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Level structures on parahoric torsors and complete integrability

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Pith's one-line read The paper proves that a moduli space of parahoric Higgs torsors with level structures is an algebraically completely integrable Hamiltonian system, with Gaudin, KP elliptic solitons, and Calogero–Moser as symplectic leaves.

desk verdict An ambitious unification of parahoric Hitchin systems with a genuine new notion of D-level structure, but the central Poisson-descent proof has a reversed differential in Lemma 5.21 and incoherent dual complexes in 5.5; as written the main theorem does not follow. read the letter →

arxiv 2506.12302 v1 pith:WCT5IBM3 submitted 2025-06-14 math.AG math.SGnlin.SI

Georgios Kydonakis , Lutian Zhao This is my paper
classification math.AGmath.SGnlin.SI MSC 14D1514H4014L1537J9937K1053D17
keywords parahorictorsorHiggsbundlelevelstructurelogarithmicfieldmomentmapLagrangianfibrationcompletelyintegrablesystemGaudinmodel
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

Parahoric $\mathcal{G}_{\boldsymbol{\theta}}$-torsors are principal bundles over a curve whose local structure at marked points is prescribed by weights in the Bruhat–Tits building. The paper adds a $D$-level structure: at each marked point one fixes a section into the quotient of the parahoric group by its pro-unipotent radical, a framing-type condition that refines the parahoric data. A moduli space of stable pairs is constructed, and a level group acts on it with a moment map computed as the sum of coresidues of the Higgs field at the divisor. From this, the paper derives a Poisson structure on the moduli space of logahoric Higgs torsors and shows that its Hitchin fibration has abelian-torsor generic fibers. The payoff is that classical integrable systems with regular singularities—Gaudin, periodic KP elliptic solitons, and the elliptic Calogero–Moser system—appear as symplectic leaves of one geometric integrable system.

What carries the argument

The paper's central object is a $D$-level structure on a parahoric $\mathcal{G}_{\boldsymbol{\theta}}$-torsor: a choice of section $\eta_x : D_x \to \mathcal{G}_{\theta_x}/\mathcal{G}^+_{\theta_x}$ at each marked point, where $\mathcal{G}^+$ is the pro-unipotent radical; this generalizes Seshadri's level structures and interacts coherently with Bruhat–Tits data. The argument is carried by the level group $G_D = (\prod_j L_{\theta_j})/Z$, whose action on $U(X, \mathcal{G}_{\boldsymbol{\theta}})$ lifts to the cotangent bundle with moment map given by the direct sum of coresidues of the Higgs field at $D$. Passing from the leveled moduli space $M_{LH}(X, \mathcal{G}_{\boldsymbol{\theta}})$ to $M_H(X, \mathcal{G}_{\boldsymbol{\theta}})$ via the forgetful map transfers the symplectic structure to a Poisson structure, using the leveled deformation complex $K_{E,\varphi,\eta}$ and its asserted quasi-isomorphism to the logahoric complex $E(\mathfrak{g}^+) \to E(\mathfrak{g}) \otimes K(D)$. The Hitchin fibration is abelianized through cameral covers and generalized Prym varieties, whose fibers act on the level sets; duality between the differential of this action and the differential of the Hitchin map makes the fibers Lagrangian.

What would settle it

Compute the hypercohomology dimensions of the leveled deformation complex $K_{E,\varphi,\eta}$ and the logahoric complex $K_{E,\varphi}$ for a concrete case—say $G=\mathrm{SL}(2,\mathbb{C})$, $X=\mathbb{P}^1$, a single marked point with Iwahori level data—and compare them; any dimension mismatch in degree 1 would disprove the quasi-isomorphism Lemma 5.21 and, with it, the Poisson-structure theorem that feeds Theorem 6.24.

