{"id":"29b5a0f9-776a-4f6a-8d38-8f5c9765f2f8","arxiv_id":"2603.16545","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Every admissible graded-polarizable VMHS of type (0,0),(-1,0),(0,-1),(-1,-1) arises up to isogeny from a 1-motive, yielding a relative Deligne equivalence under mild assumptions.","lead":"The paper proves that every admissible variation of mixed Hodge structures of a specific type comes, up to isogeny, from a 1-motive over a complex base scheme. It answers André’s question on geometric origin and gives a relative form of Deligne’s equivalence between 1-motives and such variations.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"The central claim rests on cleanly combining André’s abelian-case description with a new toric analysis over general S; failure of that gluing collapses both geometric origin and the relative Deligne equivalence.","rationale":"The reader correctly isolated the dependence on André’s abelian description plus the new toric analysis as the single load-bearing hinge; the abstract itself flags this combination as the method. Without the body one cannot confirm that the gluing of relative extension classes and sections works over general S, so the geometric-origin claim and the relative Deligne equivalence remain unchecked. No independent inconsistency or circularity appears in the abstract (the Hodge types match those of Deligne 1-motives, the question is natural, and the external André input is not self-forcing). The verdict therefore stays UNVERDICTED with low confidence pending the concrete full-text check above. The global Mumford–Tate statement is secondary and inherits the same dependence.","tokens_in":2166,"tokens_out":599,"duration_ms":25042,"concrete_test":"Obtain the full text; locate the exact citation/theorem of André’s abelian input that is invoked. Verify whether it applies verbatim over general S or only after reduction to the generic fibre/fibres. Then inspect the paper’s toric-analysis construction (the map realizing sections of the semi-abelian scheme) and check that it yields an algebraic 1-motive over S whose enriched realization matches the input VMHS up to a uniformly controlled isogeny, with no extra base-change or monodromy obstruction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract obtains the geometric-origin statement (every torsion-free graded-polarizable admissible VMHS of type (0,0),(-1,0),(0,-1),(-1,-1) arises up to isogeny from a 1-motive over S) and the relative equivalence by “combining André’s description of the abelian case with a new analysis of the toric part” via a Hodge-theoretic interpretation of sections of semi-abelian varieties. For this to work, the toric analysis must supply a torus and lattice of sections that glue to André’s abelian data to produce a 1-motive over the connected smooth finite-type C-scheme S whose enriched Hodge realization recovers the given VMHS up to isogeny. Over a general base this requires relative extension classes, algebraicity of sections, and uniform control of monodromy/graded-polarizability/admissibility. If André’s input is absolute-only or does not supply the needed relative data, or if the toric part fails to produce an algebraic map over S, both the origin claim and the equivalence (even up to isogeny) fail. The same gluing underpins the “suitable assumptions” for the strict equivalence.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript claims that every torsion-free, graded-polarizable, admissible variation of mixed Hodge structures of type (0,0), (-1,0), (0,-1), (-1,-1) over the complex analytic space associated to a connected smooth finite-type C-scheme S arises, up to isogeny, from a 1-motive over S, answering a question of André on geometric origin. The route is a Hodge-theoretic interpretation of sections of semi-abelian varieties obtained by combining André’s description of the abelian case with a new analysis of the toric part. As a consequence, under suitable assumptions on S and on the lattices and tori, the enriched Hodge realization induces an equivalence of categories (a relative Deligne equivalence); in general the statement holds only up to isogeny. The paper also introduces the global Mumford–Tate group of a 1-motive over S and identifies its neutral connected component with the Mumford–Tate group of the generic fiber.","tokens_in":2434,"tokens_out":969,"duration_ms":18608,"significance":"If the arguments hold, the paper supplies a positive answer to André’s geometric-origin question for admissible VMHS of 1-motive type and a relative analogue of Deligne’s equivalence over C. That would be a substantial contribution to the interface of 1-motives and variations of mixed Hodge structures, clarifying how algebraic 1-motives over bases relate to admissible VMHS of the indicated type. The global Mumford–Tate construction is a natural and useful addition. The abstract indicates reliance on André’s prior abelian-case work together with new toric analysis; if that combination is cleanly executed over general S, the result is of clear interest to the field.","major_comments":[{"comment":"Only the abstract is available for this review, so the load-bearing combination of André’s abelian-case description with the claimed new toric analysis cannot be checked. Both the geometric-origin statement (up to isogeny) and the relative Deligne equivalence rest on producing, over