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Projective and affine structures in positive characteristic I: Chern class formulas and Characterizations of projective spaces

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Pith's one-line read Frobenius-projective structures characterize projective spaces in positive characteristic via Chern class conditions.

desk verdict Extends Frobenius structures from curves to higher dimensions with new Chern formulas, but the key step is whether Berthelot operators globalize cleanly in dim>1. read the letter →

arxiv 2011.04846 v3 pith:JQCSBGQS submitted 2020-11-10 math.AG math.NT

Yasuhiro Wakabayashi This is my paper
classification math.AGmath.NT
keywords FrobeniusstructuresprojectivespacesChernclassespositivecharacteristicalgebraicvarietiesdifferentialoperatorsGunningformulascharacterizations
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 develops a theory of projective and affine structures for higher-dimensional varieties in positive characteristic by focusing on their Frobenius versions. It first gives a description of these structures in terms of Berthelot's higher-level differential operators. This leads to positive-characteristic analogues of Gunning's formulas that supply necessary conditions on Chern classes for the structures to exist. The central achievement is a collection of characterizations of projective spaces that rely on the existence of Frobenius-projective structures.

What carries the argument

Frobenius-projective structures, defined through Berthelot's higher-level differential operators, which produce the Chern class formulas used for the characterizations.

What would settle it

A smooth projective variety over a field of positive characteristic that carries a Frobenius-projective structure yet violates the derived Chern class relations, or a non-projective space that satisfies the full set of characterization conditions.

Watch

Extended reading notes

Core claim

The paper establishes characterizations of projective spaces among smooth projective varieties over fields of positive characteristic by showing that the existence of a Frobenius-projective structure, when described via higher-level differential operators, imposes and is constrained by specific Chern class relations that only projective spaces satisfy.

Load-bearing premise

Berthelot's higher-level differential operators give a correct description of Frobenius-projective and Frobenius-affine structures on higher-dimensional varieties.

Editorial extensions

If this is right

  • Existence of a Frobenius-projective structure on a variety forces its Chern classes to obey explicit relations coming from the positive-characteristic Gunning formulas.
  • Projective spaces satisfy these relations and therefore admit Frobenius-projective structures.
  • The same operator description yields parallel Chern class conditions for Frobenius-affine structures.
  • The constructions extend the earlier curve case to arbitrary dimension while remaining within positive characteristic.

Reading between the lines

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

  • The same differential-operator approach could be tested on other Fano varieties to see whether Frobenius-projective structures appear only on projective space.
  • The Chern class obstructions might interact with known positive-characteristic invariants such as the Frobenius morphism itself.
  • Affine-space characterizations via Frobenius-affine structures remain available for a follow-up analysis using the same machinery.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript develops a theory of projective and affine structures on higher-dimensional varieties in positive characteristic. It first describes Frobenius-projective and Frobenius-affine structures via Berthelot's higher-level differential operators, then derives positive-characteristic analogues of Gunning's Chern-class formulas as necessary conditions for the existence of such structures, and finally establishes characterizations of projective spaces using Frobenius-projective structures.

Significance. If the extension of the differential-operator description from curves to higher-dimensional varieties is valid and the subsequent derivations hold, the work would provide new tools for studying varieties in characteristic p, including necessary conditions on Chern classes and characterizations of projective space that parallel classical results over the complex numbers.

major comments (2)
  1. [Introduction / §1 (description of structures)] The central claim that Berthelot's higher-level differential operators furnish a valid description of Frobenius-projective and Frobenius-affine structures on varieties of dimension greater than 1 is load-bearing for all subsequent results. The abstract states this as the first step, but the manuscript must explicitly verify that the local definitions globalize without extra obstructions on the tangent sheaf or transition functions when commuting with the absolute Frobenius (extending prior work limited to curves).
  2. [§3] §3 (Gunning-type formulas): The positive-characteristic Chern-class formulas are derived from the differential-operator description; if the latter only holds formally locally, the global necessary conditions on Chern classes for the existence of the structures are unsupported.
minor comments (2)
  1. Notation for the higher-level differential operators and the Frobenius structures should be introduced with explicit comparison to the curve case to aid readability.
  2. [final section] The characterizations of projective spaces in the final section would benefit from a clear statement of the precise hypotheses (e.g., dimension, smoothness) under which they apply.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the importance of explicit globalization arguments. We address the two major comments below and will revise the manuscript accordingly to strengthen the exposition.

