Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

Logarithmic Non-Abelian Hodge Theory for curves in prime characteristic

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read In prime characteristic, the logarithmic de Rham and Dolbeault moduli stacks are isomorphic after twisting by a pseudo-torsor over an Artin-Schreier-type Galois cover of the Hitchin base.

desk verdict Important and likely-correct paper with a real bug in Lemma 3.2 and a mismatch in Corollary 5.13; deserves peer review after a fix. read the letter →

arxiv 2501.09850 v1 pith:2K7Q53GF submitted 2025-01-16 math.AG

Mark Andrea de Cataldo , Siqing Zhang This is my paper · ORCID
classification math.AG MSC 14H6014D2014F1014G17
keywords logarithmicnon-abelianHodgetheoryprimecharacteristicHiggsbundlesflatconnectionsArtin-SchreierHitchinfibrationmodulistacksintersectioncohomology
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 a logarithmic analogue in prime characteristic of the Non-Abelian Hodge Theorem for curves: instead of adding parabolic or parahoric boundary data as in the complex case, the logarithmic de Rham and Dolbeault moduli stacks are related by twisting with a pseudo-torsor over a Galois cover of the Hitchin base. The cover is generated by an Artin-Schreier-type map on residues, and the twist is an isomorphism over a nonempty open region that dominates the Hitchin base. A semistable version maps degree-d Dolbeault moduli to degree-pd de Rham moduli and descends to adequate moduli spaces in the cases that matter. For G=GL_r this yields geometric information about the de Rham-Hitchin morphism and a cohomological embedding of intersection cohomology, which becomes an isomorphism when r and d are coprime and p>r. If correct, the result gives a purely characteristic-p description of tame boundary behavior without extra boundary structures.

What carries the argument

The load-bearing mechanism is the Artin-Schreier-type residue map AS: c → c given by a ↦ $a^{{[p]}}$ − a, which relates the characteristic polynomial of the residue of a log-connection to the characteristic polynomial of the residue of its p-curvature: if A is the residue matrix, the p-curvature residue is governed by $A^{{p}}$ − A. Pulling back the Hitchin base along this map produces the variety of residues cA = A(ω_{C1}(D1)) ×_{Γ(D1,c_{D1})} Γ(D,c_D), through which the de Rham-Hitchin morphism factors. Over cA, the paper builds a pseudo-torsor H under the base-changed Picard stack \tilde{P}; its objects are J_{a_p}-log-connections whose p-curvature is the Frobenius pullback of the canonical section τ(a1). A Lie-theoretic lemma, stating that for a regular element x the Artin-Schreier preimage inside Lie(I_x) injects into c under the Chevalley quotient, is what makes H a well-defined \tilde{P}-pseudo-torsor and lets the twisted product H ×^{\tilde{P}} \tilde{M}_{Dol} map isomorphically to M_{dR} over the open image.

What would settle it

Compute, for a pair (G,p) allowed by the paper's standing assumption (p not dividing the order of the Weyl group, or G a product of general linear groups), the Artin-Schreier preimage of the canonical section τ(a) inside the centralizer of a regular element x; if two distinct elements A,B in that preimage have equal Chevalley images χ(A)=χ(B), then the key Lie-theoretic lemma is false and the Log-p-NAHT construction would fail.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1, the Log-p-NAHT: under the standing assumptions on (p,G), there is a non-empty pseudo-torsor H, smooth over the Artin-Schreier-type base cA, and a morphism of stacks n: H ×^{\tilde{P}} \tilde{M}_{Dol}(C1,D1) → M_{dR}(C,D) that is an isomorphism over the nonempty open image of H in cA, with that image surjecting onto the Hitchin base A(ω_{C1}(D1)). The paper also establishes the semistable version, where degree-d Dolbeault objects correspond to degree-pd de Rham objects, and, for GL_r or low-height reductive groups, the correspondence descends to adequate moduli spaces. A striking consequence in the vector-bundle case is that the de Rham-Hitchin morphism acquires connected, equidimensional fibers over the refined base, and that intersection cohomology of the Dolbeault moduli embeds into that of the de Rham moduli, becoming an isomorphism when r is coprime to d and p>r.

Load-bearing premise

The pseudo-torsor H is well-defined and smooth only if the Artin-Schreier map on the regular centralizer of a regular element is injective after composing with the Chevalley quotient; if that Lie-theoretic lemma fails, the definition of H as a pseudo-torsor, and with it the whole correspondence, collapses.

