Pith. sign in

REVIEW 3 major objections 4 minor 34 references

Rank three representations of Painleve systems: III. Dolbeault structure, spectral correspondence

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read The rank-3 JKT and rank-2 Painleve moduli spaces are the same hyperkaehler manifold, via spectral correspondence and algebraic Nahm transformation.

desk verdict A Dolbeault-side step forward for four of the six Painleve cases, with a load-bearing 'easy to check' in the two maximally twisted cases that a referee should force out before accepting. read the letter →

arxiv 2509.00418 v1 pith:JZRWLP3L submitted 2025-08-30 math.AG

Miklos Eper , Szilard Szabo This is my paper · ORCID
classification math.AG MSC 14D2014H7034M5553C26
keywords Painleveequationsrank3LaxrepresentationsmeromorphicHiggsbundlesDolbeaultmodulispacesalgebraicNahmtransformspectralcorrespondencehyperkaehlerisometryellipticfibrations
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 proves that the rank-3 Lax/Higgs-bundle representations of the six Painleve equations and the familiar rank-2 representations are not merely related but literally describe the same hyperkaehler manifold. In the Dolbeault complex structure, the authors construct an algebraic symplectic isomorphism between the two moduli spaces using spectral data: each meromorphic Higgs bundle is replaced by a spectral sheaf on a specially blown-up surface, and both families share the same spectral curves. Combined with the de Rham description from the companion paper, this upgrades the isomorphism to a hyperkaehler isometry. A reader should care because it unifies two apparently different integrable systems and yields explicit elliptic fibrations whose fibers at infinity are of affine Dynkin type eD4, eD5, eE6, eE7 and eE8 in the six cases.

What carries the argument

The spectral sheaf S(E,theta), defined by the exact sequence 0 -> p*(E tensor K(D)^-1) -> p*E tensor O(1) -> S -> 0 on the Hirzebruch surface F1, is the central object: the Beauville-Narasimhan-Ramanan correspondence identifies the Higgs bundle with this sheaf. The surface Z* is built as a sequence of nine blow-ups resolving the base locus of the pencil spanned by the spectral curve and a fixed divisor C2, followed by blowing down the (-1)-section at infinity. The algebraic Nahm transform then carries the spectral sheaf to a rank-2 Higgs bundle on the dual projective line, and the paper proves this map preserves the holomorphic symplectic forms by identifying both with the Mukai form on the

What would settle it

Compute the base-point resolution of the pencil generated by the spectral curve and C2 for the JKT IVa and JKT I cases at the triply tangent point. If the ramification degree of the spectral curve at infinity is not 3, or if resolving the base locus produces an exceptional configuration different from the blow-ups used in Sections 3.4 and 3.7, then the spectral-sheaf identification with the Mukai moduli space breaks and the symplectic isomorphism of Theorem 1.1 fails.

Watch

Extended reading notes

Core claim

Theorem 1.1 states that the algebraic Nahm transformation gives a holomorphic symplectic isomorphism between each moduli space M^{JKT*}_{Dol} of rank-3 meromorphic Higgs bundles with prescribed irregular types and the corresponding rank-2 Painleve moduli space M^{P*}_{Dol}, both in the Dolbeault complex structure. Theorem 1.2 then states that Fourier-Laplace transformation induces a hyperkaehler isometry between the rank-2 irregular connection moduli spaces of the Painleve equations and the rank-3 JKT moduli spaces. In concrete terms, the two representations are one and the same hyperkaehler 4-manifold, with the Hitchin fibration realized as an elliptic fibration on a resolved surface whose

Load-bearing premise

The central claim depends on the assertion, stated as 'it is easy to check' in the proof of Theorem 3.4, that in the maximally twisted cases JKT IVa and JKT I the spectral curve ramifies to degree 3 at infinity and that the classical resolution of the pencil's base locus coincides exactly with the blow-up sequences written down in Sections 3.4 and 3.7.

