{"id":"521af6b5-66e5-44b9-a73d-4487b451a24a","arxiv_id":"1908.05110","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A single real torus inside a log Calabi-Yau complement (or a Calabi-Yau degeneration) computes the odd weight filtration, yielding P=W type results for rational surfaces and K3 surfaces.","lead":"A new geometric proof shows that the odd part of the weight filtration on the cohomology of a log Calabi-Yau complement is exactly the kernel of restriction to a single real torus, and an analogous statement holds for Calabi-Yau degenerations. The paper uses this to prove P=W type equalities for rational surfaces with nodal anticanonical boundary and for K3 surfaces.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Kollár chain condition in Theorem 2.13 is false as stated: for (P^2, smooth cubic), Bezout forbids a rational curve meeting the cubic transversely in exactly two points, so the homotopy of profound tori is not justified.","rationale":"The reader's weakest-assumption pick — Kollár's Theorem 10 and the chain-to-homotopy Lemma 2.12 — is exactly the right location. I agree that this imported theorem is the least secure step. However, my concern is more specific than the reader's: the theorem as paraphrased in the paper is not merely unverified; it is false in a concrete example. The P^2 smooth cubic case shows that no rational curve can meet the boundary in exactly two transverse points, by Bezout. This exposes a gap in the proof of Theorem 2.14. I do not think this disproves the main theorem. The example has only one profound torus, and for the cases with multiple profound tori the chain condition may be recoverable from a weaker connectivity statement, perhaps using topological arcs or subarcs of rational curves rather than transversality in exactly two points. The degeneration half of the paper is not directly affected by this particular example, but it imports the same Kollár machinery in Theorem 3.11, so the same verification is needed there. Because the central claim is plausible and the identified flaw is a repairable gap in the proof rather than a demonstrated counterexample, the conditional verdict remains appropriate. No change from the reader's CONDITIONAL assessment is warranted.","tokens_in":22281,"tokens_out":41612,"duration_ms":453142,"concrete_test":"Check the exact statement of Kollár [21, Theorem 10] in the published paper. Independently test the stated chain condition on X = P^2, Y a smooth cubic: for a rational curve of degree e, the intersection number with Y is 3e, so it cannot intersect Y transversely in exactly two points. This settles that Theorem 2.13 as formulated is false if it is meant to cover J1 = J2 or any two points of one connected minimal stratum. If Kollár's theorem instead gives only connectivity of minimal strata in the stratified sense, verify whether Lemma 2.12 can be applied to the subarcs of the rational curves between consecutive intersections with Y^j; if so, the single-torus conclusion survives with a revised proof, otherwise Theorem 2.14 lacks support.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single-torus theorem reduces to Theorem 2.14, whose key step is the claim that all profound tori are homotopic. This is derived from Theorem 2.13, a paraphrase of Kollár [21, Theorem 10]. As stated, Theorem 2.13 asserts that distinct minimal strata Y_J are connected by rational curves in Y^{j-1}, each intersecting Y^j transversally in exactly two points. That assertion is false in a simple log Calabi--Yau example: take X = P^2 and Y a smooth cubic curve. Then (X,Y) is snc, K_X + Y = O, and delta = 1; the unique minimal stratum is Y itself. Applying Theorem 2.13 with J1 = J2 = {1} would require connecting any two points of the cubic by rational curves in X = P^2, each meeting Y transversely in exactly two points. But a rational curve of degree e in P^2 has intersection number 3e with a smooth cubic, so it cannot meet Y in only two transverse points (total local multiplicity 2 versus 3e). Thus the cited theorem cannot have exactly the content used here. The damage is not immediate: for this example there is only one profound torus, and for delta >= 2 the chain property may hold or may be replaceable by a weaker topological statement. However, the reduction from the direct sum in Theorem 2.11 to a single torus in Theorem 2.14 is the load-bearing step, and the proof of that reduction relies on a misstated import. A corrected argument could use subarcs of rational curves between consecutive intersection points with Y^j, since Lemma 2.12 only needs a cylinder in the real oriented blowup, but this repair is not present in the text.