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Constructing stable Hilbert bundles via Diophantine approximation

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Pith's one-line read For every compact Riemann surface of positive genus and every irrational θ, a sequence of stable bundles whose slopes are the even convergents of θ completes to an indecomposable holomorphic Hilbert bundle with a Hermitian–Einstein metric.

desk verdict The analytic construction is substantial and mostly convincing, but the proof of indecomposability in Section 8 has a real gap. read the letter →

arxiv 2501.15784 v3 pith:TUUXLU73 submitted 2025-01-27 math.DG math-phmath.AGmath.MP

Yucheng Liu , Biao Ma This is my paper · ORCID
classification math.DGmath-phmath.AGmath.MP MSC 14H6053C0781T13
keywords HilbertbundlesHermitian–EinsteinmetricsslopestabilityDiophantineapproximationcontinuedfractionsRiemannsurfacesDonaldsonfunctionalsecondfundamentalform
topics Yang-Mills Mass Gap
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

On any compact Riemann surface of genus at least one, this paper builds infinite-rank holomorphic Hilbert bundles that carry Hermitian–Einstein metrics. The construction starts from a sequence of finite-rank stable bundles whose slopes are the even convergents of an irrational number θ, and takes a colimit in a controlled way. The paper proves that this colimit can be fiberwise completed to a separable Hilbert bundle whose limiting metric is still Hermitian–Einstein and whose holomorphic structure is indecomposable. This matters because it extends the Kobayashi–Hitchin correspondence from finite-rank bundles to a natural class of infinite-rank objects, and it gives an analytic mechanism, Diophantine approximation feeding into Donaldson's functional, by which irrational slopes are realized geometrically.

What carries the argument

The engine is the notion of a well-approximating sequence of stable bundles, where each finite-rank member is a good approximation of the colimit in the sense that every proper subsheaf sees a slope gap that later members only widen. This converts Diophantine approximation, the even convergents of θ, into a hypothesis on slope gaps. On the analytic side the proof uses three rank-uniform estimates: a Weitzenböck estimate controlling the second fundamental form of a Hermitian–Einstein subbundle pointwise by its L² norm; convexity and lower-semicontinuity of Donaldson's functional to force convergence of restricted Hermitian–Einstein metrics; and Uhlenbeck's gauge-fixing lemma with constants independent of rank, followed by the Koszul–Malgrange theorem to put a holomorphic structure on the completed bundle.

What would settle it

Compute, for the sequence E0→E1→... on an elliptic curve whose slopes are the even convergents of θ, the second fundamental forms β_i of E0 inside Ei taken with respect to the Hermitian–Einstein metrics. If sup_X |β_i|² exceeds C‖β_i‖²_{L²} for a constant C independent of i, then the Weitzenböck estimate and the convergence argument built on it fail; this is a concrete rank-uniformity check that settles the load-bearing estimate.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: on any compact Riemann surface X with g(X) > 0, for any irrational θ, take a colimit E∞ of a sequence of slope-stable bundles whose slopes are the even convergents of θ, with injective holomorphic bundle maps. Then E∞, after completion with respect to a limiting Hermitian metric, becomes an indecomposable holomorphic separable Hilbert bundle (E, H∞) admitting a Hermitian–Einstein metric. The proof actually establishes the stronger Theorem 1.7: any well-approximating sequence of stable bundles with limiting slope θ admits such a completion. The paper also shows that the rational analogue fails, so irrationality of the slope is essential to the construction.

Load-bearing premise

The load-bearing premise is a rank-uniform pointwise bound on the second fundamental form of a Hermitian–Einstein subbundle: on a curve, sup_X |β|² ≤ C‖β‖²_{L²} with C depending only on X, which fails to generalize because in higher dimensions the full curvature tensor enters and no a priori bound is available.

Editorial extensions

If this is right

  • A Kobayashi–Hitchin correspondence holds for these infinite-rank bundles: the limiting Hermitian–Einstein metric is a limit of finite-rank Hermitian–Einstein metrics, and each Ek sits inside E as a holomorphic subbundle.
  • Every irrational θ yields an indecomposable, projectively flat holomorphic Hilbert bundle over X, so its monodromy gives an irreducible projective unitary representation of the fundamental group of X on a separable Hilbert space.
  • Rational slopes are excluded: the analogous colimit for rational θ cannot be completed to an indecomposable Hermitian–Einstein Hilbert bundle, so irrationality of the slope is essential.
  • The inverse-limit version of the construction produces Hermitian–Einstein Hilbert bundles together with holomorphic surjections onto the finite-rank bundles, setting up rigged-Hilbert-bundle structures in the sense of Gel'fand triples.
  • For complex elliptic curves the same initial data yields noncommutative tori by three routes, categorical, monodromic, and mirror-symmetric, and the paper expects these routes to give Morita-equivalent noncommutative tori.

