Pith. sign in

REVIEW 4 major objections 4 minor 1 cited by

Polynomial stability conditions for vector bundles: Positivity, equivariance and blow-ups

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read On toric varieties, P-positivity reduces to a finite test set, and a blow-up sign criterion builds new positive bundles.

desk verdict Useful, honest paper with real toric finiteness and blow-up results, but the equivariance theorem rests on a one-line citation that the authors should make into a proof. read the letter →

arxiv 2506.23842 v1 pith:JJCDZINC submitted 2025-06-30 math.AG

Rémi Delloque , Achim Napame , Carlo Scarpa , Carl Tipler This is my paper · ORCID
classification math.AG MSC 14J6014M2553C0714E05
keywords P-criticalconnectionsPδ-positivityZ-criticalequationsBridgelandstabilityconditionstoricvarietiesequivariantblow-upsvectorbundles
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

This paper builds a common framework for curvature equations on holomorphic vector bundles whose solutions are described by a polynomial equation in the curvature, the P-critical equations, which include the Hermitian-Yang-Mills, deformed Hermitian-Yang-Mills, Monge-Ampère, and J-equations. It then gives two practical tools for the positivity condition that obstructs solutions: on smooth projective toric varieties, Pδ-positivity of a reflexive sheaf is equivalent to the same inequalities on the finitely many torus orbit closures, so the test set does not depend on the polynomial or the bundle. A uniform version of Pδ-positivity is preserved by pulling back along a point blow-up exactly when a short list of sign conditions on the deformation coefficients holds. This yields new examples of positive tangent bundles on toric surfaces and a Fano threefold, and a slope-unstable bundle that still admits P-critical connections in an asymptotic regime.

What carries the argument

The machinery consists of the P-critical equation $P(F(\nabla)) = \sum_{j=0}^n \gamma_j \wedge F(\nabla)^j/j! = 0$ with closed coefficient forms $\gamma_j$, together with the numerical quantities $P_V(E) = (\sum_j [\gamma_j] \smallsmile \mathrm{ch}_j(E)) \smallfrown [V]$ used to define Pδ-positivity: $\delta_k P_V(E) > 0$ for every $k$-dimensional subvariety. The proof of the equivariance theorem rests on Brion's result that effective cycles on a smooth projective variety with a connected solvable affine group action are rationally equivalent to positive rational combinations of invariant effective cycles. The finiteness in the toric case comes from the orbit-cone correspondence, which identifies orbit closures with cones in the fan. For the blow-up theorem the key object is the pseudo-effective cone $\mathrm{Eff}^k(X)$ and its behaviour under the cohomological decomposition of a point blow-up, $c = \pi^*\pi_*c + (-1)^{n-k} l(c)\mathrm{cl}(D)^{n-k}$, which converts positivity of all cycles into positivity of the pulled-back classes plus open conditions on the coefficients $\varepsilon_j$.

What would settle it

Take a smooth projective toric threefold and a reflexive sheaf E that satisfies $\delta_k P_V(E) > 0$ for every torus orbit closure V but fails Pδ-positivity on some non-invariant subvariety; a direct intersection-theoretic computation of $P_W(E)$ for that W would disprove Corollary 1.10. Alternatively, on a point blow-up where the sign conditions $(-1)^k\delta_k\varepsilon_{n-k}>0$ hold with small $|\varepsilon_k|$, finding a class $c \in \mathrm{Eff}^k(\tilde{X})$ with $\delta_k \tilde{P}_{\varepsilon,c}(\pi^*E) \le 0$ would disprove Theorem 1.13.

Watch

Extended reading notes

Core claim

The central claim is that the numerical condition Pδ-positivity, which for suitable polynomials is the expected obstruction to solving P-critical equations, can be checked equivariantly and in a finite way. Theorem 1.9 proves that for a connected solvable affine algebraic group action, a reflexive sheaf is (strongly) Pδ-positive if and only if the same sign inequalities hold on invariant subvarieties; on toric varieties the invariant subvarieties to test are exactly the closures of torus orbits, a finite collection determined by the fan. Theorem 1.13 shows that if E is uniformly Pδ-positive on X, then its pullback to the blow-up of a point is P̃ε,δ-positive exactly when ε0 = (−1)^n ch_n(E)·[X]/rk(E) ε_n and (−1)^k δ_k ε_{n−k} > 0 for 1 ≤ k ≤ n−1, provided the coefficients are small enough. The paper also introduces equivariant P-stability and proves it agrees with asymptotic P-stability for polynomials adapted to torsion-free sheaves, which is then used to exhibit new positive and stable bundles.