Watch

Extended reading notes

Core claim

The central claim is Theorem 6.24: for a smooth complex algebraic curve $X$, a reduced effective divisor $D$, and any connected complex reductive group $G$, the moduli space $M_H(X, \mathcal{G}_{\boldsymbol{\theta}})$ of logahoric $\mathcal{G}_{\boldsymbol{\theta}}$-Higgs torsors carries a Poisson structure, and the parahoric Hitchin map $h_{\boldsymbol{\theta}} : M_H(X, \mathcal{G}_{\boldsymbol{\theta}}) \to A_{\boldsymbol{\theta}}$ fibers it by abelian torsors, making it an algebraically completely integrable Hamiltonian system. The Poisson structure comes from a forgetful map from the leveled moduli space $M_{LH}(X, \mathcal{G}_{\boldsymbol{\theta}})$, which is an open subset of the cotangent bundle of the level-structured moduli space $U(X, \mathcal{G}_{\boldsymbol{\theta}})$; the level group $G_D$ acts there with a canonical moment map sending a Higgs field to its coresidues at $D$. Generic Hitchin fibers are identified with generalized Prym varieties attached to cameral covers, hence are abelian torsors, and the symplectic leaf foliation is described through the quotient $A_{\boldsymbol{\theta}}/A^+_{\boldsymbol{\theta}} \cong \mathfrak{g}^*_D/\!/G_D$. The framework is shown to generalize Beauville's integrable system and to recover the Gaudin model, the space of $\Lambda$-periodic KP elliptic solitons, and the elliptic Calogero–Moser system as symplectic leaves.

Load-bearing premise

The whole integrable-system result rests on the claim that deforming a parahoric Higgs torsor together with its level structure is equivalent to deforming the underlying logahoric Higgs torsor; the written proof of that equivalence uses an inclusion in the wrong direction, so the claim is not established unless that step is repaired.

Editorial extensions

If this is right

  • The moduli space $M_H(X, \mathcal{G}_{\boldsymbol{\theta}})$ is Poisson and $h_{\boldsymbol{\theta}}$ is an algebraically completely integrable Hamiltonian system whose generic fibers are abelian torsors.
  • The symplectic leaf foliation of the smooth locus refines the fibration $q \circ eH_{\boldsymbol{\theta}} : M_H(X, \mathcal{G}_{\boldsymbol{\theta}})^{\mathrm{sm}} \to A_{\boldsymbol{\theta}}/A^+_{\boldsymbol{\theta}}$, with a unique symplectic leaf of maximal dimension in each fiber.
  • For $X = \mathbb{P}^1$ with Iwahori level data, logarithmic Higgs fields take the Gaudin Lax form and the Hitchin Hamiltonians generate the classical Gaudin model.
  • For an elliptic base curve and $G = \mathrm{SL}(r,\mathbb{C})$, the coadjoint orbit of $\mathrm{diag}(-1,\ldots,-1,r-1)$ defines a symplectic leaf that carries the space of $\Lambda$-periodic KP elliptic solitons with its integrable structure.
  • The elliptic Calogero–Moser system embeds as a symplectic leaf through the Hurtubise–Markman construction, and for the root system $A_{r-1}$ this leaf is symplectically isomorphic to the KP leaf.

