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The deformed Vortex equations and equivariant stability conditions

T0 review · 2 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read On vortex bundles, dHYM solvability is equivalent to Z-stability

desk verdict First non-perturbative existence–stability correspondence for higher-rank dHYM in a genuinely matrix-valued setting; honestly scoped, but Prop 5.5 and the §6.2 estimate chain need referee scrutiny before it can be called complete. read the letter →

arxiv 2607.17459 v2 pith:X5QHH5IT submitted 2026-07-20 math.DG math.AG

Tristan C. Collins , Yukai Zhang This is my paper · ORCID
classification math.DGmath.AG MSC 53C0714J6014F0535J60
keywords deformedHermitian-Yang-MillsequationdHYMvortex-typebundlesequivariantvectorZ-stabilityBridgelandstabilityaprioriestimatesdimensionalreduction
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 establishes an existence–stability correspondence for a genuinely matrix-valued, fully nonlinear PDE: an SU(2)-equivariant holomorphic vector bundle of vortex type over X×P¹ admits a solution of the higher-rank deformed Hermitian-Yang-Mills system if and only if it is Z-stable, a stability notion built from a central charge and tested against torsion quotients as well as subbundles. This is one of the first non-perturbative results of this kind for the higher-rank dHYM equation. Along the way, the paper classifies all solutions with Re Z_X(E) ≤ 0, rules out solutions for a range of phase parameters, and proves that Bridgeland stability implies Z-stability, so Bridgeland-stable vortex bundles automatically admit solutions. The proof reduces the four-dimensional dHYM system to a coupled 'deformed vortex' system on a Riemann surface and then runs a continuity method with a priori estimates. If correct, the result gives concrete evidence that Bridgeland-type stability is the right algebraic counterpart to the higher-rank dHYM equation.

What carries the argument

The key mechanism is dimensional reduction: SU(2)-equivariance turns the dHYM system on X×P¹ into the 'deformed vortex equations' on a holomorphic triple (E₁, E₂, Φ) over X, where rk E₂ = 1 and Φ is a holomorphic section of Hom(E₂, E₁). The proof then uses a continuity path interpolating from a classical vortex-type equation at t = 0 to the full deformed system at t = 1, supported by two blow-up/contradiction arguments that extract a destabilizing subtriple from a degenerating sequence of metrics, followed by a topological degree argument to obtain existence. The parameter bound tan(θ̂) ≤ σ/4π enters in exactly two places: in proving necessity of stability for type-B subtriples and in the C⁰

What would settle it

Compute the limit inequality (6.19) for a degenerating sequence of Z-stable triples with tan(θ̂) ≤ σ/4π on a flat torus: if the integrated (IIa)+(IIb)+cos(θ̂)E_UY is negative, the C⁰ bound collapses. More directly, try to construct a Z-stable vortex triple whose normalized metric ratio h₁h₂⁻¹ blows up along the continuity path.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: for a compact Riemann surface X and X̄ = X×P¹ with Kähler form ω = ω_X ⊕ (σ/2π)ω_FS, any irreducible SU(2)-equivariant holomorphic vortex-type bundle E with Im(Z_X(E)) > 0 and phase Θ(E) in (0, arctan(σ/4π)] ∪ [π/2, π] admits a solution of the deformed Hermitian-Yang-Mills system (1.1)+(1.2) if and only if E is SU(2)-equivariantly Z-stable. The paper also shows that Z-stability is implied by Bridgeland stability in the equivariant derived category, yielding Corollary 1.3: Bridgeland-stable vortex bundles admit equivariant solutions. A notable feature is that torsion sheaves supported on P¹ fibers genuinely appear as destabilizing quotients, so the stability

Load-bearing premise

The argument collapses if the blow-up limit inequality (6.19) in the C⁰ estimate fails: it needs the integrated sum (IIa)+(IIb)+cos(θ̂)E_UY to be asymptotically non-negative to rule out a diverging metric ratio, and without that bound stability cannot be converted into existence.

Editorial extensions

If this is right

  • Bridgeland-stable vortex bundles admit equivariant solutions of the dHYM system (Corollary 1.3).
  • In the regime Re Z_X(E) ≤ 0, solvability and Z-stability are completely classified: only rank-2 bundles with E₁ ≃ E₂ and non-vanishing Φ solve, and for these existence is equivalent to stability.
  • There are no solutions at all when tan(θ̂) ∈ [−σ/2π, 0).
  • Torsion quotients supported on P¹ fibers can destabilize, so any higher-rank dHYM stability theory must test these objects, not just subbundles.
  • Z-stability is practical: it reduces to checking a small class of saturated subtriples and fiber-supported quotients.