Load-bearing premise

The load-bearing premise is Brion's theorem, invoked in the proof of Theorem 1.9, that on a smooth projective variety with a connected solvable affine algebraic group action, every effective cycle is rationally equivalent to a positive rational combination of invariant effective cycles; if this fails for the relevant cycle classes, the equivariant equivalence and hence the finite toric test set collapse, and the blow-up theorem's proof further relies on Lemma 4.5 describing the pseudo-effective cones of a point blow-up.

Editorial extensions

If this is right

  • On any smooth projective toric variety, Pδ-positivity of a reflexive sheaf becomes a finite computation using only the fan: check $\delta_k P_V(E) > 0$ on torus orbit closures, independent of the polynomial P and of the bundle.
  • The blow-up theorem supplies a machine for constructing positive bundles: starting from a uniformly positive E, every small perturbation vector ε satisfying the displayed sign conditions yields a positive bundle on the blow-up.
  • Combined with the large-volume-limit correspondence for Z-critical equations, equivariant P-stability provides a practical route to proving existence of P-critical metrics for toric sheaves.
  • The examples show that positivity is compatible with slope instability: the rank-3 tangent bundle on $\mathbb{P}(O_{\mathbb{P}^2} \oplus O_{\mathbb{P}^2}(1))$ is asymptotically P-stable but not slope polystable, and hence carries P-critical connections in the asymptotic regime.
  • The P-functional is convex around P-positive critical metrics, giving local uniqueness of solutions up to holomorphic automorphism.

Reading between the lines

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

  • The finite-test philosophy of Corollary 1.10 may extend beyond toric varieties to other varieties with a solvable group action whose invariant cycles generate the Chow group; spherical varieties are a natural next class, but the paper only proves the toric statement.
  • The sign conditions $(-1)^k\delta_k\varepsilon_{n-k}>0$ suggest a chamber structure in the space of polynomial coefficients around each positive bundle; one could compute these chambers algorithmically for toric examples and compare them with walls of the Bridgeland stability space.
  • The example of a P-positive dHYM metric that is not Pδ-positive implies that for higher rank the correct numerical criterion likely needs the positivity of $P'_Y(Q)$ for all torsion-free quotients, as the paper has flagged; testing this on the toric examples would settle whether the stronger condition is also sufficient in practice.
  • The blow-up criterion may be iterated: successive point blow-ups with appropriately chosen ε give a tower of positive bundles, suggesting a construction of positive bundles on large families of rational varieties starting from a single toric seed.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper introduces P-critical equations for Hermitian metrics on holomorphic vector bundles, a slight generalization of the Z-critical equations of Dervan–McCarthy–Sektnan. It gives the moment-map interpretation, a variational description, and a local uniqueness result. The main new results are an equivariance theorem for Pδ-positivity (Theorem 1.9), a finite toric test set for Pδ-positivity (Corollary 1.10), an equivariant asymptotic stability theorem (Theorem 1.11), and a blow-up criterion for uniformly Pδ-positive bundles (Theorem 1.13). Applications include Z-positivity of tangent and anticanonical bundles on Hirzebruch surfaces and on a toric Fano 3-fold, and an example of an asymptotically P-stable but slope-unstable tangent bundle.

