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Bridgeland stability conditions extend to projective families and carry proper relative moduli of semistable objects.

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load-bearing objection Solid foundational extension: relative Bridgeland stability + proper moduli for projective families, hinging on a clean algebraicity result for (P^{1})^{n}.

arxiv 2607.28411 v1 pith:GMFLSLGS submitted 2026-07-30 math.AG

Stability conditions and moduli spaces on projective families

Chunyi Li , Zhiyu Liu , Ziqi Liu , Emanuele Macr\`i , Alexander Perry , Paolo Stellari , Xiaolei Zhao This is my paper
classification math.AG MSC 14F0814D2018G80
keywords Bridgeland stability conditionsrelative moduli spacesprojective familiesmass-Hom boundstilt-stabilitylocal Calabi-YauBondal-Orlov theoremderived categories
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.

Bridgeland stability conditions pick out stable objects in derived categories of coherent sheaves, but until recently they were hard to construct even on single projective varieties, and their moduli spaces were not known to be well-behaved in general. This paper builds a continuous family of geometric stability conditions on any flat projective family X over a mild base S, relative to a relatively ample class. The resulting relative stability conditions make the moduli functors of relatively semistable objects into algebraic stacks of finite type that are quasi-proper over S; in characteristic zero they admit proper good moduli spaces. The same circle of ideas yields mass-Hom bounds on the distinguished component, recovers tilt-stability constructions on surfaces and many threefolds, produces stability conditions on supported categories of total spaces of certain vector bundles (including local Calabi–Yau varieties), and gives a short moduli-theoretic proof of the Bondal–Orlov reconstruction theorem.

Core claim

For a flat projective morphism X→S satisfying mild hypotheses and a relatively ample numerical class H, there is a nonempty distinguished connected component of the relative stability manifold that contains geometric stability conditions. Consequently the moduli functors of relatively semistable objects are algebraic stacks of finite type and quasi-proper over S, and in characteristic zero admit proper good moduli spaces.

What carries the argument

Pullback and pushforward of stability conditions along finite morphisms, controlled by the Bayer property and filtration/cofiltration properties of the maps. The construction starts from known stability conditions on powers of an elliptic curve, pushes them to (P¹)ⁿ and then Pⁿ, and pulls them back to X; over a base the key step is that certain stability conditions on (P¹_S)ⁿ are algebraic and therefore form genuine relative stability conditions.

Load-bearing premise

The relative construction rests on certain stability conditions on products of projective lines being algebraic for small parameters, so that openness, boundedness and Harder–Narasimhan structures hold uniformly over the base.

What would settle it

Verify on a nontrivial base (a curve or the spectrum of a DVR) that the pushed-forward stability conditions on (P¹_S)ⁿ satisfy the relative axioms—openness of geometric stability, boundedness of moduli, and existence of relative Harder–Narasimhan structures. Failure of properness of the resulting moduli spaces or of gluing of filtrations would refute the main theorem.

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

If this is right

  • Every stability condition in the distinguished component on a projective scheme over a field of characteristic zero has proper moduli spaces of semistable objects.
  • Mass-Hom bounds hold throughout that component, giving an independent route to proper moduli when the variety is smooth.
  • Supported derived categories of total spaces of globally generated or ample dual bundles, including all local Calabi–Yau varieties, admit stability conditions.
  • A smooth projective variety with ample or anti-ample canonical bundle is recovered as a moduli space of stable skyscraper sheaves for suitable stability conditions in the distinguished component.
  • Tilt-stability constructions on complex surfaces and on threefolds satisfying the BMT conjecture land in the same distinguished component and therefore inherit mass-Hom bounds and proper moduli.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same finite-morphism reduction may extend to twisted derived categories or smooth proper Deligne–Mumford stacks once suitable finite covers or exceptional collections are available.
  • Projectivity of the moduli spaces (beyond properness) is left open; the natural relatively nef divisor on the good moduli space is the obvious candidate to test.
  • Full support with respect to the entire numerical Grothendieck group, rather than a quotient lattice, remains out of reach by these methods for many varieties.

