Pith. sign in

REVIEW 4 major objections 3 minor 40 references

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

T0 review · 4 major / 3 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 Parabolic bases yield a characteristic-p parabolic non-abelian Hodge correspondence in all dimensions.

desk verdict Promising parabolic-bases formalism, but the main correspondence theorem currently rests on an explicitly deferred cocycle check. read the letter →

arxiv 2501.13775 v1 pith:ERYHADIK submitted 2025-01-23 math.AG

Xiaojin Lin This is my paper · ORCID
classification math.AG MSC 14A2114G1714H3014J6014H60
keywords parabolicbasesλ-connectionsHiggsbundlesflatconnectionsinverseCartiertransformationexponentialtwistHiggs-deRhamflowpositivecharacteristic
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

This paper sets out to prove the positive-characteristic non-abelian Hodge correspondence for parabolic bundles: flat parabolic connections with nilpotent $p$-curvatures and residues correspond to parabolic Higgs bundles with nilpotent Higgs fields and residues on the Frobenius twist. The tool is a new local description, parabolic bases, in which a basis of a vector bundle is twisted by fractional powers of local equations of the boundary divisor. With this description the paper constructs the parabolic inverse Cartier and Cartier transformations by exponential twisting, and Theorem 4.6 states the resulting equivalence between $\mathrm{MIC}^{p-1}_{lf,*}(X,D)$ and $\mathrm{HIG}^{p-1}_{lf,*}(X',D')$ in arbitrary dimension. The paper also proves functoriality of the correspondence under pullback and pushforward, gives a rank-2 algorithm for the parabolic Higgs-de Rham flow operator, and simplifies a conjecture about Beauville numbers. If correct, this gives a uniform way to pass between the two sides of parabolic Hodge theory in characteristic $p$.

What carries the argument

The load-bearing object is the parabolic basis: for a smooth log variety $(X,D)$, local sections are replaced by $\{e_i/s^{\alpha_i}\}$, where $s$ is a local equation of the divisor and $\alpha_i$ are rational weights whose denominators are coprime to $p$. This makes parabolic structures into ordinary transition data, so the paper can reuse the exponential-twist machine from [14]: local connections $\nabla_{\mathrm{par},i}=\nabla_{\mathrm{can},i}+\frac{d\tilde{F}_i^*}{p}\circ F^*\theta_{\mathrm{par},i}$ are glued by transition functions $\exp(h_{ij}F^*\theta_{\mathrm{par}})F^*\varphi_{ij,\mathrm{par}}$. The second main mechanism is a parabolic Cartier descent theorem adapted from the classical proof: a strong parabolic connection with vanishing $p$-curvature descends to a parabolic vector bundle. In the rank-2 flow computation, the Deligne-Illusie class $\kappa$ of the $W_2(k)$-lifting enters through $\xi=F^*\theta\cup\kappa$, relating the extension class of the inverse Cartier output to the cup product of the pulled-back Higgs field with $\kappa$.

What would settle it

Work out the cocycle condition for the transition data $\exp(h_{ij}F^*\theta_{\mathrm{par}})F^*\varphi_{ij,\mathrm{par}}$ on a triple intersection for a parabolic Higgs bundle with at least two rational weights; a failure would mean the inverse Cartier transform is not a well-defined functor and Theorem 4.6 is false. Alternatively, in the rank-2 setting, compute the extension class $\xi$ for $\mathbb{P}^1$ minus four points and check whether $\xi=F^*\theta\cup\kappa$; any mismatch refutes Theorem 4.11.

Watch

Extended reading notes

Core claim

The central discovery is that a parabolic bundle can be described entirely by local 'parabolic bases', so the parabolic non-abelian Hodge correspondence becomes a local gluing problem. The paper proves the category of vector bundles with parabolic bases is tensor equivalent to the usual category of parabolic bundles, and extends this to $\lambda$-connections, including adjusted and strong variants. Using parabolic bases, it defines the parabolic inverse Cartier and Cartier transformations by exponential twisting, in parallel with [14]. The headline result, Theorem 4.6, is an equivalence of categories between $\mathrm{MIC}^{p-1}_{lf,*}(X,D)$ and $\mathrm{HIG}^{p-1}_{lf,*}(X',D')$: flat parabolic connections whose $p$-curvatures and residues are nilpotent of level at most $p-1$ correspond to parabolic Higgs bundles on the Frobenius twist with nilpotent Higgs fields and residues. The paper also shows this construction agrees with the earlier curve-level parabolic transform, is compatible with pullback and pushforward for branched coverings, and derives the rank-2 extension-class formula $\xi=F^*\theta\cup\kappa$, which underlies an algorithm for the parabolic Higgs-de Rham flow operator.

