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Quasi-canonical lifting of projective varieties in positive characteristic

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Pith's one-line read Prime-to-p finite covers pass Frobenius liftability down to the base variety.

desk verdict 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. read the letter →

arxiv 2506.01345 v1 pith:Z5KGJ2CE submitted 2025-06-02 math.AG math.AC

Ryo Ishizuka , Kazuma Shimomoto This is my paper · ORCID
classification math.AGmath.AC MSC 13A3513B0513B3514G45
keywords quasi-canonicalliftingcanonicalFrobeniusWittvectorsfiniteétaledescentalgebraizationofformalschemesordinaryabelianvarietieslogarithmictangentbundle
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $X$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Section 4.2, proof of Corollary 4.4, first paragraph] 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.
  2. [Section 4.1, proof of Main Theorem 2, algebraization step] 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.
  3. [Section 4.2, proof of Corollary 4.4, coequalizer construction] 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].
minor comments (4)
  1. [Section 4.2, proof of Corollary 4.4] 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.
  2. [Section 4.1, proof of Main Theorem 2] 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}.
  3. [Introduction, after the statement of Main Theorem 2] 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).
  4. [Throughout] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation detected; the descent theorem is structurally independent, with only non-load-bearing self-citations and a non-circular citation-support caveat in Corollary 4.4.

full rationale

Walking the derivation chain, Main Theorem 2 does not assume its conclusion: hypothesis (♮) concerns a finite étale cover Z admitting a quasi-canonical lifting and vanishing cohomology over W(k), while the conclusion concerns X and the descended morphism f~. The descent is proved inside the paper using Lemma 4.2 (splitting BΩ^1 by a normalized trace using the prime-to-p degree), Proposition 2.14 (ascent and uniqueness for finite étale covers with Frobenius lifts), Main Theorem 1 (algebraization), and the external deformation result [1, Variant 3.3.2]. No step re-identifies the conclusion with an input by construction, and no fitted parameter is renamed as a prediction. The only clearly self-referential citation, [30], is an announced application and is not load-bearing; [46]–[48] are standard references. The genuine risk in Corollary 4.4 (Section 4.2) is not circularity but citation support: the proof says 'By the existence of canonical lifting for ordinary Abelian varieties (see [39] and [40]) and by [7, Proposition 4.12], we have a finite étale surjection A→X whose mod-p reduction is identified with A→X', whereas the paper's own preamble credits [7, Proposition 4.12] only with a flat lifting of X. If the cited proposition does not also lift the quotient morphism, the construction of the lifted coequalizer in diagram (4.7) lacks a starting point. That would be a missing-support or correctness issue, not a self-referential reduction. Accordingly the circularity score is minimal.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper relies on standard deformation theory and algebraization theorems from the cited literature. The central descent uses the normalized trace splitting, whose prime-to-p degree hypothesis is explicitly stated. The paper introduces no free parameters and no new geometric or physical entities.

assumptions (5)
  • standard math Deformation theory of flat liftings via the cotangent complex, including the extension criterion of Zdanowicz [53, Theorem A.4] and Illusie [29, Théorème 2.1.7].
    Invoked in Main Theorem 1 and Proposition 2.14 to extend flat liftings one level at a time.
  • standard math Grothendieck algebraization for p-adic formal schemes and formal morphisms, as in Illusie [28, Corollary 8.4.7] and the Stacks Project [50, Tag 089A, Tag 0A42].
    Used to pass from compatible systems {X_n} to projective flat schemes over W(k).
  • standard math Mehta-Srinivas equivalence: global Frobenius splitting, ordinarity, and W2(k) Frobenius liftability coincide under trivial canonical sheaf and a finite étale cover with trivial canonical sheaf.
    Imported as Lemma 2.2 from [39] and [3]; used to identify canonical liftable objects.
  • domain assumption For a finite étale morphism of constant degree d prime to p, the normalized trace map gives a splitting of O_Y -> f_* O_Z.
    Proved in Lemma 3.3 and used in Lemma 4.2; the prime-to-p assumption is essential and is responsible for the descent.
  • standard math Nakkajima's theorem that a Frobenius lift over W2(k) forces ordinarity.
    Used in Lemma 2.10 to show quasi-canonical liftings are ordinary.

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Pith. "Pith review of Quasi-canonical lifting of projective varieties in positive characteristic." pith.science (2026). https://pith.science/paper/Z5KGJ2CE

@misc{pith2026250601345,
  author       = {Pith},
  title        = {Pith review of: Quasi-canonical lifting of projective varieties in positive characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z5KGJ2CE}},
  note         = {Machine review of arXiv:2506.01345}
}
abstract

The main aim of this article is to give new classes of smooth projective varieties over characteristic $p>0$ that admit flat liftings over the Witt vectors together with additional data (logarithmic structure and the Frobenius morphism) by showing a descending property of such Frobenius liftability. We establish a refined form of the classical result due to Mehta-Srinivas on the existence of canonical liftings. For this purpose, we also establish a result on the algebraization of certain $p$-adic formal schemes.

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Forward citations

Cited by 1 Pith paper

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  1. {\delta}-rings, perfectoid towers, and lim Cohen-Macaulay sequences

    math.AC 2025-09 conditional novelty 6.0 of 10

    Perfectoid towers over complete local domains yield lim Cohen-Macaulay sequences, and new perfectoid towers are constructed, including the first with p-torsion.

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