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Polynomial Bridgeland Stability Conditions on the Category of Coherent Sheaves

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Pith's one-line read For polynomial Bridgeland central charges adapted to coherent sheaves, asymptotic stability is exactly a lexicographic comparison of generalized slope vectors, independent of the charge.

desk verdict A genuinely useful characterization of asymptotic Z-stability under adapted stability vectors, but the Section 2.2 HN/JH generality is oversold and the dHYM counterexample needs a sign-convention fix. read the letter →

arxiv 2506.21939 v2 pith:ERWQVUDA submitted 2025-06-27 math.AG

Rémi Delloque This is my paper · ORCID
classification math.AG MSC 14F0514J6053C07
keywords BridgelandstabilityconditionspolynomialcoherentsheavesGiesekerHarder–NarasimhanfiltrationasymptoticZ-stabilityZ-criticalequationdeformedHermitianYang–Mills
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 broad class of stability conditions on the category of coherent sheaves that behave like Gieseker stability. It defines a stability notion adapted to sheaves of dimension $d$ by requiring that any proper subsheaf whose quotient has smaller dimension has strictly smaller slope, and asserts that this inequality is enough to guarantee Harder–Narasimhan and Jordan–Hölder filtrations. The main theorem concerns the polynomial Bridgeland central charges introduced by Bayer: when the stability vector is adapted in the same sense, a subsheaf $F$ asymptotically destabilizes $E$ exactly when a finitely computed slope vector $\mu^{U,[\omega]}(F)$ is lexicographically larger than $\mu^{U,[\omega]}(E)$. A consequence is that asymptotic stability becomes independent of the concrete choice of charge. The paper then connects this algebraic criterion to gauge theory: the deformed Hermitian Yang–Mills central charge yields a counterexample to a conjecture of Dervan–McCarthy–Sektnan, and Leung's almost Hermitian–Einstein equation appears as a special case of the $Z$-critical equation.

What carries the argument

The central object is the generalized slope vector $\mu^{U,[\omega]}(E) = (\mu_0^{U,[\omega]}(E), \dots, \mu_n^{U,[\omega]}(E))$, where $\mu_i^{U,[\omega]}(E)$ is $+\infty$ for $i < \mathrm{codim}(E)$ and $\deg_i^{U,[\omega]}(E)/\mathrm{Rk}(E)$ otherwise, with $\deg_i^{U,[\omega]}(E) = \mathrm{ch}_i^U(E) \cup [\omega]^{n-i}$ and $\mathrm{Rk}(E) = \mathrm{ch}_{\mathrm{codim}(E)}^U(E) \cup [\omega]^{n-\mathrm{codim}(E)} > 0$. The key identity is Equation (4): the coefficient of $\epsilon^p$ in $\Im(Z_\epsilon(E)\overline{Z_\epsilon(F)})$ is a sum over $j$ of $\Im(\rho_{n-j}\overline{\rho_{n-p+j}})$ times degree products. Under the adaptation condition $\Im(\rho_d \overline{\rho_i}) > 0$, all coefficients below $p = c + c' + 1$ vanish, where $c$ is the common codimension of $E$ and $F$ and $c'$ is the last index at which their slope vectors agree; the first non-vanishing coefficient is $\Im(\rho_{n-c}\overline{\rho_{n-c'-1}})\,\mathrm{Rk}(E)\,\mathrm{Rk}(F)\,(\mu_{c'+1}^{U,[\omega]}(F)-\mu_{c'+1}^{U,[\omega]}(E))$, which has a fixed positive sign. This identity is what converts asymptotic destabilization into a purely combinatorial lexicographic comparison of intersection numbers. The second piece of machinery is the adaptation condition itself (Definition 2.2), an inequality $\mu(F) < \mu(E)$ for subsheaves $F$ with lower-dimensional quotient; the paper asserts that this reproduces the Huybrechts–Lehn maximal-destabilizing-subsheaf argument and hence yields Harder–Narasimhan and Jordan–Hölder filtrations.

