{"id":"7297ba6f-d396-4b3f-99ea-022d66818417","arxiv_id":"2412.05927","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new 'categorical residue' construction shows that the formal punctured neighborhood of infinity of a smooth A-infinity category inherits a weak proper Calabi-Yau structure of one dimension lower.","lead":"This paper constructs a categorical residue that transfers Calabi-Yau structures from an A-infinity category to its formal punctured neighborhood of infinity. It uses this to prove chain-level Calabi-Yau structures on Rabinowitz Fukaya categories and on singularity categories.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.7's nondegeneracy claim is asserted but not proved: Proposition 3.19 reduces to the pairing π∞ and calls it nondegenerate without argument.","rationale":"The reader's weakest assumption concerns the external quasi-equivalence Φ used in Theorem 1.2, but the core algebraic engine is Theorem 1.7, whose proof contains an unproven nondegeneracy assertion. If Theorem 1.7 fails, both main applications fail. The gap is not a disagreement with consensus; it is an internal missing step: Proposition 3.19's proof stops at 'which is nondegenerate' without deriving this from the stated lemmas. Because the construction of π∞ involves the cup product, which is not generally injective, the nondegeneracy is not immediate. The proposed concrete test with a simple smooth non-proper CY category would either produce a counterexample or at least force an explicit argument. This does not change the reader's conditional verdict, but it sharpens the condition: the author must supply a complete proof of nondegeneracy of π∞, not merely of its agreement with the residue pairing.","tokens_in":63844,"tokens_out":26946,"duration_ms":254825,"concrete_test":"Specialize Theorem 1.7 to C = Perf(k[x]) (smooth, non-proper, CY dimension 1) with the unit Hochschild cycle as σ. Compute the residue pairing (3.68) on the Yoneda object of k[x] in bC∞ = QCoh(k[x])/Perf(k[x]) using the explicit formulas (3.52), (3.56), (3.67). If the pairing is degenerate on some cohomology class, Theorem 1.7 is false; if it is nondegenerate, it confirms this instance but does not repair the missing general argument. Either way, the computation forces the proof of Proposition 3.19 to be made explicit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem 1.7 rests on Proposition 3.19, which asserts that the residue pairing (3.68) is nondegenerate on cohomology. The proof reduces the residue pairing to the pairing π∞ (3.56) and then states 'which is nondegenerate' after (3.73), but no proof of the nondegeneracy of π∞ is given. π∞ is the composition bC∞(X,Y)⊗bC∞(Y,X) --⊔--> CC*(C^op,Z⊗Z) --∩σ--> CC*_{-n}(C^op,Z⊗Z) --πZ_*--> k[1-n]. While πZ_* is a nondegenerate pairing of bimodules (Cor 3.14) and ∩σ is a quasi-isomorphism (Lemma 2.28), the first map ⊔ is the cup product and may have a kernel; nondegeneracy of the total composition does not follow from nondegeneracy of the final factor. The proof of Prop 3.19 shows the residue pairing agrees with π∞ on cohomology, but does not show that π∞ induces an isomorphism H^*bC∞(X,Y) → H^{n-1-*}bC∞(Y,X)^∨. This is precisely the Poincaré duality conclusion of Theorem 1.7, so the gap is load-bearing for all applications.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an abstract algebraic construction, the categorical residue, which takes a weak smooth Calabi-Yau structure of dimension n on a smooth A-infinity category C and produces a weak proper Calabi-Yau structure of dimension n-1 on the categorical formal punctured neighborhood of infinity bC∞. The central claim is Theorem 1.7. The paper then applies this result in two settings: to prove that the Rabinowitz Fukaya category RW(X) of a non-degenerate Liouville manifold with c1(X)=0 carries a weak proper Calabi-Yau structure (Theorem 1.2), and to prove that Orlov's singularity category of a proper Gorenstein scheme of finite type with trivialized canonical bundle carries a proper Calabi-Yau structure (Theorem 1.6). Along the way the paper develops a framework