Editorial extensions

If this is right

  • If the central claim is correct, all six Painleve systems have rank-2 and rank-3 representations that are the same hyperkaehler manifold, not merely diffeomorphic or birational.
  • The Hitchin fibration of each Dolbeault moduli space is an elliptic fibration whose fiber at infinity is of type I0*, I1*, eE6, eE6, eE7, or eE8 for the cases JKT VI, V, IVa, IVb, II, I respectively.
  • The algebraic Nahm transformation and the Fourier-Laplace transformation of D-modules agree up to the non-abelian Hodge diffeomorphism, giving a commutative diagram linking Dolbeault and de Rham descriptions.
  • The proof strategy, which relies on geometric spectral data rather than analytic estimates with harmonic spinors, is expected by the authors to carry over to more general hyperkaehler isometries between non-abelian Hodge moduli spaces.
  • The elliptic fibrations constructed here provide an explicit model of the moduli spaces as integrable systems, with the fiber at infinity matching the Painleve cases studied earlier.

Reading between the lines

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

  • Going beyond the paper: if the isometry is genuinely hyperkaehler, the twistor families of the two moduli spaces must agree, so the Painleve isomonodromy data already encodes the JKT twistor family.
  • The same pencil-and-blown-up-surface construction should extend to other root-system leaves of the eE6 type once suitable local irregular forms are fixed, a direction the paper flags but does not carry out.
  • One could test the identification numerically by comparing the j-invariants of the elliptic fibrations constructed here with the pole divisors of Painleve tau functions in the six cases.
  • The spectral-sheaf description suggests a direct categorification: the equivalence of derived categories of Higgs bundles and spectral sheaves should imply an isomorphism of their deformation functors, yielding the symplectic isomorphism without case-by-case blow-up checks.
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

3 major / 4 minor

Summary. The paper claims to prove Theorem 1.1: the algebraic Nahm transformation defines an algebraic, holomorphic-symplectic isomorphism between the rank-3 JKT Dolbeault moduli spaces M^{JKT*}_{Dol} and the corresponding rank-2 Painlevé Dolbeault moduli spaces M^{P*}_{Dol}. The proof proceeds by constructing, for each of the six cases, a log-Calabi-Yau surface Z^* as a blow-up of the Hirzebruch surface F_1 followed by a blow-down, and using a Beauville-Narasimhan-Ramanan (BNR) spectral correspondence (Theorem 3.4) to identify the JKT moduli space with a Mukai moduli space of sheaves on Z^* (Sections 3.2--3.7, 4.1). The symplectic forms are compared in Proposition 4.9 via explicit Čech cocycles and Serre duality. Together with the de Rham/Fourier-Laplace comparison in Proposition 4.6 and the companion paper [14], the paper concludes that the two representations of each Painlevé system are hyperkähler isometric (Theorem 1.2). The paper also describes the induced elliptic fibrations and identifies the fiber at infinity as I_0^*, I_1^*, \tilde E_6, \tilde E_7, \tilde E_8.

Significance. If the main theorem is correct, it is a substantial result: it provides a geometric proof that the rank-2 and rank-3 representations of all six Painlevé systems are the same hyperkähler manifold, extending the previously known Painlevé VI case. The proof strategy is attractive and largely coordinate-free: the symplectic comparison in Section 4.5 is worked out in real detail with explicit Čech representatives, and the explicit elliptic fibrations and fiber types give concrete structural information. The paper also has notable strengths: local normal forms are listed explicitly, the Yoneda/Serre-duality comparison is spelled out, and the dependence on the companion paper [14] is clearly flagged. However, the central BNR identification in the maximally twisted cases rests on several uncomputed 'easy to check' assertions about ramification and base-point resolution. Because these assertions are load-bearing for Theorems 3.4, 1.1, and hence 1.2, the current version is not yet fully rigorous.

