{"id":"59d0a0d1-3122-4e7e-9c44-abc7f3d78d75","arxiv_id":"2501.16916","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For log smooth rigid analytic varieties, Kummer pro-étale B_dR cohomology is isomorphic to log de Rham cohomology, and logarithmic B_dR^+ cohomology gives degeneration of Hodge-Tate and Hodge-log de Rham spectral sequences when proper.","lead":"This paper computes p-adic cohomologies of logarithmic rigid analytic spaces, matching Kummer pro-étale cohomology of B_dR with logarithmic de Rham cohomology. It introduces a logarithmic B_dR^+ cohomology and proves that Hodge-Tate and Hodge-de Rham spectral sequences degenerate for proper log smooth spaces.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The one-line proof of Lemma 5.16 is the load-bearing gap: the log Hodge–Tate isomorphism R^iλ_*Ô_X ≃ Ω^{i,log}_X(-i) is used to split Rν_*Ô_X and count dimensions in Theorem 5.20, but the log analogue of [Sch13b, Lemma 3.24] is not actually supplied.","rationale":"The reader's weakest_assumption is Lemma 5.16, and my stress test confirms that this is the right place to put pressure. I read the proof of Theorem 5.20 as a chain: Lemma 5.16 gives the graded pieces R^iλ_*Ô_{X_prokét} ≃ Ω^{i,log}_X(-i); Proposition 5.15 converts this into a description of R^iλ_*Ô; Proposition 5.22 and Proposition 5.25 turn a B_dR^+/t^2-lift into a splitting of Rν_*Ô; and Theorem 5.20 then uses that splitting to identify the E_2-page and to count dimensions. The lemma is the only step whose proof consists of a bare citation to the non-log setting; everything later inherits its content. I checked the surrounding text for alternative routes. The comparison theorem 4.6 and the B_dR^+ comparison theorem 5.19 are built on [DLLZ23a], [DLLZ23b], and Bosco's work and are reasonably supported; the local Poincaré lemmas in Section 3 are quoted from [DLLZ23a]. The log Hodge–Tate statement, however, is not quoted from a published source in the form used here: [DLLZ23b, Theorem 6.2.1] gives a primitive comparison, not the full R^iλ_* identification, and the proof of Proposition 5.15 explicitly says 'same argument ... with a log-version ... stated below', so all the weight falls on Lemma 5.16. I do not see an internal contradiction, numerical fitting, or data manipulation; the concern is proof debt rather than observed falsehood. I also note that Proposition 5.21(3) is sketched rather than fully proved, but that input is more likely to be fillable from the deformation-theoretic arguments in [BMS18], whereas Lemma 5.16 contains the genuinely new logarithmic content. Therefore the appropriate verdict remains CONDITIONAL, with the condition being a complete proof of Lemma 5.16.","tokens_in":27060,"tokens_out":22770,"duration_ms":226656,"concrete_test":"Write out the proof of Lemma 5.16 in the local model case X = Spa(C<|T_1,…,T_d|>) with log structure given by T_1⋯T_r = 0, using the pro-Kummer cover obtained by extracting p-power roots of the T_i. On this cover, compute the Čech 1-cocycle representing the boundary image of δ(e_i) in R^1λ_*Ô_{X_prokét}(1), and check that the resulting map ⊕_{i=1}^r O_X δ(T_i) → R^1λ_*Ô_{X_prokét}(1)^▼ is an isomorphism. Then verify that exterior powers of this map give isomorphisms for all i≥2. Cross-check the trivial-log case r=0 against [Sch13b, Lemma 3.24]. If the computation yields a nonzero cokernel, or if the exterior powers are not isomorphisms, then Theorem 5.20(ii) fails as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive condition for Theorem 5.20 is the logarithmic Hodge–Tate isomorphism of Lemma 5.16, proved by the single sentence 'Similar to [Sch13b, Lemma 3.24]'. This lemma is not a peripheral restatement: Proposition 5.15 uses it to identify R^iλ_*Ô_{X_prokét} with Ω^{i,log}_X(-i), Proposition 5.25 uses that identification to split Rν_*Ô_{X_prokét}, and