{"id":"4f9f68f1-79b7-47d4-becb-7f8c37efc587","arxiv_id":"2506.01345","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A descent theorem: a finite étale cover with a quasi-canonical lifting, of degree prime to p, forces the base variety to have a canonical lifting over the Witt vectors.","lead":"This paper proves new families of geometric shapes, projective varieties over fields of positive characteristic, can be lifted to flat families over the Witt vectors while preserving a Frobenius map and a logarithmic structure. A generalist should care because canonical liftings are one of the main bridges from characteristic p problems to characteristic zero arithmetic geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 4.4's existence of a W(k)-lift of the quotient map A→X is asserted by citation; if [7, Prop 4.12] only lifts X and not the cover, the main new examples for Main Theorem 2 lack a proof.","rationale":"The paper's Main Theorem 2 is a conditional descent theorem; its hypothesis (♮) is strong, and the genuinely new unconditional classes come from Corollary 4.4. The proof of Corollary 4.4 has two delicate steps: lifting the quotient map A→X over W(k) and lifting the equivalence relation with Frobenius compatibility. The first is delegated to [7, Prop 4.12]; the second is sketched by an adjustment argument using [39, Appendix, Theorem 1(2)] and Proposition 2.14. My reading of the adjustment argument is plausible if the lifted relation R is already known to exist; the weak point is precisely the citation. I am not claiming the theorem is false; I am claiming the current proof is not self-contained at the point where the paper's main examples are produced. The reader's verdict (CONDITIONAL) is appropriate: if the cited proposition contains the asserted morphism lift, no change; otherwise Corollary 4.4 needs a repaired argument. The condition (♮) being strong is not itself a flaw, so I do not move the verdict.","tokens_in":26285,"tokens_out":19573,"duration_ms":220748,"concrete_test":"Locate [7, Proposition 4.12] and check its exact statement. If it asserts 'X is liftable' rather than 'the finite étale cover A→X lifts to A→X over W(k)', then run the omitted coequalizer check: let A be the canonical lift of an ordinary abelian variety A and let R be the finite étale lift over A of R=A×_X A given by Proposition 2.14; verify that the two lifted projections σ_1,σ_2:R→A satisfy the groupoid/fiber-product axioms and that the fppf quotient A/R is representable by a projective W(k)-scheme. This directly tests whether Corollary 4.4 can be repaired without the cited morphism lift.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Corollary 4.4 is the main unconditional supplier of hypothesis (♮), and its proof contains the sentence: 'By the existence of canonical lifting for ordinary Abelian varieties ... and by [7, Proposition 4.12], we have a finite étale surjection A→X whose mod-p reduction is identified with A→X.' This is a load-bearing external assertion: it claims not only that the quotient X is liftable over W(k), but that the quotient morphism A→X lifts integrally, compatibly with the canonical lift of A. Liftability of X does not by itself imply liftability of the G-action or of the finite étale equivalence relation R=A×_X A⇒A. The proof then uses the lifted relation R (from Proposition 2.14), adjusts σ_i via [39, Appendix, Theorem 1(2)], and invokes uniqueness to get σ_i=σ'_i; but if [7, Prop 4.12] does not contain the morphism-lifting statement, the existence of a W(k)-lift of the quotient map has not been established, and the coequalizer construction in (4.7) has no starting data. Since Corollary 4.4 is what verifies (♮) for the new examples (finite étale quotients of ordinary abelian varieties), the scope of Main Theorem 2 rests on this citation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies flat liftings of smooth projective varieties in characteristic p>0 to the Witt vectors W(k), together with lifts of Frobenius and logarithmic structures. Main Theorem 1 is an algebraization/descent statement: given a surjective finite étale morphism X→Y of degree prime to p, if the cover side has extendable flat liftings, one obtains a p-adic formal lifting of Y, and if the cover side algebraizes projectively, then so does Y together with the morphism. Main Theorem 2 converts this into a descent statement for quasi-canonical liftings: under condition (♮) (a finite étale cover Z of X of prime-to-p degree admitting a quasi-canonical lifting, plus vanishing of H^0 and H^1 of T_Z(-log D_Z)⊗BΩ^1_Z), the pair (X,D) admits the unique canonical lifting and the cover lifts compatibly. Corollary 4.4 applies the theorem to