of Hochschild invariants with bimodule coefficients, canonical pairing systems, and operations ⊔ and ⊓, and studies an open-closed map relating Hochschild homology of the wrapped Fukaya category with Rabinowitz Floer cohomology.","tokens_in":64096,"tokens_out":7455,"duration_ms":77008,"significance":"If Theorem 1.7 is correct, the paper provides a conceptually clean transfer principle: boundary-type Calabi-Yau structures arise canonically from smooth Calabi-Yau structures, with explicit chain-level formulas. This would substantially strengthen earlier cohomology-level statements for Rabinowitz Fukaya categories and would give a new conceptual proof of Calabi-Yau structures on singularity categories. The paper also contains useful algebraic infrastructure for Hochschild operations with multi-module coefficients. However, the current manuscript leaves a load-bearing nondegeneracy claim in Proposition 3.19 essentially unproved, and several key technical lemmas are delegated to 'straightforward computations'. Because the main applications inherit this gap, the central claims are only conditionally established.","major_comments":[{"comment":"The nondegeneracy assertion is not proved. The proof shows that the residue pairing agrees on cohomology with the pairing π∞,X,Y defined in (3.56), and then states 'which is nondegenerate' after (3.73). The map π∞ is a composition bC∞(X,Y)⊗bC∞(Y,X) → CC*(C^op,Z⊗C^op Z) → CC*_{-n}(C^op,Z⊗C^op Z) → k, where the first map is ⊔, the middle map is capping with σ, and the last is πZ_*. Even if πZ_* is nondegenerate on the length-zero subspace and capping with σ is a quasi-isomorphism, nondegeneracy of the composite does not follow from nondegeneracy of one factor: the intermediate compositions may annihilate cohomology classes. A direct argument is needed that for every nonzero [c]∈H^*(bC∞(X,Y)) there exists [d]∈H^{n-1-*}(bC∞(Y,X)) with π∞([c],[d])≠0, for example by showing that π∞ is chain-homotopic to the tautological pairing (2.133)/(5.72). This gap is load-bearing: Theorem 1.7, and consequently Theorem 1.2 through Lemma 6.3 and Theorem 1.6 through Section 4.2, depend on it.","section":"§3.4, Proposition 3.19"},{"comment":"Several technically central arguments are omitted. Lemma 3.13 verifies the canonical pairing system needed to define the map πZ_* via Proposition 2.35, and Lemma 3.15 supplies the cyclic associativity used in Lemma 3.17 to prove that the residue is a chain map. Both are dismissed as 'straightforward computation'. Given the paper's own emphasis on sign conventions, these computations should be written out or relegated to an appendix. Similarly, in the proof of Proposition 8.7, after equation (8.53) the claim that the composition 'induces the same map as the inverse of the linear dual of OC' is asserted 'because of the way we count rigid elements'; this is a nontrivial identification of chain maps and needs a precise justification if Theorem 1.10 is to be substantiated.","section":"§3.3, Lemmas 3.13 and 3.15; §8.3, Proposition 8.7"}],"minor_comments":[{"comment":"Theorem 1.6 states that D^b_sg(X) has a 'proper Calabi-Yau structure', but the proof in §4.2 concludes only a 'weak proper Calabi-Yau structure'. The terminology should be harmonized, since Definition 2.27 distinguishes weak and strong proper Calabi-Yau structures.","section":"§1.2 and §4.2"},{"comment":"The displayed sign contains a typo: '|x1||x2+|x1|+|x2|' should read '|x1||x2|+|x1|+|x2|'.","section":"Equation (2.134)"},{"comment":"There are two entries labeled [S]: Saito's 'Period mapping associated to a primitive form' and Shklyarov's 'Calabi-Yau structures on categories of matrix factorizations'. This creates ambiguity in citations and should be corrected.","section":"References"},{"comment":"The word 'nondegenenrate' should be 'nondegenerate'.","section":"§1.1"},{"comment":"The phrase 'by Lemma we have 2.28' should read 'by Lemma 2.28'.","section":"§3.