{"id":"eb8133b5-1331-4e8e-8fa8-25948f94c239","arxiv_id":"2502.04148","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A new category of Hodge microsheaves is defined via gluing mixed Hodge modules, and it reproduces Hain's loop Hodge structure on P^n and gives a mixed-geometric proof of Etgü–Lekili Koszul duality.","lead":"The authors introduce a Hodge-theoretic version of microsheaves, a tool that decorates symplectic geometry with Hodge structures, and use it to give a new geometric explanation of two known results: the Hodge structure on loops of complex projective space and the Koszul duality of Ginzburg algebras for A_n plumbings. The framework opens a path toward a 'mixed geometry' understanding of wrapped Fukaya categories.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The saturated-system dependence is load-bearing: the A_n proof requires Hodge lifts of unipotent local systems (A_s, B_s, P_s, Q_s) to lie in the chosen saturated subcategory M, but Example 2.19(2) as stated may not contain them.","rationale":"The reader's weakest_assumption correctly identified the saturated-system dependence as the key load-bearing premise. I refine this: the issue is not merely non-canonicity in principle, but a concrete risk that the specific saturated system used in the A_n plumbing proof (Example 2.19(2) glued via Fourier transforms) may not contain the unipotent local systems and Hodge lifts required by the Appendix. The paper's own Lemma 9.74–9.78 asserts these lifts exist in M(V), but the verification is computational and not systematically tied to the morphism spaces of Example 2.19. This is a checkable gap, and if it fails, Corollary 8.11 would not follow as stated. I do not dispute the extensive explicit computations in the appendix or the independent support from known results (EL17, GPS24a, HW08); these are real. I also note the secondary dependence on Arai's separate preprint for Theorem 6.6, but the saturated-system concern is more central because it affects the framework itself and the A_n application. My read does not change the reader's conditional verdict: the paper should be accepted only conditionally on verifying that the saturated system indeed contains the objects used in the main constructions, or on weakening the claims to be relative to the chosen saturated system.","tokens_in":79404,"tokens_out":44448,"duration_ms":449766,"concrete_test":"Check whether the Hodge lifts used in §9.6 lie in the saturated model. Concretely, take M(V) to be the restriction of Example 2.19(2) to the affine chart V, and test whether the rank-2 unipotent local system A_2[1] (the intermediate extension of the Jordan-block local system L_2 on C^*) is contained in the pretriangulated closure of M^{pre} by computing its extension class in Ext^1_{MHM}(C_V[1], C_V[1]) and comparing with the image of Q·[P^1] under restriction. If the extension class is zero after restricting [P^1] to V, then A_2[1] is not in M(V); one should then recompute Corollary 8.11 using an enlarged saturated system that contains A_s, B_s, P_s, Q_s. If the resulting Adams-graded endomorphism algebra B changes, the theorem is choice-dependent and the central claim needs qualification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction (Definition 4.7) defines the Hodge microsheaf category µM^c_L(X) only after choosing a saturated non-full subcategory M^c_{(j,J)} of √MHM^c on each chart. All Hodge endomorphism computations in the two main theorems are performed inside this chosen M. For the A_n plumbing (Corollary 8.11), the proof in Appendix §9.6 requires that the building blocks A_s, B_s, P_s, Q_s (unipotent monodromic mixed Hodge modules with Jordan-block local systems) and the morphisms f_s of Definition-Lemma 9.65 lie in M(V), the restriction of the saturated model of Example 2.19(2). But Example 2.19(2) starts with only constant sheaves C_{P^1}[1] and skyscrapers C_i, with morphism spaces spanned by identity, the fundamental class [P^1], and the canonical maps u_i, ι_i. Nontrivial unipotent local systems such as A_2[1] are extensions of constant sheaves whose extension class in H^1 of the open chart is invisible after restricting [P^1] to the affine chart; it is not shown that these objects or the required Hodge lifts lie in the pretriangulated closure of M^{pre}. If some required lift or morphism is not in M, then H^{∞,H}_j is not an object of the saturated category used in Lemma 8.6–8.11, and the Adams grading of the endomorphism algebra B may not match the Ginzburg algebra. The paper asserts (Lemma 9.74–9.78) that the necessary Hodge enhancements exist, but the verification is by 'one can check' statements and direct diagrams rather than a systematic proof that the gluing restricts to M. This is a concrete, checkable gap in the central claim, distinct from the secondary reliance on Arai [Ara] for Theorem 6.6.