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Restricted stability conditions on threefolds match the double-tilt construction and prove the weak BMT conjecture.

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2026-07-11 13:29 UTC pith:W2XNKXJ2

load-bearing objection Clean identification of restricted Li hearts with double-tilt hearts on threefolds, proving weak BMT for large a; solid once the Liu–Li families are granted.

arxiv 2607.04788 v1 pith:W2XNKXJ2 submitted 2026-07-06 math.AG

Stability conditions on threefolds

Yiran Cheng , Soheyla Feyzbakhsh This is my paper
classification math.AG MSC 14F0814J3018E30
keywords Bridgeland stability conditionsthreefoldsdouble-tilt constructionBayer–Macrì–Toda conjectureLi conditionrestriction of stability conditionsproducts of elliptic curves
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper shows that certain Bridgeland stability conditions on projective space restrict to smooth projective subvarieties, and that when the subvariety is a threefold those restricted hearts are exactly the ones obtained by the double-tilt construction of Bayer–Macrì–Toda. As a direct consequence it proves a weak form of the Bayer–Macrì–Toda conjecture: a numerical inequality that controls the Chern characters of tilt-semistable objects. The argument reverses the usual logic: instead of first proving geometric inequalities in order to construct stability conditions, it starts from a purely categorical family of stability conditions on products of elliptic curves, descends them to projective space, restricts them, and reads the desired inequalities off the resulting stability conditions. Readers who care about moduli spaces of sheaves or Donaldson–Thomas invariants therefore obtain both a concrete description of the relevant hearts and a new supply of Castelnuovo-type bounds without having to re-prove them from geometry.

Core claim

For a smooth polarized threefold (X,H) and a larger than an explicit constant a(X,H) depending only on an embedding of X into projective space, the heart of the restricted stability condition coincides with the double-tilted heart obtained from coherent sheaves by the Bayer–Macrì–Toda construction; consequently every object that is semistable of slope zero with respect to the second tilt satisfies the strict inequality that constitutes the weak BMT conjecture.

What carries the argument

The Li condition on stability conditions on P^n: a phase inequality under tensoring by O(m) that guarantees the restriction of the stability condition along any closed immersion X↪P^n is again a Bridgeland stability condition, together with an explicit upper bound on the Bridgeland distance between horizontal translates of Liu’s stability conditions on products of elliptic curves that verifies the Li condition for large a.

Load-bearing premise

The whole construction rests on the existence of a continuous, group-invariant family of Bridgeland stability conditions on products of elliptic curves that descend to projective space; if that family fails to exist in the required range, the restriction and the identification of hearts both collapse.

What would settle it

Exhibit a smooth polarized threefold and a value of a larger than a(X,H) for which either the restricted heart fails to equal the double-tilted heart, or a tilt-semistable object of slope zero violates the strict inequality ech_3 < (a^{2}/2)H^{2} ech_1.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper studies a two-parameter family of Bridgeland stability conditions on P^n obtained by Li’s descent of Liu’s (Z/2Z)^n ⋊ S_n-invariant stability conditions on products of elliptic curves. It identifies a range of the parameter a (controlled by a new invariant m(X ↪ P^n) and the derived a(X,H)) for which these conditions satisfy the Li phase condition and therefore restrict to Bridgeland stability conditions on any smooth projective subvariety X ⊂ P^n. For a smooth polarized threefold (X,H) and a > a(X,H), the authors prove that the restricted heart A^X_{a,b} coincides with the Bayer–Macrì–Toda double-tilt heart (Coh(X)^{Z_b^{(1)}})^{Z_{a,b}^{(2)}} (Theorem 1.2 / Theorem 3.5). As a consequence they obtain the weak BMT inequality: any Z^{(2)}_{a,b}-semistable object in the first tilt with vanishing imaginary part of the central charge satisfies the strict Castelnuovo-type bound on the third Chern character (Corollary 1.3 / 3.7). An additional section derives a BMT(S)-type inequality under the assumption that the input families exist for real parameters.

Significance. If correct, the result settles the weak Bayer–Macrì–Toda conjecture for all smooth polarized threefolds in a large open range of the stability parameters, and does so by a method that reverses the classical logic: categorical constructions of stability conditions (Liu–Li–Polishchuk) are used to deduce geometric inequalities rather than the other way around. The identification of the restricted heart with the double-tilt heart is a clean structural statement of independent interest. The phase-control estimates via Bridgeland’s generalized metric (Theorem 2.4) and the hyperplane-section reduction that compares torsion pairs (Proposition 3.3, Lemma 3.6) are carefully written and appear reusable. The paper is explicit about the size restriction a > a(X,H) and about the distinction between the weak and strong BMT conjectures.

minor comments (5)
  1. Clarify the relationship between the two numerical thresholds: Theorem 2.11 / Corollary 2.9 require a ≳ (n/π)m(X ↪ P^n), while a(X,H) is defined with the larger coefficient 1/2. A short remark that a > a(X,H) is a convenient sufficient bound (also matching Lemma 3.4) would help the reader.
  2. Section title “stability conditions on smooth projective v arieties” contains a stray space; page 7 “As s result” should be “As a result”. A global proofreading pass would catch these.
  3. The definition of m(X ↪ P^n) (Definition 2.10) takes a minimum over all resolutions of the indicated form. It would be useful to record, even briefly, that the minimum is attained (or that any resolution giving a value close to the Castelnuovo–Mumford regularity already yields a usable bound).
  4. Section 3.3 explicitly assumes the real-parameter extension of Theorem 2.1, deferred to the forthcoming work [LLL+]. A one-sentence flag at the beginning of §3 that the main theorems (1.1–1.3) are unconditional for rational (a,b) while §3.3 is conditional would make the logical status of each statement completely transparent.
  5. In the display of Z^{(2)}_{a,b} and the slope ν_{a,b} (around (24)–(32)), the same combination of Chern characters appears with slightly different normalizations; a single consistent notation for the twisted Chern character ech^{bH} throughout §3 would reduce cognitive load.