Reading between the lines

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

  • Beyond the paper: if the coresidue moment-map description is correct, symplectic leaves of the Poisson moduli space are parameterized by coadjoint orbits in $\mathfrak{g}^*_D$, so varying the parahoric weights $\boldsymbol{\theta}$ should interpolate between the Gaudin and Calogero–Moser phase spaces.
  • Beyond the paper: the same level-structure construction could be applied to non-generically split parahoric torsors via the $\Gamma$-equivariant cover formalism that the paper only cites as background, which would extend the integrable-system result to ramified covers.
  • Beyond the paper: the explicit coresidue formula suggests a computational recipe—choose residue coadjoint orbits, build the corresponding Higgs fields, and read off Hamiltonians from invariant polynomials—that could produce new integrable systems for higher-genus curves or non-trivial orbits, as the paper notes in closing.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The paper introduces a notion of D-level structure of parahoric type on parahoric G_theta-torsors over a smooth projective curve, constructs a moduli space U(X,G_theta) of stable such pairs, and defines a level group G_D acting on it. It then claims that the lifted action on the cotangent T*U(X,G_theta) is Hamiltonian with moment map given by coresidues of the Higgs field at D, and that a forgetful map from the leveled moduli space M_LH(X,G_theta) to the logahoric Higgs moduli M_H(X,G_theta) is Poisson. The generic fibers of the parahoric Hitchin fibration are asserted to be abelian torsors, yielding an algebraically completely integrable Hamiltonian system. The paper closes with examples recovering the Gaudin model, KP elliptic solitons, and the elliptic Calogero-Moser system as symplectic leaves. The central claim of the paper is Theorem 6.24, which asserts that h_theta: M_H(X,G_theta) -> A_theta is an algebraically completely integrable Hamiltonian system, with the symplectic leaf foliation described by the corestriction moment map.

Significance. If the main theorem is correct, the paper would provide a unified geometric framework for several known integrable systems with regular singularities, with a canonical moment-map description of the symplectic leaves. The definition of D-level structures of parahoric type is original, and the explicit description of the moment map as a sum of coresidues is attractive and potentially useful. The paper also makes a serious attempt to connect the level-structure geometry of Markman and Seshadri with the logahoric Higgs moduli of the authors' earlier work. However, the proof of the central Poisson-descent statement is not complete as written: the quasi-isomorphism in Lemma 5.21 has a direction error, the dual complexes in Section 5.5 are not the Serre duals of the tangent complexes, and the proof of Theorem 5.23 relies on incompatible or self-referential inclusions. These gaps are load-bearing for Theorem 6.24, so the central claim is not yet established.