Reading between the lines

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

  • Beyond the paper: the construction suggests defining a stability condition directly on infinite-rank bundles by declaring θ-stability to be the existence of a well-approximating finite-rank filtration; if that notion is well behaved, the Hilbert bundle may be unique up to isomorphism independent of the subsequence choices.
  • Beyond the paper: on higher-dimensional Kähler manifolds the same metric-limiting argument should go through for sequences of Hermitian–Einstein bundles with a uniform total curvature bound; testing this on projective surfaces with bounded Chern classes would separate the curve-specific estimate from the general mechanism.
  • Beyond the paper: the Hilbert space of holomorphic sections defined in Section 9.3 can be studied as a quantization of X with θ as a parameter, and its dimension asymptotics along convergents are computable, giving numerical evidence for a semiclassical limit.
  • Beyond the paper: for elliptic curves, comparing the C*-algebras generated by the monodromy unitaries arising from different well-approximating sequences would directly test the expected Morita equivalence of the resulting noncommutative tori.
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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

2 major / 4 minor

Summary. This paper constructs holomorphic Hilbert bundles admitting Hermitian–Einstein metrics over any compact Riemann surface of genus at least one, for any irrational slope θ. The construction starts from a sequence of slope-stable finite-rank bundles whose slopes are the even convergents of θ, following [DJL24]. The authors introduce 'well-approximating sequences' and use Diophantine approximation to bound the Donaldson functional along the sequence, obtaining uniform L∞ estimates for the restricted Hermitian–Einstein metrics. They then form the smooth colimit bundle, complete the fibers with respect to the limiting metric, and use Uhlenbeck's gauge-fixing theorem and the Koszul–Malgrange theorem to give the completion a Hilbert-bundle structure with a Hermitian–Einstein connection. The main theorems (1.2 and 1.7) assert that the resulting holomorphic Hilbert bundle is indecomposable and projectively flat. The paper also discusses inverse limits, rigged Hilbert bundles, noncommutative tori, and a Hilbert space of holomorphic states.

Significance. If correct, the construction would be the first analytic proof of existence of indecomposable holomorphic Hilbert bundles with Hermitian–Einstein metrics over projective curves of positive genus, realizing irrational slopes in infinite rank and providing a bridge between Diophantine approximation, stability conditions, and noncommutative tori. The rank-independent estimates for second fundamental forms and the use of Donaldson's functional in a sequence setting are genuinely new and potentially useful beyond this paper. The paper is also commendable for clearly separating external inputs ([DJL24] for the bundle sequence, [JMS22] for Donaldson-functional properties) from the new analytic machinery (Sections 3–7, Appendix A). However, the two correctness issues detailed below prevent me from endorsing the main theorems in their current form.

