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arxiv: 1907.05770 · v1 · pith:LDMSPZ2Snew · submitted 2019-07-12 · 🧮 math.AG · math.CV· math.DG

A variational approach to the Hermitian-Einstein metrics and the Quot-scheme limit of Fubini-Study metrics

Yoshinori Hashimoto , Julien Keller This is my paper

Pith reviewed 2026-05-24 22:27 UTC · model grok-4.3

classification 🧮 math.AG math.CVmath.DG
keywords Donaldson-Uhlenbeck-Yau theoremHermitian-Einstein metricsQuot-scheme limitDonald's functionalFubini-Study metricsBergman kernelslope stability
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The pith

Assuming uniform coercivity, a new proof of the Donaldson-Uhlenbeck-Yau theorem follows from Donaldson's functional.

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

This sequel paper assumes that the coercivity of Donaldson's functional is uniform in a certain sense and uses this to give a new proof of the Donaldson-Uhlenbeck-Yau theorem. The proof shows existence of Hermitian-Einstein metrics on slope-stable holomorphic vector bundles over smooth projective varieties. The analysis stays elementary except for the asymptotic expansion of the Bergman kernel, because the prior work already established coercivity via the Quot-scheme limit of Fubini-Study metrics. A reader cares because the result ties algebraic stability directly to the existence of a canonical metric through variational methods.

Core claim

Assuming that the coercivity is uniform in a certain sense, the Donaldson-Uhlenbeck-Yau theorem holds: every slope-stable holomorphic vector bundle admits a Hermitian-Einstein metric, proved by showing that the Quot-scheme limit of Fubini-Study metrics converges to the desired metric when the functional is coercive.

What carries the argument

Uniform coercivity of Donaldson's functional on the Quot-scheme limit of Fubini-Study metrics

If this is right

  • Slope-stable bundles admit Hermitian-Einstein metrics once uniform coercivity is granted.
  • The existence proof requires only elementary analysis plus the known Bergman kernel expansion.
  • The Quot-scheme limit directly produces the Hermitian-Einstein metric from the variational setup.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Checking uniform coercivity on explicit examples such as bundles over projective space would test the assumption in concrete cases.
  • The variational reduction might apply to other canonical metric problems on Kähler manifolds.
  • If uniformity extends beyond smooth projective varieties, the theorem could reach singular or non-projective settings.

Load-bearing premise

The coercivity of Donaldson's functional holds uniformly in a certain sense.

What would settle it

Find a slope-stable bundle where Donaldson's functional fails to be uniformly coercive yet a Hermitian-Einstein metric exists, or where uniform coercivity holds but no such metric exists.

read the original abstract

This is a sequel of our paper [arXiv:1809.08425] on the Quot-scheme limit and variational properties of Donaldson's functional, which established its coercivity for slope stable holomorphic vector bundles over smooth projective varieties. Assuming that the coercivity is uniform in a certain sense, we provide a new proof of the Donaldson-Uhlenbeck-Yau theorem, in such a way that the analysis involved in the proof is elementary except for the asymptotic expansion of the Bergman kernel.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript is a sequel to arXiv:1809.08425. Assuming uniform coercivity (in a sense made precise in the prior work) of Donaldson's functional on the space of Hermitian metrics for a slope-stable holomorphic vector bundle, it gives a new proof of the Donaldson-Uhlenbeck-Yau theorem by variational methods: existence of a minimizer is obtained from the uniform coercivity, the minimizer is shown to be Hermitian-Einstein by passing to the Quot-scheme limit, and the only non-elementary analytic input is the asymptotic expansion of the Bergman kernel.

Significance. If the uniformity hypothesis holds, the argument supplies an alternative route to DUY that isolates the role of the Bergman-kernel expansion and reduces the remainder of the analysis to standard variational and Quot-scheme techniques. The manuscript explicitly credits the coercivity result to the earlier paper and does not claim to re-derive it.

major comments (1)
  1. [Abstract and §1] Abstract and §1: the existence of a minimizer for Donaldson's functional and the passage to the Quot-scheme limit both rest on the uniform coercivity statement taken from arXiv:1809.08425. Because this uniformity is not re-established or even sketched in the present manuscript, the new proof of DUY is conditional on an external result whose justification lies outside the current text.
minor comments (1)
  1. [§1] The precise sense in which coercivity is required to be uniform (e.g., with respect to which parameters or sequences) should be recalled explicitly in §1 so that a reader need not consult the prior paper to follow the logical structure.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and for recognizing the manuscript as a sequel that isolates the variational and Quot-scheme aspects of the DUY theorem under the uniform coercivity hypothesis. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the existence of a minimizer for Donaldson's functional and the passage to the Quot-scheme limit both rest on the uniform coercivity statement taken from arXiv:1809.08425. Because this uniformity is not re-established or even sketched in the present manuscript, the new proof of DUY is conditional on an external result whose justification lies outside the current text.