Reading between the lines

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

  • If the conjecture in the paper is correct, the parameter restriction tan(θ̂) ≤ σ/4π is an artifact of the analytic method rather than the geometry; the natural next test is the gap (arctan(σ/4π), π/2).
  • The results suggest that for higher-rank dHYM, Bridgeland stability rather than slope stability is the correct instability detector, because torsion sheaves carry the destabilizing information.
  • The dimensional-reduction template should extend to other symmetry groups or toric fibrations, producing analogous deformed vortex systems where stability correspondences can be tested explicitly.
  • The numerical observation that Z-positivity may fail at actual solutions implies ellipticity of the higher-rank dHYM system is a genuinely global phenomenon, not a local algebraic consequence of solvability.
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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 / 3 minor

Summary. The paper studies the higher-rank deformed Hermitian-Yang-Mills (dHYM) system (1.1)+(1.2) on the surface X = X × P^1, for SU(2)-equivariant holomorphic vector bundles of vortex type, i.e. non-split extensions 0 → p_1^*E_1 → E → p_1^*E_2 ⊗ p_2^*O_{P^1}(2) → 0. Using the SU(2)-equivariance, the system is dimensionally reduced to the deformed vortex equations (3.6)+(3.7) for a holomorphic triple (E_1,E_2,Φ). The paper defines a notion of Z-stability (Definitions 4.4/4.8), proves that it is necessary for existence of solutions in the phase range Θ(E) ∈ (0, arctan(σ/4π)] ∪ [π/2, π], and proves the converse for the sub-range tan θ̂ ≤ σ/4π by a method of continuity, a priori estimates including a Uhlenbeck-Yau style blow-up argument, and a Leray-Schauder degree argument. It also proves that Bridgeland stability in the Arcara-Bertram tilted heart implies Z-stability, and conjectures the converse. The main theorem is Theorem 1.2; Corollary 1.3 states that Bridgeland-stable vortex-type bundles admit equivariant solutions.

Significance. If the analysis is correct, this is the first non-perturbative existence–stability correspondence for the higher-rank dHYM equation in a setting where the reduced equation is a genuinely coupled nonlinear system. The paper is honest about its scope: the angle restriction tan θ̂ ≤ σ/4π is stated precisely, the open gap for the full phase range is flagged in Conjectures 4.24 and 8.6, and §8.1.2 gives a concrete parameter set showing that Z-positivity is not an algebraic consequence of the solvability plus the global bound |Φ|² ≤ 2π/σ. The reduction to holomorphic triples, the classification of solutions with Re Z_X(E) ≤ 0 (Proposition 3.10, Corollary 4.13), and the connection to Bridgeland stability are useful contributions. However, two load-bearing points are not fully secured in the submitted text: the proof of the properness estimate Proposition 5.5 is deferred to an unpublished thesis, and the key limit inequality (6.19) in Proposition 6.6 is established by a delicate double limit whose uniformity and sign tracking are not fully exposed. These gaps are fixable within the manuscript's framework, but they prevent the paper from being accepted in its present form.

major comments (2)
  1. [§5, Proposition 5.5] Theorem 5.2 — the t=0 existence result that starts the method of continuity — depends on the properness estimate (5.6) for the functional M. The proof is not included; the text states 'We refer the reader to the second author's Ph.D. thesis [93] for complete details.' This is a load-bearing input: without (5.6), the variational argument for the existence of the initial solution does not close. Since [93] is listed as 'to appear' and is not publicly verifiable, the submission does not currently contain a complete proof of Theorem 5.2. Please either supply a full proof (for example, in an appendix) or replace the appeal with a complete argument following Bradlow [8] and Garcia-Prada [47].
  2. [§6.2, Proposition 6.6, Eq. (6.19)] The C^0 bound on h_t — and hence the stability-implies-existence direction of Theorem 1.2 — rests on the limit inequality (6.19). The proof as written is not fully self-contained. The estimate (6.30) gives |(IIIc)| ≤ e^{sλ_{r'}} C + (1/(t sinθ))(IIa) + (cosθ/(t sinθ))E_UY. Substituting this into the definition of (IIa)+(IIb)+cosθ E_UY yields a lower bound whose right-hand side is negative; the desired non-negativity of (6.19) is obtained only by taking ℓ→∞ and then s→0. The text does not explicitly justify the uniformity in ε of the constants, the passage from U_ε to X\U_ε via (6.22), or the commutation of the two limits with the estimates of Lemma 6.7. Moreover, the paper itself notes after Lemma 4.23 that the analogous signed term (4.21) is the obstruction to the full phase range; the control of (IIb) here uses exactly the condition tan θ̂ ≤ σ/4π. Because this is the key a priori estim
minor comments (3)
  1. [Corollary 1.3 and §8.3] Corollary 1.3 states that Bridgeland-stable vortex-type bundles admit equivariant solutions, but it does not explicitly repeat the phase hypothesis Θ(E) ∈ (0, arctan(σ/4π)] ∪ [π/2, π] from Theorem 1.2. If 'in the setting of Theorem 1.2' is intended to carry all those assumptions, this should be stated explicitly; otherwise the corollary overclaims, since Corollary 8.8 only proves Z-stability and Theorem 1.2 requires the phase condition.
  2. [Remark 6 and §7.2] Remark 6 asserts that the hypothesis inf_X |Φ|² < 4π/(σt) is preserved along the method of continuity, but the proof of preservation relies on Lemma 6.1, whose statement itself contains that hypothesis. The piecewise-constant μ(t) construction in the proof of Theorem 1.2 is intended to handle this, but the argument would benefit from an explicit induction showing that the a priori bound obtained at step i implies the hypothesis at step i+1.
  3. [Throughout] The manuscript contains a number of presentation issues: inconsistent notation (e.g., θ vs θ̂, h vs h_t), some displayed equations with missing parentheses (e.g., around (4.12)), and references that are not fully formatted. These do not affect the mathematics but should be corrected.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction found; the equivalence proof is genuinely analytic and not forced by definition or fit. One proof is deferred to a same-author thesis, a completeness caveat rather than circularity.

