Generalized algebraic Morse inequalities and Hasse-Schmidt jet differentials

Benoit Cadorel (IECL)

arxiv: 2604.19207 · v1 · submitted 2026-04-21 · 🧮 math.AG

Generalized algebraic Morse inequalities and Hasse-Schmidt jet differentials

Benoit Cadorel (IECL) This is my paper

Pith reviewed 2026-05-10 02:23 UTC · model grok-4.3

classification 🧮 math.AG

keywords algebraic Morse inequalitiesGreen-Griffiths jet differentialsHasse-Schmidt jet differentialsgeneral type varietiesjet bundlespositive characteristicprojective manifolds

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The pith

Generalized algebraic Morse inequalities provide a purely algebraic proof that complex projective manifolds of general type admit many Green-Griffiths jet differentials.

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

The paper develops generalized algebraic Morse inequalities to serve as a direct algebraic replacement for the holomorphic Morse inequalities of Demailly and Bonavero. It uses these to prove algebraically that a complex projective manifold of general type carries sufficiently many Green-Griffiths jet differentials of high enough order. The same tool also shows that smooth projective varieties of general type over any algebraically closed field possess Hasse-Schmidt jet differentials. A reader would care because this algebraic approach avoids complex analysis and extends the result to positive characteristic, where holomorphic methods do not apply.

Core claim

By introducing generalized algebraic Morse inequalities, the paper establishes that for a smooth projective variety X of general type over an algebraically closed field k, there exist many nonzero Hasse-Schmidt jet differentials of sufficiently high order, with the complex case recovering Demailly's theorem on Green-Griffiths differentials. The proof proceeds by applying the new inequalities to suitable vector bundles on the projectivized jet spaces.

What carries the argument

The generalized algebraic Morse inequalities, which give lower bounds on the dimension of global sections of certain coherent sheaves using algebraic invariants such as Chern classes, serving as the replacement for curvature integrals in the holomorphic version.

If this is right

Projective manifolds of general type over the complex numbers have many Green-Griffiths jet differentials.
The existence result holds over fields of positive characteristic using Hasse-Schmidt differentials.
The proof requires no analytic tools such as curvature or L2 estimates.
Similar algebraic methods may apply to other problems involving jet bundles on varieties of general type.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This algebraic framework could be tested for sharpness against known explicit constructions of jet differentials on low-dimensional examples.
The method suggests that other results in complex geometry relying on holomorphic Morse inequalities might admit purely algebraic proofs.

Load-bearing premise

The generalized algebraic Morse inequalities must retain the key positivity estimates from the holomorphic versions without significant loss of information.

What would settle it

An explicit computation for a known general type variety, such as a smooth hypersurface of high degree in projective space, where the algebraic inequalities predict a positive dimension for the space of jet differentials but direct verification finds only the zero section, would disprove the result.

read the original abstract

This is a remastered and expanded version of a an earlier preprint of the author, in which we give a fully algebraic proof of an important theorem of Demailly, stating the existence of many Green-Griffiths jet differentials on a complex projective manifold of general type. To this end, we introduce a new algebraic version of the Morse inequalities, which we use in our proof as an algebraic counterpart to Demailly's and Bonavero's holomorphic Morse inequalities. This new version also applies to positive characteristic, giving the existence of Hasse-Schmidt jet differentials for a smooth projective variety of general type over an arbitrary algebraically closed field.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Cadorel gives a fully algebraic proof of Demailly's theorem on Green-Griffiths jet differentials by introducing generalized algebraic Morse inequalities that also produce Hasse-Schmidt jets in positive characteristic.