Significance. If the equivariance and blow-up theorems hold, the paper is valuable: it reduces Pδ-positivity on toric varieties to a finite, polynomial-independent list of orbit closures, and it gives a practical construction of new positive bundles by pullback along point blow-ups. The toric computations in Section 5 are explicit and reproducible, and the counterexample in Section 3.1.1 correctly shows that P-positivity of a P-critical metric does not imply Pδ-positivity. The main caveats are the heavy reliance on external theorems, especially Brion's theorem and the companion paper [17], and a few sketched arguments in Section 4 that need to be completed.

major comments (4)
  1. [Section 3.1, proof of Theorem 1.9] The proof of Theorem 1.9 consists entirely of the assertion that every effective cycle is rationally equivalent to a positive combination of G-invariant effective cycles, with a citation to Brion [4, Theorem 1.3] and [22]. Since Corollary 1.10 and several applications depend on this statement, the authors should state the precise theorem from [4] that they are using and verify that its hypotheses cover an arbitrary smooth projective variety with an action of an arbitrary connected solvable affine algebraic group. If the cited theorem is only stated for spherical varieties, the theorem as written is not proved; alternatively, the toric case needed for Corollary 1.10 could be proved directly by degenerating a cycle by a one-parameter subgroup.
  2. [Section 3.3, Theorem 1.11] Theorem 1.11 is a load-bearing result for the example in Section 5.2, but its proof is a sketch that refers to [17, Theorems 2.9 and 2.10] for the existence and uniqueness of Harder–Narasimhan and Jordan–Hölder filtrations for the relevant polynomial stability conditions. The authors should state the precise results from [17] that are being used and explain how they imply the equivariant-to-nonequivariant reduction. As written, the reader cannot verify the theorem without consulting the companion preprint.
  3. [Section 4.1, Lemma 4.5] The proof of Lemma 4.5(1) is incomplete. The construction of sṼ1 = π−1(V)∖D assumes that sṼ1 meets D properly; if V does not contain the blown-up point, then sṼ1 ∩ D is empty and the degree argument does not apply. This case should be separated and treated directly. Lemma 4.5 is used in the proof of Theorem 1.13, so a complete proof is needed. The compactness argument in Lemma 4.5(3), which considers a function taking the value −∞, should also be stated more rigorously.
  4. [Definitions 1.12, 4.3 and Proposition 4.2] The notation “Eff^k(X)” is used inconsistently for both the effective cone and its closure, the pseudo-effective cone. Proposition 4.2's compactness argument is valid only if the cone in condition (1) is closed. The current text can be read as claiming an equivalence for the open effective cone, which is false in general. Please use separate notation, for example Psef^k(X) for the pseudo-effective cone, and make clear that uniform Pδ-positivity is defined by positivity on the closure.
minor comments (4)
  1. [Corollary 1.10] The displayed condition “0(≤) < k= dim(V) < dim(X)” is garbled; it should read “0 ≤ k < dim(X)” for strong positivity and “0 < k < dim(X)” otherwise.
  2. [Section 2.1, Definition 2.1] In the definition of P-positivity, the tangent variable should be written ξ ∈ T^{0,1}_p X ⊗ End E_p rather than ξ ∈ T^{0,1}_p X × End E_p; as written the tensor product is missing.
  3. [Theorem 1.13] In the final sentence of the proof, the condition “|\varepsilon_n| < \gamma_n” should be “|\varepsilon_n| < |\gamma_n|” or the sign convention for γ_n should be stated, since γ_n may be negative.
  4. [Section 5.2] The computation checks the rank-one subsheaves F and G, but the rank-two direct sums are dismissed without comment. Since Pα is additive in direct sums, the inequality for those subsheaves follows from the rank-one inequalities; adding this one-line explanation would make the example complete.

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem 1.11's equivariant stability equivalence is delegated to the first author's companion uniqueness theorems; toric finiteness and blow-up results are independently derived.

  1. uniqueness imported from authors [Section 3.3, paragraph after Definition 3.11 (proof of Theorem 1.11)]
    "In asymptotic regimes, for asymptotic polynomial equations adapted to torsion-free sheaves, using the results from [17], we obtain Theorem 1.11. Indeed, the proof is the same as the one given by Kool in [32, Proposition 3.19], once we have the see-saw property of P-slopes (that follows from additivity of the Chern character), and the existence and uniqueness of the aforementioned filtrations [17, Theorems 2.9 and 2.10]."