major comments (5)
  1. [§5.4, Lemma 5.21] The asserted quasi-isomorphism q^i: K_{E,phi,eta} -> K_{E,phi} is not a morphism of complexes as stated. The degree-zero component q^i_0 is described as 'the natural inclusion' E(g) -> E(g+), but E(g+) is defined in Definition 2.10 as a subsheaf of E(g); there is no natural inclusion in that direction. The degree-one component is a projection onto the first summand, so the two components are not obviously compatible with the differentials. The degree-one cohomology comparison is then deferred to a 'local analysis' that is not carried out. Since Theorem 5.23 and Theorem 6.24 depend on this quasi-isomorphism to identify the deformation complex of the leveled space with K_{E,phi}, this is a load-bearing gap in the Poisson-descent argument.
  2. [§5.5, Eq. (5.22)] The displayed tangent and cotangent complexes for M_LH(X,G_theta) are written in identical form: both are E(g+) -> E(g) ⊗ K(D), with only the sign of the differential changed. Similarly, the cotangent complex of M_H(X,G_theta) is written as E(g+) -> E(g+) ⊗ K(D), while its tangent complex is E(g) -> E(g) ⊗ K(D). Under Serre duality, the dual of a two-term complex A -> B should have terms B^∨ -> A^∨ with the appropriate twists; as written, these complexes cannot be Serre duals of the corresponding tangent complexes. Consequently, the antisymmetric map ♯ in (5.22) is not obtained from a duality pairing, and the Poisson bivector on M_H(X,G_theta) is not established.
  3. [§5.5, Theorem 5.23] The proof that the forgetful map ℓ is Poisson uses inclusions that are incompatible or self-referential. The differential dℓ is said to be induced by 'the natural inclusion of complexes K_{E,phi} ↪ \bar K_{E,phi}', but K_{E,phi} has source E(g+) while \bar K_{E,phi} has source E(g); the inclusion goes in the opposite direction, if it exists. The dual differential is then said to arise from the inclusion \bar K^∨_{E,phi} ↪ \bar K^∨_{E,phi}, which is an inclusion of a complex into itself. The claimed identity (5.24) therefore does not follow from functoriality of hypercohomology. Until the correct morphisms between these complexes are specified, the statement that ℓ is a Poisson map is unproved.
  4. [§6.4, Theorem 6.14] The abelianization of the Γ-equivariant Hitchin fiber is delegated to Scognamillo's proof with the statement that it 'repeats word by word' in the presence of Γ. This is an essential step: Corollary 6.15, which identifies the generic fibers of h_theta as abelian torsors, depends on it. Lemmas 6.11 and 6.12 check compatibility of the Γ- and W-actions on T-bundles, but no argument is given for the injectivity of the assignment (P,s) ↦ T(P,s) into Prym^Γ(Y), nor for the torsor structure on the fiber. Those are precisely the contents needed for Theorem 6.24.
  5. [§7, Theorem 7.7] The notation A_theta/A^+_theta is not a quotient in the usual sense, as Remark 7.6 concedes, yet Theorem 7.7 asserts an isomorphism of affine varieties A_theta/A^+_theta ≅ g^*_D//G_D. The proof claims injectivity of res_D: B_theta/B^+_theta -> g^*_D//G_D because 'the kernel of B_theta is inside B^+_theta', but B_theta/B^+_theta is only defined if B^+_theta is a linear subspace, and the argument does not establish that the image of A_theta injects. Since Corollary 7.9 uses this isomorphism to describe the symplectic leaf foliation, this part needs a precise statement and proof.
minor comments (4)
  1. [§3.3, Theorem 3.7] The construction of the moduli space U(X,G_theta) is summarized by saying that 'all the standard arguments ... carry through unchanged' after imposing the D-level structure. Given that this moduli space is the starting point for the cotangent and moment-map construction, the authors should either state the precise Balaji-Seshadri hypotheses being used (including the role of the local type τ and the quotient group H) or provide a more detailed proof.
  2. [§4.2, Proposition 4.6] In the proof of the singularity criterion, the claim that a semistable degree-zero coherent sheaf of rank at least 2 on a projective curve satisfies h^1 ≥ 2 is asserted without proof or reference. This is not true without further hypotheses: for example, on an elliptic curve a stable rank-2 degree-0 sheaf can have h^0 = 0 and h^1 = 0. A citation or a corrected statement is needed.
  3. [§6.2, Definition 6.4] The definition of the cameral cover is written with the shorthand φ^*(...), where φ is a section of a twisted adjoint bundle. The notation is unclear and should be replaced by a fiber-product construction over the total space of t ⊗ K_X(D), with the W-equivariant map spelled out.
  4. [§5.3, Definition 5.15] The formula η ∘ μ_E(φ) ∘ η^{-1} is confusing because η is a section of a quotient sheaf, not a bundle isomorphism; the intended adjoint-conjugation by the level structure should be written more explicitly in terms of local trivializations at the divisor points.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central derivation is a geometric construction with no fitted inputs; the flagged proof gaps are correctness concerns, not circular reductions.

full rationale

The paper's main chain—constructing leveled moduli, identifying a moment map via coresidues, descending a Poisson structure, and proving abelian torsor fibers—contains no fitted parameters and no quantity that is defined in terms of its own prediction. The level-structure moduli space is an extension of Seshadri's construction, and the moment map is computed from the cotangent action rather than assumed. The Poisson structure on MH is presented as induced by the forgetful map from the symplectic leveled space, a standard Hamiltonian-reduction mechanism. Self-citations to [21], [23], and [24] are published, parameter-free prior results with stated assumptions not including the target theorem; they provide legitimate independent support for the moduli-space and correspondence inputs. Example recoveries (Gaudin, KP elliptic solitons, Calogero–Moser) explicitly invoke external theorems [22], [33] to identify known spaces with symplectic leaves, so the unification claim is an application of prior identifications rather than a circular derivation. The reader's concern about Lemma 5.21 (the degree-zero map 'E(g) → E(g+)' has the wrong direction) and the Section 5.5 cotangent-complex identifications is a substantive mathematical rigor issue: if the quasi-isomorphism fails, the Poisson structure theorem lacks proof. But that is a gap or error in an internal proof step, not a reduction of the theorem to its own inputs. Under the circularity rubric, proof gaps without circular structure do not raise the score beyond the low range.

