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REVIEW 5 major objections 5 minor 17 references

$0$-cycles and sheaves on abelian surfaces

T0 review · 5 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Theorem 1.1: a zero-cycle on an abelian surface lies in the k-th level of a new filtration exactly when its symmetrized rational-equivalence orbit has dimension at least k−d; the paper proves this and uses it to place second Chern classes o

desk verdict The central orbit criterion is ill-typed and false as written; the filtration idea and Theorem 5.2 are real, but the paper needs a major revision. read the letter →

arxiv 2607.16364 v1 pith:4XCT2RT6 submitted 2026-07-17 math.AG

Giovanni Mongardi , Gianluca Pacienza , Laura Pertusi This is my paper
classification math.AG MSC 14J4214K1214C2514F06
keywords zero-cyclesChowgroupsabeliansurfacesfiltrationrationalequivalencegeneralizedKummervarietiessecondChernclasscoisotropicsubvarieties
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

The paper introduces a filtration S^k_d(A) on the group of zero-cycles of an abelian surface A, defined by requiring z+ι(z) to be rationally equivalent to a symmetric effective cycle plus multiples of the origin. Its main theorem proves that this filtration is exactly detectable by geometry: z belongs to S^k_d(A) if and only if the set of length-d cycles rationally equivalent to z up to the involution is nonempty and has dimension at least k−d. This makes the filtration amenable to dimension counts and links it to moduli spaces of sheaves, where the second Chern class of a simple vector bundle can be read as a cycle in the symmetric-orbit picture. On this basis the paper conjectures a sharp placement for c_2(F) in the piece S_{d(v)−1}(A), proves a weaker bound for all simple vector bundles, and shows that the sharp conjecture would imply the existence of algebraically coisotropic subvarieties with constant-cycle fibers in generalized Kummer varieties.

What carries the argument

The central object is the filtration S^k_d(A) on CH_0(A), generated by the involution ι(p)=−p and the canonical origin o_A. The load-bearing identity is Theorem 1.1's equivalence between S^k_d(A) and the condition dim O^ι_z ≥ k−d, where O^ι_z is the symmetrized rational-equivalence orbit. The proof machinery includes a Riemann-Roch argument identifying S_g(A) with cycles supported on symmetric curves whose quotient has genus g; a Zariski-density result asserting that curves on which p+ι(p) ≡ 2o_A for every p are everywhere dense; and an incidence-variety parameter count that upgrades density to the dimension bound.

What would settle it

For a concrete abelian surface (say a product of two elliptic curves), test whether the set of points x for which {y∈A : y+ι(y) ≡ x+ι(x)} is Euclidean dense, as claimed by Theorem 2.4. If density fails, the Zariski-density lemma fails and Theorem 1.1 cannot hold in full generality; if it holds, compute dim O^ι_z for a cycle z supported on a general non-symmetric curve and compare with the predicted level S^k_d(A)—the first mismatch falsifies the theorem.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for any abelian surface A and integers k>d≥0, a zero-cycle z belongs to S^k_d(A)—meaning z+ι(z) ≡ y+2(k−d)o_A for some effective cycle y of degree 2d—if and only if the symmetric orbit O^ι_z = {w∈A^(d) : z+ι(z) ≡ w+ι(w)} is nonempty and has dimension at least k−d. This is not a mere reformulation: the proof passes through a geometric description of each level as cycles supported on symmetric curves whose quotients have genus d, and that description is what lets the paper compare c_2(F) with orbit dimensions. Building on it, Theorem 5.2 gives unconditionally that c_2(F) lies in S_{d(v)+r(r−2)}(A) for every simple vector bundle of rank r≥2, and Theorem 6.1 (co

Load-bearing premise

The proof of the orbit-dimension characterization rests on Lemma 4.3: the curves on which p+ι(p) is rationally equivalent to 2o_A for every point p must form a Zariski-dense family in A, and Theorem 4.4's Case 1.b also needs a point in an incidence variety that is simultaneously general and smooth for two different projections; if either of these fails, the characterization of S^k_d and everything built on it collapses.