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 / 4 minor

Summary. The paper constructs Bridgeland stability conditions on the bounded derived category of a flat projective family \(\pi:X\to S\) (Setup 8.1) with respect to a relatively ample class \(H_{X/S}\). It produces a nonempty distinguished connected component \(\mathrm{Stab}^\dagger_{H_{X/S}}(D^b(X)/S)\) containing geometric stability conditions (Theorems 6.2, 6.8, 10.3), so that the moduli functors of relatively semistable objects are algebraic stacks of finite type and quasi-proper over \(S\) and, in characteristic 0, admit proper good moduli spaces. The construction proceeds by pushing forward algebraic stability conditions from powers of elliptic curves through \((\mathbb{P}^1)^n\) to \(\mathbb{P}^n\) and then pulling back along a closed embedding of \(X\), with the relative axioms verified via base-change compatibility and the push/pull criteria of Theorem 9.1. Complementary results include mass-Hom bounds on the distinguished component (Theorem 7.5), comparison with tilt-stability on surfaces and threefolds (Propositions 7.6–7.7), stability conditions on supported categories of total spaces of certain vector bundles (Theorems 7.8–7.9), and a new proof of the Bondal–Orlov reconstruction theorem (Theorem 11.7).

Significance. The result places Bridgeland stability on essentially the same footing as Gieseker stability for projective families, supplying proper moduli spaces in a setting where they were previously unavailable even in the absolute case over an arbitrary field. The mass-Hom bounds confirm a conjecture of Halpern-Leistner–Robotis for a large class of stability conditions, the comparison with tilt-stability clarifies the relation to earlier constructions on surfaces and threefolds, and the Bondal–Orlov argument via moduli of skyscraper sheaves is clean and conceptual. The technical apparatus (Bayer property, filtration/cofiltration properties, relative push/pull) is carefully developed and of independent use.

minor comments (4)
  1. [Theorem 6.2] In the statement of Theorem 6.2 the dependence of \(a_0\) only on the Hilbert polynomial is asserted; a one-sentence reminder that the Betti numbers of the minimal free resolution are likewise controlled by the Hilbert polynomial would make the uniformity transparent.
  2. [Remark 6.3] Remark 6.3 notes that the central charge on a general scheme is not simply the exponential of the Chern character; a forward reference to the surface/threefold comparison in §7.2 would help the reader who expects the familiar formula.
  3. [§1.8] The notation \(\widetilde{\sigma}^{a,b}_X\) versus \(\sigma^{a,b}_{\mathbb{P}^n}\) is consistent but dense; a short “notation index” at the end of the introduction would improve readability.
  4. [Example 3.20] Appendix B supplies an alternative geometric argument for the filtration property of \((\mathbb{P}^1)^n\to\mathbb{P}^n\); a parenthetical pointer in Example 3.20(3) would alert the reader that two proofs are available.

Circularity Check

0 steps flagged

No significant circularity: explicit inductive construction from elliptic-curve stability via verified push/pull criteria

full rationale

The paper builds relative stability conditions by an explicit chain: continuous families σ_{a,b} on E^n (central charge (4.3), Prop. 4.5), pushforward to (P^1)^n and P^n (Thm. 5.6, (5.2)/(5.4)), pullback along closed embeddings (Thm. 6.2, (6.1)), then fiberwise assembly over S with HN/openness/boundedness checked via Thm. 9.1 and algebraicity at special (a,b) (Thm. 5.4 from Prop. 4.10 phase signs). Support property uses quadratic forms independent of the conclusion (Lem. 6.6, Rem. 5.5). Self-citations to [38], [7], [44] supply prior absolute existence and the relative framework as black boxes; they are not rewritten as the new relative statement. No parameter is fitted to data and re-predicted; no uniqueness theorem is imported solely to force the ansatz; definitions of geometric/distinguished components are not used to prove their own non-emptiness. Complementary results (mass-Hom, tilt comparison, supported categories, Bondal–Orlov) are applications of the same constructions, not circular rewrites.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 2 invented entities