    Theorem 1.11 is the paper's equivariant-versus-global asymptotic P-stability equivalence. The proof is not carried out in the paper: it stops at a citation to the first author's companion preprint [17] for the existence and uniqueness of Harder-Narasimhan/Jordan-Holder filtrations for the same class of 'adapted' polynomials. Those uniqueness theorems are exactly what force the filtrations to be equivariant and make the equivariant-to-global reduction work. They are not proved or reproduced here, and they are not machine-checked or otherwise independently verified in the present text, so the claimed equivalence reduces at its load-bearing step to an unverified self-citation rather than to an internal derivation.

full rationale

The main derivation chains for the paper's headline claims are not circular. Theorem 1.9 reduces equivariant P-delta-positivity to ordinary P-delta-positivity by invoking Brion's theorem [4, Theorem 1.3] that effective cycles are rationally equivalent to G-invariant effective cycles; Corollary 1.10 is then a direct application to toric varieties, where invariant cycles are torus orbit closures. This is an external geometric input, not a restatement of the conclusion. Theorem 1.13 is proved by explicit cohomological estimates: Lemma 4.6 computes the pulled-back polynomial evaluations, Lemma 4.5 controls pseudo-effective classes on the blow-up, and Proposition 4.2 supplies the uniform-positivity margin. The epsilon conditions in Theorem 1.13 are derived from these estimates, not fitted from the desired conclusion. The only substantive self-citation issue is Theorem 1.11, whose proof delegates the existence and uniqueness of the relevant Harder-Narasimhan/Jordan-Holder filtrations to the first author's companion preprint [17]. That is a load-bearing self-citation at the point where equivariant stability is forced to imply global stability. Because the paper's strongest constructive results, the finite toric test set and the blow-up preservation theorem, are established independently of that self-citation, the appropriate score is moderate rather than high.

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

The new theorems are not free of external inputs: the equivariance theorem relies on a deep result of Brion, the asymptotic P-stability equivalence relies on the prior Z-critical program and on the first author's companion paper, and the toric computations rely on standard toric geometry. No new particles, forces, or ad hoc constants are postulated; the epsilon deformation parameters in the blow-up theorem are variables, not fitted values.

assumptions (6)
  • domain assumption Brion's theorem [4, Thm 1.3]: effective cycles are rationally equivalent to G-invariant effective cycles for connected solvable affine algebraic groups.
    Used in the proof of Theorem 1.9; the entire equivariance reduction and toric finiteness result depends on it.
  • domain assumption Dervan-McCarthy-Sektnan large volume limit theorem [19, Thm 1.1], restated as Theorem 2.13.
    Links asymptotic P-stability to existence of P-critical metrics, used in the application in Section 5.2.
  • domain assumption Harder-Narasimhan and Jordan-Holder filtrations for adapted asymptotic polynomials, from the companion paper [17, Thms 2.9, 2.10].
    Load-bearing for Theorem 1.11: without these filtrations, equivariant asymptotic P-stability does not imply P-stability by Kool's argument.
  • domain assumption Keller-Scarpa theorems [27, Thms 1.1 and 1.5] on P-critical and P-positive metrics implying P-stability, and positivity restrictions on quotients.
    Sets up the numerical framework and motivates Definitions 1.6 and the quotient version of positivity.
  • standard math Klyachko-Perling classification of equivariant reflexive sheaves on toric varieties by filtrations, and the orbit-cone correspondence.
    Used in Lemma 3.8 and in the efficient computations of Section 5.
  • standard math Toric intersection theory as in Cox-Little-Schenck, including the cycle class isomorphism and intersection formulas for orbit closures.
    Enables computation of P_V(E) and the explicit examples.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Polynomial stability conditions for vector bundles: Positivity, equivariance and blow-ups." pith.science (2026). https://pith.science/paper/JJCDZINC

@misc{pith2026250623842,
  author       = {Pith},
  title        = {Pith review of: Polynomial stability conditions for vector bundles: Positivity, equivariance and blow-ups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JJCDZINC}},
  note         = {Machine review of arXiv:2506.23842}
}
read the original abstract

