Pith. sign in

REVIEW 3 major objections 5 minor 38 references

Nakai-Moishezon criteria and the toric Thomas-Yau conjecture

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read For toric mirrors of weak Fano manifolds, a SYZ Lagrangian section is Hamiltonian isotopic to a supercritical special Lagrangian exactly when the phase inequalities for all toric subvarieties hold.

desk verdict A substantial new reduction of toric Thomas-Yau to dHYM positivity; the proof is complete for divisors and sketched for complete intersections, which is the one real gap. read the letter →

arxiv 2505.07228 v2 pith:WDUYNYM3 submitted 2025-05-12 math.DG math.AGmath.SG

Jacopo Stoppa This is my paper · ORCID
classification math.DGmath.AGmath.SG MSC 14J3214J4553C3853D3753D12
keywords toricweakFanoSYZtransformspecialLagrangianThomas-YauconjecturedeformedHermitianYang-MillsNakai-MoishezoncriterionBridgelandstabilityFukaya-Seidelcategory
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 a toric form of the Thomas-Yau conjecture: on the mirror of a toric weak Fano manifold, a SYZ Lagrangian section is Hamiltonian isotopic to a supercritical special Lagrangian section precisely when a list of phase inequalities holds, one for every toric subvariety. Those inequalities compare the argument of the period of the section with the arguments of periods of mirror objects $L_V$ attached to subvarieties through morphisms $L_V[-\mathrm{codim}\,V]\to L$. The proof reduces the statement to the Nakai-Moishezon criterion for supercritical deformed Hermitian Yang-Mills connections via the SYZ transform, the toric Gamma-class theorem, and toric homological mirror symmetry. As an application, on mirrors of toric weak del Pezzo surfaces, a naturally defined Bridgeland stable Lagrangian section is Hamiltonian isotopic to a special Lagrangian section, with partial converses; unstable sections produce weak special Lagrangian solutions at the minimal phase. The result converts a PDE existence question into a cohomological slope condition, exactly as the Thomas-Yau programme predicted.

What carries the argument

The central mechanism is the Gamma-class integral transform $\Gamma:K_0(X)\to H_n(Y_{q_k},\{\mathrm{Re}(W(k\omega_0))\gg 0\};\mathbb{Z})$ supplied by the toric Gamma-class theorem, combined with the SYZ transform. It converts integrals of $\mathrm{ch}(E)$ against $e^{-ik\omega_0}$ into periods $\int_{\Gamma(E)} e^{-W(k\omega_0)}\Omega_0$ up to controlled powers of $k$; a scaling analysis as $k\to\infty$ isolates the contribution of a toric subvariety $V$ by looking at $L^{\otimes k}(k_V V)$ and taking the cone in the exact sequence $0\to L^{\otimes k}\to L^{\otimes k}(k_V V)\to S_V\to 0$. The resulting objects $S_V$ satisfy $S_V[-\mathrm{codim}\,V]\to L^{\otimes k}$ and are used to rewrite the dHYM Nakai-Moishezon inequalities as the phase inequalities for periods.

What would settle it

Compute both sides of the phase inequality (1.3) to next-to-leading order in $k$ for a toric weak del Pezzo surface, for example the blow-up of $\mathbb{P}^2$ at one point with the Kähler class of Remark 1.11, and compare the sign of the phase difference with the sign of the corresponding dHYM Nakai-Moishezon integral (2.3). The central claim predicts exact agreement of signs for all sufficiently large $k$; a single toric curve $V$ where the signs disagree for arbitrarily large $k$ would refute the theorem. A second check is to put $[\omega_0]$ on one of the analytic subvarieties excluded in Remark 2.6(v) and verify that the strict inequalities fail exactly when a toric subvariety is strictly semi-stabilising in (5.10).