finite étale quotients of ordinary abelian varieties, asserting quasi-canonical liftability over W(k) without a degree condition, and canonical liftability with functoriality when the degree is prime to p. The proofs combine cotangent-complex obstruction theory, a normalized trace splitting of BΩ^1, and algebraization via norms of line bundles.","tokens_in":26550,"tokens_out":18099,"duration_ms":199322,"significance":"If the cited external input is exactly as stated, the paper gives a clean conditional descent theorem with a well-isolated hypothesis, and it supplies new unconditional examples (finite étale quotients of ordinary abelian varieties), including a proof of a claim from [1] that was previously left unproved. The trace-splitting Lemma 4.2 is a useful and elegant contribution, and the norm-based algebraization argument is a sensible way to transfer projectivity. No circularity is apparent: the descent theorem reduces the statement for X to the existence of a quasi-canonical lifting on a cover Z and does not assume its conclusion. The main results would be a genuine refinement of the Mehta-Srinivas theorem, provided the load-bearing citation in Corollary 4.4 is verified and the projectivity input in Main Theorem 2 is made explicit.","major_comments":[{"comment":"The proof asserts: \"By the existence of canonical lifting for ordinary Abelian varieties ... and by [7, Proposition 4.12], we have a finite étale surjection A→X whose mod-p reduction is identified with A→X.\" This is load-bearing: it claims that the quotient map lifts integrally, not merely that the quotient X lifts over W(k). If [7, Proposition 4.12] only proves liftability of X, then the lifted equivalence relation used in (4.7) has no starting data and the unconditional examples in Corollary 4.4 are not established. Please either quote the exact statement of [7, Proposition 4.12] or give a direct proof of the existence of the lifted finite étale map A→X.","section":"Section 4.2, proof of Corollary 4.4, first paragraph"},{"comment":"The proof says \"Now as in the proof of Main Theorem 1, one can use the norm of line bundles to conclude that there is a flat proper scheme X over W(k)\". This step requires an ample line bundle on Z over W(k) (or on the formal scheme {Z_n}). Condition (♮) only assumes that Z admits a quasi-canonical lifting, which by Definition 2.3 is a flat, proper lifting but not necessarily projective. Please either add \"projective\" to condition (♮), or prove that the lifting produced by [1, Variant 3.3.2] is projective; otherwise the norm/algebraization argument lacks a necessary hypothesis.","section":"Section 4.1, proof of Main Theorem 2, algebraization step"},{"comment":"After the commutativity of (4.7) is established, the proof states that taking coequalizers gives \"a smooth projective scheme X over W(k)\". Given the caution in Remark 3.6(1) about non-effective finite étale equivalence relations, the representability of this coequalizer by a projective scheme should be justified explicitly, either by citing the same Altman-Kleiman/quotient result used in Main Theorem 1 or by observing that it follows directly from the already-given lifted morphism A→X obtained from [7, Proposition 4.12].","section":"Section 4.2, proof of Corollary 4.4, coequalizer construction"}],"minor_comments":[{"comment":"The notation σ_i is overloaded: initially σ_i denotes the projections R→A on the closed fiber, and later the same symbols denote their lifts to R→A. Please distinguish the two uses, for example by writing σ_{i,k} for the closed-fiber maps.","section":"Section 4.2, proof of Corollary 4.4"},{"comment":"When Proposition 2.14 is invoked to obtain the diagram with logarithmic data, the divisors D_{Z_n} are never defined. It would be clearer to state that D_{Z_n} = f_n^*D_n and to note that Corollary 2.15 (or a direct pullback computation) gives F_{Z,n}^*D_{Z,n} = pD_{Z,n}.","section":"Section 4.1, proof of Main Theorem 2"},{"comment":"The sentence \"The condition (♮) is fulfilled (at least over W2(k)) ...\" is potentially confusing because (♮) requires a lifting over W(k). Please clarify that the W2(k)-lifting from [1, Theorem 5.1.1], together with the vanishing of H^0 and H^1, extends uniquely to W(k).","section":"Introduction, after the statement of Main Theorem 2"},{"comment":"There are several typographical slips, including \"lifing\" in Corollary 4.4, \"V ARIETIES\" in the title, \"mortphism\" in Question 2, and \"pj\" in the displayed proof of Main Theorem 1. These should be corrected in the final version.