Editorial extensions

If this is right

  • Theorem 1.1 converts membership in the filtration into a parameter count in A^(d), so Chow-theoretic questions become dimension-theoretic ones.
  • Theorem 5.2 yields c_2(F)∈S_{d(v)+r(r−2)}(A) for every simple vector bundle of rank r≥2; when r=2 this lands exactly in S_{d(v)}(A).
  • Assuming Conjecture 1.2, Theorem 6.1 extends the sharp placement c_2(F)∈S_{d(F)−1}(A) to all H-slope stable torsion-free sheaves with symmetric first Chern class, not just locally free ones.
  • Assuming Conjecture 1.2, Theorem 1.3 produces, for every primitive positive Mukai vector v with v^2≥6 and every 0<i≤d(v)−1, an algebraically coisotropic subvariety of codimension i−1 with constant-cycle fibers inside the generalized Kummer fiber K_H(v).
  • Under Conjecture 4.8, Conjecture 1.2 is equivalent to a related conjecture on rational equivalence in generalized Kummer fibers, showing the filtration is intrinsic to that hyperkähler geometry.

Reading between the lines

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

  • A natural but unstated extension is that the orbit-dimension criterion may hold on any surface admitting an involution whose quotient is a K3-like surface; the proof only needs the Zariski-density lemma and the Riemann-Roch support description, not special abelian-surface properties beyond the quotient geometry.
  • If Conjecture 1.2 holds, the second Chern class becomes a canonical map from the moduli space of simple sheaves into the filtration, and the proof of Theorem 1.3 suggests the coisotropic subvarieties are parametrized by the dense family of symmetric curves, giving an explicit construction to look for in concrete examples.
  • The equivalence established in Proposition 7.4 indicates that a counterexample to Conjecture 1.2 would show up as an anomalous jump in orbit dimensions inside some generalized Kummer fiber, a finite-dimensional check that may be more accessible than computing CH_0(A) directly.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper introduces a filtration S^k_d(A) on the Chow group of 0-cycles of an abelian surface A, defined by the condition that z+ι(z) be rationally equivalent to a fixed effective cycle plus a multiple of the origin. It claims a geometric characterization (Theorem 1.1): for effective cycles of degree k, z lies in S^k_d(A) exactly when its symmetrized rational-equivalence orbit O^ι_z is nonempty and has dimension at least k−d. The authors further conjecture that the second Chern class of a simple vector bundle with symmetric first Chern class lies in S_{d(v)−1}(A), prove a weaker unconditional statement (Theorem 5.2), and derive conditional consequences for torsion-free sheaves and coisotropic subvarieties of generalized Kummer varieties. The paper also contains supporting results on symmetric curves, the intersection of symmetric divisor classes, and a curve-support characterization of the filtration.

Significance. If the main theorem is correct, it establishes a precise dictionary between a Chow-theoretic filtration on an abelian surface and dimensions of rational orbits, analogous to O'Grady's and Voisin's results on K3 surfaces. This would be a valuable structural tool with applications to moduli spaces of sheaves and generalized Kummer varieties. The paper also contains independently useful results, such as Proposition 2.3 (symmetric intersections lie in multiple of the origin) and the curve-support description of Lemma 3.4, whose proofs are for the most part reasonable. However, the central Theorem 1.1 is not rigorously established as written: the statement has a definitional inconsistency that makes it literally false for d=0, and the proof has several unaddressed genericity and constructibility gaps. The overall idea is promising, but the current version is not ready for publication without substantial revision.