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces a Hodge-theoretic enhancement of the category of microsheaves for holomorphic exact symplectic manifolds whose Lagrangian cores are of Fourier type. The construction proceeds by gluing categories of monodromic mixed Hodge modules via Fourier transforms, following the microlocal Riemann–Hilbert correspondence of Côté–Kuo–Nadler–Shende. The paper also defines a notion of Hodge wrapping and applies it to two concrete settings. First, for X = P^n, it claims that the saturated Hodge endomorphism algebra of the Hodge-wrapped cotangent fiber recovers Hain's mixed Hodge structure on the chains of the based loop space (Theorem 6.6). Second, for A_n plumbings of T*P^1, it constructs Hodge lifts of microlocal skyscraper sheaves and proves (Corollary 8.11) that the Adams-graded Hodge endomorphism algebra of the lifted cocores is quasi-isomorphic to the Ginzburg dga, giving a mixed-geometric proof of Etgü–Lekili's Koszul duality. The proofs rely on an explicit and lengthy appendix that describes the relevant microlocal skyscrapers and their Hodge lifts.","tokens_in":79897,"tokens_out":10756,"duration_ms":99914,"significance":"If the main theorems are correct, the paper would be a significant step toward a microlocal Hodge theory for Weinstein manifolds, providing Hodge structures on wrapped Fukaya categories and giving a conceptual explanation of Adams gradings in symplectic Koszul duality. The explicit construction of microlocal skyscraper sheaves and their Hodge lifts for A_n plumbings is a valuable technical contribution. The paper also formulates a number of conjectures that will likely guide future work. However, the foundational construction of Hodge microsheaves is only sketched, and the main applications depend on the choice of an auxiliary saturated system whose properties are not fully verified; these issues currently weaken the claims.","major_comments":[{"comment":"The category µM^c_L(X) depends on the choice of a saturated system (Definition 4.7), and the paper does not prove independence of this choice. For the A_n plumbing, the chosen saturated model is the category M of Example 2.19(2), which is the pretriangulated closure of a category generated by C_{P^1}[1] and skyscraper sheaves C_i, with morphism spaces spanned by id, [P^1], u_i, ι_i. The appendix (Lemma 9.74) asserts that the unipotent monodromic Hodge modules A_s, B_s, P_s, Q_s and the morphisms of Definition-Lemma 9.65 admit Hodge enhancements inside M(V), the restriction of M to an affine chart. This is not verified, and in fact appears false as stated: on the affine chart V, the fundamental class [P^1] restricts to zero, and the generator set has no morphism whose cone could produce the nontrivial local system A_2[1], which is an extension of C_W by C_W with extension class in H^1(W). Since the Hodge wrapping H^{∞,H}_j is built from these objects and the saturatedness of M is used to identify its Hodge endomorphism algebra with the Ginzburg algebra (Corollary 8.11), this is a load-bearing gap.","section":"§4.4, Example 4.9, §9.6"},{"comment":"The verification that the required Hodge structures exist is carried out by direct diagrams and ‘one can check’ statements rather than by a systematic proof that the gluing construction restricts to the saturated subcategories M(V). In particular, Lemma 9.76 claims that the morphisms f_s of Definition-Lemma 9.65 lift to Hodge morphisms A_s → A_s(−1), etc., but it is not shown that these lifts lie in the Hom-spaces of M(V) as defined in Example 2.19(2). If any of these lifts or morphisms is not in M, then the object H^{∞,H}_j constructed in Theorem 9.82 does not lie in µM^C(X_Γ), and the Adams grading of the algebra B computed in Lemma 8.6 and Corollary 9.83 need not coincide with the Ginzburg algebra. A complete proof needs either to show directly that these objects and morphisms are in the saturated model, or to construct a different saturated system that contains them while preserving the Adams-grading computations.","section":"§9.6, Lemmas 9.74–9.78"},{"comment":"The proof of Theorem 6.6 depends on the unpublished preprint [Ara] for the explicit sequence of geodesic-flow times and the constructibility of Φ_{t_i}(C_x) on P^n. The paper says that the argument of [Ara] can be adapted, but it does not provide the adaptation. As a result, Theorem 6.6 is conditional on an external result that has not yet appeared in a peer-reviewed venue. The manuscript should either include a complete proof of the needed statement or clearly state the theorem as conditional on [Ara].","section":"§6.3, Theorem 