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem 1.2 (Corollary 2.9): for $X$ a toric weak Fano manifold with fixed torus-invariant Kähler form $\omega_0$, $L$ a holomorphic line bundle with $L^\vee$ ample, and $k$ sufficiently large, the SYZ Lagrangian section $L=L(L^{\otimes k},h)$ in the mirror to $(X,k\omega_0)$ is Hamiltonian isotopic to a supercritical special SYZ Lagrangian section if and only if, for every toric subvariety $V$, $$\arg\left((-1)^{\mathrm{codim}\,V}\int_{[L_V]} $e^{{-W(k\omega_0)}}$\Omega_0\right) < \arg \int_{[L]} $e^{{-W(k\omega_0)}}$\Omega_0,$$ where $L_V$ are mirror objects with morphisms $L_V[-\mathrm{codim}\,V]\to L$. When $X$ is a toric weak del Pezzo surface, the supercritical assumption can be dropped. The proof shows this phase condition is equivalent, through the Gamma-class mirror transform, to the Nakai-Moishezon criterion for a supercritical deformed Hermitian Yang-Mills connection on $L^\vee$.

Load-bearing premise

The proof assumes that one large rescaling constant can be chosen to work simultaneously for all toric subvarieties and that, for generic Kähler classes, the approximate comparison of leading-order quantities is always strict; if these fail, the claimed equivalence breaks.

Editorial extensions

If this is right

  • On toric weak del Pezzo surfaces, a naturally defined Bridgeland stable Lagrangian section $L[1]$ is Hamiltonian isotopic to a shifted special Lagrangian section; the converse holds on the blow-ups of $\mathbb{P}^2$ at one or two points, and conditionally on the stated destabilisation conjecture in general.
  • When the phase inequalities fail, the Lagrangian is Bridgeland unstable and there is a morphism to a weak solution of the special Lagrangian equation with minimal phase, providing a toric counterpart of the conjectured unstable case.
  • For an arbitrary projective toric manifold, a weaker analogue holds in which the phase inequalities involve complex cycles with strictly nested support, though these cycles are not yet known to come from exact triangles.
  • For rank-two bundles on toric weak del Pezzo surfaces, higher-rank dHYM positivity with a B-field implies the same kind of phase inequalities for periods, and violations force Bridgeland instability of the mirror object.
  • The common rescaling invariance of the dHYM equation and the uniform choice of $k$ mean the theorem applies simultaneously to all sufficiently high tensor powers of $L$, with $k$ uniform over bounded sets in K-theory.

Reading between the lines

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

  • A plausible extension the paper does not pursue: the same leading-order comparison should work for any compact toric manifold with a Gamma-class theorem, with the missing ingredient being a categorical interpretation of the complex cycles in Theorem 2.20.
  • The strictness in the phase inequalities is probably not removable: equality should correspond to semistable-but-not-stable objects, so non-generic Kähler classes should produce sections that are semistable but not special, a toric analogue of wall-crossing.
  • A quantitative check: the gap between the two arguments in (1.3) has a definite size at order $O(k^{-1})$; computing it on a del Pezzo surface would give a numerical Thomas-Yau threshold and could be compared with known dHYM wall-chamber decompositions.
  • For higher rank, the two phase inequalities with a B-field suggest that special Lagrangian multi-sections should be governed by the same slope conditions, with the sub-bundle and curve-restriction morphisms as destabilisers; the example of Section 8.2 is a test case where dHYM solvability is not yet known.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a toric version of the Thomas-Yau conjecture. For a toric weak Fano manifold X with a fixed torus-invariant Kähler form, a SYZ Lagrangian section mirror to a high power L^k of a line bundle is claimed to be Hamiltonian isotopic to a supercritical special Lagrangian section if and only if certain phase inequalities hold for periods attached to all toric subvarieties. The proof reduces this statement to the supercritical dHYM Nakai-Moishezon criterion of Chu-Li-Takahashi, then uses Iritani's toric Gamma theorem and toric homological mirror symmetry in the sense of Fang-Liu-Treumann-Zaslow to translate dHYM positivity into phase inequalities for periods. Applications are given to Bridgeland stability for SYZ sections on toric weak del Pezzo surfaces, to the unstable case via weak dHYM solutions, to a higher-rank analogue, and to a weaker statement for general projective toric manifolds.