Editorial extensions

If this is right

  • The two logarithmic moduli stacks are not étale locally equivalent over the Hitchin base, but they become isomorphic after pulling back to the Artin-Schreier cover cA of residue data, with the isomorphism holding over a nonempty open region that dominates the base.
  • For GL_r the correspondence computes how degrees change: a degree-d Dolbeault object produces a degree-pd de Rham object in the semistable case, with the refined formula tracking the Euler-characteristic correction encoded in the spectral curves.
  • The semistable correspondence descends to adequate moduli spaces for GL_r and for reductive groups of low height, yielding a proper, surjective de Rham-Hitchin-residue morphism with connected and equidimensional fibers.
  • When p>r and (d,r)=1, the perverse Leray-filtered cohomology of the degree-d Dolbeault moduli space is isomorphic to that of the degree-pd de Rham moduli space, via a split injection of intersection cohomology complexes.
  • Unlike the complex-analytic logarithmic Hodge correspondence, no parabolic or parahoric structures at the punctures are needed; the boundary behavior is controlled purely by the Artin-Schreier eigenvalue map on residues.

Reading between the lines

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

  • Because the theorem holds without fixing parabolic data, a reduction-mod-p strategy might yield a generic-residue tame logarithmic Hodge correspondence over the complex numbers, a consequence the paper does not itself claim.
  • The Artin-Schreier cover structure suggests that, in the GL_r case, the difference between the de Rham and Dolbeault spectral pictures is governed by the Frobenius action on residue eigenvalues; this could be tested explicitly on Hitchin fibers over spectral curves with repeated eigenvalues.
  • The split injection of intersection cohomology complexes is stronger than an isomorphism of global cohomology groups; if it can be upgraded to a full perverse equivalence over the entire base, it would sharpen comparisons of perverse Leray filtrations in positive characteristic.
  • The residue-local nature of the Artin-Schreier mechanism suggests possible extensions to weighted poles or to simple-normal-crossing divisors in higher dimensions, although the spectral-curve arguments used here are special to curves.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper establishes a logarithmic version of the p-NAHT for a smooth projective curve C over an algebraically closed field of characteristic p. It introduces a variety of residues cA as a fiber product recording the Artin-Schreier relation between residues of log-connections and residues of their p-curvatures, constructs a stack H of special log-connections, and proves that a twisted product of H with the Dolbeault moduli stack on the Frobenius twist is isomorphic to the de Rham moduli stack over the open image of H. It also gives a semistable refinement, a GL_r degree refinement, several geometric consequences (surjectivity, connected fibers, equidimensionality, weak Abelian fibrations), and cohomological consequences for intersection cohomology.

Significance. If the central construction is repaired, this is a substantial result: it removes parahoric/parabolic data from the logarithmic p-NAHT, identifies the obstruction to an etale-local statement as the disconnected fibers of the de Rham-Hitchin morphism and the Artin-Schreier nature of the residue map, and connects to the prior works of Chen-Zhu, Schepler, Shen, and Li-Sun. The paper is careful in several places: Appendix A gives explicit residue and p-curvature identities (76)-(82), the comparisons in Section 3.3 are concrete, and the cohomological applications are tied to a full-support theorem. However, the current version contains a false Lie-theoretic lemma that is load-bearing for the pseudo-torsor structure of H, so the central theorem is not yet supported as written.

major comments (2)
  1. [Section 3.1.5, Lemma 3.2]
  2. [Section 3.1.10, Lemma 3.7]
minor comments (5)
  1. [Section 2.1.5]
  2. [Section 2.3]
  3. [Section 5.4.2, Theorem 5.11(5)]
  4. [Section 3.1.8]
  5. [Section 3.2.2 and Proposition 3.22]

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Log-p-NAHT is a genuine construction over a new base cA, built from independent inputs (Chen–Zhu, Ogus–Vologodsky, BNR, Ngo) and the authors' prior no-pole results are not assumed conclusions.