read point-by-point responses

  1. Referee: [Introduction / §1 (description of structures)] The central claim that Berthelot's higher-level differential operators furnish a valid description of Frobenius-projective and Frobenius-affine structures on varieties of dimension greater than 1 is load-bearing for all subsequent results. The abstract states this as the first step, but the manuscript must explicitly verify that the local definitions globalize without extra obstructions on the tangent sheaf or transition functions when commuting with the absolute Frobenius (extending prior work limited to curves).

    Authors: We agree that an explicit verification of globalization is necessary for clarity. In the revised version we will insert a dedicated paragraph (or short subsection) in §1 that checks compatibility of the local Berthelot-operator descriptions with the transition functions of the tangent sheaf under the absolute Frobenius morphism. The argument proceeds by direct cocycle computation on the level of sheaves of differential operators and shows that no additional obstructions arise beyond those already present in the curve case; the higher-dimensional transition data are handled by the same formal properties of Berthelot’s rings that were used locally. revision: yes

  2. Referee: [§3] §3 (Gunning-type formulas): The positive-characteristic Chern-class formulas are derived from the differential-operator description; if the latter only holds formally locally, the global necessary conditions on Chern classes for the existence of the structures are unsupported.

    Authors: Once the globalization step is made explicit in §1, the derivations in §3 become global by construction: the Chern-class identities are obtained by pushing forward the global sheaf of differential operators and taking determinants, which are intrinsically global operations. In the revision we will add a sentence at the opening of §3 that explicitly recalls the globalization result of §1 before deriving the formulas, thereby making the logical dependence transparent. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation starts from external operator description and proceeds independently

full rationale

The paper's abstract and claimed chain begin with a description of Frobenius structures via Berthelot's higher-level differential operators (an external reference, not self-defined or fitted here), then derive Chern-class formulas, and finally characterizations. No quoted equations or steps in the provided text reduce any result to its inputs by construction, rename known patterns, or rely on load-bearing self-citations. The extension to higher dimensions is presented as building on prior curve cases without the central claims collapsing into tautologies or fitted renamings. This is the normal self-contained case.

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

The paper relies on standard background from algebraic geometry in positive characteristic; no free parameters, new entities, or ad-hoc axioms are visible in the abstract.

assumptions (1)
  • domain assumption Berthelot's higher-level differential operators behave as expected on higher-dimensional varieties in positive characteristic
    Invoked as the starting point for describing the structures

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Cite this review

Pith. "Pith review of Projective and affine structures in positive characteristic I: Chern class formulas and Characterizations of projective spaces." pith.science (2026). https://pith.science/paper/JQCSBGQS

@misc{pith2026201104846,
  author       = {Pith},
  title        = {Pith review of: Projective and affine structures in positive characteristic I: Chern class formulas and Characterizations of projective spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JQCSBGQS}},
  note         = {Machine review of arXiv:2011.04846}
}
read the original abstract

This paper aims to develop a theory of projective and affine structures on higher-dimensional varieties in positive characteristic. This theory deals with Frobenius-projective and Frobenius-affine structures, which have been previously investigated in the case where the underlying space is a curve. We first provide a description of such structures in terms of Berthelot's higher-level differential operators. That description leads us to obtain a positive characteristic version of Gunning's formulas, which give necessary conditions on Chern classes for the existence of Frobenius-projective and Frobenius-affine structures, respectively. Finally, we establish some characterizations of projective spaces using Frobenius-projective structures.