Assumptions & free parameters 0 free parameters · 6 assumptions · 3 invented entities

No fitted numerical parameters appear: the weights θ, divisor D, and base curve X are input data, not parameters tuned to make a result hold. The paper's dependencies are structural: the moduli space M_H and the equivariant correspondence come from the authors' [24], the Poisson-structure program from [23] and [21], the abelianization from Faltings-Scognamillo as invoked 'word by word', and the GIT moduli template from Balaji-Seshadri [2]. The load-bearing assumptions asserted rather than proved are the quasi-isomorphism of Lemma 5.21, the Poisson-map property of Theorem 5.23, the closed-condition treatment of level structures in Theorem 3.7, and the Γ-equivariant abelianization in Theorem 6.14.

assumptions (6)
  • standard math Serre duality and hypercohomology, including the identification T*U ≅ H⁰(E(g) ⊗ K(D))
    Invoked throughout Sections 4 and 5 to identify tangent and cotangent spaces of the moduli spaces; standard background.
  • domain assumption Heinloth's uniformization and the generically split restriction for parahoric torsors
    Remarks 2.3 and 2.5 restrict the classification of torsors to generically split data, and Theorem 7.7's residue gluing uses that restriction; the paper states it will abuse terminology and treat generically split torsors as the default.
  • ad hoc to paper The D-level structure imposes a rigid closed condition, so the GIT construction of U(X, G_θ) follows Balaji-Seshadri with 'standard arguments carry through unchanged'
    Section 3.3 and Theorem 3.7; the space of formal-disk sections into the Levi quotient is not shown to be finite-type, and the GIT analysis for the level data is not given.
  • ad hoc to paper The Γ-equivariant abelianization of the generic Hitchin fiber 'repeats word by word' the proof of Scognamillo
    Section 6.4, Theorem 6.14; the W-invariance of Γ-equivariant T-bundles is established only at the level of statements, and the extension to the full Hitchin fiber is asserted.
  • ad hoc to paper The leveled deformation complex K_{E,φ,η} is quasi-isomorphic to the logahoric complex K_{E,φ}
    Lemma 5.21; the proof's degree-0 map has reversed inclusion direction and the degree-1 'local analysis' is asserted, so the claimed quasi-isomorphism is not established as written.
  • ad hoc to paper Semistable degree-zero sheaves of rank at least 2 have h¹ ≥ 2 on projective curves of any genus
    Proposition 4.6 uses this for the Chevalley-Shephard-Todd singularity argument; it is stated without proof and is false for genus 0 and 1 (e.g. O⊕O).
invented entities (3)
  • D-level structure of parahoric type on a G_θ-torsor (Definition 3.1) independent evidence
    purpose: Selects a section η_x: D_x to G_{θx}/G+_{θx} at each marked point, providing the framing that enables the level group action and the coresidue moment map.
    The framework is validated against external systems: the Gaudin Lax operator arises as a logarithmic Higgs field (Section 8.2), and the KP and Calogero-Moser leaves match the constructions of Treibich-Verdier and Hurtubise-Markman.
  • Level group G_D = (prod_j L_{θj}) / Z (Definition 5.11) independent evidence
    purpose: Acts on the moduli space of leveled torsors and, by Hamiltonian lifting, on its cotangent; its moment map is the direct sum of coresidues, and its coadjoint orbits foliate the symplectic leaves.
    The action is defined explicitly on local trivializations (Proposition 5.12), so a reader can verify the moment-map formula (Theorem 5.17) independently.
  • The quotient A_θ/A+_θ (Section 7)
    purpose: Claimed to be isomorphic to g*_D//G_D and to serve as the base of the symplectic leaf foliation via q composed with eH_θ.
    Remark 7.6 concedes that A+_θ is not a vector space and the quotient is not well-defined; Theorem 7.7's isomorphism is not supported by a dimension count.