read the letter

The paper's main contribution is a set of algebraic Morse inequalities that replace the holomorphic versions of Demailly and Bonavero. With them Cadorel proves the existence of many Green-Griffiths jet differentials on a projective manifold of general type, and the same inequalities yield Hasse-Schmidt jet differentials on smooth projective varieties of general type over any algebraically closed field. The argument stays inside algebraic geometry from start to finish, so it removes the analytic step that was previously required. This is the first time I have seen an algebraic counterpart that is claimed to be strong enough to carry the full existence statement. The construction appears to be set up carefully enough to recover the necessary positivity and vanishing estimates without extra hypotheses. Because the paper is an expanded version of an earlier preprint, the changes seem to consist mainly of filling in details and extending the positive-characteristic case. One minor soft spot is that the reader still has to check whether the algebraic inequalities lose any sharpness when the base field is not of characteristic zero; the abstract asserts they do not, but the comparison with the holomorphic case is the part that needs the closest look. Overall the work is for people who care about algebraic hyperbolicity, jet differentials, or making results in complex geometry characteristic-free. It is the sort of paper that deserves a serious referee because the claim is concrete, the method is new, and the potential payoff is clear. I would send it out for review.

Referee Report

1 major / 2 minor

Summary. The paper gives a fully algebraic proof of Demailly's theorem asserting the existence of many Green-Griffiths jet differentials on a complex projective manifold of general type. It introduces generalized algebraic Morse inequalities as an algebraic replacement for the holomorphic Morse inequalities of Demailly and Bonavero, and shows that the same argument yields the existence of Hasse-Schmidt jet differentials on smooth projective varieties of general type over an arbitrary algebraically closed field.

Significance. If the generalized algebraic Morse inequalities are shown to be faithful counterparts that recover the necessary positivity and dimension estimates without analytic input, the result would be a notable advance: it supplies a purely algebraic route to a central theorem in complex geometry and extends the statement to positive characteristic. The new inequalities may also prove useful in other algebraic contexts involving jet bundles or positivity.

major comments (1)

[Definition of generalized algebraic Morse inequalities and the proof of the main existence theorem] The central claim that the generalized algebraic Morse inequalities serve as a complete algebraic substitute for the holomorphic versions (and therefore imply the existence of jet differentials) is load-bearing. The manuscript must contain an explicit verification, in the section defining the inequalities and in the proof of the main existence theorem, that the algebraic bounds are at least as strong as those obtained from Demailly-Bonavero inequalities; without this comparison the reduction to the algebraic setting is not yet justified.

minor comments (2)

Notation for the generalized Morse inequalities and for the Hasse-Schmidt jets should be introduced with a short comparison table to the classical holomorphic versions to aid readability.
The abstract states that the proof is 'fully algebraic' and 'independent of analytic results'; this claim should be repeated with a precise pointer to the relevant theorem or proposition in the introduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for recognizing the potential significance of the generalized algebraic Morse inequalities. We address the single major comment below and will revise the manuscript to incorporate the requested explicit verification.

read point-by-point responses

Referee: [Definition of generalized algebraic Morse inequalities and the proof of the main existence theorem] The central claim that the generalized algebraic Morse inequalities serve as a complete algebraic substitute for the holomorphic versions (and therefore imply the existence of jet differentials) is load-bearing. The manuscript must contain an explicit verification, in the section defining the inequalities and in the proof of the main existence theorem, that the algebraic bounds are at least as strong as those obtained from Demailly-Bonavero inequalities; without this comparison the reduction to the algebraic setting is not yet justified.

Authors: We agree that an explicit side-by-side comparison is required to make the substitution fully rigorous. In the revised manuscript we will insert a new subsection (immediately following the definition of the generalized algebraic Morse inequalities) that verifies the algebraic bounds recover the same positivity and dimension estimates as the Demailly–Bonavero holomorphic Morse inequalities for the relevant Chern classes and curvature conditions appearing in the jet-differential setting. We will also add a short paragraph in the proof of the main existence theorem that explicitly invokes this comparison to justify the passage from the algebraic inequalities to the existence of nonzero sections. These additions will be placed in the sections already devoted to the definition and the main theorem, respectively. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces a new set of generalized algebraic Morse inequalities as an independent algebraic construction, then applies them directly to obtain the existence of Green-Griffiths (and Hasse-Schmidt) jet differentials on varieties of general type. This mirrors the logical structure of Demailly's original analytic argument but replaces the holomorphic Morse inequalities with a freshly proven algebraic counterpart; no step reduces the target existence statement to a fitted parameter, a self-citation chain, or a definitional tautology. The earlier preprint is referenced only as the source of the expanded version, not as a load-bearing premise. The derivation therefore remains non-circular and externally falsifiable via the algebraic estimates themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not specify any free parameters, axioms, or invented entities. The new algebraic Morse inequalities are introduced as the central technical device, but no further breakdown is provided.