a general connected smooth finite-type C-scheme S, a 1-motive whose enriched Hodge realization recovers a given torsion-free graded-polarizable admissible VMHS of type (0,0), (-1,0), (0,-1), (-1,-1). That requires relative extension classes, algebraicity of sections of semi-abelian varieties, and uniform control of monodromy, graded-polarizability and admissibility. Without the full text it is impossible to verify that André’s input supplies the needed relative data or that the toric analysis glues cleanly over S; failure of that gluing would collapse both central claims.","section":null},{"comment":"The abstract asserts a strict equivalence under “suitable assumptions on S and on the lattices and the tori underlying 1-motives,” and only an isogeny statement in general. The precise content of those assumptions, and whether they are natural or severely restrictive, cannot be assessed from the abstract alone; they are load-bearing for the non-isogeny equivalence and must be stated and justified in the body of the paper.","section":null},{"comment":"The identification of the neutral connected component of the newly introduced global Mumford–Tate group of a 1-motive over S with the Mumford–Tate group of the generic fiber is stated without indication of the argument. Whether this is a formal consequence of the earlier constructions or requires separate work cannot be checked without the full text.","section":null}],"minor_comments":[{"comment":"The abstract is clear on the main claims and on the dependence on André’s abelian case plus a new toric analysis. Once the full manuscript is available, standard presentation checks (notation for the enriched realization, precise definition of the global Mumford–Tate group, and explicit statement of the “suitable assumptions”) will be needed.","section":null}],"recommendation":"uncertain","confidential_remarks":"This is an abstract-only review: the full text of arXiv:2603.16545 was not available. I cannot responsibly recommend accept, minor_revision, major_revision, or reject without seeing the proofs of the gluing of André’s abelian input to the toric analysis over general S. Recommendation is therefore uncertain. If the full manuscript is supplied, the central technical risk to re-examine is exactly that gluing (relative extension classes, algebraicity of sections, monodromy/admissibility control). Scope and novelty appear appropriate for a serious journal in algebraic geometry / Hodge theory, contingent on the arguments holding."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know is that Bertolin claims every torsion-free graded-polarizable admissible VMHS of type (0,0),(-1,0),(0,-1),(-1,-1) over the analytic space of a connected smooth finite-type C-scheme S arises up to isogeny from a 1-motive over S, answering André’s geometric-origin question, and that under suitable assumptions the enriched Hodge realization is an equivalence (relative Deligne). In general it is only up to isogeny. She also defines a global Mumford–Tate group for a 1-motive over S whose neutral component matches the generic fibre’s.\n\nWhat is new is the relative statement itself and the toric-part analysis that lets her combine André’s abelian description with a Hodge-theoretic reading of sections of semi-abelian varieties. That combination is presented as the engine. The global MT group is a clean extra. Dependence on André and on classical Deligne foundations looks ordinary, not circular.\n\nThe soft spot is exactly the one the stress-test flags: without the body we cannot see whether the toric analysis supplies algebraic lattices and tori that glue to André’s relative abelian data over a general base, controlling extension classes, monodromy and admissibility. If that gluing fails, both the origin claim and the equivalence collapse. The “suitable assumptions” are also left unspecified in the abstract. Those are real but proportionate caveats for an abstract-only read; nothing in the claim looks incoherent on its face.\n\nThis is for people already working on 1-motives, admissible VMHS and relative motives. It is not a general-audience paper. It deserves a serious referee who can check the gluing and the precise hypotheses. I would send it out.","headline":"Abstract-only: clean positive answer to André on geometric origin of these VMHS plus relative Deligne equivalence, load-bearing step is the toric–abelian gluing over general S.","tokens_in":3067,"tokens_out":464,"would_cite":false,"duration_ms":12998,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C30","14F35","32G20","14K30"],"pacs":[],"model":"grok-4.5","headline":"Every torsion-free graded-polarizable admissible variation of mixed Hodge structures of type (0,0), (-1,0), (0,-1), (-1,-1) over S arises up to isogeny from a 1-motive over S.","keywords":["1-motives","admissible variations of mixed Hodge structures","enriched Hodge realization","relative Deligne equivalence","semi-abelian varieties","Mumford-Tate groups","geometric origin","graded-polarizable VMHS"],"falsifier":"Exhibit a torsion-free graded-polarizable admissible VMHS of the given type over some smooth finite-type S that is not isogenous to the enriched Hodge realization of any 1-motive over S, or show that the toric and abelian pieces fail to glue into a 1-motive on that base.","tokens_in":3007,"feed_emoji":"🔷","tokens_out":1116,"duration_ms":16928,"temperature":0.7,"pith_summary":"The