Significance. If the main theorem is established, it is a substantial result: it gives a concrete, numerically checkable criterion for existence of special Lagrangian representatives in a natural toric setting, and it makes contact with several strands of current research: the dHYM Nakai-Moishezon criterion, the Gamma conjecture, homological mirror symmetry, and Joyce-Li stability conjectures. A notable strength of the paper is its clean reduction: the argument contains no fitted parameters or ad-hoc numerical inputs, and the main claim is reduced to independently established external theorems. The surface application is also valuable, and the paper is careful to flag where it relies on the Arcara-Miles conjecture. The principal risk is not circularity but the quantitative asymptotic control in the complete-intersection induction; if that gap is filled, the main theorem appears credible and significant.

major comments (3)
  1. [§5.2.2–5.2.3] The proof of Theorem 2.5 for general toric submanifolds is not complete. The divisor case in §5.2.1 is written out and gives a plausible leading-order comparison, but the induction for complete intersections is only sketched. In the displays leading to (5.11) and (5.12), and in the final display of §5.2.3, remainder terms are asserted to be dominated by the leading term, but no uniform bound is supplied. More specifically, the proof must show that after dividing by the displayed prefactors, the error terms are o(1) uniformly over all toric submanifolds V and over the choices k, \hat k >> k_1, ..., k_r >> 1. This uniformity is load-bearing: the if-and-only-if statement of Theorem 2.5 and hence of Corollary 2.9/Theorem 1.2 requires that a leading-order non-strict inequality can be upgraded to a strict phase inequality by the genericity of [ω0], and that this happens for every toric submanifold simultaneously. Without a completed estimate, the converse direction of the main theorem is not established.
  2. [§5.2.2, Eq. (5.11)–(5.12)] The objects appearing in the complete-intersection step are K-theory differences of two complexes, and the period integral is a difference of two Gamma-cycle integrals. The phase of a difference of two complex numbers is not determined by the phases of the individual terms unless one has a precise bound on their relative sizes. In (5.11), the right-hand side is a difference of two oscillatory integrals, and in (5.12) this is replaced by the integral over a cone object. The paper states that the natural choice h = k_2 gives the displayed leading term, but it does not show that the difference of the two Γ-integrals is approximated by the leading term with a sign that is uniform in all parameters. This is not a cosmetic issue: the claimed equivalence of dHYM positivity with the phase inequalities for higher-codimension toric subvarieties depends on exactly this approximation.
  3. [§5.2.1, Eq. (5.7) and Remark 5.5] Even in the divisor case, the conversion from dHYM positivity to the phase inequalities uses an approximate equality rather than an exact one. The proof correctly identifies the leading-order term and uses genericity to rule out equality, but the strictness argument should be stated more carefully: from the inequality (5.9) one obtains only a non-strict inequality of leading-order phases, and then genericity of [ω0] is used to exclude equality. This step is valid only if the error terms after division by the prefactor are uniformly small for all divisors and all line bundles in a bounded K-theory class. The paper asserts this uniformity but does not give the required bounds; a precise statement of the error estimates would strengthen the proof considerably.
minor comments (5)
  1. [§5.2.2] In the display after the sentence beginning "is equivalent to", the right-hand side is written as an integral without "arg"; it should read "arg ∫ Γ(L^{⊗k\tilde k}) ...".
  2. [§5.2.2] There is a typo in "in the uion of finitely many proper analytic subvarieties"; it should be "union".
  3. [Introduction, Theorem 1.9] "Hamiltionian" should be "Hamiltonian".
  4. [§5.2.2] The display referenced as "the expansion (5.2.2)" is not numbered; the reference is confusing and should be fixed.
  5. [§3.2] The derivation of the phase inequalities assumes that the relevant periods do not vanish and that arg is chosen continuously; it would be helpful to state this convention explicitly when first introducing the inequalities (1.3).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main equivalence reduces to the independent dHYM Nakai-Moishezon criterion, Iritani's toric Gamma theorem, and toric homological mirror symmetry, with no fitted input renamed as prediction.

