On the Euler characteristic of weakly ordinary varieties of maximal Albanese dimension

Jefferson Baudin

REVIEW 3 major objections 4 minor 1 cited by

On the Euler characteristic of weakly ordinary varieties of maximal Albanese dimension

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that smooth proper weakly ordinary varieties of maximal Albanese dimension have nonnegative Euler characteristic, with equality outside general type and abelian fibrations in the zero case.

desk verdict A plausible and well-structured proof of the characteristic-p Green–Lazarsfeld chi>=0, conditional on two load-bearing results in the author's own preprints; the self-contained V-module core is convincing. read the letter →

arxiv 2507.01797 v1 pith:QGDLJ7XN submitted 2025-07-02 math.AG

Jefferson Baudin This is my paper · ORCID

classification math.AG MSC 14K0514G1714F17

keywords genericvanishingpositivecharacteristicCartiercrystalsweaklyordinaryvarietiesmaximalAlbanesedimensionsemistableEulerWittvectorGrauert-Riemenschneiderabelian

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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 establishes a positive-characteristic analogue of a classical inequality from complex geometry: if a smooth proper variety $X$ over an algebraically closed field of characteristic $p>0$ has maximal Albanese dimension and is weakly ordinary, then the Euler characteristic $\chi(X,\omega_X)$ is nonnegative. Here weakly ordinary means that Frobenius induces an isomorphism on every $H^i(X,\mathcal{O}_X)$. In the non-general-type case the value is forced to be zero, and whenever the Euler characteristic is zero the Albanese image is fibered by ordinary abelian varieties. The reason this is worth knowing is that both ingredients of the classical proof, generic vanishing and Grauert-Riemenschneider vanishing, fail in positive characteristic; the paper shows that a semistable, nilpotence-forgetting version of the Euler characteristic is the right invariant, and that it can be transferred from $X$ to an abelian variety by a newly introduced weak Grauert-Riemenschneider sheaf.

What carries the argument

The central object is the weak Grauert-Riemenschneider sheaf $\omega_{\pi,\mathrm{GR}}$, a canonically defined sub-Cartier crystal of $\pi_*\omega_X$ whose perfection is $(\pi_*W\omega_X)/p$. For proper generically finite $\pi$ with $X$ smooth, it satisfies $\chi_{ss}(X,\omega_X)=\chi_{ss}(Y,\omega_{\pi,\mathrm{GR}})$. The equality is obtained by combining a Witt-vector $Q_p$-Grauert-Riemenschneider vanishing theorem, which says that a fixed $p^e$ annihilates $R^i\pi_*W\omega_X$ for $i>0$, with finite-generation and derived-$p$-completeness arguments and a field-change lemma comparing Euler characteristics over $W(k)$ and its fraction field. On an ordinary abelian variety $A$, the Fourier-Mukai transform converts Cartier modules into $V$-modules, and an induction on the completed local ring $\widehat{\mathcal{O}}_{A,0}$ shows every $V$-module has nonnegative semistable Euler characteristic.

What would settle it

Find a smooth proper weakly ordinary variety of maximal Albanese dimension in characteristic $p$ with $\chi(X,\omega_X) < 0$, since Theorem A forbids one. Alternatively, exhibit a generically finite morphism from a smooth proper $X$ for which the higher direct images $R^i\pi_*W\omega_X$ are not annihilated by any fixed $p^e$; that would break the cited $Q_p$-Grauert-Riemenschneider vanishing on which the key equality rests.

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Extended reading notes

Core claim

The paper's central claim is that a smooth proper weakly ordinary variety $X$ of maximal Albanese dimension satisfies $\chi(X,\omega_X) \ge 0$, that if $X$ is not of general type then $\chi(X,\omega_X) = 0$, and that if $\chi(X,\omega_X) = 0$ then the Albanese image of $X$ is fibered by ordinary abelian varieties. The more general form, stated as Theorem B, says that if $X$ admits a generically finite morphism to an ordinary abelian variety, then its semistable Euler characteristic $\chi_{ss}(X,\omega_X)$ is nonnegative, with equality in the non-general-type case. The key bridge is an equality $\chi_{ss}(X,\omega_X) = \chi_{ss}(Y,\omega_{\pi,\mathrm{GR}})$ for proper generically finite maps, together with the statement that every Cartier module on an ordinary abelian variety has nonnegative semistable Euler characteristic.

