{"id":"9777d944-a179-475f-9fb2-7a2cbf683dd4","arxiv_id":"2508.02331","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Voisin builds a variety with representable CH_0 and no universal 0-cycle, answering Colliot-Thélène.","lead":"Claire Voisin constructs a smooth projective variety whose CH_0 group is representable yet which has no universal 0-cycle. This answers a question by Colliot-Thélène, using a Benoist-Ottem counterexample to the integral Hodge conjecture.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Abstract-only announcement: the key step is an unstated transfer of properties from the Benoist-Ottem counterexample; if that counterexample lacks the needed geometric hypotheses, the claimed separation of CH_0-representability and universal 0-cycles may fail.","rationale":"The reader's verdict of UNVERDICTED is appropriate: the abstract alone cannot support accept/reject. My stress-test does not uncover a definitive flaw; it points to the exact place where the proof must be tested. The Benoist-Ottem work guarantees a counterexample to the integral Hodge conjecture, but the paper's construction must transfer that counterexample to a variety with the two desired zero-cycle properties. That transfer is a nontrivial theorem; it is not a corollary of the counterexample's existence. The authors' phrase 'relies on' suggests a specific mechanism, but without the text the mechanism is unverifiable. My concern agrees with the reader's weakest assumption, though I emphasize that the non-algebraic class must remain non-algebraic after the construction and that the Albanese condition for representability is equally essential. Therefore the verdict remains UNVERDICTED (UNCHANGED).","tokens_in":553,"tokens_out":9558,"duration_ms":118076,"concrete_test":"Once the full text is posted, re-derive the transfer proposition (likely in Sections 3-4) and verify that the Benoist-Ottem threefold satisfies all of its hypotheses: the Albanese morphism must yield the isomorphism CH_0(X)_0 ≅ Alb(X), and the non-algebraic integral Hodge class must map to a non-algebraic class in the constructed X that is exactly the obstruction to a universal 0-cycle. If either verification fails, the announced separation is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Full text unavailable; reviewed on abstract alone. The central claim is an existence theorem, and the proof must establish two properties for the constructed variety X: representability of CH_0 and non-existence of a universal 0-cycle. The abstract says the construction 'relies on' the Benoist-Ottem counterexample to the integral Hodge conjecture. That counterexample supplies a non-algebraic integral Hodge class, but it is not automatic that this class survives the operations (products, blow-ups, correspondences) used to build X, nor that the class is the correct obstruction to a universal 0-cycle. Similarly, representability of CH_0 is a strong condition: it typically requires controlling the Albanese morphism or a decomposition of the diagonal. The Benoist-Ottem threefold, having Kodaira dimension zero, may or may not satisfy the additional hypotheses such a transfer argument needs. The abstract does not state what those hypotheses are or prove they hold. Thus the load-bearing unstated step is the transfer lemma that derives both desired properties for X from the Benoist-Ottem example. If that lemma has hidden additional assumptions, the example may either have representable CH_0 and also a universal 0-cycle, or fail to be representable. This is not a criticism of correctness but a precise identification of where the announced proof must be checked.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper announces the construction of a smooth projective variety X whose CH_0-group is representable but which admits no universal 0-cycle, thereby answering a question of Colliot-Thélène. The abstract states that the construction relies on the Benoist-Ottem counterexample to the integral Hodge conjecture. Only the abstract was available for this review; the full proof is not included.","tokens_in":846,"tokens_out":3235,"duration_ms":33774,"significance":"If the announced construction is correct, the result demonstrates that representability of the CH_0-group and existence of a universal 0-cycle are genuinely independent properties, resolving an open question in the theory of 0-cycles. The use of a known counterexample to the integral Hodge conjecture is natural, and the novelty lies in the transfer argument. The result would be a valuable contribution to the geometry of the Albanese morphism on 0-cycles. However, verification of the significance depends on the full proof, which was not available.","major_comments":[{"comment":"The abstract announces that the construction 'relies on' the Benoist-Ottem counterexample, but it does not state which geometric properties of that threefold are transferred to the new variety. In particular, the non-algebraic integral Hodge class must survive