major comments (2)
  1. [§3, Eq. (3.16) and Proposition 3.9] Equation (3.16) is not solvable as written. Taking the trace of the diagonal block in (3.4) gives tr(√−1Λβ†∧β) = −|β|²_H and, by Lemma 3.1, ∫_X |β|²_H = 2π(µ(E)−µ(S)) rkS. Hence the average of the right-hand side of (3.16) over X equals (µ(E)−µ(S)) − 2π(µ(E)−µ(S)) = (1−2π)(µ(E)−µ(S)), which is nonzero for µ(E) ≠ µ(S). Therefore no smooth φ can satisfy (3.16), and Proposition 3.9 does not provide the conformal factor used to make det(e^{φ_i}h_i) ≡ 1 in the proof of Theorem 4.1. The correct PDE has opposite signs in the µ-terms and the β-term (e.g., Δφ = 2π(µ(S)−µ(E)) + |β|²/rkS under the 2π-convention used in (3.4)); this sign error is load-bearing for the main L∞ estimate, and the inconsistency of the factor 2π between Definition 4.4 and Lemma 3.1 must also be resolved.
  2. [§8, proof of Theorem 8.3, final paragraph] The indecomposability proof contains an unjustified assertion: for a holomorphic subbundle F ⊂ E1⊕E2, it is claimed that F ≃ π1F ⊕ π2F, so that stability of F forces one projection to vanish. This isomorphism is false: the diagonal embedding of a stable bundle F into F⊕F has both projections nonzero and isomorphic to F, while π1F⊕π2F ≅ F⊕F. More generally, the graph of a nonzero holomorphic map f:E1→E2 has rank rkE1 but its two projections need not span it as a direct sum. Consequently the dichotomy 'one projection vanishes' is not a consequence of stability, and the downward-closed argument leading to a contradiction has no valid premise. Since indecomposability is explicitly claimed in Theorems 1.2 and 1.7, this gap must be repaired (or the claim weakened) before the main theorem can be accepted.
minor comments (4)
  1. [Abstract and Section 1] The abstract contains the typo 'homological countparts' for 'counterparts', and Section 1 misspells 'Narasimhan–Seshadri' as 'Narasimhan–Seshardi'.
  2. [§4, Definition 4.4] Donaldson's functional is defined using 'µ(S)Id' without the factor 2π that appears in the Hermitian–Einstein equation (3.4); the constants in Lemma 4.9 suggest a systematic missing 2π in the definition, which should be stated consistently throughout.
  3. [§5, Theorem 5.1 proof] The induction in Theorem 5.1 has an off-by-one issue: the notation d_{m,k} and the diagram use indices inconsistently (e.g., b_m = d_{m,m−1} while the first displayed line involves d_{1,0}); the argument is salvageable by relabeling, but the indices should be cleaned up.
  4. [§9.3] In the sentence 'Riemann–Roch thoerem', 'thoerem' should read 'theorem'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central analytic construction is self-contained, and the overlapping-author citations supply external categorical inputs rather than restatements of the target theorem.

full rationale

The paper's main claim, Theorem 1.2, is that a colimit of a well-approximating sequence of stable bundles admits a Hermitian-Einstein Hilbert bundle metric as a limit of finite-rank Hermitian-Einstein metrics. The derivation chain is: (i) existence of the stable-bundle sequence is obtained from the arithmetic-stability framework (Proposition 2.17, proved in the paper, with parts of the categorical Proposition 2.12 cited from [DJL24], one of whose authors is Liu); (ii) the analytic heart, Sections 3-7, proves uniform bounds for second fundamental forms and for restricted metrics using Donaldson's functional, whose properties are cited from the independent works [Don85], [Don87], and [JMS22]; (iii) the smooth and holomorphic Hilbert bundle structures in Sections 6 and 8 come from the paper's own colimit construction, Uhlenbeck gauge fixing, and elliptic regularity. The self-citations [DJL24], [Liu24], and [Liu25] supply categorical facts about stability conditions and colimits; none of them asserts the target Hermitian-Einstein Hilbert bundle result, and none of them is used to forbid alternatives or to define the conclusion into existence. The good-approximation condition (Equation 1.5) is not defined in terms of the Hermitian-Einstein conclusion; it is a Diophantine-slope condition, and the final Einstein constant theta is the limit of the input slopes, derived from curvature convergence (Theorem 7.2), not imposed as a fitted parameter. The paper itself notes that the construction depends on subsequence choices and leaves uniqueness open, which is a limitation but not a circularity. A possible gap in the indecomposability argument in Section 8 (the asserted splitting Ei ≃ π1ϕfi(Ei) ⊕ π2ϕfi(Ei) is not generally valid for holomorphic subbundles of a direct sum) is a correctness issue, not a circular reduction; no equation of the paper sets a conclusion equal to an input by construction. Therefore the derivation is not circular.

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

The central claim rests on standard theorems (Donaldson-Uhlenbeck-Yau, Uhlenbeck gauge fixing, Koszul-Malgrange) and on the prior construction of the algebraic colimit E∞ from [DJL24]. The paper introduces new definitions (well-approximating sequence, good approximation) but these are not fitted parameters or postulated entities. No new particles, forces, or ad hoc constants are introduced; the metric H∞ is constructed as a limit of existing Hermitian-Einstein metrics.