We introduce the notion of P-critical connections for hermitian holomorphic vector bundles over compact balanced manifolds: integrable hermitian connections whose curvature solves a polynomial equation. Such connections include HYM and dHYM connections, as well as solutions to higher rank Monge-Amp\`ere or J-equations, and are a slight generalisation of Dervan-McCarthy-Sektnan's Z-critical connections motivated by Bayer's polynomial Bridgeland stability conditions. The associated equations come with a moment map interpretation, and we provide numerical conditions that are expected to characterise existence of solutions in suitable cases: P-positivity and P-stability. We then provide some devices to check those numerical conditions in practice. First, we observe that P-positivity is equivalent to its equivariant version over T-varieties. In the toric case, we thus obtain an explicit finite set of subvarieties to test P-positivity on, independently on the choice of the polynomial equation. We also introduce equivariant P-stability and discuss its relation to P-stability. Secondly, we show that a uniform version of P-positivity is preserved by pulling back along a blow-up of points. We apply those results to some examples, such as blow-ups of Hirzebruch surfaces, or a Fano 3-fold.

Figures

Figures reproduced from arXiv: 2506.23842 by the authors.

Figure 1
Figure 1. Fan in R 3 Proof. Writing −KX ∼lin 3D2 + 3D5 + 2D1, we obtain (−KX) 2 = 18D2 · D5 + 8D2 · D1 + 8D1 · D5. Using D2 2 = 0 and D2 5 = 0, we get the second and the third equalities. As D2 1 = −D1 · D2 − D1 · D5, we have D2 · D2 1 = −1 and D5 · D2 1 = −1. Hence, we have (−KX) 2 · D1 = 2. □ We can now give the proof of Proposition 5.2. Proof of Proposition 5.2. By Example 3.9, we have ch(TX) = 3 − KX + D2 · D5 + 2 3 D1 · … view at source ↗

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. The deformed Vortex equations and equivariant stability conditions

    math.DG 2026-07 conditional novelty 7.0 of 10

    On vortex-type SU(2)-equivariant bundles over a product of a curve with P¹, solvability of the deformed Hermitian–Yang–Mills system is equivalent to Z-stability, and Bridgeland stability implies Z-stability.

Reference graph

Works this paper leans on

46 extracted references · 43 canonical work pages · cited by 1 Pith paper

  1. [17]

    R´ emi Delloque, Polynomial Bridgeland Stability Conditions on the Category of Coherent Sheaves, arXiv:2506.21939 [math.AG] (2025)

    work page Pith review arXiv 2025

  2. [27]

    Julien Keller and Carlo Scarpa, Z-critical equations for holomorphic vector bundles on K¨ ahler surfaces, arXiv:2405.03312 [math.DG] (2024)

    arXiv 2024

  3. [4]

    Michel Brion, Vari´ et´ es sph´ eriques et th´ eorie de Mori, Duke Math. J. 72 (1993), no. (2), 369–404

    work page 1993

  4. [22]

    MacPherson, and Bernd Sturmfels, Intersection theory on spherical varieties, J

    William Fulton, Robert Sottile F. MacPherson, and Bernd Sturmfels, Intersection theory on spherical varieties, J. Algebr. Geom. 4 (1995), no. (1), 181–193

    work page 1995

  5. [1]

    Positivity properties of the vector bundle Monge-Amp\`ere equation

    Aashirwad N. Ballal and Vamsi P. Pingali, Positivity properties of the vector bundle Monge- Amp` ere equation, arXiv:2409.00321 [math.DG] (2024), 31 pp

    work page Pith review arXiv 2024

  6. [2]

    Arend Bayer, Polynomial Bridgeland stability conditions and the large volume limit , Geom. Topol. 13 (2009), no. (4), 2389–2425

    work page 2009

  7. [3]

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

    work page 2007

  8. [5]

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

    work page 2021

Show all 46 references
  1. [6]

    Andrew Clarke, Achim Napame, and Carl Tipler, Toric sheaves and flips , Epiga (to appear); ArXiv 2309.09267 (2025), 21 pp

    2025 arXiv

  2. [7]

    Andrew Clarke and Carl Tipler, Equivariant stable sheaves and toric GIT , Proc. R. Soc. Edinb., Sect. A, Math. 153 (2023), no. 2, 385–416