Load-bearing premise

The construction assumes that the gluing verification from the non-parabolic exponential twist works unchanged when bases are twisted by fractional powers of the boundary divisor, a check the paper leaves out by saying it is 'identical'.

Editorial extensions

If this is right

  • Theorem 4.6 gives a dimension-free parabolic non-abelian Hodge correspondence: nilpotent parabolic Higgs data and nilpotent parabolic flat connections form equivalent categories whenever the base is a smooth log variety over a perfect field of characteristic $p$.
  • Corollary 4.9 shows the new exponential-twist transform agrees with the previously known curve-level parabolic inverse Cartier transform, so existing curve computations are recovered as a special case.
  • Theorem 4.7 makes the correspondence functorial: parabolic pullback and pushforward, and $G$-invariants in the Galois case, commute with the Cartier transforms for separable branched coverings with $W_2(k)$-liftings and ramification indices coprime to $p$.
  • In rank 2, the identity $\xi=F^*\theta\cup\kappa$ turns the maximal destabilizing-subbundle problem into a cohomological computation, yielding an explicit algorithm for the parabolic Higgs-de Rham flow operator.
  • For the Beauville example, periodicity is equivalent to vanishing of an explicit $p\times p$ determinant, and for each $\lambda\in\mathbb{F}_p$ exactly $p$ periodic lifts exist (Theorem A.1, Corollary A.2).

Reading between the lines

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

  • Inference (editorial): the same parabolic-bases description should reconstruct the complex quasi-projective non-abelian Hodge correspondence as well, since the hard part, writing parabolic structures as transition data, is already done; the paper lists filtered local systems only as future work.
  • Inference (editorial): the rank-2 formula $\xi=F^*\theta\cup\kappa$ suggests a general recipe for inverse Cartier transforms of split Higgs bundles: compute the Deligne-Illusie class of a $W_2(k)$-lifting and cup it with the Frobenius pullback of the Higgs field.
  • Inference (editorial): the appendix's determinant is directly computable for small primes; verifying the count in Corollary A.2 for $p=3$ or $p=5$ would give a quick numerical check of the flow-algorithm framework.
  • Inference (editorial): because parabolic bases are just frames, the method should adapt to parahoric or principal-bundle versions of the correspondence without new geometry.
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

4 major / 3 minor

Summary. The paper introduces the notion of "parabolic bases" as a local normal form for parabolic bundles, and uses it to develop a localized framework for parabolic bundles and parabolic λ-connections. It proves an equivalence between the category PVB of vector bundles with parabolic bases and the category of parabolic bundles (Theorem 2.18), extends this to an equivalence for parabolic λ-connections (Theorem 3.7), and defines pullback and pushforward functors together with a Galois BIS-type correspondence (Theorem 3.12). The main result is Theorem 4.6, which asserts an equivalence between nilpotent parabolic flat connections and nilpotent parabolic Higgs bundles in arbitrary dimension over a perfect field of characteristic p, via parabolic Cartier and inverse Cartier transformations built by exponential twisting. The final section studies the rank-2 parabolic Higgs–de Rham flow operator, gives a criterion for the maximal destabilizing subbundle, and computes a determinant in the Li–Sheng example.

Significance. If completed, the parabolic-base framework would provide a conceptually clean local description of parabolic structures and a genuinely higher-dimensional parabolic non-abelian Hodge correspondence, extending the curve case of Krishnamoorthy–Sheng. The paper's local constructions and its comparison with the BIS correspondence are useful contributions, and the explicit rank-2 computations in the appendix are of independent interest. However, the central gluing verification for the inverse Cartier transformation is explicitly deferred, and the advertised Algorithm 4.12 and the referenced Proposition 4.20 are missing. As written, Theorem 4.6 is conditional on a verification that is not supplied.