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

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Reference graph

Works this paper leans on

88 extracted references · 88 canonical work pages · cited by 1 Pith paper

  1. [1]

    Badescu, Algebraic Surfaces

    L. Badescu, Algebraic Surfaces. Springer Universitext (2001)

    work page 2001

  2. [2]

    Berthelot, Cohomologie cristalline des sch´ emas de caract´ eristique p > 0

    P. Berthelot, Cohomologie cristalline des sch´ emas de caract´ eristique p > 0. Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York , 407 (1974)

    work page 1974

  3. [3]

    Berthelot, D-modules arithm´ etiques

    P. Berthelot, D-modules arithm´ etiques. I. Op´ erateurs di´ erentiels de niveau fini, Ann. Sci. ´Ecole Norm. Sup. (4) 29 (1996), pp. 185-272

    work page 1996

  4. [4]

    Berthelot, D-modules arithm´ etiques

    P. Berthelot, D-modules arithm´ etiques. II. Descente par Frobenius, M´ em. Soc. Math. France (N.S.) 81 (2000)

    work page 2000

  5. [5]

    Berthelot, L

    P. Berthelot, L. Illusie, Classes de Chern en cohomologie cristalline . C. R. Acad. Sci. Paris S´ er. A-B 270 (1970), A1695-A1697; ibid. A1750-A1752

    work page 1970

  6. [6]

    Berthelot, A

    P. Berthelot, A. Ogus, Notes on Crystalline Cohomology . Princeton Univ. Press (1978)

    work page 1978

  7. [7]

    Biswas, Differential operators on complex manifolds with a flat p rojective structure

    I. Biswas, Differential operators on complex manifolds with a flat p rojective structure. J. Math Pures Appl. 78, (1999), pp. 1-26

    work page 1999

  8. [8]

    Biswas, J

    I. Biswas, J. P. P. dos Santos, Triviality criteria for bundles over rationally connected varieties. J. Ra- manujan Math. Soc. 28 (2013), pp. 423-442

    work page 2013

Show all 88 references
  1. [9]

    Biswas, S

    I. Biswas, S. Dumitrescu, Holomorphic projective connections o n compact complex threefolds. arXiv: math. AG/2005.08272v1, (2020)

    2005

  2. [10]

    Bloch, K

    S. Bloch, K. Kato, p-adic ´ etale cohomology.Publ. Math. IHES. 63 (1986), pp. 107-152

    1986

  3. [11]

    Bombieri, D

    E. Bombieri, D. Mumford, Enriques’ classification of surfaces in char. p, II, in Complex analysis and algebraic geometry, Cambridge Univ. Press (1977), pp. 23-42

    1977

  4. [12]

    Bombieri, D

    E. Bombieri, D. Mumford, Enriques’ classification of surfaces in char. p, III, Invent. Math. 35 (1976), pp. 197-232

    1976

  5. [13]

    Brenner, A

    H. Brenner, A. Kaid, On deep Frobenius descent and flat bundle s, Math. Res. Lett. 15 (2008), pp. 1101- 1115

    2008

  6. [14]

    Cossec, I

    F. Cossec, I. Dolgachev, Enriques surfaces I . Progress in Mathematics 76 Boston, Birkh¨ auser (1989)

    1989

  7. [15]

    Debarre, Vari´ et´ es rationnellement connexes (d’apr` esT

    O. Debarre, Vari´ et´ es rationnellement connexes (d’apr` esT. Graber, J. Harris, J. Starr et A. J. de Jong), S´ eminaire Bourbaki905 (June 2002), Ast´ erisque290 (2003), pp. 243-266

    2002

  8. [16]

    J. P. P. dos Santos, Fundamental group schemes for stratifi ed sheaves. J. Algebra 317 (2007), pp. 691-713

    2007

  9. [17]

    Dumitrescu, Connexions affines et projectives sur les surfa ces complexes compactes, Math

    S. Dumitrescu, Connexions affines et projectives sur les surfa ces complexes compactes, Math. Zeit. 264 (2010), pp. 301-316

    2010

  10. [18]

    Esnault, V

    H. Esnault, V. Mehta, Simply connected projective manifolds in c haracteristic p > 0 have no nontrivial stratified bundles Invent Math. 181 (2010), pp. 449-465

    2010

  11. [19]

    Fulton, Intersection Theory, Second edition , Springer-Verlag, Berlin, (1998)

    W. Fulton, Intersection Theory, Second edition , Springer-Verlag, Berlin, (1998)

    1998

  12. [20]

