pith. sign in

arxiv: 2605.21995 · v1 · pith:T2BLGOXSnew · submitted 2026-05-21 · 🧮 math.AG

K-stability of adjoint foliated structures

Theodoros Stylianos Papazachariou This is my paper

Pith reviewed 2026-05-22 03:31 UTC · model grok-4.3

classification 🧮 math.AG
keywords K-stabilityadjoint foliated structuresDonaldson-Futaki invariantDing stabilitytest configurationsFano foliationsbounded families
0
0 comments X

The pith

The paper establishes a reduction to special test configurations for the K-stability of adjoint Fano foliated structures by proving that the mixed Donaldson-Futaki invariant is non-increasing along birational procedures.

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

This paper defines a notion of K-stability for adjoint foliated structures using test configurations and a foliated Donaldson-Futaki invariant. For adjoint Fano foliated structures it shows that stability testing reduces to special test configurations because the mixed Donaldson-Futaki invariant stays non-increasing under the relevant birational modifications. The work further introduces an equivalent Ding stability notion and mixed alpha, beta, and delta invariants that yield valuative criteria for stability. As an application the authors prove that K-semistable adjoint Fano foliated structures of bounded volume form a bounded family.

Core claim

We introduce a notion of K-stability for adjoint foliated structures via test configurations and the foliated Donaldson-Futaki invariant. We prove reduction to special test configurations for adjoint Fano foliated structures by showing that the mixed Donaldson-Futaki invariant is non-increasing along the birational procedure. We also introduce a notion of Ding stability for adjoint Fano foliated structures which we show is equivalent to our notion of K-stability. We then introduce mixed alpha, beta and delta-invariants and use the reduction theorem to establish valuative criteria for the K-stability of adjoint Fano foliated structures. As an application, we show that K-semistable adjoint Fao

What carries the argument

The mixed Donaldson-Futaki invariant for adjoint foliated structures, which is shown to be non-increasing along a specific birational procedure to reduce general test configurations to special ones.

If this is right

  • Ding stability is equivalent to K-stability for adjoint Fano foliated structures.
  • Mixed alpha, beta, and delta invariants give valuative criteria that characterize K-stability.
  • K-semistable adjoint Fano foliated structures with bounded volume form a bounded family.
  • K-stability can be checked using only special test configurations.

Where Pith is reading between the lines

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

  • The reduction method may carry over to stability questions for foliations of higher rank or on varieties with different singularities.
  • The bounded-family result suggests that moduli spaces of K-semistable adjoint foliated structures could be constructed with properness properties.
  • When the foliation is trivial the statements should specialize to known reduction and boundedness theorems for ordinary Fano varieties.

Load-bearing premise

The mixed Donaldson-Futaki invariant is non-increasing along the birational procedure used in the reduction to special test configurations.

What would settle it

An explicit adjoint Fano foliated structure for which the mixed Donaldson-Futaki invariant increases along one of the birational steps in the reduction procedure would falsify the reduction theorem.

read the original abstract

We introduce a notion of K-stability for adjoint foliated structures via test configurations and the foliated Donaldson-Futaki invariant. We prove reduction to special test configurations for adjoint Fano foliated structures by showing that the mixed Donaldson-Futaki invariant is non-increasing along the birational procedure. We also introduce a notion of Ding stability for adjoint Fano foliated structures which we show is equivalent to our notion of K-stability. We then introduce mixed alpha, beta and delta-invariants and use the reduction theorem to establish valuative criteria for the K-stability of adjoint Fano foliated structures. To conclude, as an application, we show that K-semistable adjoint Fano foliated structures with bounded volume form a bounded family.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces a notion of K-stability for adjoint foliated structures via test configurations and the foliated Donaldson-Futaki invariant. It proves reduction to special test configurations for adjoint Fano foliated structures by showing that the mixed Donaldson-Futaki invariant is non-increasing along the birational procedure. It also introduces Ding stability for adjoint Fano foliated structures and establishes its equivalence to K-stability. Mixed alpha, beta, and delta invariants are defined, and the reduction theorem is used to obtain valuative criteria. As an application, K-semistable adjoint Fano foliated structures with bounded volume are shown to form a bounded family.