full rationale

I walked the main derivation chain. The necessity direction (§4.5) derives the angle inequalities from the deformed vortex equations by honest integration-by-parts and Bochner-type arguments (e.g. Lemma 4.20, Proposition 4.22, Lemma 4.23), not by assuming stability. The existence direction (§6–7) is a genuine Uhlenbeck–Yau blow-up argument: Proposition 6.6 produces a destabilizing type-A subtriple from the failure of the C^0 bound, with the sign analysis of (IIa),(IIb),EUY and the eigenvalue-gap Lemma 6.7; the Leray–Schauder degree argument in §7 is a legitimate fixed-point completion. Z-stability is not defined in terms of solvability: its restricted test class is justified internally, notably by Lemma 4.21 and Lemma 8.7, and the Bridgeland comparison uses the external Arcara–Bertram construction. The only in-scope caveat is Section 5: Proposition 5.5, the properness estimate for the Bradlow functional, is stated with 'We refer the reader to the second author’s Ph.D. thesis [93] for complete details.' This is load-bearing for the t=0 start of the continuity method, but it is a deferred proof / missing support, not a reduction of the theorem to its own inputs, a fitted parameter renamed as a prediction, or a uniqueness claim imported from the authors' prior work. It should be weighed as a correctness/completeness risk, not as circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The paper introduces no fitted constants: the Kähler weight σ is input data and the phase θ̂ is determined by the central charge (3.5). Load-bearing unproved-in-text items are: (a) Garcia-Prada's classification restricting to the vortex class; (b) the phase-range restriction leaving (arctan(σ/4π), π/2) open; (c) standard Uhlenbeck–Yau and Arcara–Bertram background; (d) Proposition 5.5, deferred to the thesis; and (e) the inf|Φ|² < 4π/(σt) continuity hypothesis. One new notion, Z-stability, is introduced; it has partial external grounding via the Bridgeland implication, but its test class is chosen to fit the analysis.