major comments (4)
  1. [§4.2.1, Eq. (4.7)] The proof of Theorem 4.6 is incomplete because the global gluing of the locally defined parabolic connections is asserted rather than verified. After defining the transition matrix ψ_ij = exp(h_ij F^*θ_par) F^*φ_ij,par in Eq. (4.7), the text states that "the validation procedure is identical and thus will not be reiterated here." What must be checked is the cocycle condition for the ψ_ij on triple overlaps, the compatibility of ψ_ij with the local connections ∇_par,i, and the flatness of the glued connection. In the non-parabolic case these checks use θ_j = φ_ij^{-1} θ_i φ_ij, h_ij + h_jk + h_ki = 0, and θ^2 = 0; the parabolic analogues with φ_ij,par and T = diag(s^{α_i}) are not written down. Proposition 4.1 only verifies that (4.7) is a morphism of parabolic bases by conjugation with T, not the cocycle or flatness conditions. Since Theorem 4.6 depends directly on this gluing, the main equivalence remains conditional.
  2. [§1.2 and §4.3] The promised Algorithm 4.12 is absent. The introduction states that "the main result of this work is to provide an algorithm (Algorithm 4.12) for determining the maximal destabilizing subbundle of H," and the later text refers to an algorithm adapted from Sun–Yang–Zuo. The manuscript contains only Proposition 4.12, which gives the criterion dim(ker δ) = 1, but no algorithmic procedure is actually presented. The computational claims in Appendix A therefore rest on an algorithm that is not part of the manuscript. Please either supply Algorithm 4.12 or revise the claims to match what is proved.
  3. [Appendix A, Corollary A.2] The assertion that there are exactly p liftings does not follow from the displayed determinant. The text observes that det Δ is a polynomial of degree p in λ1, but over F_p a degree-p polynomial can have anywhere from 0 to p roots, and the displayed matrix does not show that for each λ0 the polynomial has a full set of p distinct roots, or even any root. The counting claim needs a proof or a more precise statement, for example that there are at most p liftings, or that the roots are counted with multiplicity under an explicit genericity condition.
  4. [§1.2 and Appendix A] The manuscript references Proposition 4.20, but no such proposition appears in the text. The introduction says "as demonstrated in Proposition 4.20," and Appendix A says it simplifies the Li–Sheng conjecture "using Proposition 4.2 0." Since the appendix's computation of the determinant and its relation to periodicity rely on this missing statement, the proof of Theorem A.1 is incomplete as written. Please add the missing proposition or replace the references with the actual statements used.
minor comments (3)
  1. [§4.2, Theorem 4.6] The notation for the main equivalence is inconsistent: HIG^{p-1}_{lf,*} is defined on (X,D), but Theorem 4.6 states the equivalence with HIG^{p-1}_{lf,*}/(X',D'). Please clarify whether the Frobenius twist is intended and make Theorem 1.5 consistent with Theorem 4.6.
  2. [Abstract and keywords] The abstract and keyword list contain formatting artifacts and truncated text, including the fragment "oper" at the end of the keywords. Please copyedit these passages.
  3. [Theorem 4.2 proof] In the proof of the parabolic Cartier descent theorem, the operator P is written with products over d coordinates, but the factors ∇_i are not defined for multi-indices. Please define the notation used in the products.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the parabolic exponential-twist construction is an independent adaptation of external results; the deferred cocycle verification is a rigor gap, not a circular step.

full rationale

The paper's central claim (Theorem 4.6) is built from explicit functors: parabolic bases are introduced and shown equivalent to parabolic bundles (Theorem 2.18); λ-connections with parabolic bases are shown equivalent to parabolic λ-connections (Theorem 3.7); the parabolic inverse Cartier transform is then defined by applying the exponential twist to local parabolic bases and parabolic Higgs data (equations (4.4)-(4.7)). None of these steps defines its output in terms of the target equivalence: the input category HIG^{p-1}_{lf,*} is characterized by nilpotence of the Higgs field and residues, while the output conditions in MIC^{p-1}_{lf,*} (nilpotent p-curvature and residues) are proved via the twist, not assumed. The cited works are used as external methods: [14] supplies the non-parabolic exponential-twist validation, [10] supplies Cartier descent, and [11] is used only as the comparison object in Corollary 4.9. The self-citation overlap through the author's advisor Mao Sheng does not make any premise circular, because the cited results are published external arguments rather than unverified assertions tailored to this paper. The manuscript does contain an explicit omission: Section 4.2.1 states that the cocycle condition and flatness checks for the transition data (4.7) are 'identical and thus will not be reiterated here,' and Proposition 4.5 similarly defers to [14, Section 3]. This is a completeness/rigor gap that could affect the validity of Theorem 4.6, but it is not a circularity: no equation of the paper reduces to its own input, no fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors' prior work is invoked to force the construction.