major comments (3)
  1. [§4.1, proof of Theorem 3.4 (with §§3.4 and 3.7)] In the maximally twisted cases JKT IVa and JKT I, the proof of the BNR identification depends on two assertions made without computation: (i) that a single 3-by-3 Jordan block forces the spectral curve to have ramification degree 3 at infinity, and (ii) that the 'classical' resolution [16] of the pencil's base locus coincides exactly with the blow-up sequences of §§3.4 and 3.7. These are not immediate from the normal forms (JKTIVa)/(JKTI), and the stated sequences themselves create further non-transverse base points (e.g. the intersection of the multiplicity-3 exceptional divisor with C^(3) in Figure 3). If the resolutions differed, the total transform of Σ would not have class [F∞*], and the map to the Mukai moduli would not be well-defined. This is load-bearing for Theorem 3.4 and therefore for Theorem 1.1. Please provide the local intersection-multiplicity computation for the pencil a
  2. [§3, Proposition 3.10] The proof that |F∞*| is an elliptic pencil and that the total transform of Σ lies in the same linear system relies on the sentence 'One can easily check that C = F∞* satisfies this system of equations' (system (17)), together with a uniqueness claim in H2(Z*,Z). No intersection products [C]·[eF], [C]·[Ei], or [C]·[si] are computed for any of the six cases. This is particularly delicate in the cases with infinitesimally near base points (§§3.4, 3.6, 3.7), where the multiplicities of the exceptional divisors are part of the statement. Since the class [F∞*] is used to define the support condition in M*_Muk, this verification is necessary. A table of intersection numbers for all six cases would settle the issue.
  3. [§2.1.1, Theorem 2.1 and its proof] Theorem 2.1 asserts existence, smoothness, completeness, and hyperkähler structure of the moduli spaces M^{JKT*}_{Dol} in all six cases. In the twisted cases the proof says 'we will not give full details' and invokes an 'equivariant version' of [3, Theorem 5.4] that is asserted but not stated or proved. Since every later statement in the paper is a statement about these moduli spaces, the status of this foundational point should be made precise: either quote a theorem from the literature (e.g. [3,27]) with exact hypotheses, or supply the equivariant reduction in detail. As written, the existence theorem is only sketched where the paper needs it most.
minor comments (4)
  1. [Introduction, first paragraph] The text says 'Then, in the second article [13]' but the second article is listed as [14]; the citation should be corrected.
  2. [§4.5, first sentence] Typo: 'isomoprhism' should be 'isomorphism'.
  3. [Definition 4.4] The notation 'p*z dot' for the multiplication by p^*z is ambiguous; write p^*z \cdot (or p^*z \,\cdot\,) explicitly.
  4. [Table 1 and §§3.4--3.7] The notation eE6, eE7, eE8 is used without definition; using \tilde E_6, \tilde E_7, \tilde E_8 and defining the tilde notation would improve readability.

Circularity Check

1 steps flagged · score 4.0 of 10

No definitional circularity in the Dolbeault isomorphism, but the headline hyperkähler isometry is load-bearing on the authors' own in-preparation paper [14]; the maximally twisted resolution claim is an uncomputed gap, not a circular reduction.

  1. self citation load bearing [Section 1, paragraph before Theorem 1.2]
    "On the other hand, in [14] we have proved that Fourier–Laplace transformation of D-modules determines an algebraic isomorphism between corresponding de Rham moduli spaces of irregular connections with given normal forms. In Proposition 4.6 we establish that Nahm and Fourier–Laplace transformations agree, up to applying the non-abelian Hodge diffeomorphism. Combining these results, we then immediately get: Theorem 1.2."

    Theorem 1.2 is one of the two headline results advertised in the abstract: the hyperkähler isometry between rank-2 and rank-3 Painlevé moduli spaces. Its de Rham leg is not proved in this paper; it is imported from [14], an in-preparation paper by the same two authors. The argument chain for the headline statement is therefore: Theorem 1.1 (proved here) plus [14] (same authors, unpublished, not independently checkable from the present manuscript). Thus a central advertised conclusion is load-bearing on a self-citation rather than on a proof contained in the paper. This is not a definitional reduction, but it is a genuine load-bearing self-citation for one of the paper's main claims.

full rationale

The paper's core technical work—the construction of the Dolbeault moduli spaces, the spectral-correspondence identification via the surfaces Z^*, and the comparison of holomorphic symplectic forms—is a mathematical derivation that does not reduce to its own inputs by definition. There are no fitted parameters, no empirical predictions, and no quantity is renamed as a prediction after being fitted. The spectral curve is defined by the characteristic polynomial of the Higgs field, and the Hitchin fibration is an honest map; Proposition 3.10 and Lemma 3.12 give at least structural arguments, even if some intersection-theoretic checks are delegated to 'one can easily check'. The maximally twisted cases (JKT IVa, JKT I) in the proof of Theorem 3.4 rely on an uncomputed assertion that the ramification degree is 3 and that the classical base-point resolution agrees with the blow-up sequence of Sections 3.4 and 3.7. This is a verification gap—potentially serious for Theorem 3.4 and hence Theorem 1.1—but it is not circular: the ramification degree follows from the Puiseux forms in Lemma 2.3, and the resolution comparison is an external fact about a pencil, not an assumption equivalent to the conclusion. The paper also makes heavy use of the authors' prior work: [1], [13], [14], [18], [19], [31], [32], [33]. Most of these are published and are used as ordinary mathematical citations, so they do not, by themselves, make the derivation circular. The exception is [14], an in-preparation paper by the same authors, which supplies the de Rham isomorphism needed for Theorem 1.2; that is a load-bearing self-citation for a headline claim. Overall, the central Dolbeault isomorphism has independent content and is not forced by a self-citation chain, so the score is moderate rather than high.