    2023

  3. [8]

    Collins, Adam Jacob, and Shing-Tung Yau, (1,1)-forms with specified Lagrangian phase: a priori estimates and algebraic obstructions , Camb

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

    2020

  4. [9]

    Collins, Jason Lo, Yun Shi, and Shing-Tung Yau, Stability for line bundles and deformed Hermitian-Yang-Mills equation on some elliptic surfaces , arXiv:2306.05620 [math.AG] (2023)

    Tristan C. Collins, Jason Lo, Yun Shi, and Shing-Tung Yau, Stability for line bundles and deformed Hermitian-Yang-Mills equation on some elliptic surfaces , arXiv:2306.05620 [math.AG] (2023)

    2023 arXiv

  5. [10]

    Collins and Yun Shi, Stability and the deformed Hermitian-Yang-Mills equation , Surveys in differential geometry 2019

    Tristan C. Collins and Yun 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 (2022), 1–38

    2022

  6. [11]

    Collins and G´ abor Sz´ ekelyhidi,Convergence of the J-flow on toric manifolds , J

    Tristan C. Collins and G´ abor Sz´ ekelyhidi,Convergence of the J-flow on toric manifolds , J. Differ. Geom. 107 (2017), no. 1, 47–81

    2017

  7. [12]

    Correa, Deformed Hermitian Yang-Mills equation on rational homogeneous varieties, arXiv:2304.02105 [math.DG] (2023)

    Eder M. Correa, Deformed Hermitian Yang-Mills equation on rational homogeneous varieties, arXiv:2304.02105 [math.DG] (2023)

    2023 arXiv

  8. [13]

    , DHYM connections on higher rank holomorphic vector bundles over P(TP2 ), Math. Z. 308 (2024), no. 2, 27p

    2024

  9. [14]

    Cox, John B

    David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Grad. Stud. Math., vol. 124, Providence, RI: American Mathematical Society (AMS), 2011 (English)

    2011

  10. [15]

    Jyoti Dasgupta, Arijit Dey, and Bivas Khan, Stabiliy of equivariant vector bundle over toric varieties, Documenta Mathematica 25 (2020), 1787–1833

    2020

  11. [16]

    Ved Datar and Vamsi Pritham Pingali, A numerical criterion for generalised Monge-Amp` ere equations on projective manifolds, Geom. Funct. Anal.31 (2021), no. 4, 767–814. MR 4317503

    2021

  12. [18]

    1, e70219

    R´ emi Delloque,Continuity of HYM connections with respect to metric variations , Journal of the London Mathematical Society 112 (2025), no. 1, e70219

    2025

  13. [19]

    (2), 253–355

    Ruadha ´ ı Dervan, John Benjamin McCarthy, and Lars Martin Sektnan,Z-critical connections and Bridgeland stability conditions , Cambridge Journal of Mathematics 12 (2024), no. (2), 253–355. P-STABILITY, P-POSITIVITY, EQUIV ARIANCE AND BLOW-UPS 31

    2024

  14. [20]

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

    Simon K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles , Proc. London Math. Soc. (3) 50 (1985), no. 1, 1–26. MR 765366

    1985

  15. [21]

    Hao Fang and Biao Ma, On a fully nonlinear elliptic equation with differential forms , Adv. Math. 454 (2024), Paper No. 109867, 89. MR 4781476

    2024

  16. [23]

    A collection in honor of William Fulton’s 80th birthday

    Milena Hering, Benjamin Nill, and Hendrik S¨ uß,Stability of tangent bundles on smooth toric Picard-rank-2 varieties and surfaces , Facets of algebraic geometry. A collection in honor of William Fulton’s 80th birthday. Volume 2, Cambridge: Cambridge University Press, 2022, pp....