full rationale

The central morphism n is constructed, not defined, as the twisted product H ×^P MDol -> MdR, with H defined as a Cartesian pullback of the J-de Rham Hitchin morphism along the Chen–Zhu section. The Artin–Schreier cover cA is introduced from the residue/p-curvature identity res(Ψ(∇)) = res(∇)^[p] − res(∇); it is not the target of the theorem. The pseudo-torsor structure of H is proved in Prop. 3.8 from Lemma 3.7, whose content is that the residue in the fiber is unique; this is a mathematical statement, not a renaming. The main theorem's isomorphy over cA_im is proved by constructing an inverse via Frobenius descent and the dual connection; the proof does not invoke the theorem being proved. Self-citations appear (e.g., [HZ23b] for the semistable no-pole analogue, [dCGZ22] for very good splittings), but these are independent prior results with stated assumptions that do not include the log case or the Artin–Schreier base cA, and the paper explicitly adapts them rather than assuming the Log-p-NAHT. The GL_r consequences (Prop. 4.7, Thm. 4.8) are derived from the Log-p-de Rham-BNR Lemma 4.1 and Riemann–Roch degree bookkeeping, not from a fitted parameter. The cohomological embedding Theorem 5.12 uses the support theorem [MS23] and the Decomposition Theorem, both external. The only flagged issue in the supplied reader's take is a possible gap in Lemma 3.2 (regularity hypothesis); this is a correctness concern about a proof step, not a circularity, since the lemma is not true by construction and its failure would break the proof rather than make it tautological. No equation in the paper reduces to its inputs by definition, and no prediction is a renamed fit. Hence score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The central theorem does not fit parameters or postulate hidden physical entities. It builds on standard p-NAHT machinery: regular centralizers, Kostant sections, Chen-Zhu sections, Cartier descent, and the authors' earlier semistable no-pole work. The main novel objects, cA and H, are explicit constructions. The ledger exposes the group-theoretic restrictions (Context 1.4, good enough, low height) and the reliance on prior support-theoretic results for the cohomological consequences.

assumptions (6)
  • domain assumption Context 1.4: either p does not divide |W|, or G is a product of general linear groups.
    The Chevalley isomorphism c = t//W, the Kostant section, and the centralizer formalism used to define H depend on this restriction, as stated in Section 1.1.
  • domain assumption Good enough or low height assumptions: p >= h(G), or G = product of GLs; for some results G satisfies p > ht(g).
    These hypotheses guarantee nonemptiness of H^o, existence of adequate moduli spaces, and validity of support-theoretic arguments, as in Definitions 1.6 and 1.7.
  • standard math The Chen-Zhu section tau : c -> Lie(J) and the universal regular centralizer group scheme J exist and have the stated equivariance properties.
    Used to define H via the pullback tau(a1) and to identify p-curvatures with the Kostant section, following [CZ15, Section 2.3].
  • standard math Cartier descent and the p-curvature and residue identities for log-connections on torsors in Appendix A, equations (76) through (82).
    These identities control p-curvatures and residues under twisting, duals, and Frobenius pullback, and are essential to the proof of Theorem 3.12.
  • standard math The Decomposition Theorem of [BBDG18] and Ngo's support theorem apply over the relevant bases, with the Severi inequality used when p > r.
    Used in Section 5.4 to prove full support and to derive the intersection-cohomology embedding for GL_r.
  • standard math Adequate moduli spaces for the semistable stacks exist under the stated hypotheses, citing [Lan14], [HZ23a, Thm. 2.26], and related results.
    Needed for the geometric and cohomological consequences of the semistable Log-p-NAHT.
invented entities (2)
  • cA, the variety of residues independent evidence
    purpose: A fiber product over A(omega_C1(D1)) and Gamma(D1,c_D1) recording both the Hitchin image and residue adjoint orbits, so that the de Rham-Hitchin morphism factors through a space with better fiber behavior.
    Explicitly constructed in Definition 2.6 as a fiber product, with stated nonsingularity and disjoint-union decomposition; it is not a postulated or hidden object.
  • H, the pseudo-torsor of special J_{a_p}-log-connections independent evidence
    purpose: A stack over cA whose objects have prescribed p-curvature section, used to build the Log-p-NAHT morphism as a twisted product with the Dolbeault stack.
    Defined explicitly by the Cartesian diagram (31) and shown to be a pseudo-torsor in Proposition 3.8; its behavior is determined by the paper's constructions, not by an external postulate.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Logarithmic Non-Abelian Hodge Theory for curves in prime characteristic." pith.science (2026). https://pith.science/paper/2K7Q53GF

@misc{pith2026250109850,
  author       = {Pith},
  title        = {Pith review of: Logarithmic Non-Abelian Hodge Theory for curves in prime characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2K7Q53GF}},
  note         = {Machine review of arXiv:2501.09850}
}
read the original abstract

