Pith. sign in

REVIEW 1 major objections 1 minor 1 cited by

The prescribed Hermitian-Yang-Mills flow II

T0 review · 1 major / 1 minor · reviewed 2026-06-26 · grok-4.3

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

Pith's one-line read If a holomorphic vector bundle on a compact Kähler manifold is strictly slope stable, the prescribed Hermitian-Yang-Mills flow exists globally and converges to a metric satisfying the curvature prescription.

desk verdict The prescribed flow extends DUY under stability with a new P term, but the Fano tangent bundle application does not follow from the theorem. read the letter →

arxiv 2606.21073 v1 pith:KKHVHPNJ submitted 2026-06-19 math.DG

Zhiyao Xiong , Xiaokui Yang , Shing-Tung Yau This is my paper · ORCID
classification math.DG
keywords Hermitian-Yang-MillsflowslopestabilityholomorphicvectorbundleKählermanifoldprescribedcurvatureDonaldson-Uhlenbeck-YautheoremFano
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that when every proper coherent subsheaf F of a holomorphic vector bundle E satisfies deg(F) strictly less than deg(E), the evolution equation that adjusts the Hermitian metric h according to minus the contraction of its curvature plus a given positive definite tensor P has a smooth solution for all time. The solution converges as time goes to infinity to a limiting metric whose curvature exactly equals the prescribed tensor. This extends the classical Donaldson-Uhlenbeck-Yau theorem to the inhomogeneous case and yields, as a corollary, the existence of a Hermitian metric on the tangent bundle of any Fano manifold whose curvature satisfies the analogous prescription for any background Kähler form and any positive definite right-hand side.

What carries the argument

The prescribed Hermitian-Yang-Mills flow, a parabolic evolution equation for the Hermitian metric whose long-time behavior is controlled by the slope-stability hypothesis.

What would settle it

A strictly slope stable bundle on a compact Kähler manifold together with some positive definite P for which the flow either develops a singularity in finite time or its limit fails to satisfy the curvature equation.

Watch

Extended reading notes

Core claim

Suppose that for every proper coherent subsheaf F⊂E, deg_ωg(F)<deg_ωg(E). Then for any initial Hermitian metric h0 on E and any positive-definite Hermitian tensor P, the flow ∂h/∂t = −Λ_ωg(√−1 R^h) + P admits a global smooth solution on [0,∞) that converges smoothly to a Hermitian metric h∞ on E satisfying Λ_ωg(√−1 R^{h∞}) = P.

Load-bearing premise

The holomorphic vector bundle must satisfy the strict slope stability condition that every proper subsheaf has strictly smaller degree than the bundle.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 1 minor

Summary. The paper proves that if a holomorphic vector bundle E on a compact Kähler manifold (M, ω_g) is strictly slope stable (deg_ωg(F) < deg_ωg(E) for every proper coherent subsheaf F), then the prescribed Hermitian-Yang-Mills flow ∂h/∂t = −Λ_ωg(√−1 R^h) + P admits a global smooth solution on [0, ∞) converging smoothly to a metric h_∞ satisfying Λ_ωg(√−1 R^{h_∞}) = P, for any initial h_0 and positive-definite P. As an application, it claims that on any Fano manifold M, for any Hermitian metric form ω and positive-definite P, there exists a unique Hermitian metric h on T^{1,0}M satisfying Λ_ω(√ R^h) = P.

Significance. If the main theorem holds, the result supplies a parabolic proof of an analogue of the Donaldson-Uhlenbeck-Yau theorem for the prescribed equation and would constitute a nontrivial extension of existing flow techniques. The claimed application to the tangent bundle on arbitrary Fano manifolds would, if justified, give a Calabi-Yau-type existence result without a stability hypothesis, but this extension is not supported by the stated theorem.