assumptions (6)
  • standard math Donaldson-Uhlenbeck-Yau theorem: slope poly-stable finite-rank bundles over compact Kähler manifolds admit Hermitian-Einstein metrics
    Theorem 1.5, used throughout to guarantee Hermitian-Einstein metrics H_i on E_i.
  • standard math Existence of slope stable bundles with prescribed coprime rank and degree, due to Atiyah and Narasimhan-Seshadri
    Used in Proposition 2.17 to establish arithmetic stability conditions.
  • domain assumption The colimit E∞ is a locally free sheaf of infinite rank (Drinfeld vector bundle) with the stated Hom and limit properties, from [DJL24]
    Used to define E∞ and to identify its sections; the present paper does not reprove this.
  • standard math Properties of Donaldson's functional and geodesic rays from [JMS22], including lower semicontinuity under weak W^{1,2} convergence and the expansion in Theorem 3.7
    Used in Lemmas 4.9 and 4.10 and in the proof of Theorem 4.1.
  • standard math Uhlenbeck's gauge fixing theorem with constants independent of rank, proved in Appendix A for Hilbert bundles
    Used in Lemma 8.7 to produce uniform unitary frames.
  • standard math Koszul-Malgrange theorem for Banach and Hilbert bundles
    Used in Theorem 8.3 to pass from a smooth bundle with integrable (0,1)-connection to a holomorphic structure.

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Pith. "Pith review of Constructing stable Hilbert bundles via Diophantine approximation." pith.science (2026). https://pith.science/paper/TUUXLU73

@misc{pith2026250115784,
  author       = {Pith},
  title        = {Pith review of: Constructing stable Hilbert bundles via Diophantine approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TUUXLU73}},
  note         = {Machine review of arXiv:2501.15784}
}
abstract

On any complex smooth projective curve with positive genus, we construct Hilbert bundles that admit Hermitian--Einstein metrics. Our main constructive step is by investigating the arithmetic property of the upper half plane in Bridgeland's definition of stability conditions and its homological countparts. The main analytic ingredient in our proof is a notion called a well-approximating sequence of stable bundles. This notion helps us to apply the Diophantine approximation to Donaldson's functional and bound the $L^\infty$ norm of Hermitian-Einstein metrics. We further study the continuous structures, smooth structures, and holomorphic structures on such Hilbert bundles. We hope that this construction can shed some new light on the geometric background of quantum field theory.

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Works this paper leans on

51 extracted references · 46 canonical work pages

  1. [1]

    Atiyah and Raoul Bott

    Michael F. Atiyah and Raoul Bott. The Yang - Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond., Ser. A , 308:523--615, 1983

    work page 1983

  2. [2]

    Geometric quantization of Chern - Simons gauge theory

    Scott Axelrod, Steve Della, Pietra, and Edward Witten. Geometric quantization of Chern - Simons gauge theory. J. Differ. Geom. , 33(3):787--902, 1991

    work page 1991

  3. [3]

    Mapping theorems for sobolev spaces of vector-valued functions

    Wolfgang Arendt and Marcel Kreuter. Mapping theorems for sobolev spaces of vector-valued functions. Studia Mathematica , 240:275--299, 2016

    work page 2016

  4. [4]

    M. F. Atiyah. Vector bundles over an elliptic curve. Proc. London Math. Soc. (3) , 7:414--452, 1957

    1957

  5. [5]

    Bargmann

    V. Bargmann. On unitary ray representations of continuous groups. Ann. Math. (2) , 59:1--46, 1954

    work page 1954

  6. [6]

    H. Bass. Big projective modules are free. Ill. J. Math. , 7:24--31, 1963

    work page 1963

  7. [7]

    Raoul Bott and S. S. Chern. Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math. , 114(1):71--112, 1965

    work page 1965

  8. [8]

    Curvature of vector bundles associated to holomorphic fibrations

    Bo Berndtsson. Curvature of vector bundles associated to holomorphic fibrations. Ann. Math. (2) , 169(2):531--560, 2009

    work page 2009

Show all 51 references
  1. [9]

    Strict and nonstrict positivity of direct image bundles

    Bo Berndtsson. Strict and nonstrict positivity of direct image bundles. Math. Z. , 269(3-4):1201--1218, 2011

    2011

  2. [10]

    Bierstedt

    Klaus D. Bierstedt. An introduction to locally convex inductive limits. In Functional analysis and its applications ( N ice, 1986) , ICPAM Lecture Notes, pages 35--133. World Sci. Publishing, Singapore, 1988

    1986

  3. [11]

    A proof of the Ohsawa - Takegoshi theorem with sharp estimates

    Bo Berndtsson and L \'a szl \'o Lempert. A proof of the Ohsawa - Takegoshi theorem with sharp estimates. J. Math. Soc. Japan , 68(4):1461--1472, 2016

    2016

  4. [12]

    Stability conditions on triangulated categories

    Tom Bridgeland. Stability conditions on triangulated categories. Ann. of Math. (2) , 166(2):317--345, 2007

    2007

  5. [13]

    Th \'e orie de la dualit \'e et analyse harmonique dans les groupes abeliens localement compacts