Circularity Check

0 steps flagged

No significant circularity: restricted hearts equal double-tilt by independent phase/slope comparison; weak BMT inequality is a genuine consequence, not an input.

full rationale

The derivation chain starts from Liu–Li families of stability conditions on products of elliptic curves and their descent to Pn (Theorem 2.1, Theorem 2.8, cited externally), proves a phase-control estimate (Theorem 2.4 / Corollary 2.9) that enables restriction to subvarieties X (Theorem 2.11), then for threefolds identifies the restricted heart A^X_{a,b} with the double-tilted heart of BMT14 by comparing the two torsion pairs on Coh^{B_X}(X) via hyperplane-section reduction (Proposition 3.3), explicit central-charge formulae (24)–(32), and zero-dimensionality of phase-1 objects (Lemma 3.6). The weak BMT inequality (Corollary 1.3 / 3.7) follows immediately once the hearts coincide, because a Z^{(2)}-semistable object with vanishing imaginary part would otherwise contradict the support property or the phase-1 classification. No equation equates the target inequality to a fitted constant; a(X,H) is an explicit geometric invariant, not a free parameter tuned to data. The single self-citation (Che25 in Remark 2.3) is used only for sharpness of a non-load-bearing inequality and does not underwrite the main identification. The forthcoming real-parameter extension [LLL+] is flagged as an assumption for §3.3 only and is not required for the rational case of Theorems 1.2 / 3.5. The argument is therefore self-contained against its external inputs and exhibits no circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 2 invented entities

The paper is pure mathematics. It introduces two numerical invariants (m and a(X,H)) as definitions, not free parameters. All load-bearing existence statements are taken from the literature (Liu–Li–Polishchuk stability conditions, Polishchuk’s restriction criterion, Bridgeland’s metric). No new physical entities or fitted constants appear.

axioms (4)
  • domain assumption Existence of a continuous family of (Z/2Z)^n ⋊ S_n-invariant Bridgeland stability conditions σ^{(n)}_{a,b} on products of elliptic curves with the stated central charge and skyscraper stability (Theorem 2.1).
    Invoked as the starting point of the whole construction; cited from Liu21, FLZ22, LMP+25.
  • domain assumption Li’s descent theorem: the push-forward of an invariant stability condition on C^n yields a stability condition on P^n satisfying the phase inequality ϕ^±(E) ≤ ϕ^±(E ⊗ O(1)) (Theorem 2.8).
    Used to obtain σ^{P^n}_{a,b} and the subsequent restriction results.
  • domain assumption Polishchuk’s criterion for restricting a stability condition along a closed immersion (Corollary 2.2.2 of Pol07).
    Applied in the proof of Theorem 2.11 to guarantee that ι^♯σ is again a stability condition.
  • standard math Standard properties of Bridgeland’s generalized metric on Stab(D) and the support property for the lattice Λ_n.
    Used throughout §2 to bound phase differences.
invented entities (2)
  • m(X ↪ P^n) no independent evidence
    purpose: Numerical invariant measuring the maximal degree appearing in a resolution of ι_* O_X by split bundles; controls the lower bound on a for which restriction works.
    Defined in Definition 2.10; bounded by Castelnuovo–Mumford regularity; no independent geometric meaning claimed beyond the paper.
  • a(X,H) no independent evidence
    purpose: Minimal (scaled) m over all embeddings realizing the polarization H; the threshold above which the main theorems hold.
    Defined just before Theorem 1.2; purely auxiliary.

pith-pipeline@v1.1.0-grok45 · 21225 in / 2835 out tokens · 26233 ms · 2026-07-11T13:29:36.072613+00:00 · methodology

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read the original abstract

We investigate a subspace of Bridgeland stability conditions on $\PP^n$ satisfying the so-called Li condition. These are the stability conditions whose restriction to a smooth projective subvariety $X \subset \PP^n$ is again a stability condition. We then show that, when $X$ is a threefold, the restricted stability conditions coincide with those obtained via the double-tilt construction introduced by Bayer-Macr\`i-Toda. As an application, we prove the weak BMT conjecture.

Figures

Figures reproduced from arXiv: 2607.04788 by Soheyla Feyzbakhsh, Yiran Cheng.

Figure 1
Figure 1. Figure 1: BMT(S) inequality the line joining (b, w) and Πe(E) intersects Γ2 at Πe(E) and at a second point (b ′ , w′ ), which also lies above Γ1. A direct computation gives b ′ = v2 − v0w v1 − v0b and w ′ =  v2 − v0w v1 − v0b 2 − v2b − v1w v1 − v0b . By construction, E is νb ′ ,w′-semistable and satisfies (37). Hence Corollary 3.7 yields (38) at (b ′ , w′ ). Finally, translating back to the original variables (a, … view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Stability conditions and moduli spaces on projective families

    math.AG 2026-07 accept novelty 6.5

    Stability conditions exist on projective families over arbitrary bases and admit proper relative moduli spaces of semistable objects.

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