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central issue for acceptance is whether [7, Proposition 4.12] really gives a lift of the quotient morphism A→X, not merely a lift of the quotient X. This point is not verifiable from the manuscript as written, and it is load-bearing for the new examples in Corollary 4.4. The authors should either reproduce the exact statement or give a self-contained proof. The projectivity clarification in Main Theorem 2 is also necessary, though likely easy to fix. I do not see circularity or a fundamental flaw in the descent strategy."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper deserves a serious referee. The core of it is a clean descent theorem: if a finite étale cover Z→X of prime-to-p degree admits a quasi-canonical lifting over the Witt vectors, with the two relevant H's vanishing, then X inherits a canonical lifting and the cover lifts compatibly. The proof of that (Main Theorem 2) is detailed and the trace-splitting argument in Lemma 4.2 is solid. The algebraization theorem (Main Theorem 1) is also a genuine contribution, and Proposition 2.14, which shows quasi-canonicity ascends along finite étale covers, is independently useful. The logarithmic version is a real extension, not just a rerun of Mehta–Srinivas.\n\nThe soft spot is Corollary 4.4, which supplies the main new examples: finite étale quotients of ordinary abelian varieties. The proof asserts, by combining the canonical lift of A with [7, Proposition 4.12], that there is a finite étale surjection A→X over W(k) whose reduction is A→X. The paper's own introduction describes [7, Proposition 4.12] as giving a flat lifting of X, not necessarily a lifting of the quotient map. If that citation only lifts X, then the assertion is stronger than what is cited, and the coequalizer construction in (4.7) has no starting data. This is load-bearing because Corollary 4.4 is what verifies condition (♮) for the new classes. The subsequent adjustment of the σ_i via [39, Appendix, Theorem 1(2)] is plausible, but it presupposes that some integral lift of the equivalence relation R⇒A exists.\n\nThat is the one real gap I see. It is not a sign of carelessness elsewhere; the rest of the paper is written with evident care, and I found no circular reasoning. If [7, Proposition 4.12] does contain the morphism-lifting statement, the paper is essentially correct as written. If not, Corollary 4.4 needs a direct proof of the lift of the quotient map, or the examples should be scaled back to cases where that lift is known.\n\nThe audience is specialists in lifting problems, Frobenius liftings, and mixed characteristic singularity theory. I would bring it to a reading group because the descent technique is interesting and the log version is a useful tool. My recommendation to an editor: send it to peer review. Ask the referee to verify exactly what [7, Proposition 4.12] states and to require the authors to either quote the precise statement or prove the lift of A→X in Corollary 4.4. The rest of the paper is worth publishing.","headline":"A serious lifting paper whose descent theorem is well argued, but the main new examples in Corollary 4.4 depend on an external assertion about lifting the quotient map A→X that the cited result may not contain.","tokens_in":27089,"tokens_out":2934,"would_cite":true,"duration_ms":32806,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A35","13B05","13B35","14G45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Prime-to-p finite covers pass Frobenius liftability down to the base variety.","keywords":["quasi-canonical lifting","canonical lifting","Frobenius lifting","Witt vectors","finite étale descent","algebraization of formal schemes","ordinary abelian varieties","logarithmic tangent bundle"],"falsifier":"Take a smooth projective variety $Y$ over an algebraically closed field of characteristic $p>0$ that is known not to admit any flat lifting over $W_2(k)$ (for example a Serre–Godeaux type quotient). If $Y$ is shown to have a finite étale cover $Z\\to Y$ of degree prime to $p$ that admits a quasi-canonical lifting over $W(k)$ with $H^0$ and $H^1$ of $T_Z(-\\log D_Z)\\otimes B\\Omega^1_Z$ vanishing, then Main Theorem 2 would force $Y$ to lift, disproving the theorem.","tokens_in":26069,"feed_emoji":"⬆️","tokens_out":6652,"duration_ms":65421,"temperature":0.7,"pith_summary":"The paper proves that the property of having a quasi-canonical lifting—a flat lifting over the Witt vectors together with a lifting of the Frobenius morphism—descends along finite étale surjective maps whose degree is not divisible by p. Concretely, if a finite étale cover $Z$ of a smooth projective pair $(X,D)$ admits such a lifting with logarithmic structure, then $X$ itself admits the canonical lifting, with a compatible lifting of the covering map. This