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Pith. "Pith review of Level structures on parahoric torsors and complete integrability." pith.science (2026). https://pith.science/paper/WCT5IBM3

@misc{pith2026250612302,
  author       = {Pith},
  title        = {Pith review of: Level structures on parahoric torsors and complete integrability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WCT5IBM3}},
  note         = {Machine review of arXiv:2506.12302}
}
abstract

For a smooth complex algebraic curve $X$ and a reduced effective divisor $D$ on $X$, we introduce a notion of $D$-level structure on parahoric $\mathcal{G}_{\boldsymbol \theta}$-torsors over $X$, for any connected complex reductive Lie group $G$. A moduli space of parahoric $\mathcal{G}_{\boldsymbol \theta}$-torsors equipped with a $D$-level structure is constructed and we identify a canonical moment map with respect to the action of a level group on this moduli space. This action extends to a Poisson action on the cotangent, thus inducing a Poisson structure on the moduli space of logahoric $\mathcal{G}_{\boldsymbol \theta}$-Higgs torsors on $X$. A study of the generic fibers of the parahoric Hitchin fibration of this moduli space identifies them as abelian torsors and introduces new algebraically completely integrable Hamiltonian Hitchin systems in this parahoric setting. We show that this framework generalizes, among other, the integrable system of Beauville and recovers the classical Gaudin model in its simplest form, the space of periodic KP elliptic solitons and the elliptic Calogero--Moser system, thus demonstrating that the logahoric Hitchin integrable system unifies many integrable systems with regular singularities under a single geometric framework.

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Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [22]

    Calogero–Moser Systems and Hitchin Systems

    Hurtubise, J. C., and Markman, E., “Calogero–Moser Systems and Hitchin Systems”. Commun. Math. Phys. 223 (2001), 533–582

    work page 2001

  2. [33]

    in The Grothendieck Festschrift, Vol

    Treibich,A., and Verdier, J.-L., Solitons elliptiques. in The Grothendieck Festschrift, Vol. III , 437–480, Progress in Mathematics vol. 88, Birkh¨ auser Boston, Boston, MA (1990)

    work page 1990

  3. [1]

    I., and Givental, A

    Arnol’d, V. I., and Givental, A. B., Symplectic geometry . in Dynamical Systems IV, Encyclopedia of Mathematical Sciences, vol. 4 (Springer, 1990)

    work page 1990

  4. [2]

    Moduli of parahoric G-torsors on a compact Riemann surface

    Balaji, V., and Seshadri, C. S., “Moduli of parahoric G-torsors on a compact Riemann surface”.J. Algebraic Geom. 24 (2015), no. 1, 1–49

    work page 2015

  5. [3]

    Complete integrability of the parahoric Hitchin system

    Baraglia, D., Kamgarpour, M., and Varma, R., “Complete integrability of the parahoric Hitchin system”. Int. Math. Res. Not. 2019 (2019), no. 21, 6499–6528

    work page 2019

  6. [4]

    Jacobiennes des courbes spectrales et syst` emes hamiltoniens compl` etement int´ egrables

    Beauville, A., “Jacobiennes des courbes spectrales et syst` emes hamiltoniens compl` etement int´ egrables”. Acta Math. 164 (1990), no. 3–4, 211–235

    work page 1990

  7. [5]

    Un lemme de descente

    Beauville, A., and Laszlo, Y., “Un lemme de descente”. C. R. Acad. Sci. Paris 320 (1995)

    work page 1995

  8. [6]