6.6"},{"comment":"The gluing construction of the category µMHM_L(X) is only sketched. While the equivalence with µCsh_L(X) in the sheaf case is justified by comparison with the gluing description of [CKNSa], the Hodge case is described by the sentence ‘By replacing Sh with √MHM, we define µMHM_L(X)’. It is not proved that the Fourier transforms and specialization functors on monodromic mixed Hodge modules (Section 3.3) are compatible with the diagram defining the gluing, nor that the resulting category is independent of the choices of Darboux coverings and identifications. Since this construction underlies all of the paper's results, a full and rigorous definition is needed before the main theorems can be considered established.","section":"§4.4"}],"minor_comments":[{"comment":"The notation (d) for the Tate twist in Definition 2.1 is not used consistently; throughout the paper the twist is denoted (1) or (1/2). Please unify the notation.","section":"§2.1"},{"comment":"In equation (3.16), the statement ‘FL^2 = id’ should be clarified: the square is the identity as an autoequivalence of the square-root category, not as a functor on the original category.","section":"§3.3"},{"comment":"In the displayed equation of the proof, the interchange of colimit and direct sum is not justified; a brief justification or a reference would help.","section":"§5.3, Lemma 5.10"},{"comment":"The expression (F^0F^{-1})|_{m_j} ∩ (W_1F^{-1})|_{m_j} is terse; it would be helpful to spell out how the Hodge and weight filtrations on the nearby cycle ψ_{m_j}F^{-1} are induced from the filtrations on F^{-1}.","section":"§9.6, Theorem 9.82"},{"comment":"The appendix is very long (over sixty pages) and is organized into many subsubsections; a short introductory overview of the strategy and a dependency diagram among the lemmas would greatly improve readability.","section":"§9 (Appendix)"}],"recommendation":"major_revision","confidential_remarks":"The paper is ambitious and contains substantial original ideas, but the foundational construction of Hodge microsheaves is not yet rigorous, and the A_n proof depends on nontrivial facts about the saturated model that are asserted rather than proved. The dependence on Arai's unpublished preprint for the P^n theorem is a further concern. The explicit computations in the appendix are impressive and likely to be useful regardless of the outcome. Given the scope of the gaps, I cannot recommend acceptance in the current form; a thorough revision that addresses the saturated-system issue and either proves or clearly isolates the dependence on [Ara] would be needed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere is my read on arXiv:2502.04148. The paper introduces Hodge microsheaves—a Hodge-theoretic enhancement of microsheaves on holomorphic exact symplectic manifolds—and proves two concrete things: Theorem 6.6, recovering Hain's Hodge structure on chains of the based loop space for P^n, and Corollary 8.11, giving a mixed-geometric origin for the Adams grading in Etgü–Lekili's Koszul duality for A_n plumbings. The framework is genuinely new, and the explicit appendix constructing microlocal skyscrapers for plumbings is valuable; I have not seen that written out in this detail before.\n\nThe main soft spot is the saturated-system dependence. The category µM^c_L(X) is defined only after choosing a saturated non-full subcategory of mixed Hodge modules, and all the endomorphism computations happen inside that choice. For the A_n plumbing, the proof in §9.6 needs the unipotent monodromic objects A_s, B_s, P_s, Q_s—and the morphisms between them—to lie in the restricted saturated category M(V) coming from Example 2.19(2). I do not see a proof that the extension classes defining A_2[1] and its relatives are in the morphism spaces of M. Lemma 9.74 asserts these enhancements with 'one can check' and diagrams, but it never shows that the relevant cones are in the pretriangulated closure of the generators. If they are not, the Adams grading of the endomorphism algebra in Corollary 8.11 will not match the Ginzburg algebra. This is a concrete, checkable gap, and it is load-bearing for the A_n result.\n\nSecondary issues: the P^n theorem depends on Arai's separate preprint for the geodesic-flow times and constructibility, and the gluing construction in §4.4 is sketched rather than fully worked out. Neither is fatal if the saturated-system question gets resolved.\n\nWho should read this? Microlocal sheaf theorists, symplectic geometers working on wrapped Fukaya categories, and Hodge theorists will want to know it. It deserves a serious referee: the framework is important and the main theorems are checkable. I would send it to review, but the referee should be specifically asked to verify that the necessary Hodge lifts lie in the saturated category for the plumbing example.