Assumptions & free parameters 3 free parameters · 7 assumptions · 2 invented entities

The central claim rests on a small number of imported results: Biquard-Boalch hyperkahler completeness [3], the BNR spectral correspondence in its irregular version [33], Mukai's symplectic structure [28], and Mochizuki's equivariant harmonic metrics [27]. The paper's own constructions are the surfaces Z* and the Nahm transform N, which are explicit and checkable. No fitted constants appear: the ai, bi, ci, parabolic weights and generic b0 are fixed moduli data. The main verification gaps are the asserted but uncomputed base-point resolution in the maximally twisted cases and the dependence on the in-preparation companion paper [14].

free parameters (3)
  • Fixed local coefficient data (ai, bi, ci, residues tau_i) in normal forms (JKTVI)-(JKTI) = case-dependent fixed complex numbers
    These fix the symplectic leaf (irregular type and residues) of the Poisson moduli space; they are inputs defining M^JKT*_Dol, not constants fitted to any target.
  • Parabolic weights alpha^j_pi with 0 < alpha < 1 (assumption (2)) = generic values in (0,1)
    Define alpha-stability; generic choice ensures smoothness of moduli (Theorem 2.1) and is used in the L2-boundedness argument of Lemma 4.7.
  • Generic Hitchin base point b0 (parameter of the pencil, affine line in Proposition 3.2) = generic point of the affine line
    Proposition 3.5 needs a smooth spectral curve; Theorem 1.1 is established on Zariski open subsets for generic b0, per the closing remark of Section 4.5.
assumptions (7)
  • standard math BNR spectral correspondence for meromorphic Higgs bundles, irregular version of [33, Theorem 5.4]
    Theorem 3.4 imports this correspondence; the paper sketches the adaptation to the ramified rank-3 setting.
  • ad hoc to paper BNR extends to maximally twisted (single Jordan block) rank-3 cases with the stated base-point resolution, Sections 3.4 and 3.7
    Asserted in the proof of Theorem 3.4 (Section 4.1) as "it is easy to check" citing [16]; not computed in the text.
  • domain assumption Biquard-Boalch existence and completeness of hyperkahler metrics on wild Higgs bundle moduli [3], including the equivariant (twisted) version
    Theorem 2.1 is imported from [3], with the twisted-case proof only sketched ("We will not give full details"). Completeness underlies the isometry claim.
  • domain assumption Mochizuki's existence of equivariant wild harmonic metrics [27, Section 13.4]
    Cited in the proof of Theorem 2.1 for the twisted cases.
  • standard math Mukai's symplectic structure on the moduli of simple sheaves [28], applied to the log-Calabi-Yau pair (Z*, F*_infty)
    Defined via the Yoneda pairing (25); requires Ext^2 = C (Lemma 4.8), shown under smoothness of the support.
  • domain assumption Generic smoothness: spectral curves smooth and moduli smooth for generic parameters
    Used in Proposition 3.5, Theorem 2.1, and Proposition 4.9 (which restricts to the open subset with smooth spectral curve).
  • domain assumption Simpson's formulas (6) connecting Dolbeault data (alpha, nu) and de Rham data (mu, beta)
    Quoted in the proof of Theorem 2.1 to identify the de Rham counterpart of the Dolbeault moduli space; imported from the wild non-abelian Hodge theory literature.
invented entities (2)
  • The log-Calabi-Yau surfaces Z* (nine blow-ups of the Hirzebruch surface F1 followed by blow-down of sigma_infty), with fiber at infinity F*_infty of Kodaira type I*_0, I*_1, eE6, eE7, eE8 independent evidence
    purpose: Realize each Dolbeault moduli space M^JKT*_Dol as the Mukai moduli of rank-1 sheaves on Z* via the BNR correspondence and describe the Hitchin fibration as an elliptic pencil.
    The blow-up sequences are explicitly prescribed in Sections 3.2-3.7 and the claimed fiber types match the Painleve fibers of [34]; the constructions are checkable but the resolution claims are presented via figures rather than full computations.
  • Algebraic Nahm transform N (Definition 4.4), defined by N(E,theta) = (p_hat_* S(E,theta), -1/2 p_hat_* (p*z dot) dz_hat) independent evidence
    purpose: The map asserted to be the symplectic isomorphism of Theorem 1.1 and to coincide with Fourier-Laplace transformation under the non-abelian Hodge diffeomorphism (Proposition 4.6).
    The formula is explicit and its involutivity up to sign and agreement with the classical Nahm/Fourier-Laplace construction follow the arguments of [1, Theorems 10.2 and 8.5] and [31]; the target identification, however, is delegated to the in-preparation [14].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Rank three representations of Painleve systems: III. Dolbeault structure, spectral correspondence." pith.science (2026). https://pith.science/paper/JZRWLP3L