assumptions (6)
  • domain assumption Garcia-Prada's equivariant classification: an SU(2)-equivariant holomorphic structure on a vortex bundle is described by a holomorphic triple (E₁,E₂,Φ), with the curvature block form (3.3); the theorem is confined to this class.
    Invoked in §3.1 (Proposition 3.1, Definition 3.2) and used throughout; the main theorem is not proven for general SU(2)-equivariant bundles outside the vortex class.
  • domain assumption Phase-range and central-charge restrictions: Im(Z_X(E))>0 (or Im=0 with Re<0) and Θ(E) ∈ (0, arctan(σ/4π)] ∪ [π/2, π]; the intermediate range is conjectural.
    States the scope of Theorem 1.2; the paper explicitly flags (after Lemma 4.23 and in §8.2) that the excluded range requires new techniques, so the theorem does not cover the full phase circle.
  • standard math Arcara–Bertram construction of Bridgeland stability on the tilted heart of a projective surface, including the Bogomolov–Gieseker inequality used to define the heart and prove Lemma 8.7/Corollary 8.8.
    Cited background in §8.3; the stability condition exists in the literature and is not re-proved here.
  • standard math Uhlenbeck–Yau analytic machinery: inequalities (6.4)–(6.6), Lemma A.1, and the theorem that W^{1,2} projections arising from blow-up sequences define holomorphic subbundles.
    Invoked in §6.2 and Appendix A; the paper reproduces the relevant identities (A.1)–(A.5) but relies on the original result [91] for the subbundle-extraction step.
  • ad hoc to paper Properness of the Bradlow functional M (Proposition 5.5), whose proof is deferred to the second author's thesis [93].
    Load-bearing for Theorem 5.2 (t=0 solvability); the preprint contains only a sketch ('We refer the reader to the second author's Ph.D. thesis [93] for complete details'), so a key step is paid for upstream in a not-yet-available document.
  • ad hoc to paper The method-of-continuity hypothesis inf_X |Φ|² < 4π/(σt), used in Lemmas 6.1, 6.2, Corollary 6.3 and Propositions 6.4, 6.6, 6.9.
    The paper argues in Remark 6 that the bound is preserved along the path via Lemma 6.1(ii); this preservation is asserted as a continuity/induction statement rather than an independent theorem, and it is the entry hypothesis for the main C⁰ and C¹ estimates.
invented entities (1)
  • Z-stability (Definitions 4.4/4.8) with torsion class T and admissible quotients independent evidence
    purpose: The algebro-geometric stability notion engineered to match solvability of the deformed vortex system, including torsion quotients supported on P¹-fibers.
    Genuinely new notion. External handle: Bridgeland stability, constructed independently by Arcara–Bertram on the tilted heart, is shown to imply it (Corollary 8.8), and the conjectural converse is a checkable algebraic statement. However, the test class of 'admissible' quotients is explicitly tailored to the analytic machinery, so the entity is partly hypothesis-driven.

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Pith. "Pith review of The deformed Vortex equations and equivariant stability conditions." pith.science (2026). https://pith.science/paper/X5QHH5IT

@misc{pith2026260717459,
  author       = {Pith},
  title        = {Pith review of: The deformed Vortex equations and equivariant stability conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X5QHH5IT}},
  note         = {Machine review of arXiv:2607.17459}
}
abstract

We study the higher rank deformed Hermitian-Yang-Mills (dHYM) equations for $SU(2)$-equivariant holomorphic vector bundles over $X\times \mathbb{P}^1$ for $X$ a compact Riemann surface. For a class of vector bundles of vortex type, we establish the equivalence between existence of solutions to higher rank dHYM equations and an appropriate notion of algebro-geometric stability, called $Z$-stability. We show that $Z$-stability is implied by, and conjecturally equivalent to, Bridgeland stability in $D^{b}{\rm Coh}^{SU(2)}(X\times \mathbb{P}^1)$.

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Works this paper leans on

93 extracted references · 11 linked inside Pith

  1. [93]

    Zhang,The deformed vortex equations, Thesis (Ph.D.)–University of Toronto (Canada), 2026,to appear

    Y. Zhang,The deformed vortex equations, Thesis (Ph.D.)–University of Toronto (Canada), 2026,to appear. 88 Email address:tristanc@math.toronto.edu Email address:yukai.zhang@mail.utoronto.ca Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, Canada

    2026

  2. [8]

    S. B. Bradlow,Special metrics and stability for holomorphic bundles with global sections, J. Differential Geom.33(1991), no. 1, 169–213

    1991

  3. [47]

    Garc ´ ıa-Prada,Dimensional reduction of stable bundles, vortices and stable pairs, Internat

    O. Garc ´ ıa-Prada,Dimensional reduction of stable bundles, vortices and stable pairs, Internat. J. Math.5 (1994), no. 1, 1–52

    1994

  4. [1]

    ´Alvarez-C´ onsul, M

    L. ´Alvarez-C´ onsul, M. Garcia-Fernandez, O. Garc ´ ıa-Prada,Gravitating vortices, cosmic strings, and the K¨ ahler-Yang-Mills equations, Comm. Math. Phys.351(2017), no. 1, 361–385

    2017

  5. [2]

    ´Alvarez-C´ onsul, M

    L. ´Alvarez-C´ onsul, M. Garcia-Fernandez, O. Garc ´ ıa-Prada, V. P. Pingali,Gravitating vortices and the Einstein-Bogomol’nyi equations, Math. Ann.379(2021), no. 3–4, 1651–1684

    2021

  6. [3]

    ´Alvarez-C´ onsul, O

    L. ´Alvarez-C´ onsul, O. Garc ´ ıa-Prada,Dimensional reduction and quiver bundles, J. Reine Angew. Math. 556(2003), 1–46

    2003

  7. [4]

    Arcara, A

    D. Arcara, A. Bertram,Bridgeland-stable moduli spaces for K-trivial surfaces, J. Eur. Math. Soc. (JEMS) 15(2013), no. 1, 1–38

    2013

  8. [5]