For a curve C and a reductive group G in prime characteristic, we relate the de Rham moduli of logarithmic G-connections on C to the Dolbeault moduli of logarithmic G-Higgs bundles on the Frobenius twist of C. We name this result the Log-p-NAHT. It is a logarithmic version of Chen-Zhu's characteristic p Non Abelian Hodge Theorem (p-NAHT). In contrast to the no pole case, the two moduli stacks in the log case are not isomorphic etale locally over the Hitchin base. Instead, they differ by an Artin-Schreier type Galois cover of the base. In contrast to the case over the complex numbers, where some parabolic/parahoric data are needed to specify the boundary behavior of the tame harmonic metrics, no parabolic/parahoric data are needed in Log-p-NAHT. We also establish a semistable version of the Log-p-NAHT, and deduce several geometric and cohomological consequences. In particular, when G=GL_r, the Log-p-NAHT induces an embedding of the intersection cohomology of the degree d Dolbeault moduli to that of the degree pd de Rham moduli, and the embedding is an isomorphism when r is coprime to d and p>r.

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

  1. Constructing Parabolic Non-Abelian Hodge Correspondence in Positive Characteristic Using Parabolic Bases

    math.AG 2025-01 conditional novelty 5.0 of 10

    A parabolic-bases framework is used to construct the parabolic non-abelian Hodge correspondence in positive characteristic on arbitrary-dimensional log varieties, extending Krishnamoorthy-Sheng's curve-level result.

Reference graph

Works this paper leans on

85 extracted references · 67 canonical work pages · cited by 1 Pith paper

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...

  2. [2]

    Th \'e orie des topos et cohomologie \'e tale des sch \'e mas

    M Artin, A Grothendieck, and JL Verdier. Th \'e orie des topos et cohomologie \'e tale des sch \'e mas. Tomes 1--3. Lecture Notes in Mathematics , 269:270, 1973

    1973

  3. [3]

    Irreducibility of the compactified Jacobian

    Allen B Altman, Anthony Iarrobino, and Steven L Kleiman. Irreducibility of the compactified Jacobian. Real and complex singularities, Oslo 1976 , pages 1--12, 1977

    1976

  4. [4]

    Adequate moduli spaces and geometrically reductive group schemes

    Jarod Alper. Adequate moduli spaces and geometrically reductive group schemes. Algebr. Geom. , 1(4):489--531, 2014

    2014

  5. [5]

    Geometric Langlands Correspondence for D-modules in Prime Characteristic: the GL (n) Case

    Roman Bezrukavnikov and Alexander Braverman. Geometric Langlands Correspondence for D-modules in Prime Characteristic: the GL (n) Case. Pure and Applied Mathematics Quarterly , 3(1):153--179, 2007

    2007

  6. [6]

    Faisceaux pervers , volume 4

    Alexander Beilinson, Joseph Bernstein, Pierre Deligne, and Ofer Gabber. Faisceaux pervers , volume 4. Soci \'e t \'e math \'e matique de France Paris, 2018

    work page 2018

  7. [7]

    Quantization of Hitchin’s integrable system and Hecke eigensheaves, 1991

    Alexander Beilinson and Vladimir Drinfeld. Quantization of Hitchin’s integrable system and Hecke eigensheaves, 1991

    work page 1991

  8. [8]

    On complete reducibility in characteristic p

    Vikraman Balaji, Pierre Deligne, and AJ Parameswaran. On complete reducibility in characteristic p . \'E pijournal de G \'e om \'e trie Alg \'e brique , 1, 2017

    work page 2017

Show all 85 references
  1. [9]

    The prismatization of p -adic formal schemes

    Bhargav Bhatt and Jacob Lurie. The prismatization of p -adic formal schemes. arXiv preprint arXiv:2201.06124 , 2022

    2022 arXiv

  2. [10]

    N\' e ron models , volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]

    Siegfried Bosch, Werner L\" u tkebohmert, and Michel Raynaud. N\' e ron models , volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] . Springer-Verlag, Berlin, 1990

    1990

  3. [11]

    Localization of modules for a semisimple Lie algebra in prime characteristic

    Roman Bezrukavnikov, Ivan Mirkovi \'c , Dmitriy Rumynin, and Simon Riche. Localization of modules for a semisimple Lie algebra in prime characteristic. Annals of Mathematics , pages 945--991, 2008

    2008

  4. [12]

    Arnaud Beauville, M. S. Narasimhan, and S. Ramanan. Spectral curves and the generalised theta divisor. J. Reine Angew. Math. , 398:169--179, 1989

    1989

  5. [13]

    Wild character varieties, meromorphic Hitchin systems and Dynkin diagrams

    Philip Boalch. Wild character varieties, meromorphic Hitchin systems and Dynkin diagrams. Geometry and physics , 2:433--454, 2018