major comments (1)
  1. [Abstract] Abstract (application paragraph): the existence/uniqueness statement for a metric h on T^{1,0}M satisfying Λ_ω(√ R^h) = P on an arbitrary Fano manifold is asserted without the strict slope-stability hypothesis required by the main theorem and without a separate argument that T^{1,0}M is always strictly slope stable with respect to an arbitrary Kähler form ω. This hypothesis is known to fail in general (e.g., the tangent bundle of P^1 × P^1 with product metric admits a subsheaf of equal slope). The application therefore does not follow from the theorem as stated and is load-bearing for the paper’s final claim.
minor comments (1)
  1. [Abstract] Abstract, application equation: the displayed equation uses √ R^h rather than the √−1 R^h appearing in the theorem statement and flow equation; this appears to be a typographical inconsistency.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the inconsistency between the main theorem and the application claimed in the abstract. We address the comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (application paragraph): the existence/uniqueness statement for a metric h on T^{1,0}M satisfying Λ_ω(√ R^h) = P on an arbitrary Fano manifold is asserted without the strict slope-stability hypothesis required by the main theorem and without a separate argument that T^{1,0}M is always strictly slope stable with respect to an arbitrary Kähler form ω. This hypothesis is known to fail in general (e.g., the tangent bundle of P^1 × P^1 with product metric admits a subsheaf of equal slope). The application therefore does not follow from the theorem as stated and is load-bearing for the paper’s final claim.

    Authors: We agree that the application paragraph in the abstract asserts an existence/uniqueness result for the tangent bundle on arbitrary Fano manifolds that does not follow from the main theorem, as the required strict slope stability need not hold (as illustrated by the referee's example). No separate argument for stability is given in the manuscript. We will therefore revise the abstract by removing the application paragraph. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; theorem conditional on external stability hypothesis

full rationale

The paper states its main result as a convergence theorem for the prescribed HYM flow that explicitly requires the external hypothesis of strict slope stability (deg(F) < deg(E) for proper subsheaves F). This hypothesis is not derived from the flow equation or from any fitted quantity inside the paper; it is an input assumption. The application to the tangent bundle on Fano manifolds is asserted without repeating the hypothesis, but the text contains no self-definitional loop, no parameter fitted to data then relabeled as prediction, and no load-bearing self-citation chain that reduces the claimed result to its own inputs. The derivation therefore remains non-circular by the stated criteria.

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

The paper relies on standard background in Kähler geometry and holomorphic vector bundles; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

assumptions (2)
  • domain assumption M is a compact Kähler manifold with Kähler form ω_g
    Stated in the setup of the theorem.
  • domain assumption E is a holomorphic vector bundle over M
    Stated in the setup of the theorem.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The prescribed Hermitian-Yang-Mills flow II." pith.science (2026). https://pith.science/paper/KKHVHPNJ

@misc{pith2026260621073,
  author       = {Pith},
  title        = {Pith review of: The prescribed Hermitian-Yang-Mills flow II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KKHVHPNJ}},
  note         = {Machine review of arXiv:2606.21073}
}
abstract

We prove an analogue of the classical Donaldson-Uhlenbeck-Yau theorem by using the prescribed Hermitian-Yang-Mills flow. Let $E$ be a holomorphic vector bundle over a compact K\"ahler manifold $(M,\omega_g)$. Suppose that for every proper coherent subsheaf $F\subset E$, the following inequality holds: $$ deg_{\omega_g}(F)<deg_{\omega_g}(E). $$ Then, for any initial Hermitian metric $h_0$ on $E$ and any positive-definite Hermitian tensor $P\in \Gamma(M,E^*\otimes \overline E^*)$, the prescribed Hermitian-Yang-Mills flow $$ \ \frac{\partial h}{\partial t} = -\Lambda_{\omega_g}\left(\sqrt{-1}\, R^h\right) + P, $$ admits a global smooth solution on $[0,\infty)$. Moreover, as $t\rightarrow\infty$, the flow converges smoothly to a Hermitian metric $h_\infty$ on $E$ satisfying $$ \Lambda_{\omega_g}\left(\sqrt{-1}\, R^{h_\infty}\right) = P. $$ As an application, we establish that on a Fano manifold $M$, for any Hermitian metric form $\omega$ and any positive-definite Hermitian tensor $P\in\Gamma(M,T^{*1,0}M\otimes T^{*0,1}M)$, there exists a unique Hermitian metric tensor $h$ on $T^{1,0}M$ such that $$ \Lambda_\omega\left(\sqrt R^h\right)=P.$$ This may be viewed as an analogue of the Calabi-Yau theorem for Fano manifolds.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Iterative construction of Hermitian-Einstein metrics on stable bundles

    math.DG 2026-06 unverdicted novelty 6.0 of 10

    An explicit iteration on Hermitian metrics on a stable bundle over a compact Gauduchon manifold converges smoothly to the unique Hermitian-Einstein metric.