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies crucially on Theorem 1.1 of [GGV] and on the weak smooth Calabi-Yau structure from [G1], both of which are preprints by the author and collaborators. The editor may wish to verify that these external results are available in final or suitably refereed form. The main technical gap identified in the report, the nondegeneracy claim in Proposition 3.19, appears fixable but is nontrivial and should be addressed before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely new: it defines a categorical residue map on the Hochschild complex of the formal punctured neighborhood of infinity, and shows that a weak smooth Calabi-Yau structure on C induces a weak proper Calabi-Yau structure on bC-infinity. The residue construction is explicit, and the transfer theorem (1.7) is the right kind of engine for the two applications. The Rabinowitz Fukaya application upgrades the Bae-Jeong-Kim cohomology-level pairing to a chain-level one compatible with the A-infinity structure, and the singularity category application is a clean consequence. The paper is careful with signs and sets up the algebraic framework in unusual detail.\n\nThe soft spot is exactly where the stress-test lands. Proposition 3.19 is the proof that the residue pairing is nondegenerate. Its proof reduces the residue pairing to pi-infinity, calls pi-infinity nondegenerate, and stops. But pi-infinity is a composition of three maps: the cup product (square union), capping with sigma, and pi-Z-star. Capping and pi-Z-star are individually good (quasi-isomorphism and nondegenerate pairing respectively), but the cup product can have a kernel, so the total composition need not be nondegenerate. The proof does not show that the cup product is a quasi-isomorphism in this specific bimodule setup. Since Theorem 1.7 and both applications rest on Prop 3.19, this is load-bearing. It may be fixable by a spectral sequence or by showing the square-union map is an iso on the relevant cohomology, but it is not supplied.\n\nOther soft spots are minor: several lemmas are left to 'straightforward computations' (3.13, 3.15, parts of 8.7), and the symplectic theorems rely heavily on the GGV quasi-equivalence, which itself requires non-degeneracy and c1=0. Those are stated assumptions, not hidden flaws.\n\nThis paper is for specialists in symplectic and derived geometry who want chain-level Calabi-Yau structures. It deserves a serious referee, but the referee should demand a complete proof of Prop 3.19 before acceptance. The construction is worth publishing even if the gap turns out to be nontrivial, because the framework and applications are valuable.\n\nRecommendation: send to peer review, with a request to fill the gap in Prop 3.19.","headline":"Original categorical residue construction with plausible main theorems, but the load-bearing nondegeneracy claim in Prop 3.19 is asserted without proof and must be filled before the results can be trusted.","tokens_in":64646,"tokens_out":3415,"would_cite":true,"duration_ms":31796,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E45","53D40","18G80"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the categorical formal punctured neighborhood of infinity of any smooth A-infinity category with a weak smooth Calabi-Yau structure inherits a weak proper Calabi-Yau structure of dimension one less, and uses this to…","keywords":["Calabi-Yau structures","A-infinity categories","categorical formal punctured neighborhood of infinity","Rabinowitz Fukaya category","singularity category","Hochschild homology","categorical residue","Poincaré duality"],"falsifier":"Pick a smooth $A_\\infty$-category $\\mathcal{C}$ with a weak smooth Calabi-Yau structure of dimension $n$ and two objects $X,Y$; if the cohomological residue pairing $H^i(\\widehat{\\mathcal{C}}_\\infty(X,Y))\\otimes H^{n-1-i}(\\widehat{\\mathcal{C}}_\\infty(Y,X))\\to k$ is zero for some degree $i$ on a nonzero class, then $\\widehat{\\mathcal{C}}_\\infty$ is not weak proper Calabi-Yau and Theorem 1.7 would fail. Such a check could be carried out in a computable example, for instance one where the Hochschild chain complex is finite so the pairing can be evaluated explicitly.","tokens_in":63624,"feed_emoji":"🔁","tokens_out":7054,"duration_ms":63078,"temperature":0.7,"pith_summary":"This