yields new classes of characteristic-$p$ varieties that lift fully over the Witt vectors, including finite étale quotients of ordinary abelian varieties. The paper also establishes an algebraization theorem for $p$-adic formal schemes that underpins the descent.","feed_headline":"Prime-to-p covers pass Frobenius liftability to the base","feed_subtitle":"If a finite cover lifts with its Frobenius, the base variety also lifts fully over the Witt vectors—yielding new classes.","key_machinery":"The load-bearing mechanism is the normalized trace map $-\\frac{1}{d}\\mathrm{Tr}_f\\colon f_*\\mathcal{O}_Z\\to\\mathcal{O}_X$ together with its Frobenius-pushed variant, which splits the unit map $\\mathcal{O}_X\\to f_*\\mathcal{O}_Z$ exactly when the degree $d=[K(Z):K(X)]$ is prime to $p$. This splitting annihilates the cotangent-complex obstruction to extending flat liftings from $Z$ to $X$ level by level modulo $p^n$. The other pillar is Main Theorem 1, an algebraization result that uses norms of ample line bundles to promote the resulting $p$-adic formal scheme to a projective flat $W(k)$-scheme, and Lemma 4.2, which uses the same trace splitting to descend the required vanishing of $H^0$ and $H^1$ of $T(-\\log D)\\otimes B\\Omega^1$ from $Z$ to $X$.","core_discovery":"The central claim is a descending property of Frobenius liftability: for a smooth projective nc pair $(X,D)$ over an algebraically closed field $k$ of characteristic $p>0$, if a finite étale cover $Z\\to X$ of degree prime to $p$ admits a quasi-canonical lifting $(\\mathcal{Z},\\mathcal{D}_Z,\\tilde F_Z)$ over $W(k)$ and satisfies the vanishing $H^0(Z,T_Z(-\\log D_Z)\\otimes B\\Omega^1_Z)=H^1(Z,T_Z(-\\log D_Z)\\otimes B\\Omega^1_Z)=0$, then $(X,D)$ itself admits the canonical lifting $(\\mathcal{X},\\mathcal{D},\\tilde F_X)$ over $W(k)$, together with a finite étale surjective morphism $\\tilde f\\colon \\mathcal{Z}\\to \\mathcal{X}$ compatible with the Frobenius lifts. This refines the classical canonical-lifting result for ordinary varieties with trivial cotangent bundle, and the algebraization theorem (Main Theorem 1) provides the formal-scheme input needed to construct $\\mathcal{X}$ from the lifted cover.","pith_inferences":["The descent property is likely transitive: if $Y$ is a prime-to-$p$ étale quotient of $X$ and $X$ is a prime-to-$p$ étale quotient of a quasi-canonically liftable $Z$, then $Y$ should inherit the lifting by iterating the paper's argument, though this is not explicitly stated.","The normalized-trace splitting applies to any deformation problem whose obstruction groups are compatible with the unit map $\\mathcal{O}\\to f_*\\mathcal{O}$; one may therefore expect analogous descent for other lifted structures, such as $p$-divisible groups or $\\delta$-structures in mixed characteristic.","Condition ($\\natural$) is sufficient but probably not necessary: replacing the full vanishing of $H^0$ with vanishing only of the part obstructing uniqueness might enlarge the class of quasi-canonical (rather than canonical) liftings obtained by descent."],"forward_implications":["If condition ($\\natural$) holds, $(X,D)$ gets the canonical lifting over $W(k)$, not merely a flat lifting: the Frobenius and logarithmic structure lift uniquely.","Finite étale quotients of ordinary abelian varieties admit quasi-canonical liftings over the full Witt vectors; when the quotient degree is prime to $p$, the lifting is canonical and functorial in morphisms.","The prime-to-$p$ degree hypothesis is essential: a degree-$p$ finite étale quotient can fail to lift even to $W_2(k)$, as in Serre's non-liftable quotient example.","The Picard group of a canonically lifted variety is controlled: the subgroup of line bundles $L$ on $\\mathcal{X}$ with $\\tilde F_X^*(L)\\cong L^p$ restricts isomorphically onto $\\mathrm{Pic}(X)$.","The algebraization theorem ensures that if a finite étale cover of degree prime to $p$ admits a projective flat lifting over $W(k)$, then so does the quotient, without assuming cohomological vanishing on the quotient itself."],"supporting_citations":[{"why":"Supplies the classical canonical lifting theory for ordinary varieties with trivial tangent bundle, including the uniqueness criterion used to identify canonical liftings.","marker":"[39]"},{"why":"Provides the logarithmic variant of canonical liftings over $W_2(k)$ and the Variant 3.3.2 used to construct and prove uniqueness of the lifted Frobenius-compatible nc pairs.","marker":"[1]"},{"why":"Supplies the deformation-theoretic framework (deformation tuples and Theorem A.4) used to build flat liftings step by step and to control obstructions via the cotangent complex.","marker":"[53]"},{"why":"The