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a torus-fiber description of the odd part of the weight filtration. Theorem 1.1 asserts that for a simple normal crossings log Calabi-Yau pair (X,Y), if δ is the maximal number of components of Y meeting at a point, there is a real torus T of dimension δ embedded in X\\Y such that W_{2k-1}H^k(X\\Y;Q) equals the kernel of the restriction map H^k(X\\Y;Q) -> H^k(T;Q), and that an analogous statement holds for semistable Calabi-Yau degenerations with a torus of dimension δ-1 in the smooth fiber. The proof identifies the lowest weight homology classes with classes of tori obtained by generalized Leray cycles (El Zein-Némethi) or by Clemens contraction, then uses Kollár's connectedness theorem to show all tori attached to minimal strata are homotopic. The paper then proves P=W-type identifications for rational surfaces with nodal anticanonical divisor and for K3 surfaces, and formulates a conjecture for compact hyperkähler manifolds.","tokens_in":22621,"tokens_out":31699,"duration_ms":321946,"significance":"If the main theorem is correct, it provides a very concrete geometric description of the odd weight filtration in terms of a single embedded torus, with immediate dimension bounds and a bridge to the P=W conjecture via Simpson's and Auroux's conjectures. The surface and K3 applications are genuinely interesting and go beyond previously known cases (Zhang, Gross). The proof is mostly transparent and relies on classical external results; the paper also honestly identifies the special chain condition (*) that is needed. However, the correctness of the central reduction currently depends on a misstated version of Kollár's theorem and on an invalid proof of a duality statement, so the significance will be realized only after those points are repaired.","major_comments":[{"comment":"Theorem 2.13 is false as stated. Taking X=P^2 and Y a smooth cubic gives an snc log CY pair with δ=1, and for J1=J2={1} and distinct points p1,p2 the asserted rational curve C⊂P^2 would have to meet the cubic transversely in exactly two points; Bezout's theorem forces deg(C)·3 = 2, impossible. Since the proof of Theorem 2.14 uses Theorem 2.13 to conclude that all profound tori are homotopic, the statement must be corrected (for instance, by requiring J1 and J2 to index distinct minimal strata, and by checking the curve-chain assertion against Kollár's original theorem), or Lemma 2.12 should be generalized so that subarcs of a rational curve with finitely many intersection points with Y^j give the required homotopies.","section":"Theorem 2.13 and Theorem 2.14"},{"comment":"Proposition 3.3 has a proof that is inconsistent with formula (4). With ℓ=k, formula (4) gives M_{2k-1}H^k(X1;Q)=ker(N^k_k), not im(N^k_k), and for H^{2d-k} it gives M_{2d-1}H^{2d-k}=ker(N^k_{2d-k}), not M_{2d-2k-1}. The statement of Proposition 3.3 is in fact true by Poincaré duality, and Corollary 3.4 can be proved directly from formula (4) and duality, but the proof as written does not establish it; this needs to be rewritten before the Clemens-torus computation can be used.","section":"Proposition 3.3"},{"comment":"Theorem 4.5 applies Theorem 1.1(1) to a pair (X,Y) in which Y is only assumed to be reduced and nodal, not necessarily snc, and its proof uses the weight spectral sequence with E1-term H^{2p+q}(Y^{-p}) as though Y were snc. Since Theorem 1.1 was proved only for snc divisors, this application needs an explicit reduction, e.g. by passing to an snc model of the same complement via the Gross-Hacking-Keel toric model invoked in Theorem 4.1, and by checking that the torus fiber of the Lagrangian fibration is homotopic to the profound torus in that model. As written, the P=W-type conclusion for nodal anticanonical surfaces is not rigorously derived.","section":"Theorem 4.5"}],"minor_comments":[{"comment":"The expression W_{-k}H_k should read W_{-2k}H_k.","section":"Section 2.2, proof of Theorem 2.11"},{"comment":"The phrase 'p1 and p1' should be 'p1 and p2'.","section":"Definition 2.7"},{"comment":"The vector η is used in the proof without being defined; one should state that η is a class with ⟨η,β⟩≠0 (or ⟨η,β⟩=1).","section":"Proposition 4.8"},{"comment":"There is a duplicated word in 'contained in in