    2018

  6. [14]

    Lie groups and L ie algebras

    Nicolas Bourbaki. Lie groups and L ie algebras. C hapters 4--6 . Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley

    2002

  7. [15]

    Bourbaki

    N. Bourbaki. Lie Groups and Lie Algebras: Chapters 7-9 . Number v. 3; v. 7 in Actualit \'e s scientifiques et industrielles. Springer Berlin Heidelberg, 2004

    2004

  8. [16]

    Pseudo-reductive groups , volume 26

    Brian Conrad, Ofer Gabber, and Gopal Prasad. Pseudo-reductive groups , volume 26. Cambridge University Press, 2015

    2015

  9. [17]

    Un th\' e or\`eme du support pour la fibration de H itchin

    Pierre-Henri Chaudouard and G\' e rard Laumon. Un th\' e or\`eme du support pour la fibration de H itchin. Ann. Inst. Fourier (Grenoble) , 66(2):711--727, 2016

    2016

  10. [18]

    Non-abelian H odge theory for algebraic curves in characteristic p

    Tsao-Hsien Chen and Xinwen Zhu. Non-abelian H odge theory for algebraic curves in characteristic p . Geom. Funct. Anal. , 25(6):1706--1733, 2015

    2015

  11. [19]

    Geometric Langlands in prime characteristic

    Tsao-Hsien Chen and Xinwen Zhu. Geometric Langlands in prime characteristic. Compositio Mathematica , 153(2):395--452, 2017

    2017

  12. [20]

    Lectures on H iggs moduli and abelianisation

    Peter Dalakov. Lectures on H iggs moduli and abelianisation. Journal of Geometry and Physics , 118:94--125, 2017

    2017

  13. [21]

    A support theorem for the Hitchin fibration: the case of SL_n

    Mark Andrea de Cataldo. A support theorem for the Hitchin fibration: the case of SL_n . Compositio Mathematica , 153(6):1316--1347, 2017

    2017

  14. [22]

    de Cataldo

    Mark Andrea A. de Cataldo. Perverse L eray filtration and specialisation with applications to the H itchin morphism. Math. Proc. Cambridge Philos. Soc. , 172(2):443--487, 2022

    2022

  15. [23]

    The de Rham stack and the variety of very good splittings of a curve

    Mark Andrea A de Cataldo, Michael Groechenig, and Siqing Zhang. The de Rham stack and the variety of very good splittings of a curve. arXiv preprint arXiv:2211.09630, to appear in Math. Res. Lett. , 2022

    2022 arXiv

  16. [24]

    Geometry of the logarithmic Hodge moduli space (with an Appendix joint with Siqing Zhang)

    Mark Andrea A de Cataldo and Andres Fernandez Herrero. Geometry of the logarithmic Hodge moduli space (with an Appendix joint with Siqing Zhang). arXiv preprint arXiv:2211.06754 , 2022

    2022 arXiv

  17. [25]

    Frobenius semisimplicity for convolution morphisms

    Mark Andrea de Cataldo, Thomas J Haines, and Li Li. Frobenius semisimplicity for convolution morphisms. Mathematische Zeitschrift , 289(1):119--169, 2018

    2018

  18. [26]

    Topology of Hitchin systems and Hodge theory of character varieties: the case A 1

    Mark Andrea A de Cataldo, Tam \'a s Hausel, and Luca Migliorini. Topology of Hitchin systems and Hodge theory of character varieties: the case A 1. Annals of Mathematics , pages 1329--1407, 2012

    2012

  19. [27]

    Hitchin fibrations, abelian surfaces, and the P= W conjecture

    Mark de Cataldo, Davesh Maulik, and Junliang Shen. Hitchin fibrations, abelian surfaces, and the P= W conjecture. Journal of the American Mathematical Society , 35(3):911--953, 2022

    2022

  20. [28]

    Cohomology of the moduli of Higgs bundles on a curve via positive characteristic

    Mark Andrea de Cataldo, Davesh Maulik, Junliang Shen, and Siqing Zhang. Cohomology of the moduli of Higgs bundles on a curve via positive characteristic. Journal of the European Mathematical Society , pages 1--21, 2023

    2023

  21. [29]

    A cohomological nonabelian Hodge theorem in positive characteristic

    Mark Andrea de Cataldo and Siqing Zhang. A cohomological nonabelian Hodge theorem in positive characteristic. Algebraic Geometry , 9(5), 2022

