","Lefschetz theorem extended to singular varieties","New Chow group recovers Lefschetz for singular spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000239,"raw_usage":{"total_tokens":1603,"prompt_tokens":1120,"completion_tokens":483,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":736,"completion_tokens_details":{"reasoning_tokens":416}},"tokens_in":736,"tokens_out":483,"duration_ms":21402,"temperature":1.0,"reasoning_tokens":416,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:39:39.603501+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test would be to take a projective variety $X$ with two isolated singular points whose resolutions have non-isomorphic $\\mathrm{Pic}^0$ on the exceptional components and compute the inverse limit $A^1_{\\mathrm{ext}}(X)$ explicitly: the theorem asserts this limit is isomorphic to the extended group of any single resolution and that the induced map to $H^2_{\\mathrm{Hdg}}(X)$ is surjective. Exhibiting a class in $H^2_{\\mathrm{Hdg}}(X)$ not in the image, or showing the inverse limit is strictly smaller than a single-resolution extended group, would refute the central claim.","supporting_citations":[{"cited_title":"Bloch, H","cited_arxiv_id":null,"evidence_quote":"Supplies the Bloch-Gillet-Soule cycle class map as a natural transformation to Hodge cohomology, the map whose surjectivity the paper studies."},{"cited_title":"Fulton.Intersection Theory","cited_arxiv_id":null,"evidence_quote":"Provides the definition and basic properties of operational Chow groups $A^p(X)$."},{"cited_title":"Descent, Motives and K-theory.Journal f¨ ur die reine und angewandte Mathematik, 478:127–176, 1996","cited_arxiv_id":null,"evidence_quote":"Gives the description of $R^0CH^p(Y)$ as $A^p(Y)$ and the exact sequences used to derive Corollaries 2.4 and 2.5."},{"cited_title":"Chow groups, Chow cohomology, and linear varieties","cited_arxiv_id":null,"evidence_quote":"Contributes the contracted-cubic example where the usual first Chern class and BGS maps are not surjective, which motivates the extended group."},{"cited_title":"A Lefschetz (1,1) theorem for singular varieties","cited_arxiv_id":"1605.00587","evidence_quote":"Gives the earlier motivic Lefschetz (1,1)-theorem for singular varieties whose target is strictly smaller than $H^2_{\\mathrm{Hdg}}(X)$, setting up the comparison."}],"review_version":1}