    A Torelli theorem for moduli spaces of principal bundles over a curve

    Biswas, I., and Hoffmann, N., “A Torelli theorem for moduli spaces of principal bundles over a curve”. Ann. Inst. Fourier 62 (2012), no. 1, 87–106

    work page 2012

Show all 38 references
  1. [7]

    Moduli spaces of framed G-Higgs bundles and symplectic geometry

    Biswas, I., Logares, M., and Pe´ on-Nieto, A., “Moduli spaces of framed G-Higgs bundles and symplectic geometry”. Commun. Math. Phys. 376 (2020), no. 3, 1875–1908

    2020

  2. [8]

    Riemann–Hilbert for tame complex parahoric connections

    Boalch, P., “Riemann–Hilbert for tame complex parahoric connections”. Transform. Groups 16 (2011), 27–50

    2011

  3. [9]

    Symplectic geometry on moduli spaces of stable pairs

    Bottacin, F., “Symplectic geometry on moduli spaces of stable pairs”. Ann. Sci. ´Ec. Norm. Sup´ er. (4)28 (1995), no. 4, 391–433

    1995

  4. [10]

    Torsors over the punctured affine line

    Chernousov, V., Gille, P., and Pianzola, A., “Torsors over the punctured affine line”. Amer. J. Math. 134 (2012), no. 6, 1541–1583

    2012

  5. [11]

    Local types of (Γ , G)-bundles and parahoric group schemes

    Damiolini, C., and Hong, J., “Local types of (Γ , G)-bundles and parahoric group schemes”. Proc. LMS 27 (2023)

    2023

  6. [12]

    Line bundles on the moduli stack of parahoric bundles

    Damiolini, C., and Hong, J., “Line bundles on the moduli stack of parahoric bundles”. arXiv:2412.08826 (2024)

    2024

  7. [13]

    A support theorem for the Hitchin fibration: the case of GLn and KC

    de Cataldo, M. A. A., Heinloth, J., Migliorini, L., “A support theorem for the Hitchin fibration: the case of GLn and KC ”. J. Reine Angew. Math. 780 (2021), 41–77

    2021

  8. [14]

    Journ´ ees de g´ eom´ etrie alg´ ebrique d’Orsay, France, juillet 20-26, 1992

    Donagi, R., Decomposition of spectral covers . Journ´ ees de g´ eom´ etrie alg´ ebrique d’Orsay, France, juillet 20-26, 1992. Paris: Soci´ et´ e Math´ ematique de France,Ast´ erisque218, 145-175 (1993)

    1993

  9. [15]

    in Francaviglia, M

    Donagi, R., and Markman, E., Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles . in Francaviglia, M. (ed.) et al., Integrable systems and quantum groups. Lectures given at the 1st session of the Centro Internazionale Matematico Es...

    1996

  10. [16]

    Stable G-bundles and projective connections

    Faltings, G.: “Stable G-bundles and projective connections”. J. Algebr. Geom. 2 (1993), No. 3, 507–568

    1993

  11. [17]

    Gaudin Model, Bethe Ansatz and Critical Level

    Feigin, B., Frenkel, E., and Reshetikhin, N. “Gaudin Model, Bethe Ansatz and Critical Level”. Commun. Math. Phys. 166 (1994), No. 1, 27–62

    1994

  12. [18]

    On parahoric subgroups

    Haines, T., and Rapoport, M., “On parahoric subgroups”. Adv. Math. 219 (2008), no. 1, 188–198

    2008

  13. [19]

    Uniformization of G-bundles

    Heinloth, J., “Uniformization of G-bundles”. Math. Ann. 347 (2010), no. 3, 499–528

    2010

  14. [20]

    Stable bundles and integrable systems

    Hitchin, N., “Stable bundles and integrable systems”. Duke Math. J. 54 (1987), 91–114

    1987

  15. [21]