major comments (5)
  1. [Section 1, Theorem 1.1 and Definition 3.1] The set O^ι_z is defined for z∈A^{(d)} and w∈A^{(d)}, but S^k_d(A) consists of cycles of degree k. For d=0, k=1, S^1_0(A) contains points p on symmetric hyperelliptic curves (p+ι(p)≡2o_A), while A^{(0)} is a single point; then O^ι_z cannot have dimension ≥1. Even with the evident intended correction z,w∈A^{(k)}, the equality as stated for all of CH_0(A) is ill-posed because O^ι_z is only defined for effective cycles. Please reformulate the theorem for effective degree-k cycles (or define O^ι_z for arbitrary cycles via representatives) and correct the definition in Section 1.
  2. [Lemma 4.1 proof] The proof does not justify the 'consequence' that any z with z+ι(z)≡y+2(k−d)o_A, y effective of degree 2d, satisfies dim O^ι_z ≥ k−d. To construct a (k−d)-dimensional family of w with w+ι(w)≡z+ι(z), one needs y to be representable as η+ι(η) up to rational equivalence, or an additional moving-lemma argument. This is not supplied. The sentence 'with y effective 0-cycle of degree k' also seems to be a typo for degree 2d, but the missing argument is the substantive issue.
  3. [Theorem 4.4, Case 1.b] The proof asserts that by Lemma 4.3 one can choose a curve C and a point z' that is general and smooth in both Z~_C and Z~. This simultaneous genericity requires a Bertini-type argument that is not given. More seriously, the final contradiction claims that for fibers F of q, all c∈p(F) satisfy c+ι(c)=2o_A. This would follow from Lemma 4.2 only if p(F) is a symmetric curve, but symmetry of p(F) is not established. Without this step, the nontrivial inclusion in Theorem 1.1 is not proven.
  4. [Lemma 4.3] The proof applies [Voi15, Lemma 2.3] to the Kummer K3 surface S and states that the pullbacks of the curves yield curves C⊂A with p+ι(p)≡2o_A for all p∈C. The manuscript does not verify that Voisin's lemma applies to a Kummer K3 surface (whose Picard rank is large), nor that the Beauville-Voisin cycle behaves as required under the double cover. Please provide the precise statement and a proof of the pullback assertion.
  5. [Remark 2.5] This remark claims a wide strengthening of Maclean's theorem — namely that the density statement holds for every complex K3 surface — with only a reference to [CGL22, Theorem A and Proposition 2.9] and no proof. Since the remark is not used in the main argument, it either needs a complete proof/citation or should be removed. As written, it is an unsupported assertion.
minor comments (5)
  1. [Lemma 4.1 proof] The phrase 'with y effective 0-cycle of degree k' appears to be a typo; the definition requires degree 2d.
  2. [Section 1 and throughout] The expression 'dim O^ι_z' is used for a set that is typically a countable union of subvarieties. Its dimension should be defined (e.g., supremum of dimensions of irreducible components of the Zariski closure).
  3. [Theorem 1.1 / Lemma 4.1 / Theorem 4.4] Theorem 1.1 states k>d≥0, while Theorem 4.4 and Lemma 4.1 also consider k=d. Please make the range of k consistent.
  4. [Lemma 4.6] Typo: 'theta divisor passign through o_A' should be 'passing'.
  5. [Proposition 2.1] The assertion that [CGL22, Theorem A] provides an integral curve of any given geometric genus on every K3 surface is strong; please state the cited theorem precisely so the reader can verify the hypotheses.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 is a genuinely new geometric characterization, external citations are independent prior results, and conjectures are used only conditionally.