    2022

  22. [30]

    Equations diff \'e rentielles \`a points singuliers r \'e guliers

    Pierre Deligne. Equations diff \'e rentielles \`a points singuliers r \'e guliers. Lecture notes in mathematics , 163, 1970

    1970

  23. [31]

    \'E l \'e ments de g \'e om \'e trie alg \'e brique , volume 166

    Jean Dieudonn \'e and Alexandre Grothendieck. \'E l \'e ments de g \'e om \'e trie alg \'e brique , volume 166. Springer Berlin Heidelberg New York, 1971

    1971

  24. [32]

    Langlands duality for Hitchin systems

    Ron Donagi and Tony Pantev. Langlands duality for Hitchin systems. arXiv preprint math/0604617 , 2006

    2006 arXiv

  25. [33]

    Twistor Hecke eigensheaves in genus 2

    Ron Donagi, Tony Pantev, and Carlos Simpson. Twistor Hecke eigensheaves in genus 2. arXiv preprint arXiv:2403.17045 , 2024

    2024 arXiv

  26. [34]

    Rigid connections and F-isocrystals

    H \'e lene Esnault and Michael Groechenig. Rigid connections and F-isocrystals. Acta Math , 225:103--158, 2020

    2020

  27. [35]

    Moduli of flat connections in positive characteristic

    Michael Groechenig. Moduli of flat connections in positive characteristic. Math. Res. Lett. , 23(4):989--1047, 2016

    2016

  28. [36]

    Hodge theorem for the logarithmic de Rham complex via derived intersections

    M \'a rton Hablicsek. Hodge theorem for the logarithmic de Rham complex via derived intersections. Research in the Mathematical Sciences , 7:1--21, 2020

    2020

  29. [37]

    Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve

    Tam \'a s Hausel. Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve. In Geometric methods in algebra and number theory , pages 193--217. Springer, 2005

    2005

  30. [38]

    Hilbert- M umford stability on algebraic stacks and applications to G -bundles on curves

    Jochen Heinloth. Hilbert- M umford stability on algebraic stacks and applications to G -bundles on curves. \' E pijournal Geom. Alg\' e brique , 1:Art. 11, 37, 2017

    2017

  31. [39]

    The self-duality equations on a Riemann surface

    Nigel J Hitchin. The self-duality equations on a Riemann surface. Proceedings of the London Mathematical Society , 3(1):59--126, 1987

    1987

  32. [40]

    Tame parahoric nonabelian Hodge correspondence on curves

    Pengfei Huang, Georgios Kydonakis, Hao Sun, and Lutian Zhao. Tame parahoric nonabelian Hodge correspondence on curves. arXiv preprint arXiv:2205.15475 , 2022

    2022 arXiv

  33. [41]

    P= W via H\_2

    Tam \'a s Hausel, Anton Mellit, Alexandre Minets, and Olivier Schiffmann. P= W via H\_2 . arXiv preprint arXiv:2209.05429 , 2022

    2022 arXiv

  34. [42]

    On moduli stacks of G -bundles over a curve

    Norbert Hoffmann. On moduli stacks of G -bundles over a curve. In Affine flag manifolds and principal bundles , Trends Math., pages 155--163. Birkh\" a user/Springer Basel AG, Basel, 2010

    2010

  35. [43]

    TWO PROOFS OF THE P= W CONJECTURE

    Victoria Hoskins. TWO PROOFS OF THE P= W CONJECTURE. S \'e minaire BOURBAKI , page 76e, 2023

    2023

  36. [44]

    Generators for the cohomology ring of the moduli space of rank 2 H iggs bundles

    Tam\' a s Hausel and Michael Thaddeus. Generators for the cohomology ring of the moduli space of rank 2 H iggs bundles. Proc. London Math. Soc. (3) , 88(3):632--658, 2004

    2004

  37. [45]

    p -adic non-abelian Hodge theory for curves via moduli stacks

    Ben Heuer and Daxin Xu. p -adic non-abelian Hodge theory for curves via moduli stacks. arXiv preprint arXiv:2402.01365 , 2024

    2024 arXiv

  38. [46]

    Meromorphic Hodge moduli spaces for reductive groups in arbitrary characteristic

    Andres Fernandez Herrero and Siqing Zhang. Meromorphic Hodge moduli spaces for reductive groups in arbitrary characteristic. arXiv preprint arXiv:2307.16755 , 2023

    2023 arXiv

  39. [47]