A^{j-1}'.","section":"Theorem 3.11"},{"comment":"Remark 4.13 promises a proof in [17], which is listed as 'in preparation'; this should be marked as forthcoming work or removed if it is not needed for the current argument.","section":"Remark 4.13"}],"recommendation":"major_revision","confidential_remarks":"The main geometric idea is appealing and the applications are within the journal's scope. I am recommending major revision rather than rejection because the load-bearing issues (the Kollár chain condition and the Proposition 3.3 proof) appear fixable: the former by a more careful statement and use of Kollár's theorem or by a subarc argument, the latter by replacing the flawed intermediate identifications with a direct duality proof. The referee's concern about Theorem 2.13 is real and should be addressed head-on; I would ask the authors to cite the precise statement in Kollár's paper and to verify the transversal-intersection condition against that source."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this is a genuinely useful paper. The main theorem says that for an snc log Calabi-Yau pair, a single real torus of dimension equal to the maximal depth of a stratum computes the odd weight filtration, and the analogous statement holds for projective Calabi-Yau degenerations. I don't see that statement in the cited literature; it's a real consolidation of the El Zein-Némethi and Clemens torus constructions, and the P=W-type applications for rational surfaces with nodal anticanonical boundary and for elliptic K3s go beyond what Gross and Zhang had.\n\nThe proof strategy is transparent: reduce to established results, get a direct sum over profound tori, then use Kollár's connectivity theorem to show all profound tori are homotopic. I checked the stress-test objection about (P^2, smooth cubic). I don't think it lands. That pair has delta = 1 and a single profound torus; homotopy independence within one stratum is already in Definition 2.7. If Kollár's theorem is being quoted for two points on the same stratum, the statement needs a 'distinct minimal strata' qualifier, but the needed reduction doesn't require that case. So the load-bearing step survives.\n\nThe real soft spots are presentation-level. Proposition 3.3's proof has an im/ker indexing error: formula (4) gives M_{2k-1}H^k = ker(N^k_k), not im, and the proof later writes M_{2d-2k-1} where the proposition needs M_{2d-2k}. The proposition itself is standard and the intended orthogonality argument is visible, but as written the proof is wrong. A referee should require fixing that. The surface section applies the snc theorem to nodal anticanonical divisors without saying why that's allowed; a reduced nodal curve is snc, so it's fine, but one line would remove the friction. The paper leans on Kollár, Clemens, Saito, and Soldatenkov more heavily than on its own machinery, and I did not verify those imports; nothing in the text indicates a circular dependence, and the sole self-reference [17] is confined to a remark.\n\nBottom line: this deserves a serious referee. I would send it out, ask for the Proposition 3.3 cleanup and a precise statement of the Kollár import, and expect it to come back as a solid contribution for Hodge theorists and P=W people.","headline":"A clean, genuinely new single-torus theorem for log Calabi-Yau pairs and Calabi-Yau degenerations; the main idea holds up and the flaws are presentation-level, not load-bearing.","tokens_in":23198,"tokens_out":12596,"would_cite":true,"duration_ms":122827,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C30","14D07","14J28","14J32"],"pacs":[],"model":"deepseek-v4-flash","headline":"For simple normal crossings log Calabi-Yau pairs and projective Calabi-Yau degenerations, a single real torus computes every odd-degree weight filtration of cohomology.","keywords":["mixed Hodge structure","weight filtration","log Calabi-Yau pair","torus fiber","perverse Leray filtration","P=W equality","Calabi-Yau degeneration","K3 surface"],"falsifier":"Exhibit an snc log Calabi-Yau pair whose boundary has two minimal strata of different cardinalities; then the attached profound tori have different dimensions, so the kernel to any single torus cannot equal $W_{2k-1}H^k(X\\setminus Y;\\mathbb{Q})$ for all $k$, and the theorem's conclusion fails.","tokens_in":22044,"feed_emoji":"🍩","tokens_out":15110,"duration_ms":131856,"temperature":0.7,"pith_summary":"This paper proves that for a simple normal crossings