    Ballal,The supercritical deformed Hermitian Yang-Mills equation on compact projective manifolds, Illinois J

    A. Ballal,The supercritical deformed Hermitian Yang-Mills equation on compact projective manifolds, Illinois J. Math.67(2023), no. 1, 73–99

    2023

Show all 93 references
  1. [6]

    A. N. Ballal, V. P. PingaliPositivity properties of the vector bundle Monge–Amp` ere equation, J. Geom. Phys.229(2026), Paper No. 105934

    2026

  2. [7]

    Bayer, E

    A. Bayer, E. Macri, and Y. TodaBridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities, J. Algebraic Geom.23(2014), 117–163

    2014

  3. [9]

    S. B. Bradlow, G. D. Daskalopoulos,Moduli of stable pairs for holomorphic bundles over Riemann surfaces, Internat. J. Math.2(1991), no. 5, 477–513

    1991

  4. [10]

    S. B. Bradlow, G. D. Daskalopoulos,Moduli of stable pairs for holomorphic bundles over Riemann surfaces. II, Internat. J. Math.4(1993), no. 6, 903–925

    1993

  5. [11]

    Bradlow, G

    S. Bradlow, G. D. Daskalopoulos, O. Garc ´ ıa-Prada, R. Wentworth,Stable augmented bundles over Riemann surfaces, Vector bundles in algebraic geometry (Durham, 1993), London Math. Soc. Lecture Note Ser.208, 15–67, Cambridge Univ. Press, Cambridge, 1995

    1993

  6. [12]

    S. B. Bradlow, G. D. Daskalopoulos, R. A. Wentworth,Birational equivalences of vortex moduli, Topology 35(1996), no. 3, 731–748

    1996

  7. [13]

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

    1996

  8. [14]

    Bridgeland,Stability conditions on triangulated categories, Ann

    T. Bridgeland,Stability conditions on triangulated categories, Ann. of Math. (2)166(2007), no. 2, 317–345

    2007

  9. [15]

    Bunnet, A

    D. Bunnet, A. Rinc´ on-Hidalgo,Moduli of Bridgeland semistable holomorphic triples, preprint, arXiv:2102.04995. 85

    arXiv

  10. [16]

    Y. H. Chan, A. Jacob,Singularity formation along the line bundle mean curvature flow, Int. Math. Res. Not. IMRN (2025) no. 5, Paper No. rnaf037

    2025

  11. [17]

    S.-Y. A. Chang, M. J. Gursky, P. C. Yang,The scalar curvature equation on2- and3-spheres, Calc. Var. Partial Differential Equations1(1993), no. 2, 205–229

    1993

  12. [18]

    Charbonneau, G

    B. Charbonneau, G. Oliveira, R. Sena-Dias,Deformed Hermitian-Yang-Mills equation on the manifold of full flags, preprint, 2026, arXiv:2607.08622

    2026 arXiv

  13. [19]

    Chen,The J-equation and the supercritical deformed Hermitian-Yang-Mills equation, Invent

    G. Chen,The J-equation and the supercritical deformed Hermitian-Yang-Mills equation, Invent. Math.,225 (2021), no. 2, 529–602

    2021

  14. [20]

    G. Chen, K. Ghosh,On The Ellipticity of Generalised Monge-Amp` ere Equations on Vector Bundles, preprint, arXiv:2604.21273

    arXiv

  15. [21]

    X.-X. Chen, S. Donaldson, S. SunK¨ ahler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, J. Amer. Math. Soc.28(2015), no. 1, 183–197

    2015

  16. [22]

    X.-X. Chen, S. Donaldson, S. SunK¨ ahler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than2π, J. Amer. Math. Soc.28(2015), no. 1, 199–234

    2015

  17. [23]

    X.-X. Chen, S. Donaldson, S. SunK¨ ahler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches2πand completion of the main proof, J. Amer. Math. Soc.28(2015), no. 1, 235–278

    2015

  18. [24]

    Chen, L.-C

    Y.-Z. Chen, L.-C. Wu,Second order elliptic equations and elliptic systems, Translations of Mathematical Monographs174, Translated from the 1991 Chinese original by Bei Hu, American Mathematical Society, Providence, RI, 1998

    1991

  19. [25]

    J. Chu, T. C. Collins, M.-C. Lee,The space of almost calibrated(1 , 1)-forms on a compact K¨ ahler manifold, Geom. Topol.25(2021), no. 5, 2573–2619

    2021

  20. [26]

    Chu, M.-C

    J. Chu, M.-C. Lee,Hypercritical deformed Hermitian-Yang-Mills equation revisited, J. Reine Angew. Math., 801(2023), 161–172

    2023

  21. [27]