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

The paper relies on standard positive-characteristic machinery (Deligne-Illusie, Cartier descent, Kawamata-Viehweg) and on domain assumptions (local abelianity, tame ramification). The main unproven load-bearing input is the transfer of Lan-Sheng-Zuo's exponential-twist verification to the parabolic-bases presentation, which the paper explicitly defers.

assumptions (5)
  • standard math The Deligne-Illusie class κ exists from a W2(k)-lifting of (X,D) and its Cech representative h_ij satisfies the needed properties.
    Invoked in Section 4.1.1 (equation 4.1) and used in Theorem 4.11; a standard input from Ogus-Vologodsky / Deligne-Illusie.
  • domain assumption Katz's Cartier descent theorem [10, Theorem 5.1] applies to the parabolic setting as claimed.
    The proof of Theorem 4.2 is an adaptation of Katz's argument; the paper assumes the same operator P works with parabolic bases, which is the core of the descent.
  • domain assumption The Kawamata-Viehweg lemma provides a finite tamely ramified cover with ramification indices divisible by the denominators of the parabolic weights and coprime to p.
    Used in Corollary 4.9 to compare C^{-1}_{exp} with Krishnamoorthy-Sheng's C^{-1}_{par}.
  • domain assumption Parabolic bundles are locally abelian, i.e., locally decompose into direct sums of parabolic line bundles.
    Definition 2.11; the equivalence with PVB and all local computations in Sections 2-4 rely on this local splitting.
  • domain assumption All ramification indices of the covers f are non-zero in k (coprime to p).
    Assumed in Theorem 3.12 and Theorem 4.7 so that the pushforward/pullback formulas with powers 1/n make sense.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Constructing Parabolic Non-Abelian Hodge Correspondence in Positive Characteristic Using Parabolic Bases." pith.science (2026). https://pith.science/paper/ERYHADIK

@misc{pith2026250113775,
  author       = {Pith},
  title        = {Pith review of: Constructing Parabolic Non-Abelian Hodge Correspondence in Positive Characteristic Using Parabolic Bases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ERYHADIK}},
  note         = {Machine review of arXiv:2501.13775}
}
abstract

We introduce the concept of parabolic bases to establish a localized framework for parabolic bundles and parabolic $\lambda$-connections. Building on this foundation, we propose a novel method for constructing the parabolic non-abelian Hodge correspondence in positive characteristic, extending the work originally developed by Krishnamoorthy and Sheng for algebraic curves. Additionally, we investigate the rank $2$ parabolic Higgs-de Rham flow operator and present a modified version of the Sun-Yang-Zuo algorithm, specifically adapted to the parabolic setting.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 38 canonical work pages

  1. [14]

    Nonabelian Hodge theory i n positive characteristic via exponential twisting

    Lan, G., Sheng, M., and Zuo, K. Nonabelian Hodge theory i n positive characteristic via exponential twisting. Mathematical Research Letters, 22(3):859–879, 2015

    work page 2015

  2. [11]

    Periodic de Rham bundles over curves

    Krishnamoorthy, R. and Sheng, M. Periodic de Rham bundl es over curves. arXiv e-prints, arXiv:2011.03268, 2020

    work page Pith review arXiv 2011

  3. [29]

    A Nonabelian Hodge Correspondence for Principal Bundles in Positive Characteristic

    Sheng, M., Sun, H., and Wang, J. A Nonabelian Hodge Corre spondence for Principal Bundles in Positive Characteristic. Preprint, arXiv:2405.09947 [math.AG], 2 024

    work page Pith review arXiv

  4. [1]

    and Biswas, I

    Alfaya, D. and Biswas, I. Pullback and direct image of par abolic connections and parabolic Higgs bundles. International Mathematics Research Notices , 2023(22):19546–19591, 2023

    work page 2023

  5. [2]

    and Machu, F.-X

    Biswas, I. and Machu, F.-X. On the direct images of parabo lic vector bundles and parabolic connections. Journal of Geometry and Physics , 135:219–234, 2019

    work page 2019

  6. [3]