    Authors: We agree that the argument relies on the uniform coercivity result established in arXiv:1809.08425 and does not reprove or sketch it here. This dependence is intentional: the manuscript is explicitly a sequel whose purpose is to supply an alternative, largely elementary route to DUY once uniform coercivity is granted, thereby isolating the role of the Bergman-kernel expansion. The abstract and §1 already state the assumption, but we will revise both to make the dependence on the prior paper more prominent and to clarify that the coercivity statement is imported without reproof. revision: yes

Circularity Check

1 steps flagged

Uniform coercivity of Donaldson's functional taken from prior self-citation; new DUY proof conditional on it

specific steps
  1. self citation load bearing [Abstract]
    "Assuming that the coercivity is uniform in a certain sense, we provide a new proof of the Donaldson-Uhlenbeck-Yau theorem, in such a way that the analysis involved in the proof is elementary except for the asymptotic expansion of the Bergman kernel."

    The central claim (new proof of DUY) is conditional on uniform coercivity whose justification is imported from the authors' prior paper arXiv:1809.08425 and is not re-derived. Because the uniformity controls the existence of the minimizer and passage to the limit, the DUY conclusion reduces directly to the self-cited result.

full rationale

The paper is explicitly a sequel that assumes uniform coercivity (in a sense defined in the prior work) to obtain a new proof of the Donaldson-Uhlenbeck-Yau theorem. The abstract and introduction state that this uniformity is invoked but not re-established here; the remainder of the argument reduces to the Bergman kernel expansion once the coercivity assumption is granted. This matches the self-citation load-bearing pattern because the load-bearing premise controlling existence of the minimizer and the Quot-scheme limit is justified solely by a citation to the authors' own earlier paper. No independent derivation or external verification of the uniformity is supplied in the present manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof depends on the coercivity result from the cited predecessor paper and on the standard asymptotic expansion of the Bergman kernel; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Coercivity of Donaldson's functional for slope-stable holomorphic vector bundles holds uniformly
    Invoked directly from arXiv:1809.08425 to obtain the DUY conclusion.

pith-pipeline@v0.9.0 · 5609 in / 1229 out tokens · 52652 ms · 2026-05-24T22:27:38.444609+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 2 internal anchors

  1. [1]

    Berman, B

    R. Berman, B. Berndtsson, and J. Sj¨ ostrand, A direct approach to Bergman ker- nel asymptotics for positive line bundles , Ark. Mat. 46 (2008), no. 2, 197–217. MR2430724

    work page 2008

  2. [2]

    Berman, S

    R. Berman, S. Boucksom, and M. Jonsson, A variational approach to the Yau-Tian- Donaldson conjecture, arXiv preprint arXiv:1509.04561 (2015)

    work page arXiv 2015

  3. [3]

    Boucksom, T

    S. Boucksom, T. Hisamoto, and M. Jonsson, Uniform K-stability and asymptotics of energy functionals in K¨ ahler geometry , arXiv preprint arXiv:1603.01026, to appear in JEMS (2016)

    work page arXiv 2016

  4. [4]

    , Uniform K-stability, Duistermaat-Heckman measures and si ngularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 743–841. MR3669511

    work page 2017

  5. [5]

    Catlin, The Bergman kernel and a theorem of Tian , Analysis and geometry in several complex variables (Katata, 1997), 1999, pp

    D. Catlin, The Bergman kernel and a theorem of Tian , Analysis and geometry in several complex variables (Katata, 1997), 1999, pp. 1–23. M R1699887

    work page 1997

  6. [6]

    Dervan, Uniform stability of twisted constant scalar curvature K¨ ahler metrics , Int

    R. Dervan, Uniform stability of twisted constant scalar curvature K¨ ahler metrics , Int. Math. Res. Not. IMRN 15 (2016), 4728–4783. MR3564626

    work page 2016

  7. [7]

    S. K. Donaldson, A new proof of a theorem of Narasimhan and Seshadri , J. Differential Geom. 18 (1983), no. 2, 269–277. MR710055

    work page 1983

  8. [8]

    London Math

    , Anti self-dual Yang-Mills connections over complex algebr aic surfaces and stable vector bundles , Proc. London Math. Soc. (3) 50 (1985), no. 1, 1–26. MR765366

    work page 1985

  9. [9]

    , Infinite determinants, stable bundles and curvature , Duke Math. J. 54 (1987), no. 1, 231–247. MR885784

    work page 1987

  10. [10]

    Donaldson and S

    S. Donaldson and S. Sun, Gromov-Hausdorff limits of K¨ ahler manifolds and algebraic geometry, Acta Math. 213 (2014), no. 1, 63–106. MR3261011

    work page 2014

  11. [11]

    Hashimoto, Quantisation of extremal K¨ ahler metrics , arXiv preprint arXiv:1508.02643 (2015)

    Y. Hashimoto, Quantisation of extremal K¨ ahler metrics , arXiv preprint arXiv:1508.02643 (2015)

    work page arXiv 2015

  12. [12]