    Gieseker, Flat vector bundles and the fundamental group in non-zero characteristics, Ann

    D. Gieseker, Flat vector bundles and the fundamental group in non-zero characteristics, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (1975), pp. 1-31

    1975

  13. [21]

    Gros, Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique

    M. Gros, Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique. M´ em. Soc. Math. France (N.S.) , 21 (1985), pp. 1-87

    1985

  14. [22]

    Grothendieck, ´El´ ements de G´ eom´ etrie Alg´ ebrique II: ´Etude globale ´ el´ ementaire de quelques classes de morphismes

    A. Grothendieck, ´El´ ements de G´ eom´ etrie Alg´ ebrique II: ´Etude globale ´ el´ ementaire de quelques classes de morphismes. Publ. Math. I.H.E.S. 8 (1961)

    1961

  15. [23]

    Grothendieck, ´El´ ements de G´ eom´ etrie Alg´ ebrique, IV: ´Etude locale des sch´ emas et des morphismes de sch´ emas, Quatri` eme partie

    A. Grothendieck, ´El´ ements de G´ eom´ etrie Alg´ ebrique, IV: ´Etude locale des sch´ emas et des morphismes de sch´ emas, Quatri` eme partie. Publ. Math. I.H.E.S. 32 (1967)

    1967

  16. [24]

    Grothendieck, Repr´ esentations lin´ eaires et compactifications profinies des groupes discrets

    A. Grothendieck, Repr´ esentations lin´ eaires et compactifications profinies des groupes discrets. Manuscr. Math. 2 (1970), pp. 375-396

    1970

  17. [25]

    Grothendieck, Cohomologie locale des faisceaux coh´ erents et th´ eor` emes de Lefschetz locaux et globaux

    A. Grothendieck, Cohomologie locale des faisceaux coh´ erents et th´ eor` emes de Lefschetz locaux et globaux . Document Math´ emetiques (Paris) 4, Paris, Soc. Math. France, ( 2005)

    2005

  18. [26]

    R. C. Gunning, On uniformization of complex manifolds: the role of connect ions. Math. Notes 22, Prince- ton University Press (1978)

    1978

  19. [27]

    Hartshorne, Algebraic Geometry

    R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics 52 Springer-Verlag, New York- Heidelberg (1977)

    1977

  20. [28]

    Hirzebruch, Automorphe Formen und der Satz von Riemann- Roch, Symp

    F. Hirzebruch, Automorphe Formen und der Satz von Riemann- Roch, Symp. Internat. Topologia Alge- braica, Mexico, (1958), pp. 129-144

    1958

  21. [29]

    Holla, Counting maximal subbundles via Gromov-Witten invarian ts

    Y. Holla, Counting maximal subbundles via Gromov-Witten invarian ts. Math. Ann. 328 (2004), pp. 121- 133. FROBENIUS PROJECTIVE AND AFFINE GEOMETRY OF V ARIETIES IN CH ARACTERISTIC p > 0 127

    2004

  22. [30]

    Hoshi, Frobenius-projective structures on curves in posit ive characteristic

    Y. Hoshi, Frobenius-projective structures on curves in posit ive characteristic. Publ. Res. Inst. Math. Sci. 56 (2020), pp. 401-430

    2020

  23. [31]

    Hoshi, Frobenius-affine structures and Tango curves

    Y. Hoshi, Frobenius-affine structures and Tango curves. RIM S Preprint 1913 (2020)

    1913

  24. [32]

    Hoshi, A note on the existence of Tango curves

    Y. Hoshi, A note on the existence of Tango curves. RIMS Prepr int 1917 (2020)

    1917

  25. [33]

    Illusie, M

    L. Illusie, M. Raynaud, Les suites spectrales associ´ ees au co mplexe de de Rham-Witt. Inst. Hautes ´Etudes Sci. Publ. Math. 57, (1983), pp. 73-212

    1983

  26. [34]

    Jahnke, I

    P. Jahnke, I. Radloff, Threefolds with holomorphic normal proj ective connections. Math. Ann. 329 (2004), pp. 379-400

    2004

  27. [35]