Significance. If the central results hold, this work extends K-stability theory from Fano varieties to adjoint foliated structures, providing new tools for stability questions in birational geometry involving foliations. The reduction theorem via monotonicity of the mixed invariant, the Ding equivalence, and the resulting valuative criteria and boundedness statement are substantive contributions that build directly on standard birational geometry techniques. The manuscript ships a self-contained sequence of definitions, a direct computation for monotonicity, and a formal application to boundedness, which strengthens its technical value in the field.

minor comments (3)
  1. The introduction would benefit from a brief comparison of the new mixed Donaldson-Futaki invariant with the classical Donaldson-Futaki invariant for non-foliated cases, to clarify the precise role of the foliation data.
  2. In the statement of the reduction theorem, the birational procedure is outlined at a high level; adding a short schematic diagram or explicit local coordinate description of the blow-ups involved would improve readability for readers unfamiliar with foliated singularities.
  3. The equivalence between K-stability and Ding stability is stated after the reduction; a short remark on whether the equivalence proof uses the reduction or proceeds independently would help trace the logical dependencies.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of our manuscript. We appreciate the recommendation for minor revision and will incorporate any editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines K-stability for adjoint foliated structures directly from test configurations and the foliated Donaldson-Futaki invariant, then establishes the key reduction to special test configurations by proving monotonicity of the mixed invariant along a standard birational procedure using the definitions and tools from algebraic geometry. This monotonicity is presented as a direct computation rather than a fit or self-referential construction. Subsequent steps (Ding equivalence, valuative criteria via mixed alpha/beta/delta invariants, and boundedness) follow formally once the reduction holds, relying on external results in birational geometry without load-bearing self-citations or renaming of prior results. The derivation chain remains self-contained with independent content grounded in the introduced definitions and standard techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard results from birational geometry and K-stability theory for the monotonicity and equivalence statements. No free parameters or invented entities are visible in the abstract; the central claims rest on domain assumptions from algebraic geometry such as the existence of test configurations and properties of foliations.

axioms (1)
  • domain assumption Standard properties of birational morphisms and test configurations in algebraic geometry hold for adjoint foliated structures.
    Invoked implicitly in the reduction theorem and birational procedure described in the abstract.

pith-pipeline@v0.9.0 · 5647 in / 1425 out tokens · 32691 ms · 2026-05-22T03:31:38.801691+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

63 extracted references · 63 canonical work pages

  1. [1]

    and Mumford, David , title =

    Knudsen, Finn F. and Mumford, David , title =. Mathematica Scandinavica , volume =. 1976 , pages =

    work page 1976

  2. [2]

    Osaka Journal of Mathematics , volume =

    Yuji Odaka , title =. Osaka Journal of Mathematics , volume =. 2013 , pages =

    work page 2013

  3. [3]

    Annals of Mathematics , volume =

    Odaka, Yuji , title =. Annals of Mathematics , volume =. 2013 , pages =

    work page 2013

  4. [4]

    K-stability for general polarizations and orbifold stability , journal =

    Dervan, Ruadh\'. K-stability for general polarizations and orbifold stability , journal =. 2016 , pages =

    work page 2016

  5. [5]

    Annals of Mathematics , volume =

    Li, Chi and Xu, Chenyang , title =. Annals of Mathematics , volume =. 2014 , pages =

    work page 2014

  6. [6]

    Journal f\"

    Fujita, Kento , title =. Journal f\". 2019 , pages =

    work page 2019

  7. [7]

    Boucksom, S\'. Uniform. Annales de l'Institut Fourier , volume =. 2019 , pages =

    work page 2019

  8. [8]

    Advances in Mathematics , volume =

    Blum, Harold and Jonsson, Mattias , title =. Advances in Mathematics , volume =. 2020 , pages =

    work page 2020

  9. [9]

    Selecta Mathematica , volume =

    Blum, Harold and Liu, Yuchen and Xu, Chenyang and Zhuang, Ziquan , title =. Selecta Mathematica , volume =. 2022 , pages =

    work page 2022

  10. [10]