Load-bearing premise

The proof leans on a theorem from a companion paper, not proved here, that a fixed power of $p$ annihilates all higher direct images of the Witt canonical sheaf under a generically finite map; if that theorem failed, the Euler-characteristic bridge that carries the argument would not stand.

Editorial extensions

If this is right

Every smooth proper weakly ordinary variety of maximal Albanese dimension has $\chi(X,\omega_X) \ge 0$, so a negative canonical Euler characteristic is an obstruction to weak ordinarity in this class.
For non-general-type such varieties, $\chi(X,\omega_X) = 0$, making a positive value a general-type detector within this class.
Vanishing of $\chi(X,\omega_X)$ forces the Albanese image to be fibered by ordinary abelian varieties, giving a structure theorem parallel to the characteristic-zero case.
For smooth proper threefolds not of general type admitting a generically finite map to an ordinary abelian variety, the higher direct images $R^i a_*\omega_X$ are nilpotent Cartier modules for $i>0$, so Grauert-Riemenschneider vanishing holds up to nilpotence.
The weak GR sheaf equality makes the semistable Euler characteristic a well-behaved invariant under generically finite maps even where classical Grauert-Riemenschneider vanishing fails.

Reading between the lines

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

Editorial inference: if the companion $Q_p$-Grauert-Riemenschneider vanishing theorem could be replaced by the weaker statement that the $p^\infty$-torsion of higher pushforwards is annihilated by a fixed $p$-power, as the paper's Remark 3.12 suggests, then the construction of $\omega_{\pi,\mathrm{GR}}$ and the Euler-characteristic transfer would extend to a broader class of proper morphisms.
Editorial inference: the same semistable-Euler-characteristic technology could be used to attack the paper's open question about whether non-weakly-ordinary varieties of maximal Albanese dimension can have negative $\chi(X,\omega_X)$; a systematic search over explicit small-characteristic surfaces or threefolds could settle the first unknown cases.
Editorial inference: the paper's remark after Proposition E points toward a stronger statement, namely that any Cartier module on an abelian variety with no simple factor of $p$-rank zero has $\chi_{ss} \ge 0$; if that statement holds, Theorem B would extend from ordinary abelian varieties to a much larger class, including many non-ordinary isogeny factors.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear [Colleague],

The one thing to know: this paper genuinely proves — modulo a load-bearing cited theorem — the positive-characteristic version of Green–Lazarsfeld's chi(X,omega_X) >= 0 for weakly ordinary varieties of maximal Albanese dimension, plus the structural consequences (non-general-type gives chi=0, and chi=0 gives the Albanese image fibered by ordinary abelian varieties). That is a real step in char-p generic vanishing, not a repackaging.

What is new here is worth crediting. The weak Grauert–Riemenschneider sheaf omega_{pi,GR} is a new object, and the equality chi_ss(X,omega_X)=chi_ss(Y,omega_{pi,GR}) (Cor. 3.16) is a nice bridge statement. The V-module section (4) is essentially self-contained: Proposition 4.2.1, the torsion lemma, and the induction on dimension are worked out and convincing. I did not find circularity, and the non-general-type Lemma 5.2 is an honest adaptation of Chen–Hacon.

The soft spot is exactly what the stress-test flags: Theorem 3.1 (Q_p-GR vanishing: p^e kills R^i pi_* W omega_X for i>0) is cited from the author's own [Bau25d] and is used twice — once for derived p-completeness of pi_*W omega_X in Lemma 3.11, and again for the cohomological comparison in Cor. 3.16 that transfers the Euler characteristic from X to the abelian variety. If that theorem has extra hypotheses or a gap, Theorems A–D do not follow from this paper alone. The author does not even sketch the proof here; Remark 3.12 says the full statement is needed later. That is not a flaw in the internal logic, but it is a real verification debt. A referee should demand that [Bau25d] be checked or included before this is regarded as fully established. The citation to [Bau25b] in the proof of Theorem 5.8 is a similar, smaller debt.

I do not think the paper overclaims: it is open about the ordinarity assumptions, and the failure examples in Example 5.5 are there. The V-module result is the kind of clean lemma that could stand on its own.