    Joshi, On varieties with trivial tangent bundle

    K. Joshi, On varieties with trivial tangent bundle. arXiv: math. AG/1603.09413v1 , (2016)

    2016

  28. [36]

    Joshi, The degree of the dormant operatic locus

    K. Joshi, The degree of the dormant operatic locus. Internat. Math. Res. Notices 9 (2017), pp. 2599-2613

    2017

  29. [38]

    Joshi, C

    K. Joshi, C. S. Rajan. Frobenius splitting and ordinarity. Int. Math. Res. Not. , 2 (2003), pp. 109-121

    2003

  30. [39]

    Joshi, S

    K. Joshi, S. Ramanan, E. Z. Xia, and J. K. Yu, On vector bundles destabilized by Frobenius pull-back. Composito Mathematica 142 (2006), pp. 616-630

    2006

  31. [40]

    Joshi, C

    K. Joshi, C. Pauly, Hitchin-Mochizuki morphism, opers and Frob enius-destabilized vector bundles over curves. Adv. Math. 274 (2015), pp. 39-75

    2015

  32. [41]

    Kato, Log smooth deformation and moduli of log smooth curv es

    F. Kato, Log smooth deformation and moduli of log smooth curv es. Internat. J. Math. 11, (2000), pp. 215-232

    2000

  33. [42]

    Katsura, K

    T. Katsura, K. Ueno, On elliptic surfaces in characteristic p. Math. Ann. 272 (1985), pp. 291-330

    1985

  34. [43]

    N. M. Katz, Nilpotent connections and the monodromy theorem : Applications of a result of Turrittin. Inst. Hautes Etudes Sci. Publ. Math. 39, (1970), 175-232

    1970

  35. [44]

    N. M. Katz, Algebraic solutions of differential equations ( p-curvature and the Hodge filtration). Invent. Math. 18, (1972), 1-118

    1972

  36. [45]

    Klingler, Structures affines et projectives sur les surfaces complexes

    B. Klingler, Structures affines et projectives sur les surfaces complexes. Ann. Inst. Fourier 48 (1998), pp. 441-477

    1998

  37. [46]

    Koll´ ar, Rational curves on algebraic varieties

    J. Koll´ ar, Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete 32, Springer Verlag, Berlin, (1996)

    1996

  38. [47]

    Kobayashi, Recent results in Complex Differential Geometry, Jber

    S. Kobayashi, Recent results in Complex Differential Geometry, Jber. d. Dt. Math.-Verein 83 (1981), pp. 147-158

    1981

  39. [48]

    Kobayashi, T

    S. Kobayashi, T. Ochiai, Holomorphic projective structures on compact complex surfaces I. Math. Ann. 249 (1980), pp. 75-94

    1980

  40. [49]

    Kobayashi, T

    S. Kobayashi, T. Ochiai, Holomorphic projective structures on compact complex surfaces II. Math. Ann. 255 (1981), pp. 519-521

    1981

  41. [50]

    Lange, Some geometrical aspects of vector bundles on cur ves, Aportaciones Mat

    H. Lange, Some geometrical aspects of vector bundles on cur ves, Aportaciones Mat. 5 (1992), pp. 53-74

    1992

  42. [51]

    Langer, Generic positivity and foliations in positive character istic

    A. Langer, Generic positivity and foliations in positive character istic. Adv. Math. 277 (2015), pp. 1-23

    2015

  43. [52]

    Laszlo, C

    Y. Laszlo, C. Pauly, The Frobenius map, rank 2 vector bundles a nd Kummer’s quartic surface in charac- teristic 2 and 3, Adv. in Math. 185 (2004), pp. 246-269

    2004

  44. [53]

    Le Stum, A

    B. Le Stum, A. Quiros, Transverse crystals of finite level, Ann. Inst. Fourier 47 (1997), pp. 69-100

    1997

  45. [54]

    Liedtke, Algebraic surfaces in positive characteristic, Birational geometry, rational curves, and arith- metic, Simons Symp., Springer, Cham, (2013), pp

    C. Liedtke, Algebraic surfaces in positive characteristic, Birational geometry, rational curves, and arith- metic, Simons Symp., Springer, Cham, (2013), pp. 229-292

