Extended operational Chow group and Lefschetz (1,1)-theorem

Ananyo Dan; Inder Kaur

REVIEW 3 major objections 5 minor 8 references

Extended operational Chow group and Lefschetz (1,1)-theorem

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2506.13220 v2 pith:XQDX3ECS submitted 2025-06-16 math.AG

Ananyo Dan , Inder Kaur This is my paper · ORCID

classification math.AG MSC 14C1514C3032S3532G20

keywords Lefschetz(11)-theoremextendedoperationalChowgroupBloch-Gillet-SoulecycleclassmapHodgeclassesisolatedsingularitiesmixedstructureresolutionofrationalsurface

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $p=1$.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

There 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.

Who 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[3.1, Lemma 3.1 and Definition 3.2] 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.
[2.2, Corollary 2.5] 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.
[4.1, Example 4.3] 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.

minor comments (5)

[4.1, Corollary 4.2 proof] 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.
[3.2, title] The heading 'Lefshetz (1,1) and other properties' contains a typo; it should read 'Lefschetz'.
[Abstract and Theorem 3.4(1)] 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.
[4.2, Theorem 4.4 proof] 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.
[5, Definition 5.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the surjectivity of the extended cycle class map is derived from the classical Lefschetz (1,1)-theorem on resolutions, an independent external input, rather than from the definition of the extended group.

full rationale

The paper's central construction defines A^p_ext(phi) = {alpha in A^p(tilde X_phi) : i_phi^* cl^p(alpha) = 0} (Definition 3.2). By Corollary 2.5 this is equivalent to cl^p(alpha) lying in the image of phi^* from H^{2p}_Hdg(X), so the extended group is indeed tailored to the target Hodge cohomology. However, this definition does not assume surjectivity: it only identifies which cycles on a resolution can pull back from X. The substantive surjectivity claim in Theorem 3.4(4) is proved by invoking the classical Lefschetz (1,1)-theorem on the smooth projective resolution tilde X_phi: for any alpha in H^2_Hdg(X), phi^*alpha is a Hodge (1,1)-class on tilde X_phi, hence is c1 of a line bundle; the vanishing on the exceptional divisor follows because phi maps E to points, and the compatibility across resolutions is obtained from the isomorphism A^1_ext(X) -> A^1_ext(phi) (3.3), which uses birational invariance of Pic^0. These are independent inputs, not the conclusion of the theorem. There are no load-bearing self-citations: the references to Bloch-Gillet-Soule, Fulton, Fulton-Gillet, Kimura, Soule-Gillet, Totaro, and Arapura are external, and the Stacks Project tags are standard categorical facts. Corollary 2.5's cohomological exactness is terse but is a standard mixed-Hodge-theoretic consequence of the resolution long exact sequence for isolated singularities, and the proof does not invoke the theorem being proven. The only sense in which the construction is 'by design' is that A^p_ext is chosen as the largest group of cycles on resolutions that can map to H^{2p}_Hdg(X); this makes the containment and factorization statements (Theorem 3.4(1)-(2)) near-tautological, but the Lefschetz-type surjectivity in Theorem 3.4(4) and Theorem 4.1 retains independent content. The peripheral concerns raised about Lemma 3.1's hypotheses, Example 4.3's rational-singularity assertion, and the terseness of Corollary 2.5 are correctness or exposition issues, not circularity. No circular step can be exhibited by reducing an equation or a fitted parameter to the paper's inputs.

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

The paper's central claims rest on standard Hodge theory, resolutions of singularities, the classical Lefschetz theorem, and the existing BGS cycle class map. There are no fitted numerical parameters. The conditional higher-codimension result additionally assumes the Hodge conjecture, which is flagged rather than hidden.

assumptions (6)

standard math Every complex algebraic variety has a resolution of singularities, and isolated singularities admit resolutions whose exceptional divisors have smooth components (Hironaka).
Used throughout Section 3 and in Theorem 4.1 to form resolutions and exceptional loci.
domain assumption Mixed Hodge structure on H^*(X,Q) and the pure Hodge structure on Gr^W_{2p} H^{2p}(X,Q) behave as stated.
The paper's definition of Hodge (p,p)-classes and the exact sequences in Corollary 2.5 depend on this from the introduction and Section 2.
standard math Classical Lefschetz (1,1)-theorem for smooth projective varieties: every rational (1,1) Hodge class is the class of an algebraic cycle.
Invoked in Theorem 3.4(4) and Theorem 4.1 to lift Hodge classes on X to line bundles on resolutions.
domain assumption Existence and functoriality of the Bloch-Gillet-Soule cycle class map from operational Chow groups A^p(-) to H^{2p}_Hdg(-), as established by Bloch, Gillet, and Soule.
This is the foundation of the paper's main map; cited to Corollary 2.1.
standard math Artin's contractibility criterion for exceptional curves on smooth surfaces, including the contraction of the strict transform of a cubic.
Used in Example 4.3 and Theorem 4.4 to produce the singular surfaces X.
ad hoc to paper Hodge conjecture for smooth projective varieties, assumed only in Theorem 5.2 for the higher-codimension conditional result.
Explicitly stated as an assumption in Theorem 5.2(3); not used in the main p=1 theorems.