paper claims that the admissible mixed Hodge variations that look like the enriched Hodge realization of a 1-motive actually come from one. André already attached to every 1-motive over a smooth finite-type C-scheme S a torsion-free graded-polarizable admissible variation of mixed Hodge structures of type (0,0), (-1,0), (0,-1), (-1,-1) on the associated analytic space; the question left open was whether every such variation is of geometric origin. The author answers that question affirmatively up to isogeny by giving a Hodge-theoretic reading of sections of semi-abelian varieties: André’s earlier description of the abelian case is combined with a new analysis of the toric part. Under extra assumptions on S and on the lattices and tori, the enriched realization becomes an equivalence of categories (a relative form of Deligne’s classical equivalence); in general one obtains only an equivalence up to isogeny. As a byproduct the paper defines a global Mumford–Tate group for a 1-motive over S whose neutral connected component coincides with the Mumford–Tate group of the generic fibre.","feed_headline":"All admissible 1-motive-type Hodge variations arise geometrically","feed_subtitle":"Up to isogeny every such VMHS over a smooth base comes from a 1-motive, giving a relative Deligne equivalence","key_machinery":"The enriched Hodge realization of a 1-motive, together with a Hodge-theoretic interpretation of sections of semi-abelian varieties obtained by gluing André’s description of the abelian case to a new analysis of the toric part; this mechanism produces the geometric-origin statement and the relative Deligne equivalence.","core_discovery":"Every torsion-free, graded-polarizable, admissible variation of mixed Hodge structures of type (0,0), (-1,0), (0,-1), (-1,-1) on the complex analytic space of a connected smooth finite-type C-scheme S arises, up to isogeny, from a 1-motive over S. Under suitable hypotheses on S and on the lattices and tori, the enriched Hodge realization is an equivalence between the category of 1-motives over S and the category of such variations; in general the equivalence holds only after isogeny. The global Mumford–Tate group of a 1-motive over S has neutral connected component equal to the Mumford–Tate group of its generic fibre.","pith_inferences":["The same gluing technique for abelian and toric data may extend to other relative realizations (for example étale or crystalline) once analogous descriptions of the abelian case are available.","Failure of the equivalence without isogeny points to a precise obstruction living in the torsion of the lattices or tori; computing that obstruction would give a sharper classification.","The identification of global and generic Mumford–Tate groups suggests that monodromy of the variation is controlled by the generic fibre, which could be tested on explicit families of semi-abelian varieties."],"forward_implications":["André’s question on the geometric origin of admissible VMHS of 1-motive type receives a positive answer up to isogeny.","Under the stated hypotheses on S, lattices and tori, the enriched Hodge realization becomes an equivalence of categories, giving a relative form of Deligne’s classical equivalence over C.","In general the equivalence holds only after isogeny, so the categories of 1-motives and of such VMHS coincide up to isogeny.","The global Mumford–Tate group of a 1-motive over S is well-defined and its neutral connected component recovers the Mumford–Tate group of the generic fibre."],"fun_headline_variants":["Admissible 1-motive-type VMHS arise from 1-motives up to isogeny","Relative Deligne equivalence for 1-motives and admissible VMHS","Every such graded-polarizable admissible VMHS comes from a 1-motive","Enriched Hodge realization of 1-motives is essentially surjective","Global Mumford-Tate of a 1-motive matches that of its generic fibre"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The argument stands or falls on the claim that André’s earlier description of the abelian case can be cleanly combined with the new analysis of the toric part over a general base S.","fun_headline_variants_meta":{"raw":{"variants":["Admissible 1-motive-type VMHS arise from 1-motives up to isogeny","Relative Deligne equivalence for 1-motives and admissible VMHS","Every such graded-polarizable admissible VMHS comes from a 1-motive","Enriched Hodge realization of 1-motives is essentially surjective","Global Mumford-Tate of a 1-motive matches that of its generic fibre"]},"model":"grok-4.5","effort":"low","cost_usd":0.005986,"raw_usage":{"total_tokens":1687,"prompt_tokens":938,"num_sources_used":0,"completion_tokens":111,"cost_in_usd_ticks":59860000,"prompt_tokens_details":{"text_tokens":938,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":638,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":938,"tokens_out":111,"duration_ms":5408,"temperature":1.0,"reasoning_tokens":638,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T23:41:13.820226+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a torsion-free graded-polarizable admissible VMHS of the given type over some smooth finite-type S that is not isogenous to the enriched Hodge realization of any 1-motive over S, or show that the toric and abelian pieces fail to glue into a 1-motive on that base.","supporting_citations":[],"review_version":1}