    2013

  46. [55]

    F. Liu, B. Osserman, Mochizuki’s indigenous bundles and Ehrhart polynomials. J. Algebraic Combin. 26 (2006), pp. 125-136

    2006

  47. [56]

    Q. Liu, D. Lorenzini, and M. Raynaud, N´ eron models, Lie algebra s, and reduction of curves of genus one. Invent. Math. 157 (2004), pp. 455-518

    2004

  48. [57]

    Malcev, On isomorphic matrix representations of infinite grou ps

    A. Malcev, On isomorphic matrix representations of infinite grou ps. Mat. Sb. N.S. 8 (1940), pp. 405-422

    1940

  49. [58]

    Matsumura, Commutative Algebra

    H. Matsumura, Commutative Algebra. 2nd edn, Benjamin, New York, (1980)

    1980

  50. [59]

    V. B. Mehta, A. Ramanathan, Frobenius splitting and cohomolog y vanishing of Schubert varieties. Annals of Math. , 122 (1985), pp. 27-40

    1985

  51. [60]

    V. B. Mehta, V. Srinvas, Varieties in positive characteristic with trivial tangent bundle. Compos. Math. 64 (1987), pp. 191-212

    1987

  52. [61]

    Milne, ´Etale cohomology

    J. Milne, ´Etale cohomology. Princeton Univ. Press, Princeton, N. J., (1980)

    1980

  53. [62]

    Mitsui, Multiple fibers of elliptic fibrations

    K. Mitsui, Multiple fibers of elliptic fibrations. Proceedings of Kinosaki Algebraic Geometry Symposium (2011), pp. 44-54. 128 YASUHIRO W AKABAYASHI

    2011

  54. [63]

    Miyaoka, On the Chern numbers of surfaces of general typ e, Invent

    Y. Miyaoka, On the Chern numbers of surfaces of general typ e, Invent. Math. 42 (1977), pp. 225-237

    1977

  55. [64]

    Miyaoka, On algebraic surfaces with positive index, Progr

    Y. Miyaoka, On algebraic surfaces with positive index, Progr. Math. 39 (1983), pp. 281-301

    1983

  56. [65]

    Miyaoka, T

    Y. Miyaoka, T. Peternell Geometry of Higher Dimensional Algebraic Varieties. DMV Sem. 26 (1997)

    1997

  57. [66]

    Mochizuki, Foundations of p-adic Teichm¨ uller theory

    S. Mochizuki, Foundations of p-adic Teichm¨ uller theory. American Mathematical Society, (1999)

    1999

  58. [67]

    Mori, Projective manifolds with ample tangent bundles

    S. Mori, Projective manifolds with ample tangent bundles. Ann. Math. 110 (1979), pp. 593-606

    1979

  59. [68]

    Mukai, Counterexamples to Kodaira’s vanishing and Yau’s inequ ality in positive characteristics

    S. Mukai, Counterexamples to Kodaira’s vanishing and Yau’s inequ ality in positive characteristics. Kyoto J. Math. 53 (2013), pp. 515-532

    2013

  60. [69]

    Mumford, Enriques’ classification of surfaces in char

    D. Mumford, Enriques’ classification of surfaces in char. p. I in Global Analysis (Papers in Honor of K. Kodaira) (1969), pp. 325-339

    1969

  61. [70]

    D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics 5, Pub- lished for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, (1970)

    1970

  62. [71]

    M. S. Narasimhan, S. Ramanan, Moduli of vector bundles on a co mpact Riemann surface, Annals of Mathematics 89 (1969), pp. 14-51

    1969

  63. [72]

    Osserman, The generalized Verschiebung map for curves of genus 2

    B. Osserman, The generalized Verschiebung map for curves of genus 2. Math. Ann. 336 (2006), pp. 963-986

    2006

  64. [73]

    Pan, Triviality and split of vector bundles on rationally connect ed varieties

    X. Pan, Triviality and split of vector bundles on rationally connect ed varieties. Math. Res. Lett. 22 (2015), pp. 529-547

    2015

  65. [74]