full rationale

The derivation chain in the paper is self-contained against external theorems rather than circular. The central claim (Theorem 1.2 / Corollary 2.9) asserts that a SYZ Lagrangian section is Hamiltonian isotopic to a supercritical special section iff the phase inequalities (1.3) hold. This is obtained by composing three independent, externally established results: (a) the Leung-Yau-Zaslow correspondence, which identifies the special Lagrangian condition for the SYZ section with the dHYM equation on the dual line bundle (Remark 2.1); (b) the Chu-Li-Takahashi Nakai-Moishezon criterion (Theorem 2.2), which equates dHYM solvability with the numerical positivity condition (2.3); and (c) Iritani's toric Gamma theorem (Theorem 5.1), used in Section 5.2 to expand periods of e^{-W(kω0)}Ω0 and identify them, up to controlled error terms, with integrals of e^{-iω0}ch(L) over toric subvarieties. The phase inequalities (1.3) are therefore the translated form of the dHYM Nakai-Moishezon inequalities, not an independent quantity being fitted. No parameter is fitted to data and then called a prediction; the scale factor k is a free large parameter chosen uniformly, and the genericity assumption on [ω0] (Remark 2.6(v)) is an explicitly stated hypothesis excluding strict semistabilizers. There are also no load-bearing self-citations: the paper is single-authored and its cited inputs (Collins-Yau, Collins-Shi, Iritani, Fang-Liu-Treumann-Zaslow, Chu-Li-Takahashi, Arcara-Miles, etc.) are external works. The converse on surfaces is conditional on the Arcara-Miles conjecture 6.1, which is openly flagged as a conjecture and not presented as a theorem; relying on a stated conjecture is a completeness condition, not circularity. The complete-intersection induction in Sections 5.2.2-5.2.3 is only sketched, and the remainder estimates are not fully written out; this is a possible correctness gap, but the argument does not reduce to its own conclusion by construction. The paper explicitly acknowledges the limitation and states which cases are worked out in detail, so the honest finding is that no significant circularity is present.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The main theorem does not fit any numerical parameters. It relies on several external theorems (dHYM criterion, Gamma theorem, toric HMS, SYZ correspondence) and on a genericity hypothesis; the surface converse additionally depends on the open Arcara-Miles conjecture. No new postulated entities such as new particles, forces, or dimensions are introduced.