    Parabolic bundles as orbifold bundles

    Biswas, I. Parabolic bundles as orbifold bundles. Duke Mathematical Journal , 88(2):305–325, 1997

    work page 1997

  7. [4]

    de Cataldo, M. A. and Zhang, S. Logarithmic Non-Abelian H odge Theory for curves in prime characteristic. Preprint, arXiv:2501.09850 [math.AG], 2025. URL: https:/ /arxiv.org/abs/2501.09850

    arXiv 2025

  8. [5]

    and Zhu, X

    Chen, T.-H. and Zhu, X. Non-abelian Hodge theory for alge braic curves in characteristic p. Geometric and Functional Analysis, 25(6):1706–1733, 2015. DOI: 10.1007/s00039-015-0343-6

    work page doi:10.1007/s00039-015-0343-6 2015

Show all 40 references
  1. [6]

    and Joyner, S

    Dhillon, A. and Joyner, S. Pullback of parabolic bundles and covers of P1 \ { 0, 1, ∞} . Michigan Mathematical Journal, 61(1):199–224, 2012

    2012

  2. [7]

    Donaldson, S. K. Twisted harmonic maps and the self-dual ity equations. Proceedings of the London Mathe- matical Society, 55:127–131, 1987

    1987

  3. [8]

    and Groechenig, M

    Esnault, H. and Groechenig, M. Rigid connections and F-i socrystals. Acta Mathematica , 225(1):103–158, 2020

    2020

  4. [9]

    Iyer, J. N. N. and Simpson, C. T. A relation between the par abolic Chern characters of the de Rham bundles. Mathematische Annalen , 338(2):347–383, 2007

    2007

  5. [10]

    Katz, N. M. Nilpotent connections and the monodromy the orem: Applications of a result of Turrittin. Publications Math´ ematiques de l’IH´ES, 39:175–232, 1970

    1970

  6. [12]

    and Majumder, S

    Kumar, M. and Majumder, S. Parabolic bundles in positiv e characteristic. Journal of the Ramanujan Math- ematical Society, 33(1):1–36, 2018

    2018

  7. [13]

    Bogomolov’s inequality for Higgs sheaves in positive characteristic

    Langer, A. Bogomolov’s inequality for Higgs sheaves in positive characteristic. Inventiones Mathematicae , 199(3):889–920, 2015

    2015

  8. [15]

    Semistable Higgs bundles , periodic Higgs bundles and representations of algebraic fundamental groups

    Lan, G., Sheng, M., and Zuo, K. Semistable Higgs bundles , periodic Higgs bundles and representations of algebraic fundamental groups. Journal of the European Mathematical Society , 21(10):3053–3112, 2019

    2019

  9. [16]

    Uniformization of p-adic curves via Higgs-de Rham flows

    Lan, G., Sheng, M., Yang, Y., and Zuo, K. Uniformization of p-adic curves via Higgs-de Rham flows. Journal f¨ ur die Reine und Angewandte Mathematik, 747:63–108, 2019

    2019

  10. [17]

    and Sun, H

    Li, M. and Sun, H. Tame parahoric nonabelian Hodge corre spondence in positive characteristic over algebraic curves. Selecta Mathematica, 30(4):36, 2024

    2024

  11. [18]

    and Sheng, M

    Li, M. and Sheng, M. Characterization of Beauville’s nu mbers via Hodge theory. International Mathematics Research Notices, 2022(21):17260–17281, 2022

    2022

  12. [19]

    Parabolic Fontaine-Falti ngs modules and parabolic Higgs-de Rham flows

    Liu, Z., Yang, J., and Zuo, K. Parabolic Fontaine-Falti ngs modules and parabolic Higgs-de Rham flows. Preprint, arXiv:2309.10449 [math.AG], 2023

    2023 arXiv

  13. [20]

    and Yokogawa, K

    Maruyama, M. and Yokogawa, K. Moduli of parabolic stabl e sheaves. Mathematische Annalen, 293(1):77–99, 1992

    1992

  14. [21]

    Mehta, V. B. and Seshadri, C. S. Moduli of vector bundles on curves with parabolic structures. Mathematische Annalen, 248:205–239, 1980

    1980

  15. [22]

    Kobayashi-Hitchin correspondence for t ame harmonic bundles and an application

    Mochizuki, T. Kobayashi-Hitchin correspondence for t ame harmonic bundles and an application. Ast´ erisque, 309, 2006