log Calabi-Yau pair, the odd-degree pieces of the weight filtration on the cohomology of the complement are exactly the kernel of restriction to one real torus, whose dimension is the codimension of the deepest stratum of the boundary. The same statement holds for the monodromy weight filtration of a projective Calabi-Yau degeneration, with a torus one dimension lower. The point is not just that one such torus exists but that it is canonical up to homotopy, so a single probe detects the weight filtration at every degree. This makes the weight filtration concretely computable and yields P=W-type equalities: on rational surfaces with nodal anticanonical boundary and on K3 surfaces, the perverse Leray filtration of a torus fibration coincides with the weight filtration.","feed_headline":"One torus captures the weight filtration","feed_subtitle":"For log Calabi-Yau pairs, odd-weight cohomology equals the part a single torus cannot see.","key_machinery":"The mechanism is the profound torus: for each minimal stratum $Y_J$ of the boundary, take a point and loop around each of the $j$ components meeting there. The load-bearing step is Lemma 2.12: a rational curve contained in the $(j-1)$-stratum and meeting the $j$-strata transversally in exactly two points produces a homotopy between the two associated profound tori. A cited theorem on log Calabi-Yau pairs guarantees chains of such rational curves between every pair of minimal strata, so all profound tori are homotopic; together with the generalized Leray cycles this turns the direct sum of tori in Theorem 2.11 into a single torus. For degenerations, the same picture is built from the contraction map, whose preimage of a point in a deep stratum is a torus of dimension $\\delta-1$, and the same curve-chain argument identifies all of those tori.","core_discovery":"The central claim is Theorem 1.1: if $(X,Y)$ is an snc log Calabi-Yau pair and $\\delta$ is the maximal number of components of $Y$ meeting nontrivially, then there is a real torus $T\\subset X\\setminus Y$ such that $W_{2k-1}H^k(X\\setminus Y;\\mathbb{Q})=\\ker(H^k(X\\setminus Y;\\mathbb{Q})\\to H^k(T;\\mathbb{Q}))$ for all $k$; locally $T$ is $\\{(\\varepsilon e^{i\\theta_1},\\ldots,\\varepsilon e^{i\\theta_\\delta},0,\\ldots,0)\\}$. For a projective Calabi-Yau degeneration, the analogous torus has dimension $\\delta-1$ and computes the monodromy weight filtration $M_{2k-1}H^k(X_1;\\mathbb{Q})$ in the same way. The proof shows that every torus attached to a stratum of the boundary can be deformed into a torus attached to a minimal stratum, and that in the Calabi-Yau case all these minimal-stratum tori are homotopic, because chains of rational curves connecting the minimal strata induce homotopies of the corresponding tori. Thus the a priori large collection of tori collapses to a single torus, and the weight filtration is visible as a single kernel.","pith_inferences":["The collapse to one torus is essentially a property of the dual complex of the boundary being connected by rational curves; one could test the same rational-curve-chain condition on other families of log Calabi-Yau pairs to predict where a single torus should exist.","For compact holomorphic symplectic manifolds with Lagrangian torus fibrations, the K3 argument suggests that the monodromy operator built from an isotropic fiber class and a transverse class should reproduce the perverse Leray filtration; a numerical check on known examples would separate the cases where this holds from those where it fails.","The surface theorem suggests a two-dimensional converse: if a log Calabi-Yau surface admits a Lagrangian torus fibration whose general fiber is homotopic to the profound torus, then perverse Leray equals weight, so constructing such fibrations on other rational surfaces would extend the result."],"forward_implications":["For any log Calabi-Yau pair, the dimension of each odd-weight graded piece is bounded by $\\binom{\\delta}{k}$, and the single torus controls the equality cases.","On a rational surface with a reduced nodal anticanonical divisor, the Lagrangian torus fibration satisfies $P_i = W_{2i} = W_{2i+1}$ for its perverse Leray and weight filtrations, giving a P=W-type equality.","On a K3 surface with an elliptic fibration, there is a semistable degeneration whose monodromy weight filtration matches the perverse Leray filtration of the fibration.","If a Betti moduli space admits a log Calabi-Yau compactification with $\\delta = \\dim$, Theorem 1.1(1) produces a real torus of the expected dimension computing the odd weight filtration, exactly the kind of object the P=W conjecture predicts at highest weight."],"supporting_citations":[{"why":"Constructs the generalized Leray cycles and proves they generate the weight filtration on homology.","marker":"[12]"},{"why":"Supplies the theorem that minimal strata of an snc log Calabi-Yau pair have equal size and are linked by chains of rational curves.","marker":"[21]"},{"why":"Describes, via the contraction map, the tori in the smooth fiber whose classes span the image of the monodromy operator.","marker":"[4]"},{"why":"Shows a rational surface with nodal anticanonical divisor is obtained from a toric pair by blowups, enabling the surface fibration construction.","marker":"[16]"},{"why":"Provides almost toric fibrations on such blowups, giving the Lagrangian torus fibration whose generic fiber is a profound torus.","marker":"[31]"},{"why":"Constructs the semistable degeneration of K3 surfaces whose monodromy logarithm is a multiple of the prescribed operator, supplying the degeneration side of the K3 theorem.","marker":"[29]"},{"why":"Gives the decomposition-theorem formula expressing the perverse Leray filtration as kernels of restriction to smooth fibers.","marker":"[7]"},{"why":"Establishes the decomposition theorem for proper holomorphic morphisms, underlying the perverse filtration computation.","marker":"[24]"}],"fun_headline_variants":["One real torus encodes the entire weight filtration","Many tori collapse to one: weight filtration revealed","A lone torus computes odd cohomology","Single torus suffices for log CY weight filtration","One torus yields P=W type results"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on a theorem that every two minimal strata of an snc log Calabi-Yau boundary have the same number of components and are connected by chains of rational curves meeting the deepest strata transversally; if that is false, the profound tori are not necessarily homotopic and one torus would not compute the weight filtration.","fun_headline_variants_meta":{"raw":{"variants":["One real torus encodes the entire weight filtration","Many tori collapse to one: weight filtration revealed","A lone torus computes odd cohomology","Single torus suffices for log CY weight filtration","One torus yields P=W type results"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00057,"raw_usage":{"total_tokens":2682,"prompt_tokens":916,"completion_tokens":1766,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":532,"completion_tokens_details":{"reasoning_tokens":1693}},"tokens_in":532,"tokens_out":1766,"duration_ms":20607,"temperature":1.0,"reasoning_tokens":1693,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:24:11.825392+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit an snc log Calabi-Yau pair whose boundary has two minimal strata of different cardinalities; then the attached profound tori have different dimensions, so the kernel to any single torus cannot equal $W_{2k-1}H^k(X\\setminus Y;\\mathbb{Q})$ for all $k$, and the theorem's conclusion fails.","supporting_citations":[{"cited_title":"Elzein and A","cited_arxiv_id":null,"evidence_quote":"Constructs the generalized Leray cycles and proves they generate the weight filtration on homology."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that minimal strata of an snc log Calabi-Yau pair have equal size and are linked by chains of rational curves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Describes, via the contraction map, the tori in the smooth fiber whose classes span the image of the monodromy operator."},{"cited_title":"Gross, P","cited_arxiv_id":null,"evidence_quote":"Shows a rational surface with nodal anticanonical divisor is obtained from a toric pair by blowups, enabling the surface fibration construction."},{"cited_title":"Symington","cited_arxiv_id":null,"evidence_quote":"Provides almost toric fibrations on such blowups, giving the Lagrangian torus fibration whose generic fiber is a profound torus."},{"cited_title":"de Cataldo and L","cited_arxiv_id":null,"evidence_quote":"Gives the decomposition-theorem formula expressing the perverse Leray filtration as kernels of restriction to smooth fibers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the decomposition theorem for proper holomorphic morphisms, underlying the perverse filtration computation."}],"review_version":1}