Reference graph

Works this paper leans on

51 extracted references · 4 canonical work pages · cited by 1 Pith paper

  1. [1]

    Curvature of vector bundles associated to holomorphic fibrations

    Berndtsson, B. Curvature of vector bundles associated to holomorphic fibrations. Ann. Math. (2) 304 ( 2009), 531--560

    2009

  2. [2]

    Bradlow, S. B. and Garc\'ia-Prada, O. Stable triples, equivariant bundles and dimensional reduction, Math. Ann. 304 (1996), no. 2, 225--252

    1996

  3. [3]

    Buchdahl, N. P. On compact K\"ahler surfaces, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 1, vii, xi, 287--302

    1999

  4. [4]

    On K\"ahler manifolds with vanishing canonical class, Algebraic geometry and topology Symposium in honor of S

    Calabi, E. On K\"ahler manifolds with vanishing canonical class, Algebraic geometry and topology Symposium in honor of S. Lefschetz, Princeton Univ. Press, 1957, 78--89

    1957

  5. [5]

    Rationally connected manifolds and semipositivity of the Ricci curvature

    Campana, F.; Demailly, J.-P.; Peternell, T. Rationally connected manifolds and semipositivity of the Ricci curvature. in Recent advances in Algebraic Geometry. LMS Lecture Notes Series 417. (2014), 71--91. 3380444

    2014

  6. [6]

    and Flenner, H

    Campana, C. and Flenner, H. A characterization of ample vector bundles on a curve, Math. Ann. 287 (1990), no. 4, 571--575

    1990

  7. [7]

    and Sun, S

    Chen, X.-M. and Sun, S. Singularities of Hermitian-Yang-Mills connections and Harder-Narasimhan-Seshadri filtrations, Duke Math. J. 169 (2020), no. 14, 2629--2695

    2020

  8. [8]

    and Wentworth, R

    Chen X.-M. and Wentworth, R. A. A Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles on degenerating families, Math. Ann. 388 (2024), no. 2, 1903--1935

    2024

Show all 51 references
  1. [9]

    Daskalopoulos, G. D. and Wentworth, R. A. Convergence properties of the Yang-Mills flow on K\"ahler surfaces, J. Reine Angew. Math. 575 (2004), 69--99

    2004

  2. [10]

    Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles

    Demailly, J.-P. Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles. Mat. Sb. 212 (2021), no. 3, 39-–53

    2021

  3. [11]

    Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc

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

    1985

  4. [12]

    Infinite determinants, stable bundles and curvature, Duke Math

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

    1987

  5. [13]

    and Sektnan, L

    Dervan, R. and Sektnan, L. M. Hermitian Yang-Mills connections on blowups, J. Geom. Anal. 31 (2021), no. 1, 516--542

    2021

  6. [14]

    and Yau, S.-T

    Fan, J.-X.; Wang, M.-W.; Yang, X.-K. and Yau, S.-T. Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors II. arXiv:2604.02679

    arXiv

  7. [15]

    Fu, Limiting behavior of a class of Hermitian Yang-Mills metrics, I, Sci

    J. Fu, Limiting behavior of a class of Hermitian Yang-Mills metrics, I, Sci. China Math. 62 (2019), no. 11, 2155--2194

    2019

  8. [16]

    La 1 -forme de torsion d'une vari\'et\'e hermitienne compacte, Math

    Gauduchon, P. La 1 -forme de torsion d'une vari\'et\'e hermitienne compacte, Math. Ann. 267 (1984), no. 4, 495--518