    Chu, M.-C

    J. Chu, M.-C. Lee, R. Takahashi,A Nakai–Moishezon type criterion for supercritical deformed Hermitian– Yang–Mills equation, J. Differential Geom.126(2024), no. 2, 583–632

    2024

  22. [28]

    T. C. Collins, A. Jacob, S.-T. Yau, (1 , 1)forms with specified Lagrangian phase: a priori estimates and algebraic obstructions, Camb. J. Math.8(2020), no. 2, 407–452

    2020

  23. [29]

    T. C. Collins, J. Lo, Y. Shi, and S.-T. Yau,Stability for Line Bundles and Deformed Hermitian-Yang-Mills Equation on Some Elliptic Surfaces, preprint, arXiv:2306.05620v2

    arXiv

  24. [30]

    T. C. Collins, Y. Shi,Stability and the deformed Hermitian-Yang-Mills equation, Surveys in differential geometry 2019. Differential geometry, Calabi-Yau theory, and general relativity. Part 2, Surv. Differ. Geom., 24, 1–38

    2019

  25. [31]

    T. C. Collins, D. Xie, S.-T. Yau,The deformed Hermitian-Yang-Mills equation in geometry and physics, Geometry and physics. Vol. I, 69–90, Oxford Univ. Press, Oxford, 2018

    2018

  26. [32]

    T. C. Collins, S.-T. Yau,Moment maps, nonlinear PDE and stability in mirror symmetry, I: geodesics, Ann. PDE7(2021), no. 1, Paper No. 11

    2021

  27. [33]

    T. C. Collins, and S.-T. YauMoment maps, nonlinear PDE, and stability in Mirror Symmetry, arXiv:1811.04824, preprint

    arXiv

  28. [34]

    E. M. Correa,DHYM connections on higher rank holomorphic vector bundles over P(TP2 ), Math. Z.308 (2024), no. 2

    2024

  29. [35]

    Datar, V

    V. Datar, V. P. Pingali,A numerical criterion for generalised Monge-Amp` ere equations on projective manifolds, Geom. Funct. Anal.31(2021), no. 4

    2021

  30. [36]

    Datar, R

    V. Datar, R. Mete, J. Song,Minimal slopes and bubbling for complex Hessian equations, Adv. Math.491 (2026), Paper No. 110865, 84

    2026

  31. [37]

    Delloque, A

    R. Delloque, A. Napame, C. Scarpa, C. Tipler,Polynomial stability conditions for vector bundles: Positivity, equivariance and blow-ups, preprint, 2025, arXiv:2506.23842

    2025 arXiv

  32. [38]

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

    J.-P. Demailly,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

  33. [39]

    Demailly, and M

    J.-P. Demailly, and M. P˘ aunNumerical characterization of the K¨ ahler cone of a compact K¨ ahler manifold, Ann. of Math. (2)159(2004), no. 3, 1247–1274. 86

    2004

  34. [40]

    Dervan, J

    R. Dervan, J. B. McCarthy, L. M. Sektnan, Z-critical connections and Bridgeland stability conditions, Camb. J. Math.12(2024), no. 2

    2024

  35. [41]

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

    1985

  36. [42]

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

    1983

  37. [43]

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

    1987

  38. [44]

    Douglas,Dirichlet branes, homological mirror symmetry, and stability, Proceedings of the International Congress of Mathematicians, Vol

    M. Douglas,Dirichlet branes, homological mirror symmetry, and stability, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 395–408, Higher Ed. Press, Beijing, 2002

    2002

  39. [45]

    Douglas, B

    M. Douglas, B. Fiol, and C. R¨ omelsbergerStability and BPS branes, J. High Energy Phys. (2005), no. 9, 006

    2005

  40. [46]

    Garcia-Fernandez, V

    M. Garcia-Fernandez, V. P. Pingali, C. Yao,Gravitating vortices with positive curvature, Adv. Math.388 (2021), Paper No. 107851, 42

    2021

  41. [48]

    Ghosh,vortex type equations on compact Riemann surfaces, Differential Geom

    K. Ghosh,vortex type equations on compact Riemann surfaces, Differential Geom. Appl.93(2024)

    2024

  42. [49]

    Guan,Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math

    B. Guan,Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math. J.,163(2014), 1491-1524

    2014

  43. [50]

    Gualtieri,Generalized complex geometry, Ann

    M. Gualtieri,Generalized complex geometry, Ann. of Math. (2)174(2011), no. 1, 75–123

    2011

  44. [51]

    X. Han, X. Jin,Chern number inequalities of deformed Hermitian-Yang-Mills metrics on four dimensional K¨ ahler manifolds, Manuscripta Math.174(2024), no. 3–4, 963–972