Theorem 5.20 uses the splitting both to identify the E_2-page of the Hodge–Tate spectral sequence and to obtain the dimension equality for the Hodge–log de Rham degeneration. The one-sentence proof omits the two points that are genuinely log-specific: first, the Kummer sequence 0→Z_p(1)→lim_{←×p}ν^{-1}M^{gp}→ν^{-1}M^{gp}→0 gives a boundary map M^{gp}→R^1λ_*Z_p(1), and one must prove that this map extends OX-linearly to all of Ω^{1,log}_X and is an isomorphism there; second, one must prove that exterior powers of this map give the corresponding identifications for all i≥2, not just the first Hodge–Tate map. Neither point follows formally from the smooth case, because the boundary map records the log monodromy at the divisor and its OX-linearity uses the log derivation δ:M^{gp}→Ω^{1,log}_X. The rest of Section 5.4 is conditional on this lemma: if the log Hodge–Tate map is only an injection with nontrivial cokernel, the splitting in Proposition 5.25 fails and the dimension count in Theorem 5.20 does not go through. The B_dR^+/t^2-lifting proposition is also compressed, but it is a deformation-theoretic input more likely to be fillable from [BMS18], whereas Lemma 5.16 contains the genuinely new log content.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops logarithmic p-adic cohomology for log smooth rigid analytic varieties. In the first part, it computes the Kummer pro-étale cohomology of B_dR and B_dR^+ for log smooth rigid analytic varieties over a discretely valued field, with coefficients in a filtered module with integrable log connection (Theorem 4.6), generalizing Bosco's theorem to the log setting. In the second part, for X over C, it introduces a logarithmic B_dR^+-cohomology RΓ_{B_dR^+}(X) by applying the Lη_f functor to Rλ_*B_dR^+ (Definition 5.7), proves that this theory deforms log de Rham cohomology (Theorem 5.13), establishes a log de Rham–étale comparison (Theorem 5.19), and states the main degeneration theorem (Theorem 5.20): for proper log smooth X of dimension d, the Hodge–log de Rham spectral sequence degenerates at E_1, the Hodge–Tate spectral sequence degenerates at E_2, and H^i_{B_dR^+}(X) is finite free over B_dR^+.","tokens_in":27532,"tokens_out":9984,"duration_ms":90274,"significance":"If the main results are fully established, this is a substantial extension of Scholze's p-adic Hodge theory for proper smooth rigid analytic varieties to the logarithmic setting, with potential applications to semistable comparison theorems and to the cohomology of open varieties via compactification. The paper's structure is commendable: it reduces many statements to established results of Scholze, Bosco, Diao–Lan–Liu–Zhu, Katz, and Kiehl, and the use of condensed mathematics is consistent. The construction of logarithmic B_dR^+-cohomology via Lη_f is a natural and promising framework. No free parameters or fitted constants enter the arguments; all claims are meant to be derived from external proven results. However, the central degeneration theorem depends on two assertions whose proofs are not supplied in adequate detail, in particular the logarithmic Hodge–Tate isomorphism of Lemma 5.16 and the existence of log smooth B_dR^+/t^2-lifts in Proposition 5.21(3). These gaps are load-bearing for the main theorem and need to be completed before the central claims can be regarded as established.","major_comments":[{"comment":"The proof of Lemma 5.16 is a single sentence, 'Similar to [Sch13b, Lemma 3.24]', but this lemma is load-bearing: Proposition 5.15 uses it to identify R^iλ_*Ô^+_X with Ω^{i,log}_X(−i), Proposition 5.25 uses that identification to split Rν_*Ô^+_X, and Theorem 5.20 uses the splitting to identify the E_2-page of the Hodge–Tate spectral sequence and to count dimensions. The log-specific content is not a formal consequence of the smooth case: one must prove that the boundary map M^gp_X → R^1λ_*Z_p(1) induced by the Kummer sequence extends O_X-linearly to all of Ω^{1,log}_X and is an isomorphism, and that its exterior powers give the identifications for all i ≥ 2. As written, the proof omits these steps; if the log Hodge–Tate map is only an injection with nontrivial cokernel, the splitting in Proposition 5.25 fails and the dimension count in Theorem 5.20 does not go through. A complete proof of Lemma 5.16 is required.","section":"Section 5.2, Lemma 5.16"},{"comment":"Proposition 5.21(3) asserts that every proper log smooth rigid analytic variety over C admits a log smooth B_dR^+/t^2-lift, and this is used in Proposition 5.25 and Theorem 5.20. The proof is a sketch in several respects. First, it identifies the log structure of X with the compactifying log structure induced by the trivial locus U = X^tr; this requires justification for arbitrary log smooth X over C, not only those étale locally modeled on toric charts. Second, the deformation-theoretic argument using a versal deformation of the closed immersion D → X needs to be made precise: the compatibility of the versal deformation with the log structure and the existence of a formally smooth classifying map are asserted rather than proved. Third, the claim that B_dR^+/t^2 → C has a 'natural F̄-structure' is not immediate: B_dR^+ has a natural W(k)-structure, but an embedding of an algebraic closure F̄ of F into B_dR^+ compatible with θ is not automatic and must be checked explicitly. Since this lifting is a key input to the splitting of Rν_*Ô^+_X and to the proof of Theorem 5.20, a complete proof is needed.","section":"Section 5.4.1, Proposition 5.21(3)"}],"minor_comments":[{"comment":"The phrase 'a complete discrete valuation ting' should read 'ring'.","section":"Introduction, opening line"},{"comment":"The notation H^i_{B_dR^+}(X/B_dR^+) is nonstandard and appears to conflict with the notation H^i_{B_dR^+}(X) used elsewhere; please make consistent.","section":"Theorem 5.20"},{"comment":"The displayed formula 'dX i=1 dim_C H^j(X, Ω^{i,log}_X)' should be a sum ∑_{i=1}^d, and the variable appears to be missing in several places.","section":"Proof of Theorem 5.20"},{"comment":"The map is constructed as a splitting of Rν_*Ô^+_X, but the proof invokes Rλ_*Ô^+_X; the relationship between the two morphisms of sites should be made explicit.","section":"Proposition 5.25"},{"comment":"The phrase 'a (non filtered!) natural quasi-isomorphisms' should be singular: 'a natural quasi-isomorphism'.","section":"Corollary 1.3"},{"comment":"There is a typo 'beteween' in the proof; also the notation X is reused for both the rigid variety and the formal scheme, which is confusing.","section":"Section 5.4.1, Proposition 5.21(3)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a promising contribution and the gaps identified are not obviously fatal, but they are not merely presentation issues. Lemma 5.16 and Proposition 5.21(3) are essential inputs to Theorem 5.20; if the author can supply complete proofs of these two points, the central claims of the paper would be well supported. The present version, however, is not yet publication-ready because the main degeneration theorem is conditional on unproved assertions. I would encourage the editor to ask for a revision that supplies the missing arguments, especially the full proof of the logarithmic Hodge–Tate isomorphism."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — this is a substantial extension of the Bosco/DLLZ23a program, and the overall architecture is sound. The genuinely new content: a Kummer pro-étale computation of B_dR and B_dR^+ for log smooth rigid spaces over a discretely valued field, a log B_dR^+ cohomology via Lη_t, and — conditional on one key lemma — log Hodge–Tate and Hodge–log de Rham degeneration for proper log smooth rigid spaces. This is not a restatement of a prior theorem; the log statements are new.