\n\nBest,\n[You]","headline":"New Hodge microsheaf framework with real applications; the A_n plumbing proof has a checkable gap about whether the needed unipotent local systems actually belong to the chosen saturated category.","tokens_in":80452,"tokens_out":8322,"would_cite":true,"duration_ms":75148,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F08","32S35","53D37"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper defines Hodge microsheaves and shows they recover the classical Hodge structure on chains of based loop spaces and the Koszul duality of Ginzburg algebras for A_n plumbings.","keywords":["Hodge microsheaves","mixed Hodge modules","microlocal sheaf theory","wrapped Fukaya category","saturated mixed structure","Koszul duality","Ginzburg algebra","based loop space Hodge structure"],"falsifier":"Compute the Hodge endomorphism algebra of the wrapped cotangent fiber for $\\mathbb{P}^2$ under two different saturated systems: if the resulting Hodge numbers or weights differ, the claimed independence of the choice fails. Alternatively, test the predicted degree-weight pairs $(0,0),(1,2),(4,6),(5,8),\\dots$ for the homology of the based loop space of $\\mathbb{P}^2$ against an independent bar-construction calculation.","tokens_in":79207,"feed_emoji":"🌀","tokens_out":9962,"duration_ms":89247,"temperature":0.7,"pith_summary":"This paper introduces Hodge microsheaves, a Hodge-theoretic enhancement of the microsheaf category for holomorphic exact symplectic manifolds whose Lagrangian core is of Fourier type. The aim is to equip wrapped objects, especially cotangent fibers and cocores, with mixed Hodge data, so that Hodge-theoretic invariants become accessible at the microlocal level. If the construction is right, two known results gain a mixed-geometric origin: the mixed Hodge structure on chains of based loop spaces of algebraic varieties, and the Koszul duality of Ginzburg algebras attached to A_n plumbings. The paper proves both in the stated cases, with the loop-space theorem for projective space depending on an explicit geodesic-flow computation supplied by a separate preprint.","feed_headline":"Hodge microsheaves encode loop Hodge structure and Koszul duality","feed_subtitle":"A microlocal lift of wrapped sheaves carries Hodge data and recovers the loop-chain and Ginzburg-algebra gradings.","key_machinery":"The engine is a saturated mixed structure on microsheaf categories. For a Lagrangian core of Fourier type, the paper assigns to each stratum a saturated non-full subcategory of mixed Hodge modules, then glues these with Fourier transforms and specialization functors to form $\\mu\\mathrm{MC}_L(X)$; saturation means $\\bigoplus_d \\mathrm{Hom}_{\\widehat{D}}(M, N(d)) \\cong \\mathrm{Hom}_D(F(M), F(N))$, so mixed Hodge gradings survive as Adams gradings on endomorphism algebras. Hodge wrapping is then computed by taking an inductive system of mixed Hodge modules whose underlying sheaves are the images of the wrapped object under successive Reeb-flow quantizations.","core_discovery":"The central claim is that a saturated category of Hodge microsheaves $\\mu\\mathrm{MC}_L(X)$ can be constructed for any holomorphic exact symplectic manifold with a Lagrangian core $L$ of Fourier type, by gluing mixed Hodge modules along normal-bundle strata with Fourier and specialization functors. Saturation guarantees that the endomorphism algebra of a Hodge-lifted object carries an Adams-type grading equal to the total Tate-twisted Hom in the mixed category. The paper then proves two applications: for $X=\\mathbb{P}^n$, the saturated Hodge endomorphism algebra of the Hodge-wrapped cotangent fiber matches the classical mixed Hodge structure on the chains of the based loop space; for $A_n$ plumbings of $T^*\\mathbb{P}^1$, the Adams-graded Ginzburg algebra is isomorphic to the Hodge endomorphism algebra of the lifted cocores, giving a mixed-geometric proof of the known Koszul duality.","pith_inferences":["If the saturated-system choice can be shown canonical, the Hodge microsheaf category would become an intrinsic invariant of the holomorphic exact symplectic manifold, not just of the chosen core and auxiliary subcategories; a natural first test is comparing $\\mathbb{P}^n$ weights across different saturated systems.","The same gluing prescription may extend to other conical symplectic resolutions, where it would give mixed-geometric proofs of Koszul duality for category $\\mathcal{O}$ beyond the $A_n$ plumbing case; the paper's relative-core example already points in that direction.","The dependence on an explicit geodesic-flow sequence suggests