paper proves a categorical analogue of a classical topological fact: the boundary of a compact manifold with boundary satisfying Poincaré–Lefschetz duality is itself a closed manifold satisfying Poincaré duality one dimension lower. The analogue says that if an $A_\\infty$-category $\\mathcal{C}$ is smooth and carries a weak smooth Calabi-Yau structure of dimension $n$, then its categorical formal punctured neighborhood of infinity $\\widehat{\\mathcal{C}}_\\infty$ carries a weak proper Calabi-Yau structure of dimension $n-1$. The proof produces an explicit Hochschild residue map and verifies that the induced pairing is nondegenerate. Because the wrapped Fukaya category of a Liouville manifold is smooth and Calabi-Yau, this yields a weak proper Calabi-Yau structure on the Rabinowitz Fukaya category, and similarly yields a proper Calabi-Yau structure on Orlov's singularity category. The result upgrades previously known cohomology-level duality statements to chain-level structures compatible with the $A_\\infty$-structure.","feed_headline":"A residue makes each Calabi-Yau category's boundary Calabi-Yau too","feed_subtitle":"Rabinowitz Fukaya categories and singularity categories inherit proper duality one dimension lower.","key_machinery":"The load-bearing object is the categorical formal punctured neighborhood of infinity, $\\widehat{\\mathcal{C}}_\\infty$, defined as the essential image of the Yoneda embedding into the category of Calkin modules $\\operatorname{Fun}(\\mathcal{C}^{\\mathrm{op}}, \\mathrm{Ch}_k/\\mathrm{Perf}_k)$; it is nonzero exactly when $\\mathcal{C}$ fails to be proper. The argument also relies on the canonical pairing systems of Section 2.8 and the cap product with the smooth Calabi-Yau cycle $\\sigma$, which is a quasi-isomorphism by Lemma 2.28. These combine into the residue map $\\operatorname{res}$, a degree $1-n$ chain map out of Hochschild chains of $\\widehat{\\mathcal{C}}_\\infty$, whose induced pairing is the graded-symmetric, nondegenerate residue pairing.","core_discovery":"The central claim is Theorem 1.7: for any smooth $A_\\infty$-category $\\mathcal{C}$ with a weak smooth Calabi-Yau structure of dimension $n$, the categorical formal punctured neighborhood $\\widehat{\\mathcal{C}}_\\infty$ admits a weak proper Calabi-Yau structure of dimension $n-1$, given by the residue $\\operatorname{res}\\colon CC_{n-1}(\\widehat{\\mathcal{C}}_\\infty)\\to k$. The residue is built from a canonical pairing system on the bimodule cones that form the morphism spaces of $\\widehat{\\mathcal{C}}_\\infty$, capped with the smooth Calabi-Yau cycle $\\sigma$; Proposition 3.19 proves the induced pairing on cohomology is nondegenerate. Theorem 1.2 and Theorem 1.6 are the concrete payoffs: the Rabinowitz Fukaya category of a non-degenerate Liouville manifold with $c_1(X)=0$ is weak proper Calabi-Yau of dimension $n-1$, and Orlov's singularity category of a Gorenstein scheme of finite type with trivial canonical bundle is proper Calabi-Yau of dimension $n-1$.","pith_inferences":["The theorem suggests that being Calabi-Yau is hereditary: the categorical boundary of any smooth Calabi-Yau category is proper Calabi-Yau one dimension down, so categories that are 'almost proper' still have well-defined Poincaré duality at their boundary.","If the residue lifts to an $S^1$-invariant chain map, the weak proper structure would become strong, which the author expects would connect to pre-Calabi-Yau structures on the wrapped Fukaya category.","The same formal construction may extend to $\\mathbb{Z}/2$-graded settings, which would give a conceptual explanation for the proper Calabi-Yau structure on matrix factorization categories via Orlov's equivalence.","The pairing-respecting open-closed map suggests that the categorical residue pairing and the tautological pairing on Rabinowitz Floer cohomology should agree at chain level; checking this explicitly would test the compatibility of the algebraic and Floer-theoretic formalisms."],"forward_implications":["Theorem 1.2: the Rabinowitz