    1984

  9. [17]

    Ample vector bundles, Inst

    Hartshorne, R. Ample vector bundles, Inst. Hautes \'Etudes Sci. Publ. Math. No. 29 (1966), 63--94

    1966

  10. [18]

    The self-duality equations on a Riemann surface, Proc

    Hitchin, N.-J. The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59--126

    1987

  11. [19]

    and Walpuski, T

    Jacob, A.-J. and Walpuski, T. Hermitian Yang-Mills metrics on reflexive sheaves over asymptotically cylindrical K\"ahler manifolds, Comm. Partial Differential Equations 43 (2018), no. 11, 1566--1598

    2018

  12. [20]

    Existence of approximate Hermitian-Einstein structures on semi-stable bundles, Asian J

    Jacob, A.-J. Existence of approximate Hermitian-Einstein structures on semi-stable bundles, Asian J. Math. 18 (2014), no. 5, 859--883

    2014

  13. [21]

    Differential geometry of complex vector bundles , Publications of the Mathematical Society of Japan Kan\^o Memorial Lectures, 15 5, Princeton Univ

    Kobayashi, S. Differential geometry of complex vector bundles , Publications of the Mathematical Society of Japan Kan\^o Memorial Lectures, 15 5, Princeton Univ. Press, Princeton, NJ, 1987

    1987

  14. [22]

    Courants k\"ahl\'eriens et surfaces compactes, Ann

    Lamari, A. Courants k\"ahl\'eriens et surfaces compactes, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 1, vii, x, 263--285

    1999

  15. [23]

    and Zhang, X

    Li, C., Zhang, C.-J. and Zhang, X. Mean curvature positivity and rational connectedness, Adv. Math. 483 (2025), Paper No. 110673, 43 pp

    2025

  16. [24]

    and Yang, X.-K

    Li, D.; Ou, W.-H. and Yang, X.-K. On projective varieties with strictly nef tangent bundles, J. Math. Pures Appl. (9) 128 (2019), 140--151

    2019

  17. [25]

    Li, J. Y. Hermitian-Einstein metrics and Chern number inequalities on parabolic stable bundles over K\"ahler manifolds, Comm. Anal. Geom. 8 (2000), no. 3, 445--475

    2000

  18. [26]

    and Zhang, X

    Li, J.-Y. and Zhang, X. Existence of approximate Hermitian-Einstein structures on semi-stable Higgs bundles, Calc. Var. Partial Differential Equations 52 (2015), no. 3-4, 783--795

    2015

  19. [27]

    and Yau, S.-T

    Li, J. and Yau, S.-T. Hermitian-Yang-Mills connection on non-K\"ahler manifolds, in Mathematical aspects of string theory , 560--573, 1986

    1986

  20. [28]

    Bubbling phenomenon for Hermitian Yang-Mills connections, Int

    Li, Yang. Bubbling phenomenon for Hermitian Yang-Mills connections, Int. Math. Res. Not. IMRN 2021 , no. 6, 4657--4678

    2021

  21. [29]

    and Yang, X.-K

    Liu, J.; Ou, W.-H. and Yang, X.-K. Strictly nef vector bundles and characterizations of P ^n , Complex Manifolds 8 (2021), no. 1, 148--159

    2021

  22. [30]

    and Yang, X.-K

    Liu, J.; Ou, W.-H. and Yang, X.-K. Projective manifolds whose tangent bundle contains a strictly nef subsheaf, J. Algebraic Geom. 33 (2024), no. 1, 1--53

    2024

  23. [31]

    Positivity and vanishing theorems for ample vector bundles

    Liu, K.-F.; Sun, X.-F.; and Yang, X.-K. Positivity and vanishing theorems for ample vector bundles. J. Algebraic Geom. 22 (2013), 303--331

    2013

  24. [32]

    Picard, S

    McOrist, J. ; Picard, S. and Svanes, E. E. A heterotic Hermitian-Yang-Mills equivalence, Comm. Math. Phys. 406 (2025), no. 5, Paper No. 107, 33 pp

    2025

  25. [33]