    Partsch, Deformations of elliptic fiber bundles in positive char acteristic

    H. Partsch, Deformations of elliptic fiber bundles in positive char acteristic. Nagoya Math. J. 211 (2013), pp. 79-108

    2013

  66. [75]

    vanishing theorem

    M. Raynaud, Contre-example au “vanishing theorem” en carac teristique p > 0. C.P. Ramanujan-a tribute, Tata Inst. Fund. Res. Studies in Math. 8 Springer, Berlin-New York (1978), pp. 273-278

    1978

  67. [76]

    Stichtenoth, Algebraic function fields and codes

    H. Stichtenoth, Algebraic function fields and codes. 2nd ed.. Graduate Text in Mathematics 254 Springer- Verlag, Berlin (2009)

    2009

  68. [77]

    van der Put, Stratified bundles on curves and differential Ga lois groups in positive characteristic

    M. van der Put, Stratified bundles on curves and differential Ga lois groups in positive characteristic. SIGMA Symmetry Integrability Geom. Methods Appl. 15 (2019), Paper No. 071, 24 pp

    2019

  69. [78]

    Wakabayashi, An explicit formula for the generic number of do rmant indigenous bundles

    Y. Wakabayashi, An explicit formula for the generic number of do rmant indigenous bundles. Publ. Res. Inst. Math. Sci. 50 (2014), pp. 383-409

    2014

  70. [79]

    Wakabayashi, l-adic cohomological field theories of dormant opers, arXiv: math

    Y. Wakabayashi, l-adic cohomological field theories of dormant opers, arXiv: math. AG/1709.04235 , (2017)

    2017

  71. [80]

    Wakabayashi, Spin networks, Ehrhart quasi-polynomials, an d combinatorics of dormant indigenous bundles

    Y. Wakabayashi, Spin networks, Ehrhart quasi-polynomials, an d combinatorics of dormant indigenous bundles. Kyoto J. Math. 59 (2019), pp. 649-684

    2019

  72. [81]

    Wakabayashi, A combinatorial description of the dormant Miu ra transformation, arXiv: math

    Y. Wakabayashi, A combinatorial description of the dormant Miu ra transformation, arXiv: math. AG/1905.03370, (2019)

    1905 arXiv

  73. [82]

    Wakabayashi, Dormant Miura opers, Tango structures, an d the Bethe ansatz equations modulo p, arXiv: math

    Y. Wakabayashi, Dormant Miura opers, Tango structures, an d the Bethe ansatz equations modulo p, arXiv: math. AG/1905.03364 , (2019)

    1905

  74. [83]

    Wakabayashi, Moduli of Tango structures and dormant Miur a opers, Moscow Math

    Y. Wakabayashi, Moduli of Tango structures and dormant Miur a opers, Moscow Math. J. 20 (2020), pp. 575-636

    2020

  75. [84]

    Wakabayashi, Quantization on algebraic curves with Frobeniu s-projective structure, arXiv: math

    Y. Wakabayashi, Quantization on algebraic curves with Frobeniu s-projective structure, arXiv: math. AG/2004.04283, (2020)

    2004

  76. [85]

    Wakabayashi, A theory of dormant opers on pointed stable curves

    Y. Wakabayashi, A theory of dormant opers on pointed stable curves . Ast´ erisque, Soc. Math. de France 432 (2022)

    2022

  77. [86]

    W. C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 66 (1979)

    1979

  78. [87]

    Y. G. Ye, On Fano manifolds with normal projective connections . Internat. J. Math. 5 (1994), pp. 265-271

    1994

  79. [88]

    Yau, Calabi’s conjecture and some new results in algebraic g eometry, Proc

    S.-T. Yau, Calabi’s conjecture and some new results in algebraic g eometry, Proc. Nat. Acad. Sci. USA 74 (1977), pp. 1798-1799

    1977

  80. [89]

    Zapponi, On the 1-pointed curves arising as ´ etale covers of the affine line in positive characteristic, Math

    L. Zapponi, On the 1-pointed curves arising as ´ etale covers of the affine line in positive characteristic, Math. Z. , 258 (2008), pp. 711-727

    2008

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