@misc{pith2026250900418,
  author       = {Pith},
  title        = {Pith review of: Rank three representations of Painleve systems: III. Dolbeault structure, spectral correspondence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JZRWLP3L}},
  note         = {Machine review of arXiv:2509.00418}
}
read the original abstract

We prove that there exists a holomorphic symplectic isomorphism between the rank 2 and 3 representations of the Painleve systems in the Dolbeault complex structure, and give explicit descriptions of the corresponding elliptic fibrations. This, combined with the de Rham description given in part II, implies that the corresponding moduli spaces are hyperKaehler isometric to each other.

Figures

Figures reproduced from arXiv: 2509.00418 by the authors.

Figure 1
Figure 1. Pencil arrangement for JKT V I case. of a double section Q of p and a section L1 of p, again for simplicity. As the notation suggests, the proper trasform of Q in CP 2 is a quadric.) C2 = F1 + 2F2 + 3σ∞, and we have 5 base points. Apply three blow-ups at the points of F1 ∩ (Q + L1), while two infinitely close blow-ups at a1 = F2 ∩ L1 (in these base points, the curves meet transversely). At a0, we need to apply four … view at source ↗
Figure 2
Figure 2. Pencil arrangement for JKT V case [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Pencil arrangement for JKT IV a case. 3.5 Construction of Z IV b In this case C1 is as in Section 3.2, while C2 = 3F + 3σ∞, where the F fiber of the ruling is counted with multiplicity three. The points F ∩ Li = ai−1 (i = 1, 2, 3) are the base points, where the curves meet transversely. We apply three infinitesimally close blow-ups in each of them. The result is visualized in [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Pencil arrangement for JKT IV b case. 3.6 Construction of Z II Here we have C1 as in Section 3.3, and C2 = 3F +3σ∞. At a1 = F ∩L1, where the two curves are transverse, we apply three blow-ups and get a section. At a0, the curves are doubly tangent to each other. Here a…
Figure 5
Figure 5. Figure 5: Pencil arrangement for JKT II case. Summary JKT system polar divisor of θ type of irregular singularity at ∞ Dynkin diagram of F ∗ ∞ JKT V I D = {0} + 2{∞} untwisted I ∗ 0 (or De4) JKT V D = {0} + 2{∞} minimally twisted I ∗ 1 (or De5) JKT IV a D = {0} + 2{∞} maximally …
Figure 6
Figure 6. Figure 6: Pencil arrangement for JKT I case. 4.1 Spectral sheaf Definition 4.1. The spectral sheaf is the coherent sheaf S(E,θ) of OF1 -modules defined by the short exact sequence 0 → p ∗ (E ⊗ KX(D) −1 ) ζIE−ξp∗θ −−−−−→ p ∗ E ⊗ OF1|X(1) → S(E,θ) → 0. (18) Remark 4.2. The support…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [34]

    Szab ´o: Perversity equals weight for Painlev´ e spaces,Adv

    Sz. Szab ´o: Perversity equals weight for Painlev´ e spaces,Adv. Math. 383, 2 (2021), 107667. doi: 10.1016/j.aim.2021.107667

    work page arXiv 2021

  2. [16]