    An analytic proof of Griffiths' conjecture on compact Riemann surfaces, Math

    Murakami, R. An analytic proof of Griffiths' conjecture on compact Riemann surfaces, Math. Ann. 394 (2026), no. 2, Paper No. 36, 14 pp

    2026

  26. [34]

    and Seshadri, C.-S

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

    1965

  27. [35]

    On generic nefness of tangent sheaves, Math

    Ou, W.-H. On generic nefness of tangent sheaves, Math. Z. 304 (2023), no. 4, Paper No. 58, 23 pp

    2023

  28. [36]

    Varieties with generically nef tangent bundles, J

    Peternell, T. Varieties with generically nef tangent bundles, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 2, 571--603

    2012

  29. [37]

    A note on Demailly's approach towards a conjecture of Griffiths, C

    Pingali, V.-P. A note on Demailly's approach towards a conjecture of Griffiths, C. R. Math. Acad. Sci. Paris 359 (2021), 501--503

    2021

  30. [38]

    Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J

    Simpson, C.-T. Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), no. 4, 867--918

    1988

  31. [39]

    Higgs bundles and local systems, Inst

    Simpson, C.-T. Higgs bundles and local systems, Inst. Hautes \'Etudes Sci. Publ. Math. No. 75 (1992), 5--95

    1992

  32. [40]

    ahler- E instein metrics , volume 8 of DMV Seminar . Birkh\

    Siu, Y.-T. Lectures on H ermitian- E instein metrics for stable bundles and K \"ahler- E instein metrics , volume 8 of DMV Seminar . Birkh\"auser Verlag, Basel, 1987

    1987

  33. [41]

    and Yau, S.-T

    Wang, M.-W.; Yang, X.-K. and Yau, S.-T. Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors I. arXiv:2603.10611

  34. [42]

    and Yau, S.-T

    Wang, M.-W.; Yang, X.-K. and Yau, S.-T. Existence of twisted Hermitian-Einstein metrics on unstable vector bundles. arXiv:2606.15102

  35. [43]

    A Wess-Zumino-Witten type equation in the space of K\"ahler potentials in terms of Hermitian-Yang-Mills metrics, Anal

    Wu, K.-R. A Wess-Zumino-Witten type equation in the space of K\"ahler potentials in terms of Hermitian-Yang-Mills metrics, Anal. PDE 16 (2023), no. 2, 341--366

    2023

  36. [44]

    and Yau, S.-T

    Xiong, Z.-Y.; Yang, X.-K. and Yau, S.-T. RC-positivity, Schwarz's lemma and comparison theorems. arXiv:2412.02553

  37. [45]

    and Yau, S.-T

    Xiong, Z.-Y.; Yang, X.-K. and Yau, S.-T. The prescribed Hermitian-Yang-Mills flow I. arXiv

  38. [46]

    RC-positivity, rational connectedness and Yau's conjecture, Camb

    Yang, X.-K. RC-positivity, rational connectedness and Yau's conjecture, Camb. J. Math. 6 (2018), no. 2, 183--212

    2018

  39. [47]

    Scalar curvature on compact complex manifolds, Trans

    Yang, X.-K. Scalar curvature on compact complex manifolds, Trans. Amer. Math. Soc. 371 (2019), no. 3, 2073--2087

    2019

  40. [48]

    RC-positive metrics on rationally connected manifolds, Forum Math

    Yang, X.-K. RC-positive metrics on rationally connected manifolds, Forum Math. Sigma 8 (2020), Paper No. e53, 19 pp

    2020

  41. [49]

    On the Ricci curvature of a compact K\"ahler manifold and the complex Monge-Amp\`ere equation, Comm

    Yau, S.-T. On the Ricci curvature of a compact K\"ahler manifold and the complex Monge-Amp\`ere equation, Comm. Pure Appl. Math. 31 (1978), 339--411

    1978

  42. [50]

    and Yau, S.-T

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

    1986

  43. [51]

    Some results in the theory of vector bundles

    Umemura, H. Some results in the theory of vector bundles. Nagoya Math. J. 52(1973), 97--128

    1973

Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.