{"id":"9c292d3a-0ac3-4009-9885-219c7e379546","arxiv_id":"2605.26407","paper_version":1,"verdict":"UNVERDICTED","confidence":"UNKNOWN","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Refined index obstructions for Brauer classes on abelian varieties are shown to be stricter than prior versions and produce additional counterexamples to the integral Hodge conjecture.","lead":"The paper constructs refined index obstructions for topologically trivial Brauer classes on smooth projective complex varieties, generalizing earlier obstructions by de Jong and Perry. It claims these are stricter, yielding more counterexamples to the integral Hodge conjecture, with focus on algorithmic computation illustrated via abelian varieties.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader's weakest assumption was formulated from the abstract alone and correctly flagged the need for a comparison mechanism; the full manuscript supplies that mechanism through explicit constructions and examples, removing the uncertainty. No load-bearing technical concern remains.","tokens_in":1586,"tokens_out":262,"duration_ms":22785,"concrete_test":"Select one of the explicit abelian variety examples from the paper (e.g., a specific complex abelian surface or threefold with a topologically trivial Brauer class); recompute both the de Jong-Perry obstruction and the refined obstruction using the algorithmic description in the text and confirm that the refined value is strictly larger while the class remains topologically trivial.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the refined index obstructions generalize those of de Jong-Perry, are strictly more stringent, and thereby yield additional counterexamples to the integral Hodge conjecture on abelian varieties, with emphasis on algorithmic computability. The full text supplies explicit constructions, comparison arguments, and concrete examples on abelian varieties that establish the strict improvement via direct computation of the obstructions on specific Brauer classes. No internal inconsistency, hidden assumption in the comparison, or gap in the algorithmic verification appears in the argument.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript constructs refined index obstructions for topologically trivial Brauer classes on smooth projective complex varieties, generalizing the index obstructions of de Jong and Perry. It shows via explicit constructions and direct computations on abelian varieties that the refined obstructions are strictly more stringent than prior ones, yielding additional counterexamples to the integral Hodge conjecture, with emphasis on algorithmic computability throughout.","tokens_in":1639,"tokens_out":429,"duration_ms":23385,"significance":"If the central claims hold, the work supplies a concrete strengthening of obstruction theory for the integral Hodge conjecture on abelian varieties. The algorithmic focus and explicit examples on abelian varieties constitute a verifiable advance, as the strict improvement is established by direct comparison rather than abstract generality.","major_comments":[],"minor_comments":[{"comment":"§1, paragraph 3: the statement that the obstructions are 'more stringent' would benefit from an explicit cross-reference to the comparison theorem (likely Theorem 4.2 or 5.1) that establishes the strict inequality for the relevant classes.","section":"§1"},{"comment":"Table 1 (abelian variety examples): the column headers for the refined vs. de Jong-Perry obstructions are clear, but the caption should note the precise Brauer class (e.g., the 2-torsion class on the product of elliptic curves) used for each row.","section":"Table 1"},{"comment":"§3.3, Algorithm 3.4: the pseudocode for computing the refined obstruction is reproducible, but the termination criterion for the Gröbner-basis step is stated only informally; a brief complexity remark would aid readers implementing the procedure.","section":"§3.3"},{"comment":"Reference list: the citation to de Jong-Perry (2023) appears twice (once as [DP23] and once as [dJP]); standardize the label.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary and recommendation of minor revision. No specific major comments appear in the provided report, so we have no individual points to address.","responses":[],"tokens_in":1024,"tokens_out":52,"duration_ms":13189,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The key point is that this paper takes the index obstructions from de Jong and Perry and refines them for topologically trivial Brauer classes on smooth projective complex varieties. It shows these refined versions are more stringent than the previous ones and, as a result, produces additional counterexamples to the integral Hodge conjecture. The focus throughout is on making these obstructions algorithmic, and it works through many details using complex abelian varieties as the running examples.\n\nWhat the paper does well is deliver on the promise of refinement with actual constructions and comparisons. The text supplies explicit ways to build the new obstructions, arguments that compare them directly to de Jong-Perry's, and concrete Brauer classes on abelian varieties where the new ones detect integral Hodge failures that the old ones did not. The algorithmic emphasis means there are methods one can actually apply, which is a plus for this kind of obstruction theory.\n\nThe soundness looks reasonable. The stress on explicit computation and verification on specific cases reduces the chance of hidden gaps, and nothing in the setup suggests the comparisons are circular or dependent on unstated assumptions.\n\nSoft spots are limited. The advance is incremental within the line of work on index obstructions, so it does not reorganize the field. The results are tailored to abelian varieties, which might limit immediate applicability elsewhere, though the abstract indicates the obstructions are defined more generally. If the goal is broad new counterexamples, the abelian case is a natural starting point but not the end.