    Hashimoto and J

    Y. Hashimoto and J. Keller, Quot-scheme limit of Fubini–Study metrics and Donald- son ’s functional for bundles , arXiv preprint arXiv:1809.08425 (2018)

    work page arXiv 2018

  13. [13]

    Huybrechts and M

    D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves , Second, Cambridge Mathematical Library, Cambridge University Pre ss, Cambridge, 2010. MR2665168

    work page 2010

  14. [14]

    det” and “Div

    F. F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div” , Math. Scand. 39 (1976), no. 1, 19–

    work page 1976

  15. [15]

    Kobayashi, Differential Geometry of Holomorphic Vector Bundles , Math

    S. Kobayashi, Differential Geometry of Holomorphic Vector Bundles , Math. Seminar Notes (by I. Enoki), 1982. in Japanese

    work page 1982

  16. [16]

    15, Princeton University Pre ss, Princeton, NJ; Prince- ton University Press, Princeton, NJ, 1987

    , Differential geometry of complex vector bundles , Publications of the Mathe- matical Society of Japan, vol. 15, Princeton University Pre ss, Princeton, NJ; Prince- ton University Press, Princeton, NJ, 1987. Kanˆ o Memorial L ectures, 5. MR909698

    work page 1987

  17. [17]

    L¨ ubke, Stability of Einstein-Hermitian vector bundles , Manuscripta Math

    M. L¨ ubke, Stability of Einstein-Hermitian vector bundles , Manuscripta Math. 42 (1983), no. 2-3, 245–257. MR701206

    work page 1983

  18. [18]

    L¨ ubke and A

    M. L¨ ubke and A. Teleman, The Kobayashi-Hitchin correspondence, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR1370660

    work page 1995

  19. [19]

    Ma and G

    X. Ma and G. Marinescu, Holomorphic Morse inequalities and Bergman kernels , Progress in Mathematics, vol. 254, Birkh¨ auser Verlag, Bas el, 2007. MR2339952 (2008g:32030)

    work page 2007

  20. [20]

    S. T. Paul, CM stability of projective varieties , arXiv preprint arXiv:1206.4923 (2012)

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  21. [21]

    , Stable Pairs and Coercive Estimates for The Mabuchi Functio nal, arXiv preprint arXiv:1308.4377 (2013)

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  22. [22]

    D. H. Phong and J. Sturm, Stability, energy functionals, and K¨ ahler-Einstein metrics, Comm. Anal. Geom. 11 (2003), no. 3, 565–597. MR2015757

    work page 2003

  23. [23]

    Y. T. Siu, Lectures on Hermitian-Einstein metrics for stable bundles and K¨ ahler- Einstein metrics , DMV Seminar, vol. 8, Birkh¨ auser Verlag, Basel, 1987. MR90 4673 HERMITIAN–EINSTEIN METRICS 33

    work page 1987

  24. [24]

    Sz´ ekelyhidi,The partial C 0-estimate along the continuity method , J

    G. Sz´ ekelyhidi,The partial C 0-estimate along the continuity method , J. Amer. Math. Soc. 29 (2016), no. 2, 537–560. MR3454382

    work page 2016

  25. [25]

    Tian, Canonical metrics in K¨ ahler geometry , Lectures in Mathematics ETH Z¨ urich, Birkh¨ auser Verlag, Basel, 2000

    G. Tian, Canonical metrics in K¨ ahler geometry , Lectures in Mathematics ETH Z¨ urich, Birkh¨ auser Verlag, Basel, 2000. Notes taken by Meike Akveld. MR1787650

    work page 2000

  26. [26]

    Uhlenbeck and S.-T

    K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles , Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S257– S293. Frontiers of the mathematical sciences: 1985 (New Yor k, 1985). MR861491

    work page 1986

  27. [27]

    On the existence of Hermitian- Yang-Mills connections in stable vector bundles

    , A note on our previous paper: “On the existence of Hermitian- Yang-Mills connections in stable vector bundles” [Comm. Pure Appl. Mat h. 39 (1986), S257– S293; MR0861491 (88i:58154)] , Comm. Pure Appl. Math. 42 (1989), no. 5, 703–707. MR997570

    work page 1986

  28. [28]

    Wang, Balance point and stability of vector bundles over a project ive manifold , Math

    X. Wang, Balance point and stability of vector bundles over a project ive manifold , Math. Res. Lett. 9 (2002), no. 2-3, 393–411. MR1909652

    work page 2002

  29. [29]

    , Canonical metrics on stable vector bundles , Comm. Anal. Geom. 13 (2005), no. 2, 253–285. MR2154820 Yoshinori Hashimoto Department of Mathematics, Tokyo In- stitute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan. Email address: hashimoto@math.titech.ac.jp Julien Keller Aix Marseille Universit ´ e, CNRS, Centrale Marseille, Institut de...

    work page 2005