Who is this for: people working in positive-characteristic birational geometry and generic-vanishing theory. It deserves a serious referee. My recommendation: send it out, with the explicit instruction that the referee verify the statements in [Bau25d] and [Bau25b] that the argument relies on. The core of the paper is solid; the dependency is exposed, not hidden.

Best,

Referee Report

3 major / 4 minor

Summary. The paper proves positive characteristic analogues of the Green–Lazarsfeld and Chen–Hacon theorems for smooth proper varieties of maximal Albanese dimension. Theorem A (Corollary 5.7) asserts that a weakly ordinary such variety satisfies χ(X, ω_X) ≥ 0; Theorem C asserts equality when X is not of general type; and Theorem D asserts that if χ(X, ω_X) = 0, then the Albanese image of X is fibered by ordinary abelian varieties. The strategy is to replace classical Grauert–Riemenschneider vanishing by a Q_p–GR vanishing theorem cited from the author's preprint [Bau25d], to construct a weak Grauert–Riemenschneider sheaf ω_{π,GR} whose semistable Euler characteristic agrees with that of ω_X (Corollary 3.16), and to prove directly that every Cartier module on an ordinary abelian variety has nonnegative semistable Euler characteristic (Corollary 4.2.5). The V-module part of the proof in Section 4 is self-contained and appears sound; the main verification gap is the external dependency on [Bau25d] and, for Theorem D, on results cited from [Bau25b].

Significance. If the cited Q_p–GR vanishing and the supporting results in [Bau25b,c] are correct, the main results are significant: they extend classical Euler-characteristic bounds to positive characteristic under weak ordinarity, introduce a weak Grauert–Riemenschneider sheaf that is likely to be of independent interest, and include explicit counterexamples showing that the ordinarity hypotheses are necessary. The induction proving Proposition 4.2.1 is clean and gives a genuinely new statement about V-modules on ordinary abelian varieties. The paper also formulates concrete open questions. The main caveat is that the load-bearing vanishing theorem is not proved in this manuscript.

major comments (3)

[§3, Theorem 3.1] Theorem 3.1, the Q_p–Grauert–Riemenschneider vanishing theorem cited from [Bau25d], is load-bearing and is neither proved nor sketched here. It is used in Lemma 3.11 (the i=1 case) to prove that π_*Wω_X is derived p-complete, and in Corollary 3.16 (all i>0) to obtain the equality χ_ss(Y, ω_{π,GR}) = χ_ss(X, ω_X). If [Bau25d] proves this statement only under additional hypotheses, Corollary 3.16, and therefore Theorems A–D, would not follow from the arguments given. The manuscript should either include a proof of Theorem 3.1 or state precisely the hypotheses and give a fully checkable reference for the exact statement used.
[§5, Theorem 5.8 and Proposition 5.9] Theorem D depends on Proposition 5.9, cited from [Bau25b] as 'To appear', in the form that a normal variety Y of maximal Albanese dimension with V^0_inj(a_*ω) ≠ bA has Albanese image fibered by abelian varieties. This statement is used without proof, and it is as central to Theorem D as the V-module result is to Theorem A. Please provide a proof or a precise, checkable statement of this result, or else state clearly that Theorem D is conditional on [Bau25b].
[§5, Lemma 5.2] Lemma 5.2, used for Theorem C, relies on [Bau25c, Theorem 4.3] for the nonvanishing H^0(Y, ω_Y) ≠ 0 for varieties of maximal Albanese dimension. This is another same-author preprint, and the manuscript should indicate exactly which hypotheses are needed (for instance, whether weak ordinarity or ordinarity of the Albanese is assumed) so that the reader can verify that the lemma applies to the normal canonical variety Y appearing in the proof.

minor comments (4)

[§1.1, Theorem D] The phrase 'ordinary abelian varietes' is a typo and should read 'ordinary abelian varieties'.
[§5, proof of Theorem 5.3] In the sentence 'it is enough to show that H^0(A, a_*ω_X ⊗ L) = 0 for some L ∈ Pic^0(X)', the Picard group should be Pic^0(A), not Pic^0(X).
[§3, proof of Proposition 3.5] The reference '[Bau25d, Remark 2.11]' appears to be a typo for '[Bau25d, Remark 2.2.11]'.
[§1.2, paragraph on Grauert–Riemenschneider] The phrase 'An important (any maybe surprising) feature' should be 'An important (and perhaps surprising) feature'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the Euler-characteristic inequality is proved from V-modules, and the cited Qp-GR vanishing is an independent load-bearing input, not an assumption of the conclusion.