    Henri Cartan and Roger Godement. Th \'e orie de la dualit \'e et analyse harmonique dans les groupes abeliens localement compacts. Ann. Sci. \'E c. Norm. Sup \'e r. (3) , 64:79--99, 1947

    1947

  6. [14]

    Noncommutative geometry

    Alain Connes. Noncommutative geometry . Springer, 1994

    1994

  7. [15]

    Bergman bundles and applications to the geometry of compact complex manifolds

    Jean-Pierre Demailly. Bergman bundles and applications to the geometry of compact complex manifolds. arXiv preprint arXiv:2003.04003 , 2020

    2003 arXiv

  8. [16]

    Continuum envelopes on Fargues--Fontaine curves and elliptic curves

    Heng Du, Qingyuan Jiang, and Yucheng Liu. Continuum envelopes on Fargues--Fontaine curves and elliptic curves. arXiv preprint arXiv:2404.04551 , 2024

    2024 arXiv

  9. [17]

    S. K. Donaldson. A new proof of a theorem of Narasimhan and Seshadri . J. Differ. Geom. , 18:269--277, 1983

    1983

  10. [18]

    S. K. Donaldson. Anti self-dual Yang Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. (3) , 50:1--26, 1985

    1985

  11. [19]

    S. K. Donaldson. Infinite determinants, stable bundles and curvature. Duke Math. J. , 54:231--247, 1987

    1987

  12. [20]

    S. K. Donaldson. Boundary value problems for Yang - Mills fields. J. Geom. Phys. , 8(1-4):89--122, 1992

    1992

  13. [21]

    Michael R. Douglas. Dirichlet branes, homological mirror symmetry, and stability. In Proceedings of the I nternational C ongress of M athematicians, V ol. III ( B eijing, 2002) , pages 395--408. Higher Ed. Press, Beijing, 2002

    2002

  14. [22]

    Infinite-dimensional vector bundles in algebraic geometry: an introduction

    Vladimir Drinfeld. Infinite-dimensional vector bundles in algebraic geometry: an introduction. In The unity of mathematics. In honor of the ninetieth birthday of I. M. Gelfand. Papers from the conference held in Cambridge, MA, USA, August 31--September 4, 2003. , pages 263--30...

    2003

  15. [23]

    Courbes et fibr \'e s vectoriels en th \'e orie de Hodge \(p\) -adique , volume 406 of Ast \'e risque

    Laurent Fargues and Jean-Marc Fontaine. Courbes et fibr \'e s vectoriels en th \'e orie de Hodge \(p\) -adique , volume 406 of Ast \'e risque . Paris: Soci \'e t \'e Math \'e matique de France (SMF), 2018

    2018

  16. [24]

    Trudinger

    David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order , volume 224 of Grundlehren Math. Wiss. Springer, Cham, 1977

    1977

  17. [25]

    A solution of an \(L^ 2 \) extension problem with an optimal estimate and applications

    Qi'an Guan and Xiangyu Zhou. A solution of an \(L^ 2 \) extension problem with an optimal estimate and applications. Ann. Math. (2) , 181(3):1139--1208, 2015

    2015

  18. [26]

    Nigel J. Hitchin. Flat connections and geometric quantization. Commun. Math. Phys. , 131(2):347--380, 1990

    1990

  19. [27]

    Geodesic rays in the Donaldson--Uhlenbeck--Yau theorem

    Mattias Jonsson, Nicholas McCleerey, and Sanal Shivaprasad. Geodesic rays in the Donaldson--Uhlenbeck--Yau theorem. arXiv preprint arXiv:2210.09246 , 2022

    2022 arXiv

  20. [28]

    J. L. Koszul and B. Malgrange. Sur certaines structures fibr \'e es complexes. Archiv der Mathematik , 9(1):102--109, 1958

    1958

  21. [29]

    The convenient setting of global analysis , volume 53

    Andreas Kriegl and Peter W Michor. The convenient setting of global analysis , volume 53. American Mathematical Soc., 1997

    1997

  22. [30]

    Stability structures, motivic D onaldson- T homas invariants and cluster transformations

    Maxim Kontsevich and Yan Soibelman. Stability structures, motivic D onaldson- T homas invariants and cluster transformations. arXiv preprint arXiv:0811.2435 , 2008

    2008 arXiv

  23. [31]

    Nicolaas H. Kuiper. The homotopy type of the unitary group of hilbert space. Topology , 3(1):19--30, 1965

    1965

  24. [32]