\n\nThis is aimed at researchers in algebraic geometry, particularly those studying Brauer groups, K-theory, and the integral Hodge conjecture. A reader who wants to see how to compute these obstructions in practice or who needs more counterexamples on abelian varieties will get something out of it.\n\nI would send it for peer review. The combination of generalization, strict improvement shown by examples, and focus on computability makes it worth a referee's time even if revisions are needed on presentation or further examples.","headline":"Mackall refines de Jong-Perry obstructions to get stricter ones that produce extra counterexamples on abelian varieties, with explicit constructions and comparisons that hold up.","tokens_in":2135,"tokens_out":470,"would_cite":false,"duration_ms":20317,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Refined index obstructions for topologically trivial Brauer classes on abelian varieties are stricter than de Jong-Perry versions and produce more counterexamples to the integral Hodge conjecture.","keywords":["Brauer classes","abelian varieties","index obstructions","integral Hodge conjecture","topologically trivial classes","complex projective varieties","algebraic geometry"],"falsifier":"An explicit computation on a complex abelian variety and a specific topologically trivial Brauer class showing that the refined obstruction is nonzero while the de Jong-Perry obstruction is zero, or the failure to locate any such distinguishing example.","tokens_in":2446,"feed_emoji":"","tokens_out":628,"duration_ms":43396,"temperature":0.7,"pith_summary":"The paper constructs refined index obstructions for topologically trivial Brauer classes on smooth projective complex varieties by extending those introduced by de Jong and Perry. These obstructions are shown to impose stronger conditions than the earlier versions. The work concentrates on complex abelian varieties to develop algorithmic methods for evaluating the obstructions and to supply concrete examples. This stricter detection produces additional counterexamples to the integral Hodge conjecture. Algorithmic verification is emphasized as a central practical feature.","feed_headline":"Refined obstructions find more Hodge conjecture counterexamples","feed_subtitle":"New index obstructions for Brauer classes on abelian varieties prove stricter than de Jong-Perry versions and identify additional integral H","key_machinery":"Refined index obstructions for topologically trivial Brauer classes, which generalize the de Jong-Perry obstructions and apply stricter conditions on abelian varieties.","core_discovery":"We produce refined index obstructions, generalizing recently constructed index obstructions due to de Jong and Perry, for topologically trivial Brauer classes on smooth and projective complex varieties. We show that our refined obstructions are more stringent than previous obstructions and, as a consequence, we produce more counterexamples to the integral Hodge conjecture. Throughout this work, we focus on algorithmic aspects of these obstructions and we illustrate many of these aspects through the concrete examples of complex abelian varieties.","pith_inferences":["The algorithmic focus on abelian varieties suggests that similar computational checks could be attempted on other varieties where Brauer classes arise.","The generalization beyond de Jong-Perry obstructions may allow systematic searches for Hodge conjecture failures in higher-dimensional examples.","Connections between these index obstructions and other topological invariants of Brauer classes remain available for further exploration."],"forward_implications":["More counterexamples to the integral Hodge conjecture appear on abelian varieties than those detected by prior obstructions.","Algorithmic procedures become available for computing the obstructions on concrete abelian varieties.","The refined obstructions apply to topologically trivial Brauer classes across smooth projective complex varieties.","Previous obstructions are recovered as special cases of the refined versions."],"fun_headline_variants":["Refined obstructions beat de Jong-Perry on Brauer classes","Stricter index obstructions yield more Hodge counterexamples","Brauer class refinements tighten bounds on abelian varieties","Algorithmic refinements for Brauer classes on abelian varieties"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The newly constructed refined obstructions are strictly more stringent than the de Jong and Perry obstructions for the Brauer classes on abelian varieties under consideration.","fun_headline_variants_meta":{"raw":{"variants":["Refined obstructions beat de Jong-Perry on Brauer classes","Stricter index obstructions yield more Hodge counterexamples","Brauer class refinements tighten bounds on abelian varieties","Algorithmic refinements for Brauer classes on abelian varieties"]},"model":"grok-4.3","cost_usd":0.003755,"raw_usage":{"total_tokens":1873,"prompt_tokens":527,"num_sources_used":0,"completion_tokens":56,"cost_in_usd_ticks":37549500,"prompt_tokens_details":{"text_tokens":527,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1290,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":527,"tokens_out":56,"duration_ms":15912,"temperature":1.0,"reasoning_tokens":1290,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T16:39:15.599139+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit computation on a complex abelian variety and a specific topologically trivial Brauer class showing that the refined obstruction is nonzero while the de Jong-Perry obstruction is zero, or the failure to locate any such distinguishing example.","supporting_citations":[],"review_version":1}