\n\nWhat the paper does well: Theorem 4.6 is carefully structured, reducing to established Poincaré lemma and local descriptions from DLLZ23a. The comparison in 5.19 is a clean consequence of the Lη_t formalism, and the deformation interpretation in 5.13 is sensible. The citation pattern is fine; reliance on Scholze, Bosco, and DLLZ23a is legitimate, not circular.\n\nThe soft spot is exactly where the stress-test lands. Lemma 5.16 is the load-bearing log Hodge–Tate isomorphism, and its proof is one sentence: 'Similar to [Sch13b, Lemma 3.24].' That is not enough. The smooth-case argument does not formally give the log version; you need to prove that the boundary map M^gp → R^1ν_*Z_p(1) extends OX-linearly to Ω^{1,log}_X and is an isomorphism there, and that exterior powers give the higher-i identifications. Those points are log-specific and they are used in Proposition 5.15, in the splitting of Rν_*Ô_X in Proposition 5.25, and in the dimension count of Theorem 5.20. If the log Hodge–Tate map is only injective, the splitting fails and the degeneration argument does not go through. So the main theorem is conditional on an unproved lemma.\n\nA second soft spot is Proposition 5.21(3), the B_dR/t^2-lifting in the proper case. It is a long sketch rather than a full proof, but it is more plausibly fillable from BMS18/Guo23 deformation theory; still, the proper case needs details.\n\nThere are no fabricated results, no fitted parameters, and no circularity. The paper is honest about its limits and explicitly states that a more comprehensive study is forthcoming.\n\nFor whom: specialists in p-adic Hodge theory for log rigid spaces. A serious referee should engage; the paper deserves peer review. But the referee should demand a complete proof of Lemma 5.16 and a fuller treatment of 5.21(3) before acceptance. I would not cite it in the current form.","headline":"Substantial and mostly well-built log generalization of the Bosco/DLLZ23a program, but the load-bearing log Hodge–Tate lemma is delegated to a one-line 'similar' proof and needs to be supplied before the degenerations can be trusted.","tokens_in":28033,"tokens_out":2513,"would_cite":false,"duration_ms":22280,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F30","14G22","14F40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes logarithmic p-adic Hodge theory for proper log smooth rigid analytic varieties, proving Hodge–Tate and Hodge–log de Rham degeneration via a new logarithmic B_dR^+ cohomology theory.","keywords":["p-adic Hodge theory","log rigid analytic varieties","Kummer pro-étale cohomology","B_dR cohomology","Hodge–Tate spectral sequence","log de Rham cohomology","period sheaves","condensed mathematics"],"falsifier":"Verify Lemma 5.16 explicitly for the one-dimensional toric log smooth example X = Spa(C⟨N⟩, O_C⟨N⟩), the log unit disk with log structure at the origin: if the natural map from M^gp_X to $R^{1}$λ_*Ô_{X_{prokét}}(1)▼ is not an isomorphism, or if the higher direct images differ from $Ω^{{i,log}}$_X(-i), Theorem 5.20 collapses. Alternatively, compute the Hodge–Tate spectral sequence for a proper log smooth curve over C and check whether it degenerates at E_2; a counterexample would directly falsify the theorem.","tokens_in":26785,"feed_emoji":"","tokens_out":7506,"duration_ms":63036,"temperature":0.7,"pith_summary":"The paper extends p-adic Hodge theory from smooth rigid analytic varieties to log smooth ones, which carry a well-behaved boundary structure. Its main claim is that for a proper log smooth rigid analytic variety X over the p-adic completion C, the Hodge–log de Rham spectral sequence and the Hodge–Tate spectral sequence both degenerate at their first page, and that the logarithmic B_dR^+ cohomology