    Introduction to diophantine approximations

    Serge Lang. Introduction to diophantine approximations. Addison- Wesley Series in Mathematics . Reading , Mass . etc.: Addison - Wesley Publishing Company . VIII , 83 p. (1966)., 1966

    1966

  25. [33]

    Filtrations and torsion pairs in A bramovich- P olishchuk's heart

    Yucheng Liu. Filtrations and torsion pairs in A bramovich- P olishchuk's heart. Peking Math. Journal, published online , April, 2024

    2024

  26. [34]

    Different hearts on elliptic curves

    Yucheng Liu. Different hearts on elliptic curves. Acta Mathematica Sinica, English Series, published online , January, 2025

    2025

  27. [35]

    Direct images, fields of Hilbert spaces, and geometric quantization

    L \'a szl \'o Lempert and R \'o bert Szoke. Direct images, fields of Hilbert spaces, and geometric quantization. Commun. Math. Phys. , 327(1):49--99, 2014

    2014

  28. [36]

    Malgrange and R

    B. Malgrange and R. Narasimhan. Lectures on the Theory of Functions of Several Complex Variables . Number Nr. 13 in Lectures on mathematics and physics: Mathematics. Tata Institute of Fundamental Research, 1958

    1958

  29. [37]

    A partial regularity result for a class of stationary Yang - Mills fields in high dimension

    Yves Meyer and Tristan Rivi \`e re. A partial regularity result for a class of stationary Yang - Mills fields in high dimension. Rev. Mat. Iberoam. , 19(1):195--219, 2003

    2003

  30. [38]

    J. P. May and J. Sigurdsson. Parametrized homotopy theory , volume 132 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2006

    2006

  31. [39]

    D. Mumford. Projective invariants of projective structures and applications. Proc. Int . Congr . Math . 1962, 526-530 (1963)., 1963

    1963

  32. [40]

    M. S. Narasimhan and C. S. Seshadri. Stable and unitary vector bundles on a compact Riemann surface. Ann. Math. (2) , 82:540--567, 1965

    1965

  33. [41]

    Polishchuk

    A. Polishchuk. Classification of holomorphic vector bundles on noncommutative two-tori. Doc. Math. , 9:163--181, 2004

    2004

  34. [42]

    Topology of infinite-dimensional manifolds

    Katsuro Sakai. Topology of infinite-dimensional manifolds . Springer Monogr. Math. Cham: Springer, 2020

    2020

  35. [43]

    The unitary group in its strong topology

    Martin Schottenloher. The unitary group in its strong topology. Advances in Pure Mathematics , 8(05):508, 2018

    2018

  36. [44]

    Carlos T. Simpson. Constructing variations of Hodge structure using Yang - Mills theory and applications to uniformization. J. Am. Math. Soc. , 1(4):867--918, 1988

    1988

  37. [45]

    H. H. Schaefer and M. P. Wolff. Topological vector spaces , volume 3 of Graduate Texts in Mathematics . Springer-Verlag, New York, second edition, 1999

    1999

  38. [46]

    A Singularity removal theorem for Yang-Mills fields in higher dimensions

    Terence Tao and Gang Tian. A Singularity removal theorem for Yang-Mills fields in higher dimensions . J. Amer. Math. Soc , 17(3):557--593, 2004

    2004

  39. [47]

    Uhlenbeck

    Karen K. Uhlenbeck. Connections with L^ p bounds on curvature . Communications in Mathematical Physics , 83(1):31 -- 42, 1982

    1982

  40. [48]

    Uhlenbeck and S

    K. Uhlenbeck and S. T. Yau. On the existence of Hermitian - Yang - Mills connections in stable vector bundles. Commun. Pure Appl. Math. , 39:257--293, 1986

    1986

  41. [49]

    Uhlenbeck and S

    K. Uhlenbeck and S. T. Yau. A note on our previous paper: On the existence of Hermitian Yang - Mills connections in stable vector bundles. Commun. Pure Appl. Math. , 42(5):703--707, 1989

    1989

  42. [50]

    A curvature formula associated to a family of pseudoconvex domains

    Xu Wang. A curvature formula associated to a family of pseudoconvex domains. Ann. Inst. Fourier , 67(1):269--313, 2017

    2017

  43. [51]

    A Dirichlet Problem in Noncommutative Potential Theory

    Kuang-Ru Wu. A Dirichlet Problem in Noncommutative Potential Theory . Michigan Mathematical Journal , 69(2):369 -- 380, 2020

    2020

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