the construction to obstruct a universal 0-cycle, and the Albanese morphism of the constructed variety must be controlled to ensure representability of CH_0. These properties are not automatic consequences of the counterexample's existence, so the full text must supply a transfer lemma that verifies both. As it stands, the abstract alone does not provide enough information to check the central claim.","section":"Abstract"},{"comment":"Representability of CH_0 is a strong condition, typically requiring either a decomposition of the diagonal or precise control of the Albanese morphism. The abstract gives no indication of how the construction achieves this, nor which hypotheses on the Benoist-Ottem threefold (e.g., on its Albanese map or cycle classes) are needed. The full text should explicitly state and verify these hypotheses; otherwise the announced example may fail to have representable CH_0 or may accidentally admit a universal 0-cycle.","section":"Abstract"}],"minor_comments":[{"comment":"The phrase 'no universal 0-cycle' is ambiguous: it could mean no universal 0-cycle in the sense of Voisin, or no family of 0-cycles that specializes to every fiber. The full text should define the term precisely when first used.","section":"Abstract"},{"comment":"The abstract refers to 'our investigation' without a reference; the full text should cite the earlier work and clarify which results are assumed.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"For the editor: The submitted material is only an abstract, so I cannot verify the proof. The claimed result is significant and the approach is plausible, but the load-bearing transfer step from the Benoist-Ottem counterexample is not stated. I recommend obtaining the full manuscript before making a decision. I see no evidence of circularity or improper citation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe one thing to know: if the construction works, this answers Colliot-Thélène's question and gives the first example where representability of CH_0 does not force a universal 0-cycle. That is a real step, not a tweak.\n\nWhat is genuinely new: the combination. Benoist-Ottem produced a counterexample to the integral Hodge conjecture; Voisin's abstract says she builds on it to get this separation. That is not in the prior literature, and the question was explicitly open. The author also continues her own earlier program on the Albanese morphism and 0-cycles, which is honest framing, not circularity.\n\nWhat the paper does well, even at this level: it states the problem precisely and identifies the needed external input. The construction is not self-referential; it leans on an independent, now-accepted result.\n\nThe soft spot is the missing proof. The abstract says only that the construction 'relies on' Benoist-Ottem. To establish representability of CH_0 and absence of a universal 0-cycle, Voisin must show certain properties survive the transfer: the Albanese map, cycle class computations, and the obstruction from a non-algebraic integral Hodge class. None of those are automatic. If the Benoist-Ottem threefold doesn't satisfy an additional hypothesis, the example could collapse or accidentally admit a universal 0-cycle. That's the load-bearing step, and we can't check it from the abstract.\n\nThat said, this is Claire Voisin. Her previous work on the same circle of ideas is reliable, and I'd bet the transfer works. But 'bet' isn't 'referee report'.\n\nWho should read this: anyone working on zero-cycles, universal 0-cycles, or the integral Hodge conjecture. It will be a standard reference if it holds.\n\nMy recommendation: send it to a good journal and let a referee check the transfer carefully. This deserves peer review, not a desk rejection. If the proof is as clean as the abstract suggests, it's a solid contribution.","headline":"Voisin's abstract-only announcement of a variety separating representable CH_0 from universal 0-cycles is new and likely true; the proof's transfer from Benoist-Ottem needs careful checking.","tokens_in":1277,"tokens_out":2599,"would_cite":true,"duration_ms":27888,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C25","14C30","14K30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A smooth projective variety can have representable CH_0 but no universal 0-cycle.","keywords":["CH_0 group","representable Chow group","universal 0-cycle","Albanese morphism","integral Hodge conjecture","zero-cycles","smooth projective variety"],"falsifier":"On the specific variety constructed in the paper, check whether the transferred integral cohomology class lies in the image of the cycle class map; if it does, a universal 0-cycle likely exists, contradicting the paper's conclusion.","tokens_in":404,"feed_emoji":"📐","tokens_out":13527,"duration_ms":144658,"temperature":0.7,"pith_summary":"This paper constructs a smooth projective variety whose zero-cycle group CH_0 is representable: the Albanese map identifies the degree-zero part of CH_0 with the Albanese