it defines is a finite free B_dR^+-module. This yields a logarithmic de Rham–étale comparison theorem, filling the missing logarithmic analogue of the classical rigid-analytic comparison. If correct, it provides a foundation for semistable and syntomic comparison theorems in p-adic geometry.","feed_headline":"Log Hodge–Tate degeneration proven for p-adic rigid varieties","feed_subtitle":"Introduces logarithmic B_dR^+ cohomology and proves the log de Rham–étale comparison for proper log smooth spaces.","key_machinery":"The central object is the logarithmic B_dR^+ cohomology RΓ_{B_dR^+}(X) := RΓ_{ét,cond}(X, Lη_t Rλ_* B_dR^+), where Lη_t is the décalage functor that adjusts the Hodge filtration. The key structural input is the logarithmic Hodge–Tate isomorphism R^i λ_* Ô_{X_{prokét}} ≃ $Ω^{{i,log}}$_X(-i), stated as Lemma 5.16 and asserted to be analogous to a known lemma in the smooth case. The splitting is achieved by choosing a log smooth B_dR^+/$t^{2}$-lift of X and proving that the lift gives a torsor whose class splits the Hodge–Tate map, so that Rν_*Ô_X decomposes into shifted logarithmic differentials; the degeneration theorems then follow by dimension counting.","core_discovery":"The paper introduces a logarithmic B_dR^+ cohomology RΓ_{B_dR^+}(X) for log smooth rigid analytic varieties over C, defined by applying the décalage functor Lη_t to Rλ_*B_dR^+ on the Kummer pro-étale site. Theorem 5.19 shows that after inverting t this cohomology is canonically isomorphic to H^i_{két}(X,Q_p)⊗_{Q_p}B_dR, and that it agrees with the known comparison isomorphism when X descends to a discretely valued field. For X proper and log smooth, Theorem 5.20 proves that the Hodge–log de Rham spectral sequence $E_1^{{ij}}$=H^j(X,$Ω^{{i,log}}$_X) degenerates at E_1, that the Hodge–Tate spectral sequence $E_2^{{ij}}$=H^j(X,$Ω^{{i,log}}$_X)(-j) degenerates at E_2, and that H^i_{B_dR^+}(X) is finite free over B_dR^+. The proof obtains a splitting of Rν_*Ô_X as ⊕_i $Ω^{{i,log}}$_X(-i)[-i] using a log smooth B_dR^+/$t^{2}$-lift of X, then counts dimensions via the primitive comparison theorem.","pith_inferences":["Supplying a full proof of Lemma 5.16 would likely allow the same framework to cover almost proper and punctured rigid spaces, extending the logarithmic Riemann–Hilbert correspondence beyond the proper case.","The splitting of Rν_*Ô_X via a B_dR^+/t^2-lift is probably independent of the chosen lift up to homotopy; one can test whether the induced Hodge–Tate filtration on H^k_{két}(X,Q_p) is independent of that choice.","The same Lη_t machinery should define logarithmic syntomic cohomology, and the paper announces this as a sequel; a natural test is whether the syntomic comparison holds for proper log smooth X without the discretely valued descent assumption.","A concrete check for a proper log smooth curve over C (for instance, the projective line with log structure at several points) would verify that the Hodge–Tate degeneration recovers known computations of log de Rham cohomology, testing the dimension-counting step."],"forward_implications":["For proper log smooth rigid varieties, the log de Rham–étale comparison (Theorem 5.19) holds, giving a filtered isomorphism between Kummer étale cohomology with B_dR coefficients and the new B_dR^+ cohomology.","The Hodge–log de Rham spectral sequence degenerates at E_1, so logarithmic de Rham cohomology of proper log smooth X is finite-dimensional and admits a Hodge decomposition in the p-adic sense.","The Hodge–Tate spectral sequence degenerates at E_2, producing a Hodge–Tate decomposition for log smooth proper rigid analytic varieties.","H^i_{B_dR^+}(X) is finite free over B_dR^+, so Theorem 5.13 describes B_dR^+ cohomology as a deformation of log de Rham cohomology that specializes to it modulo t.","When the log structure comes from a