    Semistable Non Abelian Hodge theorem in positive characteristic

    Andres Fernandez Herrero and Siqing Zhang. Semistable Non Abelian Hodge theorem in positive characteristic. arXiv preprint arXiv:2310.09923 , 2023

    2023 arXiv

  40. [48]

    Nilpotent orbits in representation theory

    Jens Carsten Jantzen. Nilpotent orbits in representation theory. Lie theory: Lie algebras and representations , pages 1--211, 2004

    2004

  41. [49]

    Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin

    Nicholas M Katz. Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin. Publications Math \'e matiques de l'IH \'E S , 39:175--232, 1970

    1970

  42. [50]

    Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 1989

    Kazuya Kato. Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 1989

    1988

  43. [51]

    Lie group representations on polynomial rings

    Bertram Kostant. Lie group representations on polynomial rings. American Journal of Mathematics , 85(3):327--404, 1963

    1963

  44. [52]

    Semistable modules over L ie algebroids in positive characteristic

    Adrian Langer. Semistable modules over L ie algebroids in positive characteristic. Doc. Math. , 19:509--540, 2014

    2014

  45. [53]

    Bogomolov’s inequality for Higgs sheaves in positive characteristic

    Adrian Langer. Bogomolov’s inequality for Higgs sheaves in positive characteristic. Inventiones mathematicae , 199(3):889--920, 2015

    2015

  46. [54]

    Moduli spaces of semistable modules over Lie algebroids

    Adrian Langer. Moduli spaces of semistable modules over Lie algebroids. arXiv preprint arXiv:2107.03128 , 2021

    2021 arXiv

  47. [55]

    Bogomolov’s inequality and Higgs sheaves on normal varieties in positive characteristic

    Adrian Langer. Bogomolov’s inequality and Higgs sheaves on normal varieties in positive characteristic. Journal f \"u r die reine und angewandte Mathematik (Crelles Journal) , 2024(810):1--48, 2024

    2024

  48. [56]

    Algebraic geometry and arithmetic curves , volume 6

    Qing Liu. Algebraic geometry and arithmetic curves , volume 6. Oxford University Press on Demand, 2002

    2002

  49. [57]

    Champs alg \'e briques , volume 39

    G \'e rard Laumon and Laurent Moret-Bailly. Champs alg \'e briques , volume 39. Springer, 2018

    2018

  50. [58]

    Le lemme fondamental pour les groupes unitaires

    G \'e rard Laumon and Bao Ch \^a u Ng \^o . Le lemme fondamental pour les groupes unitaires. Annals of mathematics , pages 477--573, 2008

    2008

  51. [59]

    Tame Parahoric Nonabelian Hodge Correspondence in Positive Characteristic over Algebraic Curves, 2024

    Mao Li and Hao Sun. Tame Parahoric Nonabelian Hodge Correspondence in Positive Characteristic over Algebraic Curves, 2024

    2024

  52. [60]

    Abelian unipotent subgroups of reductive groups

    George J McNinch. Abelian unipotent subgroups of reductive groups. Journal of Pure and Applied Algebra , 167(2-3):269--300, 2002

    2002

  53. [61]

    Sub-principal homomorphisms in positive characteristic

    George J McNinch. Sub-principal homomorphisms in positive characteristic. Mathematische Zeitschrift , 244:433--455, 2003

    2003

  54. [62]

    Algebraic groups: the theory of group schemes of finite type over a field , volume 170

    James S Milne. Algebraic groups: the theory of group schemes of finite type over a field , volume 170. Cambridge University Press, 2017

    2017

  55. [63]

    Kobayashi-Hitchin correspondence for tame harmonic bundles and an application

    Takuro Mochizuki. Kobayashi-Hitchin correspondence for tame harmonic bundles and an application. arXiv preprint math/0411300 , 2004

    2004 arXiv

  56. [64]

    Fourier--Mukai and autoduality for compactified Jacobians

    Margarida Melo, Antonio Rapagnetta, and Filippo Viviani. Fourier--Mukai and autoduality for compactified Jacobians. I. Journal f \"u r die reine und angewandte Mathematik (Crelles Journal) , 2019(755):1--65, 2019

    2019

  57. [65]

    A criterion for the existence of logarithmic connections on curves over a perfect field

    S Manikandan and Anoop Singh. A criterion for the existence of logarithmic connections on curves over a perfect field. Indian Journal of Pure and Applied Mathematics , 53(2):330--339, 2022

    2022

  58. [66]

    Cohomological --independence for moduli of one-dimensional sheaves and moduli of Higgs bundles