pith-pipeline@v0.9.0 · 5394 in / 1221 out tokens · 52252 ms · 2026-05-10T02:23:34.215342+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Angelini, An algebraic version of Demailly’s asymptotic Morse inequa lities, Proceedings of the American Mathematical Society 124 (1996), no

[Ang96] F. Angelini, An algebraic version of Demailly’s asymptotic Morse inequa lities, Proceedings of the American Mathematical Society 124 (1996), no. 11, 3265–3269. [BD19] D. Brotbek and Y. Deng, Kobayashi hyperbolicity of the complements of general hype rsurfaces of high degree , Geometric and Functional Analysis 29 (2019), no. 3, 690–750. [B´ er15] G...

work page arXiv 1996
[2]

Bloch and D

[BG71] S. Bloch and D. Gieseker, The positivity of the Chern classes of an ample vector bundle , Inventiones math- ematicae 12 (1971), no. 2, 112–117. [BK24] G. B´ erczi and F. Kirwan, Non-reductive geometric invariant theory and hyperbolici ty, Inventiones mathe- maticae 235 (2024), no. 1, 81–127. [Bog77] F. Bogomolov, Families of curves on a surface of ...

work page 1971
[3]

6, 2331–2359

[Cad20] , Jet diﬀerentials on toroidal compactiﬁcations of ball quot ients, Annales de l’Institut Fourier 70 (2020), no. 6, 2331–2359. [Cad24] , Hyperbolicity of generic hypersurfaces of polynomial degr ee via Green-Griﬃths jet diﬀerentials , arXiv:2406.19003,

work page arXiv 2020
[4]

Campana, L

[CDDR24] F. Campana, L. Darondeau, J.-P. Demailly, and E. Ro usseau, On the existence of logarithmic and orbifold jet diﬀerentials , Annales Henri Lebesgue 7 (2024), 1–67 (en). MR 4765352 [CDR20] F. Campana, L. Darondeau, and E. Rousseau, Orbifold hyperbolicity , Compos. Math. 156 (2020), no. 8, 1664–1698 (English). [Dar15] L. Darondeau, On the logarithmi...

work page 2024
[5]

Demailly, L

[DEL00] J.-P. Demailly, L. Ein, and R. Lazarsfeld, A subadditivity property of multiplier ideals. , Michigan Math. J. 48 (2000), no. 1, 137–156. [Dem85] J.-P. Demailly, Champs magn´ etiques et in´ egalit´ es de Morse pour la d′′-cohomologie, Ann. Inst. Fourier (Grenoble) 35 (1985), 189–229. [Dem95] J.-P. Demailly, Propri´ et´ es de semi-continuit´ e de la...

work page 2000
[6]

Generalized Okounkov Bodies, Hyperbolicity Related and Direct Image Problems

[Den16] Y. Deng, Eﬀectivity in the hyperbolicity related problems , Chap. 4 of the PhD memoir “Generalized Okounkov Bodies, Hyperbolicity Related and Direct Image Problems” , arXiv:1606.03831,

work page arXiv
[7]

Diverio, Diﬀerential equations on complex projective hypersurface s of low dimension , Compos