full rationale

The derivation chain contains no step in which a claimed output is identical, by construction, to an input. The main inequality is reduced to Corollary 4.2.5, which is proved from first principles in Section 4: Proposition 4.2.1 establishes chi_ss(V) >= 0 for V-modules on the completed local ring k[[x1,...,xg]] by induction on g, using Lemma 4.2.2 to choose coordinates fixed by the Verschiebung via the external fact [Mum08, Cor. p.143] and Lemma 4.2.4 to discard torsion. The bridge from X to the abelian variety is Corollary 3.16, whose equality chi_ss(X,omega_X) = chi_ss(A,omega_{a,GR}) is proved by comparing W(k)- and k-valued Euler characteristics; it relies on Theorem 3.1 (Qp-GR vanishing), quoted from the author's earlier preprint [Bau25d]. That theorem is a parameter-free vanishing statement whose hypotheses do not include any of the target inequalities, so importing it is a verification dependency rather than a circular input. The weak GR sheaf omega_{pi,GR} is not defined to have the same Euler characteristic as omega_X; it is constructed as a stabilised sub-Cartier crystal in Proposition 3.5, and the equality of Euler characteristics is then derived in Corollary 3.16. Likewise Theorem D's use of Proposition 5.9 from [Bau25b] is an external support result, not an assumption of the conclusion. No fitted parameter is relabelled as a prediction, and no self-citation is used to rule out alternatives by fiat. Score 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper is a proof-based algebraic geometry paper, so there are no fitted numerical parameters and no empirical entities. The main unproved inputs are prior theorems, several of which are by the author of the present paper. The weak GR sheaf is a genuinely new mathematical object introduced to carry the semistable Euler characteristic across a generically finite morphism.

assumptions (4)

domain assumption Qp-Grauert-Riemenschneider vanishing (Theorem 3.1, [Bau25d])
Load-bearing: used in Lemma 3.11 and Corollary 3.16 to equate chi_ss(X, omega_X) with chi_ss(Y, omega_{pi,GR}). Not proved or independently verified in this preprint.
domain assumption Fourier-Mukai equivalence and generic vanishing for Cartier and V-modules from [Bau25c], [HP16], [HP22]
Used in Lemma 4.1.9 and Theorem 4.1.8 to transfer Cartier-module Euler characteristics to V-modules on the dual abelian variety.
domain assumption Artinianness of Cartier crystals ([BB11])
Used in Proposition 3.5 to force stabilization of the descending chain of sub-Cartier crystals and hence define omega_{pi,GR}.
domain assumption Base field k is algebraically closed and sometimes uncountable
Throughout the paper k is algebraically closed; uncountability is invoked in Lemma 5.2 and Theorem 5.8 for 'very general' arguments.

invented entities (1)

weak Grauert-Riemenschneider sheaf omega_{pi,GR}
purpose: A canonically defined sub-Cartier crystal of pi_* omega_X whose semistable Euler characteristic equals chi_ss(X, omega_X), used to transfer the problem to an abelian variety.
Introduced in Definition 3.6 as a new mathematical object. It has no falsifiable empirical handle; its justification is the theorem that (pi_* W omega_X)/p is isomorphic to its perfection.

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Pith. "Pith review of On the Euler characteristic of weakly ordinary varieties of maximal Albanese dimension." pith.science (2026). https://pith.science/paper/QGDLJ7XN

@misc{pith2026250701797,
  author       = {Pith},
  title        = {Pith review of: On the Euler characteristic of weakly ordinary varieties of maximal Albanese dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QGDLJ7XN}},
  note         = {Machine review of arXiv:2507.01797}
}

abstract

We show that a smooth proper weakly ordinary variety $X$ of maximal Albanese dimension satisfies $\chi(X, \omega_X) \geq 0$. We also show that if $X$ is not of general type, then $\chi(X, \omega_X) = 0$ and the Albanese image of $X$ is fibered by abelian varieties. The proof uses the positive characteristic generic vanishing theory developed by Hacon-Patakfalvi, as well as our recent Witt vector version of Grauert-Riemenschneider vanishing.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Generic vanishing theory in positive characteristic
math.AG 2025-07 conditional novelty 5.0 of 10

The paper proves an equivalence between Cartier crystals and V-crystals on dual abelian varieties and derives H^0(X,ω_X)≠0, with S^0(X,ω_X)≠0 in the ordinary case, for normal proper varieties of maximal Albanese dimension.