The paper works entirely inside standard derived algebraic geometry and Bridgeland’s axiomatics. No empirical free parameters appear. Load-bearing background is the existence of stability conditions on powers of elliptic curves (Liu), the theory of relative stability conditions and moduli (Bayer–Lahoz–Macrì–Nuer–Perry–Stellari), and Polishchuk’s inducing results for t-structures; these are cited and used as black boxes. The only paper-specific devices are the distinguished component Stab† and the Bayer property, both defined explicitly from the constructions.

axioms (6)
  • standard math Bridgeland’s definition of stability conditions, support property, and deformation theorem (Theorem 2.4)
    Used throughout as the ambient framework; cited from [20].
  • domain assumption Existence of the continuous family σ_{a,b} on E^n for (a,b)∈R_{>0}×R (Theorem 4.2 / Proposition 4.5, extending Liu)
    Starting point of the whole inductive construction; proved by base change from the algebraically closed case plus isogenies.
  • domain assumption Relative stability conditions and proper good moduli spaces as developed in [7] (Definitions 8.17, Theorems 8.19, 8.23)
    The target notion of ‘stability condition over S’ and the existence of proper moduli once the four axioms are verified are taken from [7].
  • domain assumption Setup 8.1 technical hypotheses on X and S (noetherian Nagata, finite Krull dimension, quasi-projective over affine)
    Needed for the moduli stacks and HN structures of [7] to apply; stated explicitly.
  • domain assumption Filtration and cofiltration properties of the quotient maps E^n→(P^1)^n→P^n and of closed immersions (Examples 3.20, 3.23)
    Used to verify the Bayer-property hypotheses that let pushforward and pullback preserve slicings (Propositions 3.21, 3.24).
  • standard math Polishchuk’s results on inducing t-structures and slicings (Appendix A, extending [57])
    Technical engine for pull/push of slicings; proved in the appendix with minor extensions.
invented entities (2)
  • Distinguished connected component Stab†_{H_X}(D^b(X)) / Stab†_{H_{X/S}}(D^b(X)/S) no independent evidence
    purpose: Canonical connected component containing the explicitly constructed geometric stability conditions eσ^{a,b}, independent of embedding once the ample class is fixed
    Defined in Definitions 5.9, 6.4 and after Theorem 10.3; shown independent of embedding by Theorem 6.8. No independent experimental handle; pure mathematical definition.
  • Bayer property (P ⪯ P⊗L[l]) no independent evidence
    purpose: Simplified numerical criterion that guarantees the technical conditions for pushforward/pullback of slicings
    Definition 3.16; used throughout §§3–6 and 9. Purely definitional convenience.

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read the original abstract

We extend the construction of stability conditions on projective schemes over a field to projective families over an arbitrary base, and prove that they admit proper relative moduli spaces of semistable objects. We also prove a number of complementary results: the existence of mass-Hom bounds for these stability conditions, as conjectured by Halpern-Leistner and Robotis; a comparison with tilt-stability on surfaces and threefolds; a construction of stability conditions on the supported derived category of total spaces of certain vector bundles, including all local Calabi--Yau varieties; and a simple new proof of Bondal and Orlov's reconstruction theorem.

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Works this paper leans on

63 extracted references · 9 linked inside Pith

  1. [1]

    Alonso Tarr ´ ıo, L., Jerem ´ ıas L´ opez, A., Souto Salorio, M., Construction of t-structures and equivalences of derived categories,Trans. Amer. Math. Soc.355(2003), 2523–2543

    2003

  2. [2]

    Alper, J., Halpern-Leistner, D., Heinloth, J., Existence of moduli spaces for algebraic stacks,Invent. Math. 234(2023), 949–1038

    2023

  3. [3]

    Anno, R., Bezrukavnikov, R., Mirkovi´ c, I., Stability conditions for Slodowy slices and real variations of stability,Mosc. Math. J.15(2015), 187–203, 403

