[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"mzKLsJ8uvF":3},"# BridgelandStability\n\nAI-assisted formalization in Lean 4 / Mathlib of the main results of\nBridgeland's\n[Stability conditions on triangulated categories](https://annals.math.princeton.edu/2007/166-2/p01)\n(Annals of Mathematics, 2007).\n\n## What Bridgeland stability conditions are\n\nInspired by Douglas's work on Π-stability in string theory, Bridgeland\nshowed that the classical stability package — central charges, semistable\nobjects, Harder-Narasimhan filtrations — can be lifted from abelian categories\nto triangulated categories. A Bridgeland stability condition on a triangulated\ncategory `D` consists of a central charge `Z : K₀(D) → ℂ` and a slicing\n`P(φ)` into semistable objects of each phase, subject to compatibility and\nHarder-Narasimhan existence axioms.\n\nThe headline result is that the space `Stab(D)` of all such conditions is\nitself a complex manifold, with local charts given by the central charge.\nThe theory has since become one of the most active areas in modern\nmathematics. It provides infrastructure for wall-crossing in birational\ngeometry, Donaldson-Thomas theory, and the study of moduli spaces of sheaves\non surfaces and threefolds.\n\n## Main results\n\nThe formalization covers Sections 2–7 of the paper, culminating in:\n\n- **Theorem 1.2** — the central charge map is a local homeomorphism on each\n  connected component of `Stab(D)`.\n- **Corollary 1.3** — connected components of `Stab_N(D)` are\n  finite-dimensional complex manifolds.\n\nBoth are proved for an arbitrary surjective class map `v : K₀(D) →+ Λ` and\nspecialized to the identity (Theorem 1.2) and the numerical quotient\n(Corollary 1.3). The formalization works in class-map generality as in\nBayer–Macrì–Stellari [8, Appendix A] and Bayer–Lahoz–Macrì–Nuer–Perry–Stellari\n[11], where the central charge factors through a surjection `v : K₀(D) →+ Λ`;\nthe classical `Stab(D)` is the specialization `v = id`.\n\n## Documentation\n\n- [**Project website**](https://mattrobball.github.io/BridgelandStability/) —\n  informal mathematical descriptions of each declaration, with a table of\n  paper statements that currently have exact formal analogs.\n- [**API docs**](https://mattrobball.github.io/BridgelandStability/api/) —\n  auto-generated Lean API documentation (doc-gen4).\n- [**Comparator Manual**](https://mattrobball.github.io/BridgelandStability/comparator/) —\n  independent verification of the formal statements against their source,\n  listing the trusted base you must audit to trust the proof.\n\nThe root import is [`BridgelandStability.lean`](BridgelandStability.lean).\n\n## Project charter\n\nThis is an experiment in AI-assisted formalization.\n\n**Rule 1: Humans write no Lean code beyond Mathlib.** The goal is to study how\nfar a human-led effort can push AI-assisted formalization on top of Mathlib.\n\n**Rule 2: Getting Lean to accept a proof script is not success.** The end state\nhas to be Mathlib quality: correct abstractions, reusable lemmas, sane names,\ncontrolled imports, and proofs that could plausibly survive code review and\nupstreaming.\n\n## Selected references\n\n1. Michael R. Douglas,\n   [D-branes, Categories and N=1 Supersymmetry](https://arxiv.org/abs/hep-th/0011017),\n   J. Math. Phys. 42 (2001), 2818-2843.\n2. Tom Bridgeland,\n   [Stability conditions on triangulated categories](https://annals.math.princeton.edu/2007/166-2/p01),\n   Annals of Mathematics 166 (2007), 317-345.\n3. Tom Bridgeland,\n   [Stability conditions on K3 surfaces](https://ora.ox.ac.uk/objects/uuid:577d12a4-1dd8-497b-af5d-bc52623a2ac1),\n   Duke Mathematical Journal 141 (2008), 241-291.\n4. Daniele Arcara and Aaron Bertram,\n   [Bridgeland-stable moduli spaces for K-trivial surfaces](https://ems.press/journals/jems/articles/4662),\n   Journal of the European Mathematical Society 15 (2013), 1-38.\n5. Arend Bayer and Emanuele Macrì,\n   [Projectivity and birational geometry of Bridgeland moduli spaces](https://www.ams.org/jams/2014-27-03/S0894-0347-2014-00790-6/),\n   Journal of the American Mathematical Society 27 (2014), 707-752.\n6. Arend Bayer and Emanuele Macrì,\n   [MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations](https://doi.org/10.1007/s00222-014-0501-8),\n   Inventiones mathematicae 198 (2014), 505-590.\n7. Tom Bridgeland and Ivan Smith,\n   [Quadratic differentials as stability conditions](https://link.springer.com/article/10.1007/s10240-014-0066-5),\n   Publications mathematiques de l'IHES 121 (2015), 155-278.\n8. Arend Bayer, Emanuele Macrì, and Paolo Stellari,\n   [The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds](https://doi.org/10.1007/s00222-016-0665-5),\n   Inventiones mathematicae 206 (2016), 869-933.\n9. Arend Bayer,\n   [A short proof of the deformation property of Bridgeland stability conditions](https://arxiv.org/abs/1606.02169),\n   preprint, 2016.\n10. Marcello Bernardara, Emanuele Macrì, Benjamin Schmidt, Xiaolei Zhao,\n    [Bridgeland Stability Conditions on Fano Threefolds](https://epiga.episciences.org/3255),\n    EpiGA 1 (2017), article 8.\n11. Chunyi Li,\n    [On stability conditions for the quintic threefold](https://link.springer.com/article/10.1007/s00222-019-00888-z),\n    Inventiones mathematicae 218 (2019), 301-340.\n12. Arend Bayer, Martí Lahoz, Emanuele Macrì, Howard Nuer, Alexander Perry, Paolo Stellari,\n    [Stability conditions in families](https://link.springer.com/article/10.1007/s10240-021-00124-6),\n    Publications mathematiques de l'IHES 133 (2021), 157-325.\n13. Soheyla Feyzbakhsh and Richard P. Thomas,\n    [Rank r DT theory from rank 1](https://www.ams.org/jams/2023-36-03/S0894-0347-2023-00922-5/),\n    Journal of the American Mathematical Society 36 (2023), 795-826.\n14. Soheyla Feyzbakhsh and Richard P. Thomas,\n    [Rank r DT theory from rank 0](https://projecteuclid.org/journals/duke-mathematical-journal/volume-173/issue-11/Rank-r-DT-theory-from-rank-0/10.1215/00127094-2023-0085.short),\n    Duke Mathematical Journal 173 (2024), 2063-2116.\n15. Soheyla Feyzbakhsh, Naoki Koseki, Zhiyu Liu, Nick Rekuski,\n    [Stability conditions on Calabi-Yau threefolds via Brill-Noether theory of curves](https://arxiv.org/abs/2509.24990),\n    preprint, 2025.\n16. Chunyi Li,\n    [A Remark on Stability Conditions on Smooth Projective Varieties](https://arxiv.org/abs/2601.22994),\n    preprint, 2026.\n",1780242003379]