Reference graph

Works this paper leans on

10 extracted references · 6 canonical work pages · cited by 1 Pith paper

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Generic vanishing theory in positive characteristic

Available at arXiv:2507.00771. [Bau25d] J. Baudin. A Grauert–Riemenschneider vanishing theorem for Witt canonical sheaves. arXiv e-print: arXiv:2506.14647v1 ,

work page Pith review arXiv
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A Grauert-Riemenschneider vanishing theorem for Witt canonical sheaves

Available at arXiv:2506.14647. [BB11] M. Blickle and G. B¨ ockle. Cartier modules: finiteness results. J. Reine Angew. Math., 661:85– 123,

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Available at arXiv:2312.13456. 22 JEFFERSON BAUDIN [BKR25] J. Baudin, T. Kawakami, and L. R¨ osler. On Grauert–Riemenschneider vanishing for Cohen– Macaulay schemes of klt type. arXiv e-print: arXiv:2506.21381v1 ,

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Available at arXiv:2506.21381. [BPRZ25] J. Baudin, Zs. Patakfalvi, L. R¨ osler, and M. Zdanowicz. On GorensteinQp-rational threefolds and fourfolds. arXiv e-print: arXiv:2506.15491v1 ,

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On Gorenstein $\mathbb{Q}_p$-rational threefold and fourfold singularities

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Available at arXiv:2306.05378. [Bau25a] J. Baudin. Duality between Wn-Cartier crystals and perverse Z/pnZ-sheaves. arXiv e-print: arXiv:2506.13346v1,

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [5]

Generic vanishing theory in positive characteristic

Available at arXiv:2507.00771. [Bau25d] J. Baudin. A Grauert–Riemenschneider vanishing theorem for Witt canonical sheaves. arXiv e-print: arXiv:2506.14647v1 ,

work page Pith review arXiv

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A Grauert-Riemenschneider vanishing theorem for Witt canonical sheaves

Available at arXiv:2506.14647. [BB11] M. Blickle and G. B¨ ockle. Cartier modules: finiteness results. J. Reine Angew. Math., 661:85– 123,

work page Pith review arXiv

[3] [8]

22 JEFFERSON BAUDIN [BKR25] J

Available at arXiv:2312.13456. 22 JEFFERSON BAUDIN [BKR25] J. Baudin, T. Kawakami, and L. R¨ osler. On Grauert–Riemenschneider vanishing for Cohen– Macaulay schemes of klt type. arXiv e-print: arXiv:2506.21381v1 ,

arXiv

[4] [9]

[BPRZ25] J

Available at arXiv:2506.21381. [BPRZ25] J. Baudin, Zs. Patakfalvi, L. R¨ osler, and M. Zdanowicz. On GorensteinQp-rational threefolds and fourfolds. arXiv e-print: arXiv:2506.15491v1 ,

arXiv

[5] [10]

On Gorenstein $\mathbb{Q}_p$-rational threefold and fourfold singularities

Available at arXiv:2506.15491. [CH01] J. A. Chen and C. D. Hacon. Pluricanonical maps of varieties of maximal Albanese dimension. Math. Ann., 320(2):367–380,

work page Pith review arXiv

[6] [2008]

With appendices by C. P. Ramanujam and Y. Manin, Corrected reprint of the second (1974) edition. [Sch22] C. Schnell. The Fourier-Mukai transform made easy. Pure Appl. Math. Q. , 18(4):1749–1770,

work page 1974

[7] [2021]

Para-abelian varieties and Albanese maps

Available at arXiv:2101.10829. [LZ04] A. Langer and T. Zink. De Rham-Witt cohomology for a proper and smooth morphism. J. Inst. Math. Jussieu , 3(2):231–314,

work page Pith review arXiv

[8] [2023]

[Bau25a] J

Available at arXiv:2306.05378. [Bau25a] J. Baudin. Duality between Wn-Cartier crystals and perverse Z/pnZ-sheaves. arXiv e-print: arXiv:2506.13346v1,

arXiv