    2024

  45. [52]

    F. R. Harvey, H. B. Lawson Jr.Calibrated geometries, Acta Math., vol. 148 (1982), 47–157

    1982

  46. [53]

    Hull,Compactifications of the heterotic superstring, Phys

    C.M. Hull,Compactifications of the heterotic superstring, Phys. Lett. B178(1986), no. 4, 357–364

    1986

  47. [54]

    Jacob,The Deformed Hermitian-Yang-Mills Equation and Level Sets of Harmonic Polynomials, preprint, arXiv:2204.01875

    A. Jacob,The Deformed Hermitian-Yang-Mills Equation and Level Sets of Harmonic Polynomials, preprint, arXiv:2204.01875

    arXiv

  48. [55]

    Jacob, N

    A. Jacob, N. Sheu,The deformed Hermitian-Yang-Mills equation on the blowup of Pn, Asian J. Math.26 (2022), no. 6, 847–864

    2022

  49. [56]

    Jacob, S.-T

    A. Jacob, S.-T. Yau,A special Lagrangian type equation for holomorphic line bundles, Math. Ann.369 (2017), no. 1-2, 869–898

    2017

  50. [57]

    Jaffe, C

    A. Jaffe, C. Taubes,Vortices and monopoles, Progress in Physics, vol. 2, Structure of static gauge theories, Birkh¨ auser, Boston, MA (1980)

    1980

  51. [58]

    Joyce,Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow, EMS Surv

    D. Joyce,Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow, EMS Surv. Math. Sci.2(2015), no. 1, 1–62

    2015

  52. [59]

    Keller, C

    J. Keller, C. Scarpa, Z-critical equations for holomorphic vector bundles on K¨ ahler surfacesMath. Ann. 395(2026), no. 1

    2026

  53. [60]

    L. Lara, H. S´ a Earp,Asymptotically Z-stable bundles over projective surfaces, preprint, 2026, arXiv:2604.20264

    2026 arXiv

  54. [61]

    N. C. Leung,Einstein type metrics and stability on vector bundles, J. Differential Geom.45(1997), no. 3, 514–546

    1997

  55. [62]

    N. C. Leung, S.-T. Yau, E. ZaslowFrom special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai, Adv. Theor. Math. Phys.4(2000), no. 6, 1319–1341

    2000

  56. [63]

    LiOn stability conditions for the quintic threefold, Invent

    C. LiOn stability conditions for the quintic threefold, Invent. Math.218(2019), no. 1, 301–340

    2019

  57. [64]

    Li,Stability conditions in (complex) algebraic geometry: construction and application, Proceedings of the International Consortium of Chinese Mathematicians 2019

    C. Li,Stability conditions in (complex) algebraic geometry: construction and application, Proceedings of the International Consortium of Chinese Mathematicians 2019. Vol. 2 of 2, 1315–1330, Int. Press, Somerville, MA, 2024

    2019

  58. [65]

    Macr ` ı, B

    E. Macr ` ı, B. Schmidt,Lectures on Bridgeland stability, Moduli of curves, Lect. Notes Unione Mat. Ital., vol. 21, 139–211, Springer, Cham, 2017

    2017

  59. [66]

    Mari˜ no, R

    M. Mari˜ no, R. Minasian, G. Moore, and A. StromingerNonlinear instantons from supersymmetric p-branes, J. High Energy Phys. (2000), no. 1 87

    2000

  60. [67]

    Mart ´ ınez-Romero, A

    E. Mart ´ ınez-Romero, A. Rinc´ on-Hidalgo, A. R¨ uffer,Bridgeland stability conditions on the category of holomorphic triples over curves, preprint, 2020, arXiv:1905.04240

    2020 arXiv

  61. [68]

    Martucci, P

    L. Martucci, P. Smyth,Supersymmetric D-branes and calibrations on general N = 1backgrounds, J. High Energy Phys. (2005), no. 11

    2005

  62. [69]

    J. B. McCarthy,Stability conditions and canonical metrics, Thesis (Ph.D.)–Imperial College, London (U.K.), 2023, arXiv:2302.04966

    2023 arXiv

  63. [70]

    Mete,Singularity formation in co-dimension one of the dHYM cotangent flow on blow up of CP3 at a point, Math

    R. Mete,Singularity formation in co-dimension one of the dHYM cotangent flow on blow up of CP3 at a point, Math. Res. Lett.33(2026), no. 1, 213–244

    2026

  64. [71]

    Minasian, A

    R. Minasian, A. Tomasiello,Variations on stability, Nuclear Phys. B631(2002), no. 1-2, 43–65

    2002

  65. [72]