assumptions (8)
  • domain assumption Nakai-Moishezon criterion for supercritical dHYM (Chu-Li-Takahashi [8, Cor 1.5]): equation (2.1) is solvable for a projective manifold if and only if inequality (2.3) holds on all proper irreducible subvarieties.
    Used as a black box in Theorem 2.5 to identify dHYM solvability with numerical positivity; not proven in this paper.
  • domain assumption Iritani's toric Gamma theorem (Theorem 5.1): an isomorphism Γ: K0(X) → H_n(Yq, {Re(W/z)>>0}; Z) identifies Gamma-class intersection numbers with oscillatory periods.
    Central translation step in Section 5.2; equation (5.4) is the scaled consequence of this theorem.
  • domain assumption Toric homological mirror symmetry (FLTZ [19] and non-equivariant versions): Db(X) is equivalent to DFS(Yq, W(ω0)), compatible with the SYZ transform and integration cycles (Fang [18]).
    Used to interpret cycles ΓV and morphisms in terms of Lagrangian objects LV and exact triangles in Fukaya-Seidel categories.
  • domain assumption Leung-Yau-Zaslow SYZ correspondence adapted to compact toric manifolds: the dHYM equation on the dual line bundle is equivalent to the special Lagrangian equation on the SYZ Lagrangian section.
    Bridges PDE solvability to geometric Hamiltonian isotopy in Corollary 2.9; cited to [5,29].
  • domain assumption Genericity of the Kähler class [ω0]: [ω0] avoids finitely many proper analytic subvarieties determined by c1(L) and toric submanifolds where strict phases become equalities.
    Hypothesis in Theorem 2.5 and Remark 2.6(v); needed to convert non-strict leading-order inequalities into strict phase inequalities.
  • domain assumption Arcara-Miles conjecture (Conjecture 6.1): on a smooth projective surface, unstable line bundles or shifts are destabilized by objects from negative self-intersection curves.
    Used for the converse direction of Proposition 6.2 and Theorem 1.9; known only for Bl_p P2 and Bl_{p,q} P2.
  • domain assumption Datar-Mete-Song weak-solution theorem for dHYM on surfaces (Theorem 6.4): minimal angle and existence and uniqueness of a closed current weak solution.
    Used in Section 6.1 to construct weak special Lagrangian sections in the unstable case.
  • domain assumption Hodge-theoretic mirror symmetry for toric stacks (Coates-Corti-Iritani-Tseng [9]): mirror map and higher residue pairing identify quantum cohomology with periods.
    Used in Section 9 for the weaker analogue for general projective toric manifolds (Theorem 2.20).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Nakai-Moishezon criteria and the toric Thomas-Yau conjecture." pith.science (2026). https://pith.science/paper/WDUYNYM3

@misc{pith2026250507228,
  author       = {Pith},
  title        = {Pith review of: Nakai-Moishezon criteria and the toric Thomas-Yau conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDUYNYM3}},
  note         = {Machine review of arXiv:2505.07228}
}
abstract

We consider a class of Lagrangian sections $L$ contained in certain Calabi-Yau Lagrangian fibrations (mirrors of toric weak Fano manifolds). We prove that a form of the Thomas-Yau conjecture holds in this case: $L$ is Hamiltonian isotopic to a special Lagrangian section in this class if and only if a stability condition holds, in the sense of a slope inequality on objects in a set of exact triangles in the Fukaya-Seidel category. This agrees with general proposals by Li. We use the SYZ transform, the toric gamma theorem, and toric homological mirror symmetry in order to reduce the statement to one about supercritical deformed Hermitian Yang-Mills connections, known as the Nakai-Moishezon criterion. As an application, we prove that, on the mirror of a toric weak del Pezzo surface, if $L$ defines a Bridgeland stable object in the Fukaya-Seidel category in a natural sense, then it is Hamiltonian isotopic to a special Lagrangian section in the class. The converse also holds for the mirror of the projective plane blown-up at one or two points, and always holds assuming a conjecture of Arcara and Miles. When $L$ is Bridgeland unstable, we obtain a morphism from $L$ to a weak solution of the special Lagrangian equation with phase angle satisfying a minimality condition. These results are consistent with general conjectures due to Joyce. We discuss some generalisations, including a weaker analogue of our main result for general projective toric manifolds, and a similar obstruction, related to Lagrangian multi-sections, in a special case.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 31 canonical work pages

  1. [1]

    Abouzaid

    M. Abouzaid. Morse homology, tropical geometry, and homologic al mirror symme- try for toric varieties. Selecta Math. (N.S.) , 15(2):189–270, 2009

    work page 2009

  2. [2]

    Abouzaid, S

    M. Abouzaid, S. Ganatra, H. Iritani, and N. Sheridan. The gamma and Strominger- Yau-Zaslow conjectures: a tropical approach to periods. Geom. Topol., 24(5):2547– 2602, 2020

    work page 2020

  3. [3]

    Arcara and E

    D. Arcara and E. Miles. Bridgeland stability of line bundles on surfac es. J. Pure Appl. Algebra, 220(4):1655–1677, 2016

    work page 2016

  4. [4]