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Pith. "Pith review of Extended operational Chow group and Lefschetz (1,1)-theorem." pith.science (2026). https://pith.science/paper/XQDX3ECS

@misc{pith2026250613220,
  author       = {Pith},
  title        = {Pith review of: Extended operational Chow group and Lefschetz (1,1)-theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XQDX3ECS}},
  note         = {Machine review of arXiv:2506.13220}
}

abstract

Let $X$ be a singular, projective variety. For every $p>0$, $H^{2p}(X,\mathbb{Q})$ is equipped with a mixed Hodge structure. The elements of $\mathrm{Gr}^W_{2p}H^{2p}(X,\mathbb{Q}) \cap H^{p,p} \mathrm{Gr}^W_{2p}H^{2p}(X,\mathbb{C})$ will be called Hodge (p,p)-classes. The purpose of this article, is to study the Bloch-Gillet-Soul\'{e} (BGS) cycle class map from the $p$-th operational Chow group $A^p(X)$ to the space of $(p,p)$-Hodge classes. We show that if $p=1$ and $X$ is a normal surface with at worst rational singularities, then the BGS cycle class map is surjective. This extends the Lefschetz $(1,1)$-theorem to the setup of rational surface singularities. However, the BGS map is not always surjective. For this reason we introduce extended operational Chow group $A^p_{\mathrm{ext}}(X)$ which contains the operational Chow group. We show that the BGS cycle class map extends to $A^p_{\mathrm{ext}}(X)$. Moreover, if $p=1$ and $X$ has at worst isolated singularity (not necessarily a surface), then the extended BGS map is surjective. This further extends the Lefschetz $(1,1)$-theorem to the case of isolated singularities.

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Reference graph

Works this paper leans on

8 extracted references · 7 canonical work pages

[1]

D. Arapura. A Lefschetz (1, 1) theorem for singular varieties.arXiv preprint arXiv:1605.00587, 2016

work page Pith review arXiv 2016
[2]

Bloch, H

S. Bloch, H. Gillet, and C. Soul´ e. Non-archimedean Arakelov theory.Journal of Algebraic Geometry, 4(4):427– 486, 1995

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W. Fulton.Intersection Theory. Springer Science & Business Media, 2013

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Fulton and H

W. Fulton and H. Gillet. Riemann-Roch for general algebraic varieties.Bulletin de la Soci´ et´ e math´ ematique de France, 111:287–300, 1983

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Fractional intersection and bivariant theory.Communications in algebra, 20(1):285–302, 1992

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work page 1992
[6]

Descent, Motives and K-theory.Journal f¨ ur die reine und angewandte Mathematik, 478:127–176, 1996

C Soul´ e and H Gillet. Descent, Motives and K-theory.Journal f¨ ur die reine und angewandte Mathematik, 478:127–176, 1996

work page 1996
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The Stacks Project Authors.Stacks Project.https://stacks.math.columbia.edu, 2018

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Chow groups, Chow cohomology, and linear varieties

B Totaro. Chow groups, Chow cohomology, and linear varieties. InForum of Mathematics, Sigma, volume 2. Cambridge University Press, 2014. CUNEF Universidad, C. de Leonardo Prieto Castro, 2, Moncloa - Aravaca, 28040 Madrid, Spain Email address:dan.ananyo@cunef.edu School of Mathematics & Statistics, University of Glasgow, Glasgow G12 8QQ, U.K. Email address...

work page 2014

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

D. Arapura. A Lefschetz (1, 1) theorem for singular varieties.arXiv preprint arXiv:1605.00587, 2016

work page Pith review arXiv 2016

[2] [2]

Bloch, H

S. Bloch, H. Gillet, and C. Soul´ e. Non-archimedean Arakelov theory.Journal of Algebraic Geometry, 4(4):427– 486, 1995

work page 1995

[3] [3]

Fulton.Intersection Theory

W. Fulton.Intersection Theory. Springer Science & Business Media, 2013

work page 2013

[4] [4]

Fulton and H

W. Fulton and H. Gillet. Riemann-Roch for general algebraic varieties.Bulletin de la Soci´ et´ e math´ ematique de France, 111:287–300, 1983

work page 1983

[5] [5]

Fractional intersection and bivariant theory.Communications in algebra, 20(1):285–302, 1992

S Kimura. Fractional intersection and bivariant theory.Communications in algebra, 20(1):285–302, 1992

work page 1992

[6] [6]

Descent, Motives and K-theory.Journal f¨ ur die reine und angewandte Mathematik, 478:127–176, 1996

C Soul´ e and H Gillet. Descent, Motives and K-theory.Journal f¨ ur die reine und angewandte Mathematik, 478:127–176, 1996

work page 1996

[7] [7]

The Stacks Project Authors.Stacks Project.https://stacks.math.columbia.edu, 2018

2018

[8] [8]

Chow groups, Chow cohomology, and linear varieties

B Totaro. Chow groups, Chow cohomology, and linear varieties. InForum of Mathematics, Sigma, volume 2. Cambridge University Press, 2014. CUNEF Universidad, C. de Leonardo Prieto Castro, 2, Moncloa - Aravaca, 28040 Madrid, Spain Email address:dan.ananyo@cunef.edu School of Mathematics & Statistics, University of Glasgow, Glasgow G12 8QQ, U.K. Email address...

work page 2014