What would settle it

Pick a threefold $X$ with an adapted stability vector $\rho$ and a fixed polarization, choose a pure-dimension-three sheaf $E$ and a subsheaf $F$ whose slope vectors first differ at a known entry, expand $\Im(Z_\epsilon(E)\overline{Z_\epsilon(F)})$ using Equation (4), and verify that the sign of the lowest-order non-vanishing term equals the sign of that first difference; any counterexample to this sign rule would refute Theorem 3.9.

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Extended reading notes

Core claim

The central claim is Theorem 3.9: for a smooth projective variety $X$, a polarization $[\omega]$, a twisting class $U$, and a stability vector $\rho$ adapted to sheaves of dimension $d$ (meaning $\Im(\rho_d \overline{\rho_i}) > 0$ for every $i < d$), a subsheaf $F$ of a pure-dimension-$d$ coherent sheaf $E$ asymptotically $Z$-destabilizes $E$ if and only if the generalized slope vector $\mu^{U,[\omega]}(F)$ is lexicographically $\ge \mu^{U,[\omega]}(E)$, with strict inequality exactly for strict destabilization. The slope vector is assembled from the twisted Chern character $\mathrm{ch}^U(E)=\mathrm{ch}(E)\cup U$: its $i$-th entry is $\deg_i^{U,[\omega]}(E)/\mathrm{Rk}(E) = (\mathrm{ch}_i^U(E) \cup [\omega]^{n-i})/\mathrm{Rk}(E)$, with $+\infty$ entries above the codimension. The proof computes the lowest-order non-vanishing coefficient of the polynomial $\epsilon \mapsto \Im(Z_\epsilon(E)\overline{Z_\epsilon(F)})$; the adaptedness hypothesis forces all earlier coefficients to vanish, leaving a leading term that is a positive multiple of the first entry at which the two slope vectors differ. Corollaries are that, for adapted $\rho$, asymptotic $Z$-stability coincides with the explicit lex-degree stability $P_{Z,d}$ and does not depend on the particular choice of $\rho$. The paper also exhibits, in Example 3.4, a charge for which this criterion shows $\mathcal{O}_X$ on a threefold is asymptotically destabilized by codimension-three ideal sheaves even though the flat metric solves the deformed Hermitian Yang–Mills equation, contradicting Conjecture 1.6 of Dervan–McCarthy–Sektnan.

Load-bearing premise

The load-bearing premise is that the adaptation inequality alone is enough to force the existence of a unique maximal destabilizing subsheaf, and hence Harder–Narasimhan and Jordan–Hölder filtrations, for any totally ordered $\mathbb{Q}$-vector space of degrees; the paper states this by invoking Huybrechts–Lehn without proving the boundedness and discreteness steps that their construction requires.

Editorial extensions

If this is right

  • For any charge adapted to sheaves of dimension $d$, asymptotic $Z$-stability is equivalent to $P_{Z,d}$-stability, so algebraic stability of pure-dimension-$d$ sheaves no longer depends on the particular stability vector $\rho$.
  • The deformed Hermitian Yang–Mills charge on a threefold asymptotically destabilizes $\mathcal{O}_X$ by codimension-three ideal sheaves even though the flat metric solves the dHYM equation at every scale, giving a counterexample to Conjecture 1.6 of Dervan–McCarthy–Sektnan.
  • Gieseker stability is recovered exactly as asymptotic $Z$-stability with $U = \mathrm{Td}(X)$ and an adapted, parameter-independent stability vector, extending Gieseker (Simpson) stability uniformly to all coherent sheaves.
  • With the explicit stability vector of Example 3.13, the $Z_\epsilon$-critical equation becomes Leung's almost Hermitian–Einstein equation, making Leung's Kobayashi–Hitchin correspondence a special case of the Dervan–McCarthy–Sektnan correspondence.