full rationale

The paper's central claim, Theorem 1.1, does not reduce to its inputs by construction. The filtration S^k_d(A) is defined directly in terms of rational equivalence and effective 0-cycles (Definition 3.1), while the orbit O^ι_z is defined separately as the set of points with the same z+ι(z) class (Section 4). The two inclusions in Theorem 1.1 are argued through Lemma 4.1 and Theorem 4.4, which use external results of Voisin, Maclean, Lin, and others; none of these are self-citations, and the paper does not fit any parameters or rename known results as new predictions. Conjectures 1.2 and 4.8 are explicitly labeled as conjectures and are used only to obtain conditional sharpenings (Remark 5.3, Theorem 1.3, Proposition 7.4), not to prove the main structural theorem. Some steps, such as the reliance on the Zariski-density statement imported from [Voi15, Lemma 2.3] in Lemma 4.3 and the simultaneous-genericity choice in Theorem 4.4, Case 1.b, may represent correctness risks, but that is not circularity: the supporting statements are independent of the theorem being proved. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The conditional results (Theorems 1.3, 6.1, Corollaries 5.7, 6.4, and Proposition 7.4) explicitly assume Conjecture 1.2 and sometimes Conjecture 4.8; these are the paper's own conjectures and are listed as ad hoc assumptions of the conditional part. The unconditional theorems depend on imported results from [CGL22], [Voi15], [Mac04], [Lin16], [Blo76], and [Yos01]. No free parameters are fitted; the filtration indices d,k are structural, not tuned to data.

assumptions (8)
  • domain assumption Existence of symmetric hyperelliptic curves / Zariski-dense sets of such curves on A, imported from [CGL22, Theorem A].
    Used in Proposition 2.1, Remark 2.2, Lemma 3.3, and Lemma 3.4 to produce symmetric curves with p+ι(p) ≡ 2o_A; the density claim is imported from [CGL22].
  • domain assumption [Voi15, Lemma 2.3]: on a K3 surface (here the Kummer K3 of A) the union of curves whose general point is rationally equivalent to the Beauville–Voisin cycle is dense.
    Basis of Lemma 4.3, which is used in Theorem 4.4 to pick curves C with p+ι(p) ≡ 2o_A.
  • domain assumption [Mac04, Theorem 1.2] plus the extension to arbitrary complex K3 surfaces claimed in Remark 2.5.
    Needed for Theorem 2.4 (density of {y : y+ι(y) ≡ x+ι(x)}) and for part of Corollary 5.7; the extension is asserted with a sketch.
  • ad hoc to paper Conjecture 1.2 (the paper's own conjecture) is assumed for Theorem 1.3, Theorem 6.1, Corollary 5.7, Corollary 6.4, and Proposition 7.4.
    The conditional results are explicitly labeled as depending on Conjecture 1.2; this is a stated assumption, not a hidden one.
  • ad hoc to paper Conjecture 4.8 is assumed in Remark 5.3 and Proposition 7.4 (the comparison of conjectures).
    Upgrading Theorem 5.2 to the sharp index d(v)−1 and proving the equivalence with [CLZ26, Conj. 7.2] require this additional conjecture.
  • domain assumption [Lin16, Section 2.2] computation dim(O_{j o_A}) = j−1.
    Used in Lemma 4.6 and Proposition 4.5.
  • domain assumption [Blo76, Corollary 2.4]: on an abelian variety a cycle in the kernel of the summation map satisfies z ≡ ι(z), and Chow groups are preserved under isogeny.
    Used in Proposition 4.5, Lemma 4.6, and Section 7.
  • domain assumption [Yos01, Theorems 0.1–0.2]: the Albanese map and hyperkähler structure on moduli spaces of sheaves on abelian surfaces.
    Used in Section 7 to define K_H(v) as a fiber of the Albanese morphism and to apply the hyperkähler framework.

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Pith. "Pith review of $0$-cycles and sheaves on abelian surfaces." pith.science (2026). https://pith.science/paper/4XCT2RT6

@misc{pith2026260716364,
  author       = {Pith},
  title        = {Pith review of: $0$-cycles and sheaves on abelian surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4XCT2RT6}},
  note         = {Machine review of arXiv:2607.16364}
}
abstract

We introduce a filtration on the Chow ring of an abelian surface $A$, inspired by O'Grady's filtration on K3 surfaces. We give a geometric description of the filtration, and we prove that it is deeply linked with the rational orbit of points in the generalized Kummer variety of $A$. We propose a conjecture on the second Chern class of sheaves on this abelian surface, and we provide some evidence for the conjecture and prove some of its applications.

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