    2015

  4. [4]

    Lieblich),J

    Arcara, D., Bertram, A., Bridgeland-stable moduli spaces for K-trivial surfaces (with an appendix by M. Lieblich),J. Eur. Math. Soc. (JEMS)15(2013), 1–38

    2013

  5. [5]

    Math.152(2016), 1740–1776

    Balmer, P., Dell’Ambrogio, I., Sanders, B., Grothendieck–Neeman duality and the Wirthm¨ uller isomor- phism,Compos. Math.152(2016), 1740–1776

    2016

  6. [6]

    Ann.375 (2019), 1597–1613

    Bayer, A., A short proof of the deformation property of Bridgeland stability conditions,Math. Ann.375 (2019), 1597–1613

    2019

  7. [7]

    Bayer, A., Lahoz, M., Macr ` ı, E., Nuer, H., Perry, A., Stellari, P., Stability conditions in families,Publ. Math. Inst. Hautes ´Etudes Sci.133(2021), 157–325

    2021

  8. [8]

    Bayer, A., Lahoz, M., Macr ` ı, E., Stellari, P., Stability conditions on Kuznetsov components,Ann. Sci.´Ec. Norm. Sup´ er.56(2023), 517–570

    2023

  9. [9]

    J.160 (2011), 263–322

    Bayer, A., Macr ` ı, E., The space of stability conditions on the local projective plane,Duke Math. J.160 (2011), 263–322

    2011

  10. [10]

    , Projectivity and birational geometry of Bridgeland moduli spaces,J. Amer. Math. Soc.27(2014), 707–752

    2014

  11. [11]

    Math.206(2016), 1–65

    Bayer, A., Macr ` ı, E., Stellari, P., The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds,Invent. Math.206(2016), 1–65

    2016

  12. [12]

    Algebraic Geom.23(2014), 117–163

    Bayer, A., Macr ` ı, E., Toda, Y., Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities,J. Algebraic Geom.23(2014), 117–163

    2014

  13. [13]

    Beilinson, A., Bernstein, J., Deligne, P., Gabber, O., Faisceaux pervers,Ast´ erisque100(2018), vi+180 pp

    2018

  14. [14]

    Bondal, A., Representations of associative algebras and coherent sheaves,Izv. Akad. Nauk SSSR Ser. Mat. 53(1989), 25–44

    1989

  15. [15]

    Math.125(2001), 327–344

    Bondal, A., Orlov, D., Reconstruction of a variety from the derived category and groups of autoequiva- lences,Compos. Math.125(2001), 327–344

    2001

  16. [16]

    Bondal, A., van den Bergh, M., Generators and representability of functors in commutative and noncom- mutative geometry,Mosc. Math. J.3(2003), 1–36, 258

    2003

  17. [17]

    Takahashi’s conjecture for (P 2, E) and a refined sheaves/Gromov–Witten correspondence,Duke Math

    Bousseau, P., A proof of N. Takahashi’s conjecture for (P 2, E) and a refined sheaves/Gromov–Witten correspondence,Duke Math. J.172(2023), 2895–2955

    2023

  18. [18]

    Bousseau, P., Descombes, P., Le Floch, B., Pioline, B., BPS dendroscopy on localP 2,Comm. Math. Phys. 405(2024), Paper No. 108, 98 pp

    2024

  19. [19]

    Bridgeland, T., Stability conditions on a non-compact Calabi–Yau threefold,Comm. Math. Phys.266 (2006), 715–733

    2006

  20. [20]

    , Stability conditions on triangulated categories,Ann. of Math. (2)166(2007), 317–345

    2007

  21. [21]

    J.141(2008), 241–291

    , Stability conditions on K3 surfaces,Duke Math. J.141(2008), 241–291

    2008

  22. [22]

    Bridgeland, T., Smith, I., Quadratic differentials as stability conditions,Publ. Math. Inst. Hautes ´Etudes Sci.121(2015), 155–278

    2015

  23. [23]