Reading between the lines

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

  • The Section 2.2 assertion that the Huybrechts–Lehn proof carries over verbatim lacks the boundedness and discreteness checks; the claimed Harder–Narasimhan and Jordan–Hölder existence in full generality is therefore conditional on those checks, which are automatic for the finite-dimensional degree spaces used in Section 3.
  • The dHYM counterexample indicates that any Kobayashi–Hitchin correspondence for polynomial charges needs an extra hypothesis on the subsheaves considered, such as requiring the quotient to be pure-dimensional or the sheaf to be slope-semistable, as in the Dervan–McCarthy–Sektnan theorem.
  • Because the adapted criterion is purely numerical, it is directly testable in examples: for explicit threefolds, one can compare the lexicographic slope order with the sign of $\Im(Z_\epsilon(E)\overline{Z_\epsilon(F)})$ at small $\epsilon$ to locate the threshold where destabilization sets in.
  • Since $P_{Z,d}$-stability is independent of $\rho$, moduli spaces of asymptotically $Z$-stable sheaves for adapted charges, if constructed via the Harder–Narasimhan filtrations, should coincide with moduli of $P_{Z,d}$-semistable sheaves and can be compared with the classical Gieseker moduli.
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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

3 major / 5 minor

Summary. The paper introduces a class of stability conditions on Coh(X) defined by a totally ordered Q-vector-space-valued degree morphism, calls them "adapted to coherent sheaves," and claims Harder-Narasimhan and Jordan-Holder filtrations for them in full generality. It then specializes to Bayer's polynomial Bridgeland central charges in the large-volume limit. The central result, Theorem 3.9, asserts that when the stability vector is adapted to sheaves of dimension d, a subsheaf asymptotically destabilizes a pure d-dimensional sheaf exactly when its generalized slope vector is larger in the lexicographic order. Consequences include independence of stability from the concrete choice of the adapted stability vector, identification with the explicit lex-degree stability P_{Z,d}, a realization of Gieseker/Simpson stability within this framework, and a claimed counterexample to Dervan-McCarthy-Sektnan Conjecture 1.6. The main proof is a direct asymptotic expansion of the leading coefficient of the central-charge cross product.

Significance. If the missing filtration arguments are supplied, the paper provides a useful and explicit bridge between Bayer's large-volume polynomial stability and more classical slope/Gieseker-type stability. The computation in Theorem 3.9 is structurally correct in the finite-dimensional lexicographic degree spaces that occur in Section 3, and the resulting characterization gives a concrete, checkable stability criterion. The identification of Gieseker stability with a parameter-independent polynomial central charge is a nice observation, and the proposed counterexample to a conjecture of Dervan-McCarthy-Sektnan is potentially interesting. However, the advertised general claim that adapted stability conditions admit Harder-Narasimhan and Jordan-Holder filtrations is currently unsupported, because the proof is delegated to Huybrechts-Lehn without verifying the necessary boundedness and discreteness inputs in the stated generality. The contribution is therefore conditional on repairing that gap.