Fukaya category $RW(X)$ of a non-degenerate Liouville manifold with $c_1(X)=0$ carries a weak proper Calabi-Yau structure of dimension $n-1$, not just a cohomology-level pairing.","Theorem 1.6: for a Gorenstein scheme of finite type with trivial canonical bundle, the dg enhancement of Orlov's singularity category is proper Calabi-Yau of dimension $n-1$, without assuming isolated singularities.","The residue map depends only on the homology class of the smooth Calabi-Yau cycle $\\sigma$.","The open-closed map $OC_R$ from Hochschild homology of $W$ with coefficients in $RW$ to Rabinowitz Floer cohomology is a quasi-isomorphism and respects the pairings.","Any smooth $A_\\infty$-category with a weak smooth Calabi-Yau structure has a boundary invariant $\\widehat{\\mathcal{C}}_\\infty$ that is itself weak proper Calabi-Yau, so the construction is fully general and not tied to Floer geometry."],"supporting_citations":[{"why":"Provides the quasi-equivalence $\\Phi\\colon RW(X)\\to\\widehat{W}_\\infty$ that carries the residue from the categorical punctured neighborhood to the Rabinowitz Fukaya category.","marker":"[GGV]"},{"why":"Constructs the weak smooth Calabi-Yau structure on the wrapped Fukaya category whose homology class $\\sigma$ feeds the residue construction.","marker":"[G1]"},{"why":"Introduces the categorical formal punctured neighborhood of infinity and supplies the quasi-equivalence between $D^b_{sg}(X)^{\\mathrm{op}}$ and $\\widehat{D^b\\mathrm{Coh}(X)}_\\infty$ used in Theorem 1.6.","marker":"[E]"},{"why":"Provides the definitions and homological-algebra framework for Calabi-Yau structures on $A_\\infty$-categories.","marker":"[KS]"},{"why":"Establishes the cohomology-level Calabi-Yau pairing on the Rabinowitz Fukaya category that Theorem 1.2 refines to chain level.","marker":"[BJK]"},{"why":"Supplies the statement that a trivialization of the canonical bundle is equivalent to a strong smooth Calabi-Yau structure on $D^b\\mathrm{Coh}(X)$.","marker":"[BD]"},{"why":"Gives Serre duality and the nondegenerate pairing for singularity categories in the isolated case, the setting Theorem 1.6 generalizes.","marker":"[M]"}],"fun_headline_variants":["Residue transfers Calabi-Yau to the categorical boundary","One dimension lower: residue yields proper Calabi-Yau","Punctured infinity gets Calabi-Yau from a residue","Boundary duality via residue: Calabi-Yau persists"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction requires the original category to already have a weak smooth Calabi-Yau structure, and the Rabinowitz application additionally requires the quasi-equivalence $\\Phi\\colon RW(X)\\to\\widehat{W}_\\infty$, which is established only for non-degenerate Liouville manifolds with $c_1(X)=0$.","fun_headline_variants_meta":{"raw":{"variants":["Residue transfers Calabi-Yau to the categorical boundary","One dimension lower: residue yields proper Calabi-Yau","Punctured infinity gets Calabi-Yau from a residue","Boundary duality via residue: Calabi-Yau persists"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000208,"raw_usage":{"total_tokens":1379,"prompt_tokens":893,"completion_tokens":486,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":509,"completion_tokens_details":{"reasoning_tokens":417}},"tokens_in":509,"tokens_out":486,"duration_ms":4771,"temperature":1.0,"reasoning_tokens":417,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:12:17.095565+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick a smooth $A_\\infty$-category $\\mathcal{C}$ with a weak smooth Calabi-Yau structure of dimension $n$ and two objects $X,Y$; if the cohomological residue pairing $H^i(\\widehat{\\mathcal{C}}_\\infty(X,Y))\\otimes H^{n-1-i}(\\widehat{\\mathcal{C}}_\\infty(Y,X))\\to k$ is zero for some degree $i$ on a nonzero class, then $\\widehat{\\mathcal{C}}_\\infty$ is not weak proper Calabi-Yau and Theorem 1.7 would fail. Such a check could be carried out in a computable example, for instance one where the Hochschild chain complex is finite so the pairing can be evaluated explicitly.","supporting_citations":[],"review_version":1}