Stacks Project provides the standard lemmas on étale liftings, trace maps, ampleness of norms, and algebraization that the proof invokes repeatedly.","marker":"[50]"},{"why":"Provides Grothendieck's existence theorem and algebraization results (Corollary 8.4.7) that convert the constructed $p$-adic formal schemes into flat projective schemes over $W(k)$.","marker":"[28]"},{"why":"Establishes existence of flat liftings for finite étale quotients of ordinary Abelian varieties, a base case that Corollary 4.4 upgrades to quasi-canonical liftings.","marker":"[7]"},{"why":"Supplies the canonical lifting of ordinary Abelian varieties over the Witt vectors, the input used when $Z$ is taken to be an ordinary Abelian variety.","marker":"[40]"}],"fun_headline_variants":["Prime-to-p cover lifts imply base Frobenius lifts","Frobenius liftability descends from prime-to-p covers","Base inherits Frobenius lift from prime-to-p cover","Cover Frobenius lift forces canonical lift on base","Lifting a prime-to-p cover lifts the base too"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole descent rests on the existence of a finite étale cover $Z$ of $X$, of degree not divisible by $p$, that already has a Frobenius-compatible flat lifting over the full Witt vectors and whose first two cohomology groups of $T_Z(-\\log D_Z)\\otimes B\\Omega^1_Z$ vanish; if no such cover exists, the conclusion that $X$ itself lifts can fail.","fun_headline_variants_meta":{"raw":{"variants":["Prime-to-p cover lifts imply base Frobenius lifts","Frobenius liftability descends from prime-to-p covers","Base inherits Frobenius lift from prime-to-p cover","Cover Frobenius lift forces canonical lift on base","Lifting a prime-to-p cover lifts the base too"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000757,"raw_usage":{"total_tokens":3322,"prompt_tokens":858,"completion_tokens":2464,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":474,"completion_tokens_details":{"reasoning_tokens":2381}},"tokens_in":474,"tokens_out":2464,"duration_ms":19220,"temperature":1.0,"reasoning_tokens":2381,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:45:59.537478+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a smooth projective variety $Y$ over an algebraically closed field of characteristic $p>0$ that is known not to admit any flat lifting over $W_2(k)$ (for example a Serre–Godeaux type quotient). If $Y$ is shown to have a finite étale cover $Z\\to Y$ of degree prime to $p$ that admits a quasi-canonical lifting over $W(k)$ with $H^0$ and $H^1$ of $T_Z(-\\log D_Z)\\otimes B\\Omega^1_Z$ vanishing, then Main Theorem 2 would force $Y$ to lift, disproving the theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical canonical lifting theory for ordinary varieties with trivial tangent bundle, including the uniqueness criterion used to identify canonical liftings."},{"cited_title":"Achinger, J","cited_arxiv_id":null,"evidence_quote":"Provides the logarithmic variant of canonical liftings over $W_2(k)$ and the Variant 3.3.2 used to construct and prove uniqueness of the lifted Frobenius-compatible nc pairs."},{"cited_title":"Zdanowicz, Liftability of singularities and their Frobenius morphism modulo p2, International Mathematics Research Notices 2018 (2017), 4513–4577","cited_arxiv_id":null,"evidence_quote":"Supplies the deformation-theoretic framework (deformation tuples and Theorem A.4) used to build flat liftings step by step and to control obstructions via the cotangent complex."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Stacks Project provides the standard lemmas on étale liftings, trace maps, ampleness of norms, and algebraization that the proof invokes repeatedly."},{"cited_title":"Illusie, Grothendieck’s existence theorem in formal geometry , Fundamental Algebraic Geometry, Mathematical Surveys and Monographs 123 AMS 2005","cited_arxiv_id":null,"evidence_quote":"Provides Grothendieck's existence theorem and algebraization results (Corollary 8.4.7) that convert the constructed $p$-adic formal schemes into flat projective schemes over $W(k)$."},{"cited_title":"Bernasconi, I","cited_arxiv_id":null,"evidence_quote":"Establishes existence of flat liftings for finite étale quotients of ordinary Abelian varieties, a base case that Corollary 4.4 upgrades to quasi-canonical liftings."},{"cited_title":"Messing, The crystals associated to Barsotti-Tate groups: with appl ications to Abelian schemes , Lecture Notes in Math., 264 Springer, 1972","cited_arxiv_id":null,"evidence_quote":"Supplies the canonical lifting of ordinary Abelian varieties over the Witt vectors, the input used when $Z$ is taken to be an ordinary Abelian variety."}],"review_version":1}