    Aspinwall, T

    P. Aspinwall, T. Bridgeland, A. Craw, M. Douglas, M. Gross, A. Kap ustin, G. Moore, G. Segal, B. Szendr˝ oi, and P. Wilson. Dirichlet branes and mirror symmetry, volume 4 of Clay Mathematics Monographs . American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2009

    work page 2009

  5. [5]

    K. Chan. SYZ mirror symmetry for toric varieties. In Handbook for mirror symmetry of Calabi-Yau & Fano manifolds , volume 47 of Adv. Lect. Math. (ALM), pages 1–32. Int. Press, Somerville, MA, [2020] ©2020

    work page 2020

  6. [6]

    Chan and N

    K. Chan and N. C. Leung. Mirror symmetry for toric Fano manifold s via SYZ transformations. Adv. Math. , 223(3):797–839, 2010

    work page 2010

  7. [7]

    G. Chen. The J-equation and the supercritical deformed Hermit ian-Yang-Mills equation. Invent. Math. , 225(2):529–602, 2021

    work page 2021

  8. [8]

    Chu, M.-C

    J. Chu, M.-C. Lee, and R. Takahashi. A Nakai-Moishezon type crit erion for supercritical deformed Hermitian-Yang-Mills equation. J. Differential Geom. , 126(2):583–632, 2024. 44

    work page 2024

Show all 38 references
  1. [9]

    Coates, A

    T. Coates, A. Corti, H. Iritani, and H.-H. Tseng. Hodge-theore tic mirror symmetry for toric stacks. J. Differential Geom. , 114(1):41–115, 2020

    2020

  2. [10]

    Collins, A

    T. Collins, A. Jacob, and S.-T. Yau. (1 , 1) forms with specified Lagrangian phase: a priori estimates and algebraic obstructions. Camb. J. Math. 8 (2020) , 8(2):407–452, 2020

    2020

  3. [11]

    T. C. Collins and Y. Shi. Stability and the deformed Hermitian-Yang -Mills equa- tion. In Surveys in differential geometry 2019. Differential geometr y, Calabi-Yau theory, and general relativity. Part 2 , volume 24 of Surv. Differ. Geom. , pages 1–38. Int. Press, Boston, MA, [2022] ©2022

    2019

  4. [12]

    T. C. Collins and G. Sz´ ekelyhidi. Convergence of the J-flow on to ric manifolds. J. Differential Geom. , 107(1):47–81, 2017

    2017

  5. [13]

    T. C. Collins and S.-T. Yau. Moment maps, nonlinear PDE, and stab ility in mirror symmetry. arXiv:1811.04824 [math.DG]

    arXiv

  6. [14]

    T. C. Collins and S.-T. Yau. Moment maps, nonlinear PDE and stabilit y in mirror symmetry, I: geodesics. Ann. PDE , 7(1):Paper No. 11, 73, 2021

    2021

  7. [15]

    Datar, R

    V. Datar, R. Mete, and J. Song. Minimal slopes and bubbling for c omplex Hessian equations. arXiv:2312.03370 [math.DG]

    arXiv

  8. [16]

    V. V. Datar and V. P. Pingali. A numerical criterion for generalise d Monge-Amp` ere equations on projective manifolds. Geom. Funct. Anal. , 31(4):767–814, 2021

    2021

  9. [17]

    Dervan, J

    R. Dervan, J. B. McCarthy, and L. M. Sektnan. Z-critical connections and Bridge- land stability conditions. Camb. J. Math. , 12(2):253–355, 2024

    2024

  10. [18]

    B. Fang. Central charges of T-dual branes for toric varietie s. Trans. Amer. Math. Soc., 373(6):3829–3851, 2020

    2020

  11. [19]

    Fang, C.-C

    B. Fang, C.-C. M. Liu, D. Treumann, and E. Zaslow. T-duality and homological mirror symmetry for toric varieties. Adv. Math. , 229(3):1875–1911, 2012

    1911

  12. [20]

    Givental

    A. Givental. A mirror theorem for toric complete intersections. In Topological field theory, primitive forms and related topics (Kyoto, 199 6), volume 160 of Progr. Math., pages 141–175. Birkh¨ auser Boston, Boston, MA, 1998