    Haesemeyer), eprintarXiv:2402.04605

    Canonaco, A., Neeman, A., Stellari, P., The passage among the subcategories of weakly approximable triangulated categories (with an appendix by C. Haesemeyer), eprintarXiv:2402.04605

    Pith/arXiv arXiv

  24. [24]

    Cheng, Y., Feyzbakhsh, S., Stability conditions on threefolds, eprintarXiv:2607.04788

    Pith/arXiv arXiv

  25. [25]

    Alg´ ebrique9(2025), Art

    Dell, H., Stability conditions on free abelian quotients, ´Epijournal G´ eom. Alg´ ebrique9(2025), Art. 16, 39 pp

    2025

  26. [26]

    Feyzbakhsh, S., Koseki, N., Liu, Z., Rekuski, N., Stability conditions on Calabi–Yau threefolds via Brill– Noether theory of curves, eprintarXiv:2509.24990

    arXiv

  27. [27]

    Fu, L., Li, C., Zhao, X., Stability manifolds of varieties with finite Albanese morphisms,Trans. Amer. Math. Soc.375(2022), 5669–5690. 82 C. LI, Z. LIU, Z. LIU, E. MACR `I, A. PERRY, P. STELLARI, AND X. ZHAO

    2022

  28. [28]

    Haiden, F., Katzarkov, L., Kontsevich, M., Flat surfaces and stability structures,Publ. Math. Inst. Hautes ´Etudes Sci.126(2017), 247–318

    2017

  29. [29]

    Haiden, F., Sung, B., The stability manifold ofE×E×E, eprintarXiv:2410.08028

    Pith/arXiv arXiv

  30. [30]

    Halpern-Leistner, D., Robotis, A., The space of augmented stability conditions, eprintarXiv:2501.00710

    Pith/arXiv arXiv

  31. [31]

    Hotchkiss, J., Perry, A., The period-index conjecture for abelian threefolds and Donaldson–Thomas theory, eprintarXiv:2405.03315

    Pith/arXiv arXiv

  32. [32]

    Huybrechts, D., Stellari, P., Proof of C˘ ald˘ araru’s conjecture, inModuli spaces and arithmetic geometry, 31–42, Adv. Stud. Pure Math.45, Math. Soc. Japan, Tokyo, 2006

    2006

  33. [33]

    King, A., Moduli of representations of finite-dimensional algebras,Quart. J. Math. Oxford Ser. (2)45 (1994), 515–530

    1994

  34. [34]

    Kuznetsov, A., Liu, S., Perry, A., Inducing t-structures on semiorthogonal components, eprint arXiv:2606.26193

    Pith/arXiv arXiv

  35. [35]

    Li, C., Stability conditions on Fano threefolds of Picard number 1,J. Eur. Math. Soc. (JEMS)21(2019), 709–726

    2019

  36. [36]

    Math.218(2019), 301–340

    , On stability conditions for the quintic threefold,Invent. Math.218(2019), 301–340

    2019

  37. [37]

    , A real reduction of the manifold of Bridgeland stability conditions, eprintarXiv:2506.21995

    Pith/arXiv arXiv

  38. [38]

    , A remark on stability conditions on smooth projective varieties, eprintarXiv:2601.22994

    arXiv

  39. [39]

    Li, C., Macr ` ı, E., Perry, A., Stellari, P., Zhao, X., Stability conditions on products of curves and Hilbert schemes of surfaces, eprintarXiv:2512.14207

    arXiv

  40. [40]

    Algebraic Geom.15(2006), 175–206

    Lieblich, M., Moduli of complexes on a proper morphism,J. Algebraic Geom.15(2006), 175–206

    2006

  41. [41]

    J.138(2007), 23–118

    , Moduli of twisted sheaves,Duke Math. J.138(2007), 23–118

    2007

  42. [42]

    Math.51(2007), 209–236

    Lipman, J., Neeman, A., Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor, Illinois J. Math.51(2007), 209–236

    2007

  43. [43]

    Liu, P., Stability conditions and moduli spaces on the Kuznetsov component of cubic fivefolds, eprint arXiv:2509.21454