    Hirzebruch: ¨Uber vierdimensionale Riemannsche Fl¨ achen mehrdeutiger analytischer Functionen von zwei complexen Ver¨ anderlichen,Math

    F. Hirzebruch: ¨Uber vierdimensionale Riemannsche Fl¨ achen mehrdeutiger analytischer Functionen von zwei complexen Ver¨ anderlichen,Math. Ann. 126, 2 (1953) 1–22

    work page 1953

  3. [14]

    Eper, Sz

    M. Eper, Sz. Szab ´o: Rank three representations of Painlev´ e systems: II. de Rham structure, Fourier–Laplace transforma- tion. In preparation

    work page

  4. [1]

    Aker, Sz

    K. Aker, Sz. Szab´o: Algebraic Nahm transform for parabolic Higgs bundles on P1, Geometry and Topology 18 5 (2014), 2487- 2545

    work page 2014

  5. [2]

    Beauville, M

    A. Beauville, M. S. Narasimhan, and S. Ramanan: Spectral curves and the generalised theta divisor,J. reine angew. Math. 398, (1989), 169–179

    work page 1989

  6. [3]

    Biquard, P

    O. Biquard, P. Boalch: Wild non-abelian Hodge theory on curves, Compos. Math. 140, 1 (2004), 179–204

    work page 2004

  7. [4]

    Biquard, M

    O. Biquard, M. Jardim: Asymptotic behaviour and the moduli space of doubly-periodic instantons, J. Eur. Math. Soc. (JEMS) 3, 4 (2001), 335–375

    work page 2001

  8. [5]

    Biswas, S

    I. Biswas, S. Ramanan: An Infinitesimal Study of the Moduli of Hitchin Pairs, Journal of the London Mathematical Society 49 2 (1994)

    work page 1994

Show all 34 references
  1. [6]

    P. Boalch: Wild character varieties, meromorphic Hitchin systems and Dynkin diagrams, Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin , Oxford University Press (2018)

    2018

  2. [7]

    Boalch: Geometry and braiding of Stokes data; Fission and wild character varieties, Annals of Mathematics 179, 1 (2014), 301-365

    P. Boalch: Geometry and braiding of Stokes data; Fission and wild character varieties, Annals of Mathematics 179, 1 (2014), 301-365

    2014

  3. [8]

    P. J. Braam, P. van Baal: Nahm’s Transformation for In- stantons, Commun. Math. Phys. 122, (1989), 267–280

    1989

  4. [9]

    S. A. Cherkis, A. Larra´ın-Hubach, M. A. Stern: Instan- tons on multi-Taub-NUT Spaces III: Down Transform, Com- pleteness, and Isometry, to appear in: J. Diff. Geom

  5. [10]

    Diaconescu, R

    D.-E. Diaconescu, R. Donagi, T. Pantev: BPS states, torus links and wild character varieties, Commun. Math. Phys. 359, 3 (2018), 1027–1078. 41

    2018

  6. [11]

    Donagi, E

    R. Donagi, E. Markman: Spectral covers, algebraically com- pletely integrable Hamiltonian systems, and moduli of bundles, Lecture Notes in Mathematics 1620 (1996), Springer

    1996

  7. [12]

    Eper, Sz

    M. Eper, Sz. Szab ´o: Asymptotic geometry of non-abelian Hodge theory and Riemann–Hilbert correspondence, rank three eE6 case, arXiv:2407.20638

    arXiv

  8. [13]

    Eper, Sz

    M. Eper, Sz. Szab ´o: Rank three representations of Painlev´ e systems: I. Wild character varieties, arxiv:2505.21186

    arXiv

  9. [15]

    Griffiths, J

    P. Griffiths, J. Harris: Principles of Algebraic Geometry, Wiley. (1994)

    1994

  10. [17]

    Huybrechts, M

    D. Huybrechts, M. Lehn: The geometry of moduli spaces of sheaves, Cambridge University Press (2010)

    2010

  11. [18]

    Ivanics, A

    P. Ivanics, A. Stipsicz, Sz. Szab ´o: Two-dimensional mod- uli spaces of rank 2 Higgs bundles over CP 1 with one irregular singular point, J. Geom. Phys. , 130 (2018) 184-212