major comments (3)
  1. [Section 2.2, Theorems 2.9 and 2.10] The existence and uniqueness of Harder-Narasimhan and Jordan-Holder filtrations is asserted for every adapted degree morphism Deg: K(X) -> V with V an arbitrary totally ordered Q-vector space, but the proof is delegated to [4, Sections 1.3 and 1.5] with the sentence that the proofs are 'the exact same.' This does not address the load-bearing boundedness input: Huybrechts-Lehn selects a maximal destabilizing subsheaf using discreteness and boundedness of the set of Hilbert polynomials of subsheaves of a fixed sheaf, and Definition 2.2 only controls slopes of subsheaves whose quotients have strictly smaller dimension. It does not imply that the set {mu(F)} is bounded above in V or that a maximal element exists. As written, the abstract's claim that adapted conditions admit HN and JH filtrations in this generality is not established. The Section 3 results use finite-dimensional lexicographic degree spaces and may survive a restriction of the statement, but the general theorems need either a proof of the required boundedness or an explicit restriction to degree spaces for which the Huybrechts-Lehn argument applies.
  2. [Section 3.3, after Corollary 3.12] The paper asserts without proof that the stability condition P_{Z,d} is itself adapted to sheaves of dimension d. This assertion is needed if Theorem 2.9 is to be applied to P_{Z,d} to obtain Harder-Narasimhan filtrations for this explicit stability condition. The claim is plausible, since the first non-zero degree of a lower-dimensional quotient is positive, but the verification should be written out explicitly rather than left as an unstated consequence of Theorem 3.9.
  3. [Example 3.4] The computation displayed in Example 3.4 gives the sign of Im(Z_epsilon(O_X) overline{Z_epsilon(i_* O_V)}), but the destabilizing subobject used in Definition 1.1 is the ideal sheaf I_V, not the sky-scraper sheaf i_* O_V. The conclusion that O_X is asymptotically destabilized by I_V therefore requires the additional step Im(Z(O_X) overline{Z(I_V)}) = -Im(Z(O_X) overline{Z(O_V)}) up to real terms, using additivity of the central charge. This step is straightforward, but it is essential for the claimed counterexample to [3, Conjecture 1.6] and should be stated explicitly.
minor comments (5)
  1. [Proof of Theorem 3.9] The definition of c' should be made less ambiguous: it should say that c' is the largest integer in {c, ..., n} for which mu_{c'}(F) = mu_{c'}(E), with the understanding that c' = c if the equality fails already at c+1. The current phrasing 'Let c <= c' <= n the largest integer...' is grammatically confusing and should be rewritten.
  2. [Example 2.3] The adaptation condition is stated as positivity of (Gamma_{0,k} cup [V])^{(n,n)}, but the collection is introduced as (Gamma_{k,j}) with indices 1 <= k <= n and 0 <= j <= d_k - 1. The notation should specify which component of the Gamma's is meant to pair with [V], and the index convention should be made consistent.
  3. [Lemma 3.7] In the proof of Lemma 3.7, the derivation that the coefficient b is positive is very terse. A short sentence explaining that one uses Im(rho_j overline{rho_{j+1}}) > 0, together with the already established sign of Im(rho_n overline{rho_j}), would improve readability.
  4. [Proposition 3.14(3)] The statement that the heart of the bounded t-structure is 'Coh(X) up to an even number of shifts' is imprecise. The proof should specify the exact shift, or at least state which perversity function is used and how the parity is determined by the arguments of the rho_i.
  5. [Throughout Section 3] The notation for the cross product Im(Z_epsilon(E) Z_epsilon(F)) is ambiguous without an explicit overline on the second factor. Since the argument comparison in Definition 1.1 requires the conjugate, please ensure that all such expressions are typeset consistently as Im(Z_epsilon(E) overline{Z_epsilon(F)}).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's main characterization is derived by direct computation from stated hypotheses, and all load-bearing references are external benchmarks rather than self-supporting citations.

full rationale

The paper's central result, Theorem 3.9, is not circular. It assumes the stability vector is adapted to sheaves of dimension d (Definition 3.6) and then proves, by explicit leading-term analysis of the polynomial ℑ(Zε(E)Zε(F)), that asymptotic destabilization is equivalent to lexicographic comparison of the generalized slope vectors (Eq. (4) and the subsequent coefficient computation). The conclusion is not used to define the adaptation hypothesis; it is derived from it. Corollary 3.12 (independence of the choice of ρ) is a logical consequence of Theorem 3.9, not an input. The only notable delegation is in Section 2.2, where Harder–Narasimhan and Jordan–Hölder filtration statements are asserted to follow exactly as in Huybrechts–Lehn [4]. That is an external reference, not a self-citation, and the possible gap (boundedness in arbitrary totally ordered Q-vector spaces) is a correctness or completeness concern, not a circularity. Likewise, Example 3.4 and Example 3.13 are computations against independent external results (the dHYM equation and Leung's equation), not renamed inputs. No fitted parameter is called a prediction, and no definition secretly contains the target result. Accordingly, the derivation chain is self-contained up to the stated external benchmarks, and the circularity score is 0.