    Pith/arXiv arXiv

  44. [44]

    Reine Angew

    Liu, Y., Stability conditions on product varieties,J. Reine Angew. Math.770(2021), 135–157

    2021

  45. [45]

    Liu, Z., Mao, T., Tilt-stability on singular schemes and Bogomolov–Gieseker-type inequalities, eprint arXiv:2605.13808

    Pith/arXiv arXiv

  46. [46]

    Macr ` ı, E., Stability conditions on curves,Math. Res. Lett.14(2007), 657–672

    2007

  47. [47]

    Algebraic Geom.18(2009), 605–649

    Macr ` ı, E., Mehrotra, S., Stellari, P., Inducing stability conditions,J. Algebraic Geom.18(2009), 605–649

    2009

  48. [48]

    S.,Algebraic groups, Cambridge Stud

    Milne, J. S.,Algebraic groups, Cambridge Stud. Adv. Math.170, Cambridge University Press, Cambridge, 2017

    2017

  49. [49]

    Mukai, S., Semi-homogeneous vector bundles on an Abelian variety,J. Math. Kyoto Univ.18(1978), 239–272

    1978

  50. [50]

    Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability,J. Amer. Math. Soc.9(1996), 205–236

    1996

  51. [51]

    , Derived categories and Grothendieck duality, inTriangulated categories, 290–350, London Math. Soc. Lecture Note Ser.375, Cambridge University Press, Cambridge, 2010

    2010

  52. [52]

    , Boundedt-structures on the category of perfect complexes,Acta Math.233(2024), 239–284

    2024

  53. [53]

    Orlov, D., Equivalences of derived categories and K3 surfaces,J. Math. Sci. (New York)84(1997), 1361– 1381

    1997

  54. [54]

    Math.350(2019), 877–972

    Perry, A., Noncommutative homological projective duality,Adv. Math.350(2019), 877–972

    2019

  55. [55]

    Topol.26(2022), 3055–3121

    Perry, A., Pertusi, L., Zhao, X., Stability conditions and moduli spaces for Kuznetsov components of Gushel–Mukai varieties,Geom. Topol.26(2022), 3055–3121

    2022

  56. [56]

    Reine Angew

    Piyaratne, D., Toda, Y., Moduli of Bridgeland semistable objects on 3-folds and Donaldson–Thomas invariants,J. Reine Angew. Math.747(2019), 175–219

    2019

  57. [57]

    Polishchuk, A., Constant families oft-structures on derived categories of coherent sheaves,Mosc. Math. J.7(2007), 109–134

    2007

  58. [58]

    Math.54(2014), 427–482

    , Phases of Lagrangian-invariant objects in the derived category of an abelian variety,Kyoto J. Math.54(2014), 427–482

    2014

  59. [59]

    The Stacks Project Authors,Stacks Project,https://stacks.math.columbia.edu(2026)

    2026

  60. [60]

    Math.217(2008), 2736–2781

    Toda, Y., Moduli stacks and invariants of semistable objects on K3 surfaces,Adv. Math.217(2008), 2736–2781. STABILITY CONDITIONS AND MODULI SPACES ON PROJECTIVE F AMILIES 83

    2008

  61. [61]

    Woolf, J., Some metric properties of spaces of stability conditions,Bull. Lond. Math. Soc.44(2012), 1274–1284

    2012

  62. [62]

    Wu, D., Zhang, N., Inducing stability conditions on canonical bundles over Fano varieties, in preparation

  63. [63]

    F. Enriques

    Yoshioka, K., Moduli spaces of twisted sheaves on a projective variety, inModuli spaces and arithmetic geometry, 1–30, Adv. Stud. Pure Math.45, Math. Soc. Japan, Tokyo, 2006. Mathematics Institute (WMI), University of W arwick Coventry, CV4 7AL, United Kingdom Email address:C.Li.25@warwick.ac.uk URL:https://sites.google.com/site/chunyili0401/ Department o...

    2006