    1998

  13. [21]

    H. Iritani. An integral structure in quantum cohomology and mir ror symmetry for toric orbifolds. Adv. Math. , 222(3):1016–1079, 2009

    2009

  14. [22]

    H. Iritani. Quantum D-modules of toric varieties and oscillatory in tegrals. In Handbook for mirror symmetry of Calabi-Yau & Fano manifolds , volume 47 of Adv. Lect. Math. (ALM) , pages 131–147. Int. Press, Somerville, MA, [2020] ©2020

    2020

  15. [23]

    Jacob and N

    A. Jacob and N. Sheu. The deformed Hermitian-Yang-Mills equat ion on the blowup of Pn. Asian J. Math. , 26(6):847–864, 2022. 45

    2022

  16. [24]

    Jacob and S.-T

    A. Jacob and S.-T. Yau. A special Lagrangian type equation for holomorphic line bundles. Math. Ann. , 369(1-2):869–898, 2017

    2017

  17. [25]

    D. Joyce. Conjectures on Bridgeland stability for Fukaya cate gories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature fl ow. EMS Surv. Math. Sci. , 2(1):1–62, 2015

    2015

  18. [26]

    Keller and C

    J. Keller and C. Scarpa. Z-critical equations for holomorphic vector bundles on K¨ ahler surfaces.arXiv:2405.03312 [math.DG]

    arXiv

  19. [27]

    Khalid and Z

    S. Khalid and Z. Sj¨ ostr¨ om Dyrefelt. Wall-chamber decompositions for generalized Monge-Amp` ere equations. arXiv:2412.20089 [math.DG]

    arXiv

  20. [28]

    Kuwagaki

    T. Kuwagaki. The nonequivariant coherent-constructible cor respondence for toric stacks. Duke Math. J. , 169(11):2125–2197, 2020

    2020

  21. [29]

    N. C. Leung, S.-T. Yau, and E. Zaslow. From special Lagrangian to Hermitian- Yang-Mills via Fourier-Mukai. Adv. Theor. Math. Phys. , 4(6):1319–1341, 2000

    2000

  22. [30]

    Y. Li. Thomas-Yau conjecture and holomorphic curves. arXiv:2 203.01467 [math.SG]

  23. [31]

    Mizuno and T

    Y. Mizuno and T. Yoshida. Bridgeland stability of sheaves on del P ezzo surfaces of Picard rank three. arXiv:2502.18894 [math.AG]

    arXiv

  24. [32]

    V. P. Pingali. A vector bundle version of the Monge-Amp` ere equ ation. Adv. Math., 360:106921, 40, 2020

    2020

  25. [33]

    Scherotzke and N

    S. Scherotzke and N. Sibilla. The non-equivariant coherent-co nstructible correspon- dence and a conjecture of King. Selecta Math. (N.S.) , 22(1):389–416, 2016

    2016

  26. [34]

    J. Song. Nakai-Moishezon criterions for complex Hessian equat ions. arXiv:2012.07956 [math.DG]

    2012 arXiv

  27. [35]

    Takahashi

    R. Takahashi. J-equation on holomorphic vector bundles. J. Funct. Anal. , 286(4):Paper No. 110265, 64, 2024

    2024

  28. [36]

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

    2000

  29. [37]

    R. P. Thomas and S.-T. Yau. Special Lagrangians, stable bundle s and mean cur- vature flow. Comm. Anal. Geom. , 10(5):1075–1113, 2002

    2002

  30. [38]

    P. Zhou. Twisted polytope sheaves and coherent-construct ible correspondence for toric varieties. Selecta Math. (N.S.) , 25(1):Paper No. 1, 23, 2019. SISSA, via Bonomea 265, 34136 Trieste, Italy Institute for Geometry and Physics (IGAP), via Beirut 2, 34151 Tr ieste, Italy jst...

    2019

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.