variety. Yet the variety admits no universal 0-cycle, a canonical degree-one zero-cycle whose class behaves compatibly over all field extensions. The example answers a long-standing question that asked whether representability of CH_0 forces the existence of a universal 0-cycle. If correct, the paper shows that these two properties are genuinely independent.","feed_headline":"Representable CH_0 groups can lack universal 0-cycles","feed_subtitle":"A new example answers a long-standing question about zero-cycles.","key_machinery":"Two notions carry the argument. First, the Chow group of zero-cycles CH_0(X) and its representability via the Albanese morphism alb_*: CH_0(X)^0 → Alb(X), which is required to be an isomorphism. Second, the universal 0-cycle, a distinguished degree-one cycle with good behaviour under field extensions. The paper's machinery combines these with a transfer construction based on a counterexample to the integral Hodge conjecture provided in the literature: a variety carrying an integral cohomology class that is not algebraic. That non-algebraic class is used to prevent the existence of a universal 0-cycle, while the geometric setup is arranged so that the Albanese kernel vanishes, making CH_0 representable.","core_discovery":"The paper establishes that there exists a smooth projective variety X over the complex numbers with representable CH_0-group but no universal 0-cycle. Representability means that the Albanese morphism induces an isomorphism CH_0(X)^0 ≅ Alb(X), so all degree-zero zero-cycles are accounted for by the Albanese variety. A universal 0-cycle would be a degree-one zero-cycle defined in a way that survives arbitrary base change; its absence is detected by an integral cohomology obstruction imported from a known counterexample to the integral Hodge conjecture. The construction transfers that counterexample into the zero-cycle setting while preserving representability of CH_0, thereby answering the question in the negative.","pith_inferences":["One natural next step, not treated in the abstract, is to ask whether a similar construction can produce a variety over a number field, which would connect the separation of zero-cycle properties to arithmetic questions.","If the integral cohomology obstruction is the true source of the missing universal 0-cycle, then one could test a Hodge-theoretic criterion: a variety should admit a universal 0-cycle exactly when the relevant integral cohomology classes lift to algebraic cycle classes.","The same transfer strategy might be reusable to separate other cycle-theoretic properties, such as universal triviality of CH_0 from representability, although the paper does not state this."],"forward_implications":["The long-standing question is answered negatively: representability of the CH_0-group does not imply the existence of a universal 0-cycle.","The hierarchy of zero-cycle properties gains a new separation: representable CH_0 is strictly weaker than having a universal 0-cycle.","A known counterexample to the integral Hodge conjecture now has a direct consequence in the theory of zero-cycles, not only in cycle class theory.","The geometry of the Albanese morphism on 0-cycles is shown to encode information beyond the representability criterion, so studying the Albanese map is a productive route for further examples."],"supporting_citations":[],"fun_headline_variants":["Representable CH_0 without universal 0-cycles","No universal 0-cycle for representable CH_0","CH_0 representable, yet no universal 0-cycle","Answering Colliot-Thélène: no universal 0-cycle","Universal 0-cycles absent in representable CH_0"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction relies on the cited counterexample to the integral Hodge conjecture having the specific geometric properties needed for the transfer argument; if those properties are absent, the constructed variety could fail to have representable CH_0 or could accidentally admit a universal 0-cycle.","fun_headline_variants_meta":{"raw":{"variants":["Representable CH_0 without universal 0-cycles","No universal 0-cycle for representable CH_0","CH_0 representable, yet no universal 0-cycle","Answering Colliot-Thélène: no universal 0-cycle","Universal 0-cycles absent in representable CH_0"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000169,"raw_usage":{"total_tokens":1161,"prompt_tokens":739,"completion_tokens":422,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":355,"completion_tokens_details":{"reasoning_tokens":337}},"tokens_in":355,"tokens_out":422,"duration_ms":4868,"temperature":1.0,"reasoning_tokens":337,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T05:00:28.013267+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the specific variety constructed in the paper, check whether the transferred integral cohomology class lies in the image of the cycle class map; if it does, a universal 0-cycle likely exists, contradicting the paper's conclusion.","supporting_citations":[],"review_version":1}