    2018

  12. [19]

    Ivanics, A

    P. Ivanics, A. Stipsicz, Sz. Szab ´o: Hitchin fibrations on moduli of irregular Higgs bundles and motivic wall-crossing, J. Pure Appl. Algebra, 223, 9 (2019) 3989-4064

    2019

  13. [20]

    Joshi, A

    N. Joshi, A. Kitaev, P. Treharne: On the Linearization of the Painlev´ e III-VI Equations and Reductions of the Three- Wave Resonant System, Journal of Mathematical Physics 48, 10 (2007)

    2007

  14. [21]

    Joshi, A

    N. Joshi, A. Kitaev, P. Treharne: On the Linearization of the First and Second Painlev´ e Equations,Journal of Physics A: Mathematical and Theoretical 42, 5 (2009) 42

    2009

  15. [22]

    Katzarkov, M

    L. Katzarkov, M. Kontsevich, T. Pantev: Hodge theo- retic aspects of mirror symmetry, in: From Hodge theory to in- tegrability and TQFT tt*-geometry, Proc. Sympos. Pure Math. 78, American Mathematical Society, (2008) 87–174

    2008

  16. [23]

    Keane, Sz

    C. Keane, Sz. Szab´o: Normal forms of endomorphism-valued power series, Involve: a Journal of Mathematics 11, 1 (2018), 81–94

    2018

  17. [24]

    Maciocia: Metrics on the moduli spaces of instantons over Euclidean 4-space, Commun

    A. Maciocia: Metrics on the moduli spaces of instantons over Euclidean 4-space, Commun. Math. Phys. 135, 3 (1991), 467– 482

    1991

  18. [25]

    Markman: Spectral curves and integrable systems,Compos

    E. Markman: Spectral curves and integrable systems,Compos. Math. 93, 3 (1994), 255–290

    1994

  19. [26]

    Mihajlovi´c: Elliptic Fibrations in 4-Dimensional Topology and Geometry, PhD thesis, Central European University (2022)

    S. Mihajlovi´c: Elliptic Fibrations in 4-Dimensional Topology and Geometry, PhD thesis, Central European University (2022)

    2022

  20. [27]

    Mochizuki: Wild harmonic bundles and wild pure twistor D-modules, Ast´ erisque340, Soci´ et´ e Math´ ematique de France (2011)

    T. Mochizuki: Wild harmonic bundles and wild pure twistor D-modules, Ast´ erisque340, Soci´ et´ e Math´ ematique de France (2011)

    2011

  21. [28]

    Mukai: Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent

    S. Mukai: Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77, (1984), 101–116

    1984

  22. [29]

    Mumford, T

    D. Mumford, T. Oda: Algebraic Geometry II, 73 Texts and Readings in Mathematics, Hindustan Book Agency (2015)

    2015

  23. [30]

    Nakajima: Monopoles and Nahm’s equations, in: Ein- stein metrics and Yang-Mills connections (Sanda, 1990) 145, Lecture Notes in Pure and Appl

    H. Nakajima: Monopoles and Nahm’s equations, in: Ein- stein metrics and Yang-Mills connections (Sanda, 1990) 145, Lecture Notes in Pure and Appl. Math., Dekker, (1993) 193– 211

    1990

  24. [31]

    Szab ´o: Nahm transform for integrable connections on the Riemann sphere, M´ emoires de la Soci´ et´ e Math´ ematique de France 110, (2007)

    Sz. Szab ´o: Nahm transform for integrable connections on the Riemann sphere, M´ emoires de la Soci´ et´ e Math´ ematique de France 110, (2007)

    2007

  25. [32]

    Szab´o: The Plancherel theorem for Fourier–Laplace–Nahm transform for connections on the projective line, Commun

    Sz. Szab´o: The Plancherel theorem for Fourier–Laplace–Nahm transform for connections on the projective line, Commun. Math. Phys. 338, 2 (2015), 753–769. 43

    2015

  26. [33]

    Szab ´o: The birational geometry of unramified irregular Higgs bundles on curves, International Journal of Mathematics 28, 6 (2017)

    Sz. Szab ´o: The birational geometry of unramified irregular Higgs bundles on curves, International Journal of Mathematics 28, 6 (2017)

    2017

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.