    Davesh Maulik and Junliang Shen. Cohomological --independence for moduli of one-dimensional sheaves and moduli of Higgs bundles. Geometry & Topology , 27(4):1539--1586, 2023

    2023

  59. [67]

    The P=W conjecture for GL\_n

    Davesh Maulik and Junliang Shen. The P=W conjecture for GL\_n. Annals of Mathematics , 200(2):529--556, 2024

    2024

  60. [68]

    Fibration de Hitchin et endoscopie

    Bao Ch \^a u Ng \^o . Fibration de Hitchin et endoscopie. Inventiones mathematicae , 164:399--453, 2006

    2006

  61. [69]

    Le lemme fondamental pour les alg\`ebres de L ie

    Bao Ch \^a u Ng \^o . Le lemme fondamental pour les alg\`ebres de L ie. Publ. Math. Inst. Hautes \' E tudes Sci. , (111):1--169, 2010

    2010

  62. [70]

    Crystalline prisms: Reflections on the present and past

    Arthur Ogus. Crystalline prisms: Reflections on the present and past. arXiv preprint arXiv:2204.06621 , 2022

    2022

  63. [71]

    Nonabelian Hodge theory in characteristic p

    Arthur Ogus and Vadim Vologodsky. Nonabelian Hodge theory in characteristic p. Publications math \'e matiques , 106(1):1--138, 2007

    2007

  64. [72]

    Kostant section, universal centralizer, and a modular derived Satake equivalence

    Simon Riche. Kostant section, universal centralizer, and a modular derived Satake equivalence. Mathematische Zeitschrift , 286(1):223--261, 2017

    2017

  65. [73]

    Courbes spectrales et compactifications de jacobiennes

    Daniel Schaub. Courbes spectrales et compactifications de jacobiennes. Math. Z. , 227(2):295--312, 1998

    1998

  66. [74]

    Logarithmic nonabelian Hodge theory in characteristic p

    Daniel Kenneth Schepler. Logarithmic nonabelian Hodge theory in characteristic p . University of California, Berkeley, 2005

    2005

  67. [75]

    Exemples de plongements des groupes PSL2 (Fp) dans des groupes de Lie simples

    Jean-Pierre Serre. Exemples de plongements des groupes PSL2 (Fp) dans des groupes de Lie simples. Inventiones mathematicae , 124:525--562, 1996

    1996

  68. [76]

    Compl \`e te r \'e ductibilit \'e

    J Serre. Compl \`e te r \'e ductibilit \'e . ASTERISQUE-SOCIETE MATHEMATIQUE DE FRANCE , 299:195, 2005

    2005

  69. [77]

    Geometric reductivity over arbitrary base

    Conjeevaram S Seshadri. Geometric reductivity over arbitrary base. Advances in Mathematics , 26(3):225--274, 1977

    1977

  70. [78]

    Tamely Ramified Geometric Langlands Correspondence in Positive Characteristic

    Shiyu Shen. Tamely Ramified Geometric Langlands Correspondence in Positive Characteristic . International Mathematics Research Notices , page rnae005, 02 2024

    2024

  71. [79]

    Harmonic bundles on noncompact curves

    Carlos T Simpson. Harmonic bundles on noncompact curves. Journal of the American Mathematical Society , 3(3):713--770, 1990

    1990

  72. [80]

    Carlos T. Simpson. Moduli of representations of the fundamental group of a smooth projective variety. I . Inst. Hautes \' E tudes Sci. Publ. Math. , (79):47--129, 1994

    1994

  73. [81]

    Carlos T. Simpson. Moduli of representations of the fundamental group of a smooth projective variety. II . Inst. Hautes \' E tudes Sci. Publ. Math. , (80):5--79 (1995), 1994

    1995

  74. [82]

    Springer isomorphisms in characteristic p

    Paul Sobaje. Springer isomorphisms in characteristic p. Transformation Groups , 20:1141--1153, 2015

    2015

  75. [83]

    Linear algebraic groups , volume 9

    Tonny Albert Springer. Linear algebraic groups , volume 9. Springer, 1998

    1998

  76. [84]

    Stacks Project

    The Stacks Project Authors . Stacks Project. https://stacks.math.columbia.edu

  77. [85]

    A\_1 -type overgroups of elements of order p in semisimple algebraic groups and the associated finite groups

    Donna Testerman. A\_1 -type overgroups of elements of order p in semisimple algebraic groups and the associated finite groups. Journal of Algebra , 177(1):34--76, 1995

    1995

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.