[Div08] S. Diverio, Diﬀerential equations on complex projective hypersurface s of low dimension , Compos. Math. 144 (2008), no. 4, 920–932. [Div09] , Existence of global invariant jet diﬀerentials on projecti ve hypersurfaces of high degree , Mathema- tische Annalen 344 (2009), no. 2, 293–315. [DMR10] S. Diverio, J. Merker, and E. Rousseau, Eﬀective algeb...

work page 2008
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2, Berli n: Springer, 1998 (English)

Folge, vol. 2, Berli n: Springer, 1998 (English). [GG80] M. Green and P. Griﬃths, Two applications of algebraic geometry to entire holomorph ic mappings , The Chern Symposium 1979, Proc. Internal. Sympos. Berkeley, CA , 1979 (New York), Springer-Verlag, 1980, p. 41–74. [Har77] R. Hartshorne, Algebraic geometry , 1st ed

work page 1998
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[JV18] A

ed., Graduate Texts in Mathematics, 52, Sprin ger New York, New York, NY, 1977 (eng). [JV18] A. Javanpeykar and A. Vezzani, Non-archimedean hyperbolicity and applications , https://hal.science/hal- 01901289, October

work page 1977
[10]

Kawamata, A generalization of Kodaira-Ramanujam’s vanishing theore m, Mathematische Annalen 261 (1982), no

[Kaw82] Y. Kawamata, A generalization of Kodaira-Ramanujam’s vanishing theore m, Mathematische Annalen 261 (1982), no. 1, 43–46. [Lan87] S. Lang, Introduction to complex hyperbolic spaces , Springer-Verlag, New York,

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48, Springer-Verlag, Berlin, 2004, Classical s etting: line bundles and linear series

Folge, vol. 48, Springer-Verlag, Berlin, 2004, Classical s etting: line bundles and linear series. [McQ98] M. McQuillan, Diophantine approximations and foliations , Publications Math´ ematiques de l’IH´ES 87 (1998), 121–174. [Mer15] J. Merker, Algebraic diﬀerential equations for entire holomorphic cu rves in projective hypersurfaces of general type: Optim...

work page 2004
[12]

Siu and S

[SY96] Y.-T. Siu and S. K. Yeung, Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane , Invent. Math. 124 (1996), 573–618. [Voj04] P. Vojta, Jets via Hasse-Schmidt derivations , arXiv:math/0407113, 2004

work page arXiv 1996

[1] [1]

Angelini, An algebraic version of Demailly’s asymptotic Morse inequa lities, Proceedings of the American Mathematical Society 124 (1996), no

[Ang96] F. Angelini, An algebraic version of Demailly’s asymptotic Morse inequa lities, Proceedings of the American Mathematical Society 124 (1996), no. 11, 3265–3269. [BD19] D. Brotbek and Y. Deng, Kobayashi hyperbolicity of the complements of general hype rsurfaces of high degree , Geometric and Functional Analysis 29 (2019), no. 3, 690–750. [B´ er15] G...

work page arXiv 1996

[2] [2]

Bloch and D

[BG71] S. Bloch and D. Gieseker, The positivity of the Chern classes of an ample vector bundle , Inventiones math- ematicae 12 (1971), no. 2, 112–117. [BK24] G. B´ erczi and F. Kirwan, Non-reductive geometric invariant theory and hyperbolici ty, Inventiones mathe- maticae 235 (2024), no. 1, 81–127. [Bog77] F. Bogomolov, Families of curves on a surface of ...

work page 1971

[3] [3]

6, 2331–2359

[Cad20] , Jet diﬀerentials on toroidal compactiﬁcations of ball quot ients, Annales de l’Institut Fourier 70 (2020), no. 6, 2331–2359. [Cad24] , Hyperbolicity of generic hypersurfaces of polynomial degr ee via Green-Griﬃths jet diﬀerentials , arXiv:2406.19003,

work page arXiv 2020

[4] [4]