    2006

  16. [23]

    Wild harmonic bundles and wild pure twist or D-modules

    Mochizuki, T. Wild harmonic bundles and wild pure twist or D-modules. Ast´ erisque, 340, 2011

    2011

  17. [24]

    Branched coverings and algebraic functions

    Namba, M. Branched coverings and algebraic functions. Pitman Research Notes in Mathematics Series , 161,

  18. [25]

    and Laaroussi, A

    Borne, N. and Laaroussi, A. Parabolic connections and s tack of roots. Bulletin des Sciences Math´ ematiques, 187:33, 2023

    2023

  19. [26]

    and Vologodsky, V

    Ogus, A. and Vologodsky, V. Nonabelian Hodge theory in c haracteristic p. Publications Math´ ematiques de l’IH´ES, 106:1–138, 2007

    2007

  20. [27]

    Logarithmic nonabelian Hodge theory in ch aracteristic p

    Schepler, D. Logarithmic nonabelian Hodge theory in ch aracteristic p. Preprint, arXiv:0802.1977 [math.AG], 2008

    1977 arXiv

  21. [28]

    Tamely ramified geometric Langlands correspon dence in positive characteristic

    Shen, S. Tamely ramified geometric Langlands correspon dence in positive characteristic. International Math- ematics Research Notices , 2024(7):6176–6208, 2024

    2024

  22. [30]

    and Wang, J

    Sheng, M. and Wang, J. Tensor product theorem for parabo lic λ -connections. Preprint, arXiv:2107.06624 [math.AG], 2021

    2021 arXiv

  23. [31]

    and Zuo, K

    Sheng, M. and Zuo, K. Periodic Higgs subbundles in posit ive and mixed characteristic. Preprint, arXiv:1206.4865 [math.AG], 2012

    2012 arXiv

  24. [32]

    Simpson, C. T. Constructing variations of Hodge struct ure using Yang-Mills theory and applications to uniformization. Journal of the American Mathematical Society , 1(4):867–918, 1988. 30 Xiaojin Lin

    1988

  25. [33]

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

    1990

  26. [34]

    Projective crystalline re presentations of ´ etale fundamental groups and twisted periodic Higgs-de Rham flow

    Sun, R., Yang, J., and Zuo, K. Projective crystalline re presentations of ´ etale fundamental groups and twisted periodic Higgs-de Rham flow. Journal of the European Mathematical Society , 24(6):1991–2076, 2022

    1991

  27. [35]

    Uluda˘ g, A. M. Orbifolds and their uniformization. In Arithmetic and Geometry Around Hypergeometric Functions, pages 373–406. Birkh¨ auser, 2007

    2007

  28. [36]

    Frobenius pull-back of parabolic bund les and dormant opers

    Wakabayashi, Y. Frobenius pull-back of parabolic bund les and dormant opers. Preprint, arXiv:2408.12267 [math.AG], 2024

    2024 arXiv

  29. [37]

    and Zuo, K

    Yang, J. and Zuo, K. Constructing families of abelian va rieties of GL 2-type over 4-punctured complex projective line via p-adic Hodge theory and Langlands corre spondence and application to algebraic solutions of Painleve VI equation. Preprint, arXiv:2303.09298 [math .AG], 2023

    2023 arXiv

  30. [38]

    Compactification of moduli of parabolic sh eaves and moduli of parabolic Higgs sheaves

    Yokogawa, K. Compactification of moduli of parabolic sh eaves and moduli of parabolic Higgs sheaves. Journal of Mathematics of Kyoto University , 33(2):451–504, 1993

    1993

  31. [39]

    Infinitesimal deformation of parabolic Hi ggs sheaves

    Yokogawa, K. Infinitesimal deformation of parabolic Hi ggs sheaves. International Journal of Mathematics , 6(1):125–148, 1995

    1995

  32. [40]

    Arithmetic Simpson correspondence and GL2-motivic local systems over P1\{ 0, 1, λ, ∞}

    Zuo, K. Arithmetic Simpson correspondence and GL2-motivic local systems over P1\{ 0, 1, λ, ∞} . Conference presentation, Sino-French Conference in Algebraic and Com plex Geometry, 2018. Email address : xjlin@mail.ustc.edu.cn 31

    2018

Pith tools

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