a general principle: constructibility of the flow kernels controls the existence of Hodge wrappings, so testing the paper's complex-wrapping conjecture on other Fano manifolds would delimit where the method applies.","The authors' motivation suggests a Hodge-brane version of Fukaya categories in which local systems are replaced by variations of Hodge structure; the sheaf-theoretic route taken here may be the only currently feasible way to make that idea precise."],"forward_implications":["For $X=\\mathbb{P}^n$, the wrapped cotangent fiber admits a Hodge lift whose saturated endomorphism algebra reproduces the classical mixed Hodge structure on chains of the based loop space, giving a microlocal construction of that structure.","For $A_n$ plumbings of $T^*\\mathbb{P}^1$, the lifted cocores give an Adams-graded endomorphism algebra isomorphic to the Ginzburg algebra, so the known Koszul duality acquires a mixed-geometric explanation.","Whenever an object in the saturated system admits an inductive system of mixed Hodge modules along Reeb-flow images, its Hodge wrapping exists and is computable as a colimit.","Saturated mixed structures can be defined on microsheaf categories for Lagrangian cores of Fourier type, which implies that Adams gradings in such geometric settings come from mixed Hodge theory.","For the relative core of $A_n$ plumbings, the category $\\mathcal{O}$ is recovered from Hodge microsheaves and is Koszul, matching the symplectic-duality picture."],"supporting_citations":[{"why":"supplies the theory of mixed Hodge modules from which the saturated systems are drawn.","marker":"[Sai90]"},{"why":"provides microsupport, microlocalization, and Fourier–Sato transforms used in gluing microsheaf categories.","marker":"[KS94]"},{"why":"identifies wrapped Fukaya categories with microsheaf categories, making cotangent fibers and cocores objects to be Hodge-lifted.","marker":"[GPS24a]"},{"why":"constructs the mixed Hodge structure on based loop-space chains that the $\\mathbb{P}^n$ theorem is claimed to reproduce.","marker":"[Hai87]"},{"why":"supplies the explicit geodesic-flow times and constructibility of the flow images on which the $\\mathbb{P}^n$ loop-space proof depends.","marker":"[Ara]"},{"why":"establishes the Koszul duality of Ginzburg algebras for $A_n$ plumbings that the paper reproves from mixed geometry.","marker":"[EL17]"},{"why":"gives the saturation condition and the mixed-geometry philosophy converting Hodge gradings into Adams gradings.","marker":"[BGS96]"},{"why":"defines the Fourier transform of monodromic mixed Hodge modules used in the construction of Hodge microsheaves.","marker":"[Sai22]"},{"why":"identifies wrapping with the colimit of geodesic-flow quantizations, used to detect Hodge wrappings.","marker":"[Kuo23]"}],"fun_headline_variants":["Hodge microsheaves link loop spaces and Koszul duality","Microlocal Hodge theory unifies loop space and Ginzburg algebra","Hodge sheaves on cotangent bundles yield Koszul duality","Mixed Hodge microsheaves prove Koszul duality for plumbings","Saturated Hodge microsheaves recover loop and Koszul structures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a chosen saturated system of mixed Hodge modules yields canonical Hodge structures on the wrapped objects, with the projective-space loop-space theorem additionally relying on an external preprint that supplies the explicit geodesic-flow times and constructibility of the flow images.","fun_headline_variants_meta":{"raw":{"variants":["Hodge microsheaves link loop spaces and Koszul duality","Microlocal Hodge theory unifies loop space and Ginzburg algebra","Hodge sheaves on cotangent bundles yield Koszul duality","Mixed Hodge microsheaves prove Koszul duality for plumbings","Saturated Hodge microsheaves recover loop and Koszul structures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000215,"raw_usage":{"total_tokens":1376,"prompt_tokens":839,"completion_tokens":537,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":455,"completion_tokens_details":{"reasoning_tokens":440}},"tokens_in":455,"tokens_out":537,"duration_ms":5561,"temperature":1.0,"reasoning_tokens":440,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T23:21:03.208482+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Hodge endomorphism algebra of the wrapped cotangent fiber for $\\mathbb{P}^2$ under two different saturated systems: if the resulting Hodge numbers or weights differ, the claimed independence of the choice fails. Alternatively, test the predicted degree-weight pairs $(0,0),(1,2),(4,6),(5,8),\\dots$ for the homology of the based loop space of $\\mathbb{P}^2$ against an independent bar-construction calculation.","supporting_citations":[],"review_version":1}