    Murakami, J-equations and deformed Hermitian-Yang-Mills equations on holomorphic submersions, Math

    R. Murakami, J-equations and deformed Hermitian-Yang-Mills equations on holomorphic submersions, Math. Z.312(2026), no. 3, Paper No. 91

    2026

  66. [73]

    Murakami,Weak limits of theJ-flow and the deformed Hermitian-Yang-Mills flow on K¨ ahler surfaces: boundary cases, Ann

    R. Murakami,Weak limits of theJ-flow and the deformed Hermitian-Yang-Mills flow on K¨ ahler surfaces: boundary cases, Ann. Global Anal. Geom.69(2026), no. 2, Paper No. 7

    2026

  67. [74]

    Nirenberg,Topics in nonlinear functional analysis, With a chapter by E

    L. Nirenberg,Topics in nonlinear functional analysis, With a chapter by E. Zehnder, Notes by R. A. Artino, Lecture Notes, 1973–1974, Courant Institute of Mathematical Sciences, New York University, New York, 1974

    1973

  68. [75]

    Phong,Geometric flows from unified string theories, Surveys in differential geometry 2022

    D.H. Phong,Geometric flows from unified string theories, Surveys in differential geometry 2022. Essays on geometric flows—celebrating 40 years of Ricci flow, Surv. Differ. Geom. 27, 75–102, Int. Press, Somerville, MA, 2024

    2022

  69. [76]

    V. P. Pingali,A vector bundle version of the Monge-Amp` ere equation, Adv. Math.360(2020)

    2020

  70. [77]

    Rincon Hidalgo,Bridgeland Stability Conditions on the Category of Holomorphic Triples, Thesis (Ph.D.)– Freie Universitaet Berlin (Germany), 2019

    A. Rincon Hidalgo,Bridgeland Stability Conditions on the Category of Holomorphic Triples, Thesis (Ph.D.)– Freie Universitaet Berlin (Germany), 2019

    2019

  71. [78]

    Schmidt,Counterexample to the generalized Bogomolov-Gieseker inequality for threefolds, Int

    B. Schmidt,Counterexample to the generalized Bogomolov-Gieseker inequality for threefolds, Int. Math. Res. Not. IMRN (2017), no. 8, 2562–2566

    2017

  72. [79]

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

    1988

  73. [80]

    J. P. Solomon,The Calabi homomorphism, Lagrangian paths and special Lagrangians, Math. Ann357 (2013), no. 4, 1389–1424

    2013

  74. [81]

    Song,Nakai–Moishezon criterions for complex Hessian equations, preprint 2020, arxiv:2012.07956

    J. Song,Nakai–Moishezon criterions for complex Hessian equations, preprint 2020, arxiv:2012.07956

    2020 arXiv

  75. [82]

    The Stacks project authors,The Stacks project,https://stacks.math.columbia.edu, 2026

    2026

  76. [83]

    Strominger,Superstrings with torsion, Nuclear Phys

    A. Strominger,Superstrings with torsion, Nuclear Phys. B 274 (1986), no. 2, 253–284

    1986

  77. [84]

    Sz´ ekelyhidi,Fully non-linear elliptic equations on compact Hermitian manifolds, J

    G. Sz´ ekelyhidi,Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differential Geom. 109(2018), no. 2, 337–378

    2018

  78. [85]

    Takahashi,Collapsing of the line bundle mean curvature flow on K¨ ahler surfaces, Calc

    R. Takahashi,Collapsing of the line bundle mean curvature flow on K¨ ahler surfaces, Calc. Var. Partial Differential Equations60(2021), no. 1, Paper No. 27

    2021

  79. [86]

    Takahashi,Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow, Internat

    R. Takahashi,Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow, Internat. J. Math.31(2020), no. 14

    2020

  80. [87]

    Takahashi,J-equation on holomorphic vector bundles, J

    R. Takahashi,J-equation on holomorphic vector bundles, J. Funct. Anal.286(2024), no. 4

    2024

  81. [88]

    Thaddeus,Stable pairs, linear systems and the Verlinde formula, Invent

    M. Thaddeus,Stable pairs, linear systems and the Verlinde formula, Invent. Math.117(1994), no. 2, 317–353

    1994

  82. [89]

    R. P. Thomas,Moment maps, monodromy, and mirror manifolds, Symplectic geometry and mirror symmetry (Seoul, 2000), 467–498, World Sci. Publ., River Edge, NJ, 2001

    2000

  83. [90]

    R. P. Thomas, S.-T. Yau,Special Lagrangians, stable bundles, and mean curvature flow, Comm. Anal. Geom. 10(2002), no. 5, 1075–1113

    2002

  84. [91]

    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), S257–S293

    1986

  85. [92]

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

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

    1978

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