    Tame parahoric nonabelian Hodge correspondence on curves

    Huang, P., Kydonakis, G., Sun, H., and Zhao, L., “Tame parahoric nonabelian Hodge correspondence on curves”. arXiv:2205.15475 (2022). 50 GEORGIOS KYDONAKIS AND LUTIAN ZHAO

    2022 arXiv

  16. [23]

    Poisson structures on moduli spaces of Higgs bundles over stacky curves

    Kydonakis, G., Sun, H., and Zhao, L., “Poisson structures on moduli spaces of Higgs bundles over stacky curves”. Adv. Geom. 24 (2024), no. 2, 163–182

    2024

  17. [24]

    Logahoric Higgs torsors for a complex reductive group

    Kydonakis, G., Sun, H., and Zhao, L., “Logahoric Higgs torsors for a complex reductive group”. Math. Ann. 388(3) (2024), 3183-3228

    2024

  18. [25]

    Moduli of parabolic Higgs bundles and Atiyah algebroids

    Logares, M., and Martens, J., “Moduli of parabolic Higgs bundles and Atiyah algebroids”. J. Reine Angew. Math. 649 (2010), 89–116

    2010

  19. [26]

    Spectral curves and integrable systems

    Markman, E., “Spectral curves and integrable systems”. Compositio Math. 93 (1994), no. 3, 255–290

    1994

  20. [27]

    Unrefined minimal K-types for p-adic groups

    Moy, A., and Prasad, G., “Unrefined minimal K-types for p-adic groups”. Invent. Math. 116 (1994), 393–408

    1994

  21. [28]

    Nguyen, T. Q. T., Parahoric Hitchin fibration . Ph.D. thesis, University of Edinburgh (2021)

    2021

  22. [29]

    Stable principal bundles on a compact Riemann surface

    Ramanathan, A., “Stable principal bundles on a compact Riemann surface”. Math. Ann. 213 (1975), no. 2, 129–152

    1975

  23. [30]

    G., and Semenov-Tyan-Shansky, M

    Reyman, A. G., and Semenov-Tyan-Shansky, M. A., Group theoretical methods in the theory of finite dimensional integrable systems . in: Dynamical Systems VII, Arnol’d, V. I. and Novikov, S. P. (eds.), Encyclopedia of Mathematical Sciences, vol. 16, pp. 119–193 (1987) (Russian)

    1987

  24. [31]

    An elementary approach to the abelianization of the Hitchin system for arbitrary re- ductive groups

    Scognamillo, R., “An elementary approach to the abelianization of the Hitchin system for arbitrary re- ductive groups”. Compos. Math. 110 (1998), No. 1, 17–37

    1998

  25. [32]

    S., Fibr´ es vectoriels sur les courbes alg´ ebriques

    Seshadri, C. S., Fibr´ es vectoriels sur les courbes alg´ ebriques. Ast´ erisque, vol. 96 (1982)

    1982

  26. [34]

    Vanhaecke, P., Integrable systems in the realm of algebraic geometry. 2nd ed. Lecture Notes in Mathematics

  27. [35]

    On parahoric Hitchin systems over curves

    Wang, B., “On parahoric Hitchin systems over curves”. Int. J. Math. 34 (2023), no. 13, 2350081

    2023

  28. [36]

    Global Springer theory

    Yun, Z., “Global Springer theory”. Adv. Math. 228 (2011), 266–328

    2011

  29. [37]

    thesis, COMUE Universit´ e Cˆ ote d’Azur (2017)

    Zelaci, H., Moduli spaces of anti-invariant vector bundles over curves and conformal blocks , Ph.D. thesis, COMUE Universit´ e Cˆ ote d’Azur (2017). Department of Mathematics, University of Patras University Campus, Patras 26504, Greece E-mail address : gkydonakis@math.upatras...

    2017

  30. [1638]

    x, 256 p

    Berlin: Springer. x, 256 p. (2001)

    2001

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.