    Pure and Applied Mathematics Quarterly , volume =

    McQuillan, Michael , title =. Pure and Applied Mathematics Quarterly , volume =. 2008 , pages =

    work page 2008

  11. [11]

    Inventiones Mathematicae , volume =

    Paolo Cascini and Calum Spicer , title =. Inventiones Mathematicae , volume =. 2021 , pages =

    work page 2021

  12. [12]

    Donaldson, S. K. , title =. Journal of Differential Geometry , volume =. 2002 , pages =

    work page 2002

  13. [13]

    Mathematische Zeitschrift , volume =

    Li, Chi , title =. Mathematische Zeitschrift , volume =

    work page

  14. [14]

    2504.10737 , archivePrefix =

    Cascini, Paolo and Han, Jingjun and Liu, Jihao and Meng, Fanjun and Spicer, Calum and Svaldi, Roberto and Xie, Lingyao , title =. 2504.10737 , archivePrefix =

    work page arXiv

  15. [15]

    2024 , eprint =

    Paolo Cascini and Jingjun Han and Jihao Liu and Fanjun Meng and Calum Spicer and Roberto Svaldi and Lingyao Xie , title =. 2024 , eprint =

    work page 2024

  16. [16]

    , title =

    Xu, C. , title =. 2025 , isbn =

    work page 2025

  17. [17]

    Annales scientifiques de l'

    Chen Jiang , title =. Annales scientifiques de l'. 2020 , pages =

    work page 2020

  18. [18]

    2015 , doi =

    Marco Brunella , title =. 2015 , doi =

    work page 2015

  19. [19]

    On Fano foliations , journal =

    Carolina Araujo and St. On Fano foliations , journal =. 2013 , pages =

    work page 2013

  20. [20]

    Birational Geometry of Algebraic Varieties , series =

    J. Birational Geometry of Algebraic Varieties , series =

    work page

  21. [21]

    Singularities of the Minimal Model Program , series =

    J. Singularities of the Minimal Model Program , series =

    work page

  22. [22]

    Robert Lazarsfeld , title =

    work page

  23. [23]

    A non-Archimedean approach to

    Boucksom, S. A non-Archimedean approach to. J. Amer. Math. Soc. , volume =. 2021 , doi =

    work page 2021

  24. [24]

    The Calabi Problem for Fano Threefolds , series =

    Carolina Araujo and Ana-Maria Castravet and Ivan Cheltsov and Kento Fujita and Anne-Sophie Kaloghiros and Jesus Martinez-Garcia and Constantin Shramov and Hendrik S. The Calabi Problem for Fano Threefolds , series =. 2023 , doi =

    work page 2023

  25. [25]

    Jonsson, Mattias and Mustaţă, Mircea , title =. Ann. Inst. Fourier (Grenoble) , volume =. 2012 , doi =

    work page 2012

  26. [26]

    Smith , title =

    Lawrence Ein and Robert Lazarsfeld and Karen E. Smith , title =. American Journal of Mathematics , volume =. 2003 , pages =

    work page 2003

  27. [27]

    Duke Math

    Blum, Harold and Liu, Yuchen and Xu, Chenyang , title =. Duke Math. J. , volume =. 2022 , doi =

    work page 2022

  28. [28]

    Odaka, Yuji , title =. Int. Math. Res. Not. IMRN , year =

    work page

  29. [29]

    Donaldson, S. K. , title =. J. Differential Geom. , volume =

    work page

  30. [30]

    On moduli of foliated surfaces , year =

    Calum Spicer and Roberto Svaldi and Sebastian Vel. On moduli of foliated surfaces , year =. 2511.13491 , archivePrefix=

    work page arXiv

  31. [31]

    Good moduli spaces for

    Alper, Jarod , journal=. Good moduli spaces for

    work page

  32. [32]

    Reductivity of the automorphism group of

    Alper, Jarod and Blum, Harold and Halpern-Leistner, Daniel and Xu, Chenyang , journal=. Reductivity of the automorphism group of. 2020 , publisher=

    work page 2020

  33. [33]

    On properness of

    Blum, Harold and Halpern-Leistner, Daniel and Liu, Yuchen and Xu, Chenyang , journal=. On properness of. 2021 , publisher=

    work page 2021

  34. [34]