Campana, L

[CDDR24] F. Campana, L. Darondeau, J.-P. Demailly, and E. Ro usseau, On the existence of logarithmic and orbifold jet diﬀerentials , Annales Henri Lebesgue 7 (2024), 1–67 (en). MR 4765352 [CDR20] F. Campana, L. Darondeau, and E. Rousseau, Orbifold hyperbolicity , Compos. Math. 156 (2020), no. 8, 1664–1698 (English). [Dar15] L. Darondeau, On the logarithmi...

work page 2024

[5] [5]

Demailly, L

[DEL00] J.-P. Demailly, L. Ein, and R. Lazarsfeld, A subadditivity property of multiplier ideals. , Michigan Math. J. 48 (2000), no. 1, 137–156. [Dem85] J.-P. Demailly, Champs magn´ etiques et in´ egalit´ es de Morse pour la d′′-cohomologie, Ann. Inst. Fourier (Grenoble) 35 (1985), 189–229. [Dem95] J.-P. Demailly, Propri´ et´ es de semi-continuit´ e de la...

work page 2000

[6] [6]

Generalized Okounkov Bodies, Hyperbolicity Related and Direct Image Problems

[Den16] Y. Deng, Eﬀectivity in the hyperbolicity related problems , Chap. 4 of the PhD memoir “Generalized Okounkov Bodies, Hyperbolicity Related and Direct Image Problems” , arXiv:1606.03831,

work page arXiv

[7] [7]

Diverio, Diﬀerential equations on complex projective hypersurface s of low dimension , Compos

[Div08] S. Diverio, Diﬀerential equations on complex projective hypersurface s of low dimension , Compos. Math. 144 (2008), no. 4, 920–932. [Div09] , Existence of global invariant jet diﬀerentials on projecti ve hypersurfaces of high degree , Mathema- tische Annalen 344 (2009), no. 2, 293–315. [DMR10] S. Diverio, J. Merker, and E. Rousseau, Eﬀective algeb...

work page 2008

[8] [8]

2, Berli n: Springer, 1998 (English)

Folge, vol. 2, Berli n: Springer, 1998 (English). [GG80] M. Green and P. Griﬃths, Two applications of algebraic geometry to entire holomorph ic mappings , The Chern Symposium 1979, Proc. Internal. Sympos. Berkeley, CA , 1979 (New York), Springer-Verlag, 1980, p. 41–74. [Har77] R. Hartshorne, Algebraic geometry , 1st ed

work page 1998

[9] [9]

[JV18] A

ed., Graduate Texts in Mathematics, 52, Sprin ger New York, New York, NY, 1977 (eng). [JV18] A. Javanpeykar and A. Vezzani, Non-archimedean hyperbolicity and applications , https://hal.science/hal- 01901289, October

work page 1977

[10] [10]

Kawamata, A generalization of Kodaira-Ramanujam’s vanishing theore m, Mathematische Annalen 261 (1982), no

[Kaw82] Y. Kawamata, A generalization of Kodaira-Ramanujam’s vanishing theore m, Mathematische Annalen 261 (1982), no. 1, 43–46. [Lan87] S. Lang, Introduction to complex hyperbolic spaces , Springer-Verlag, New York,

work page 1982

[11] [11]

48, Springer-Verlag, Berlin, 2004, Classical s etting: line bundles and linear series

Folge, vol. 48, Springer-Verlag, Berlin, 2004, Classical s etting: line bundles and linear series. [McQ98] M. McQuillan, Diophantine approximations and foliations , Publications Math´ ematiques de l’IH´ES 87 (1998), 121–174. [Mer15] J. Merker, Algebraic diﬀerential equations for entire holomorphic cu rves in projective hypersurfaces of general type: Optim...

work page 2004

[12] [12]

Siu and S

[SY96] Y.-T. Siu and S. K. Yeung, Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane , Invent. Math. 124 (1996), 573–618. [Voj04] P. Vojta, Jets via Hasse-Schmidt derivations , arXiv:math/0407113, 2004

work page arXiv 1996