    2022

  17. [24]

    Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves , 2 ed., Cambridge Mathematical Library, Cambridge University Press, 2010

    2010

  18. [25]

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

    2022

  19. [26]

    126 (2024), no

    Ryosuke Takahashi Jianchun Chu, Man-Chun Lee, A Nakai–Moishezon type criterion for su- percritical deformed Hermitian–Yang–Mills equation, J.Differential Geom. 126 (2024), no. 2, 583–632

    2024

  20. [28]

    Sohaib Khalid and Zakarias Sj¨ ostr¨ om Dyrefelt,The set of destabilizing curves for deformed Hermitian Yang-Mills and Z-critical equations on surfaces , IMRN 7 (2024), 5773–5814

    2024

  21. [29]

    , Wall-chamber decompositions for generalized Monge-Amp` ere equations , arXiv:2412.20089 [math.DG] (2024)

    2024 arXiv

  22. [30]

    2, 337 – 375

    Alexander Klyachko, Equivariant bundle on toral varieties , Mathematics of the USSR- Izvestiya 35 (1990), no. 2, 337 – 375

    1990

  23. [31]

    Allen Knutson and Eric Sharpe, Sheaves on toric varieties for physics , Adv. Theor. Math. Phys. 2 (1998), no. 4, 873–961 (English)

    1998

  24. [32]

    Martijn Kool, Fixed point loci of moduli spaces of sheaves on toric varieties , Adv. Math. 227 (2011), no. 4, 1700–1755 (English)

    2011

  25. [33]

    Robert Lazarsfeld, Positivity in algebraic geometry I , 1 ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol. 48, Springer- Verlag Berlin Heidelberg, 2004

    2004

  26. [34]

    Mehdi Lejmi and G´ abor Sz´ ekelyhidi,The J-flow and stability , Adv. Math. 274 (2015), 404–

    2015

  27. [35]

    thesis, Imperial College London, 2023, https://doi.org/10.25560/103792

    John Benjamin McCarthy, Stability conditions and canonical metrics , Ph.D. thesis, Imperial College London, 2023, https://doi.org/10.25560/103792

    2023 doi

  28. [36]

    Dedicata 219 (2025), no

    Damien M´ egy, Mihai Pavel, and Matei Toma, Semistability conditions defined by ample classes, Geom. Dedicata 219 (2025), no. 1, Paper No. 18, 19. MR 4852975

    2025

  29. [37]

    Achim Napame, Stability of equivariant logarithmic tangent sheaves on toric varieties of Picard rank two, Annales de la facult´ e des sciences de Toulouse (to appear); ArXiv 2111.15387 (2025), 33 pp

    2025 arXiv

  30. [38]

    Pure Appl

    Achim Napame and Carl Tipler, Toric sheaves, stability and fibrations, J. Pure Appl. Algebra 228 (2024), no. 3, 30pp

    2024

  31. [39]

    Markus Perling, Graded rings and equivariant sheaves on toric varieties , Math. Nachr. 263- 264 (2004), 181–197 (English)

    2004

  32. [40]

    Vamsi Pritham Pingali, A vector bundle version of the Monge-Amp` ere equation, Adv. Math. 360 (2020), 106921, 40. MR 4035954

    2020

  33. [41]

    Jian Song, Nakai-Moishezon criterions for complex Hessian equations , 2020

    2020

  34. [42]

    Ryosuke Takahashi, J-equation on holomorphic vector bundles , J. Funct. Anal. 286 (2024), no. 4, Paper No. 110265, 64. MR 4673614

    2024

  35. [43]

    Carl Tipler, A note on stable toric sheaves of low rank , Transformation groups (to appear); ArXiv 2307.02822 (2025), 7 pp

    2025 arXiv

  36. [44]

    Pure Appl

    Karen Uhlenbeck and Shing-Tung 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 York, 1985). MR 861491

    1986

  37. [45]

    Claire Voisin, Hodge theory and complex algebraic geometry I, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2002, Translated by Leila Schneps

    2002

  38. [46]

    1, 37 – 48

    Keiichi Watanabe and Masayuki Watanabe, The classification of fano 3-folds with torus em- beddings, Tokyo Journal of Mathematics 5 (1982), no. 1, 37 – 48. 32 R. DELLOQUE, A. NAPAME, C. SCARPA, AND C. TIPLER R´emi Delloque, Univ Brest, UMR CNRS 6205, Laboratoire de Math´ematiqu...

    1982

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.