strictly normal crossing divisor D, the logarithmic B_dR^+ cohomology of X is isomorphic to the B_dR^+ cohomology of the open complement U = X - D, connecting log and ordinary pro-étale cohomology."],"supporting_citations":[{"why":"Supplies the foundations of log adic spaces, the logarithmic differentials Ω^{i,log}_X, the Kummer pro-étale site, and the primitive comparison theorem used to identify étale cohomology with Ω^{i,log}_X-cohomology.","marker":"[DLLZ23b]"},{"why":"Provides the logarithmic Riemann–Hilbert correspondence and the local computations of graded pieces of OB_dR,log used in Sections 3 and 4.","marker":"[DLLZ23a]"},{"why":"Establishes the smooth rigid-analytic Hodge–Tate and de Rham comparison theorems that the logarithmic results here are modelled on and reduce to in the trivial-log case.","marker":"[Sch13a]"},{"why":"Its Lemma 3.24 is the asserted analogue behind Lemma 5.16, the logarithmic Hodge–Tate isomorphism on which the degeneration proof rests.","marker":"[Sch13b]"},{"why":"Sets up the condensed pro-étale cohomology formalism and provides the lemmas on Banach and condensed sheaves used throughout the paper.","marker":"[Bos23a]"},{"why":"Introduces the Lη_f period-sheaf approach to B_dR^+ cohomology that the paper adapts to the logarithmic setting.","marker":"[Bos23b]"},{"why":"Supplies the décalage Lη_f functor and the formal-model lifting result (Proposition 13.15) used in constructing B_dR^+ cohomology and log B_dR^+/t^2-lifts.","marker":"[BMS18]"},{"why":"Provides the torsor and splitting technique for the Hodge–Tate map via B_dR^+/t^2-lifts, which Proposition 5.22 adapts to the logarithmic setting.","marker":"[Heu25]"}],"fun_headline_variants":["Log B_dR^+ cohomology: Hodge-Tate degeneration for proper rigid spaces","New log de Rham-étale comparison and spectral degeneracies for rigid varieties","Kummer pro-étale cohomology of log B_dR^+ computed","Hodge-log de Rham degeneration proven for log smooth rigid spaces","Proper log smooth spaces: two spectral sequence degeneracies proven"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the main degeneration theorem rests on Lemma 5.16, which states that the higher direct images of the structure sheaf along the Kummer pro-étale site are exactly the logarithmic differentials with Tate twists; the paper does not prove this lemma, only saying it is analogous to a known result, and if the logarithmic analogue fails the degeneration argument no longer goes through.","fun_headline_variants_meta":{"raw":{"variants":["Log B_dR^+ cohomology: Hodge-Tate degeneration for proper rigid spaces","New log de Rham-étale comparison and spectral degeneracies for rigid varieties","Kummer pro-étale cohomology of log B_dR^+ computed","Hodge-log de Rham degeneration proven for log smooth rigid spaces","Proper log smooth spaces: two spectral sequence degeneracies proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000481,"raw_usage":{"total_tokens":2398,"prompt_tokens":981,"completion_tokens":1417,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":1315}},"tokens_in":597,"tokens_out":1417,"duration_ms":12617,"temperature":1.0,"reasoning_tokens":1315,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T05:38:32.916191+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Verify Lemma 5.16 explicitly for the one-dimensional toric log smooth example X = Spa(C⟨N⟩, O_C⟨N⟩), the log unit disk with log structure at the origin: if the natural map from M^gp_X to $R^{1}$λ_*Ô_{X_{prokét}}(1)▼ is not an isomorphism, or if the higher direct images differ from $Ω^{{i,log}}$_X(-i), Theorem 5.20 collapses. Alternatively, compute the Hodge–Tate spectral sequence for a proper log smooth curve over C and check whether it degenerates at E_2; a counterexample would directly falsify the theorem.","supporting_citations":[],"review_version":1}