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

The paper's results are conditional mainly on three things: (1) the adaptedness positivity hypothesis on the stability vector, which is the central modeling input and is not derived from anything; (2) imported background theorems (Huybrechts-Lehn, Bayer, Dervan-McCarthy-Sektnan) that do most of the heavy lifting in Sections 2 and 3.4; and (3) an unproved claim that the HL filtration arguments transfer verbatim to arbitrary totally ordered degree spaces. There are no fitted parameters and no invented entities.

assumptions (6)
  • standard math Grothendieck-Riemann-Roch and the fact that for a coherent sheaf supported on a d-dimensional subvariety, the lowest Chern character component is Σ rk(E|V_i)[V_i] with positive coefficients.
    Used in Section 2.1 (definition of Rk(E) as a positive integer) and Proposition 3.3 to locate the leading degree component.
  • standard math Huybrechts-Lehn theory of Harder-Narasimhan and Jordan-Holder filtrations for Gieseker stability, including the boundedness arguments that produce the maximal destabilizing subsheaf (Lemma 2.7).
    Imported verbatim in Section 2.2; the paper asserts the adaptedness condition makes the proofs go through without change but does not reprove the boundedness input.
  • standard math Bayer's construction of polynomial Bridgeland stability conditions, existence of the heart A, and the large-volume-limit theorems ([1, Prop 1.2.1, Theorem 3.2.2]).
    The definition of asymptotic Z-stability (Definition 1.1) and Proposition 3.14 use this machinery directly.
  • ad hoc to paper The positivity conditions ℑ(ρ_d ρ̄_i) > 0 for all i < d (Definition 3.6) are imposed rather than derived; they are the sharp hypothesis under which Theorem 3.9 holds.
    The entire theory in Section 3 is built on this ordering assumption; it is exactly what Example 3.4 shows is missing for the dHYM charge, so it is the paper's key modeling choice, stated but not derived.
  • domain assumption The Dervan-McCarthy-Sektnan Kobayashi-Hitchin correspondence and Conjecture 1.6 ([3, Theorem 1.1]) are taken as background for the motivation and for the claimed counterexample.
    Theorem 1.3 is quoted in the introduction; the counterexample in Example 3.4 is an assertion about the scope of an unquoted conjecture whose exact hypotheses are not reproduced.
  • ad hoc to paper That the adaptation condition (Definition 2.2) suffices to run the Huybrechts-Lehn proof verbatim for an arbitrary totally ordered Q-vector space V.
    Theorems 2.9 and 2.10 are stated in this generality with no proof; the adaptation condition does not obviously imply the boundedness/discreteness used in HL, so this is an unproved premise of the Section 2 claims.

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Pith. "Pith review of Polynomial Bridgeland Stability Conditions on the Category of Coherent Sheaves." pith.science (2026). https://pith.science/paper/ERWQVUDA

@misc{pith2026250621939,
  author       = {Pith},
  title        = {Pith review of: Polynomial Bridgeland Stability Conditions on the Category of Coherent Sheaves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ERWQVUDA}},
  note         = {Machine review of arXiv:2506.21939}
}
read the original abstract

In this short note, we provide a broad class of examples of stability conditions on the category of coherent sheaves which generalise Gieseker stability. We refer to them as "adapted to coherent sheaves" and they admit Harder--Narasimhan and Jordan--H\"older filtrations. We also study the particular case of Bayer's polynomial Bridgeland stability conditions, and their relation to the gauge theoretical counterpart introduced by Dervan--McCarthy--Sektnan.

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Forward citations

Cited by 1 Pith paper

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

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    P-critical connections generalize Z-critical connections; on toric varieties P-positivity is checked finitely, and uniform P-positivity survives point blow-ups.

Reference graph

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