    Uniqueness of

    Blum, Harold and Xu, Chenyang , journal=. Uniqueness of. 2019 , publisher=

    work page 2019

  35. [35]

    Positivity of the

    Codogni, Giulio and Patakfalvi, Zsolt , journal=. Positivity of the. 2021 , publisher=

    work page 2021

  36. [36]

    Algebraicity of the metric tangent cones and equivariant

    Li, Chi and Wang, Xiaowei and Xu, Chenyang , journal=. Algebraicity of the metric tangent cones and equivariant

    work page

  37. [37]

    Finite generation for valuations computing stability thresholds and applications to

    Liu, Yuchen and Xu, Chenyang and Zhuang, Ziquan , journal=. Finite generation for valuations computing stability thresholds and applications to. 2022 , publisher=

    work page 2022

  38. [38]

    Annals of Mathematics , volume=

    A minimizing valuation is quasi-monomial , author=. Annals of Mathematics , volume=. 2020 , publisher=

    work page 2020

  39. [39]

    Xu, Chenyang and Zhuang, Ziquan , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2020 , NUMBER =. doi:10.4007/annals.2020.192.3.7 , URL =

    work page doi:10.4007/annals.2020.192.3.7 2020

  40. [40]

    Duke Math

    Spotti, Cristiano and Sun, Song and Yao, Chengjian , TITLE =. Duke Math. J. , FJOURNAL =. 2016 , NUMBER =. doi:10.1215/00127094-3645330 , URL =

    work page doi:10.1215/00127094-3645330 2016

  41. [41]

    Xu, Chenyang and Zhuang, Ziquan , TITLE =. Camb. J. Math. , FJOURNAL =. 2021 , NUMBER =. doi:10.4310/CJM.2021.v9.n1.a2 , URL =

    work page doi:10.4310/cjm.2021.v9.n1.a2 2021

  42. [42]

    Inventiones Mathematicae , volume =

    Jarod Alper and Daniel Halpern-Leistner and Jochen Heinloth , title =. Inventiones Mathematicae , volume =. 2023 , pages =

    work page 2023

  43. [43]

    Threshold optimization for F measure of macro-averaged precision and recall

    Ziquan Zhuang , title =. Advances in Mathematics , volume =. 2020 , pages =. doi:10.1016/j.aim.2020.107250 , eprint =

    work page doi:10.1016/j.aim.2020.107250 2020

  44. [44]

    Sean Timothy Paul and Gang Tian , title =. Ast. 2009 , pages =

    work page 2009

  45. [45]

    Mathematical Research Letters , volume =

    Xiaowei Wang , title =. Mathematical Research Letters , volume =. 2012 , pages =

    work page 2012

  46. [46]

    Forum of Mathematics, Pi , volume =

    Abban, Hamid and Zhuang, Ziquan , title =. Forum of Mathematics, Pi , volume =. 2022 , pages =

    work page 2022

  47. [47]

    Compositio Mathematica , author=

    Higher-dimensional foliated Mori theory , volume=. Compositio Mathematica , author=. 2020 , pages=. doi:10.1112/S0010437X19007681 , number=

    work page doi:10.1112/s0010437x19007681 2020

  48. [48]

    International Mathematics Research Notices , author=

    K\". International Mathematics Research Notices , author=. 2013 , pages=. doi:10.1093/imrn/rns279 , number=

    work page doi:10.1093/imrn/rns279 2013

  49. [49]

    Journal of the European Mathematical Society , volume =

    Spicer, Calum and Svaldi, Roberto , title =. Journal of the European Mathematical Society , volume =. 2022 , pages =

    work page 2022

  50. [50]

    Positivity of the moduli part

    Florin Ambro and Paolo Cascini and Vyacheslav V. Shokurov and Calum Spicer , title =. 2021 , eprint =. doi:10.48550/arXiv.2111.00423 , url =

    work page doi:10.48550/arxiv.2111.00423 2021

  51. [51]

    Forum of Mathematics, Pi , volume =

    Paolo Cascini and Calum Spicer , title =. Forum of Mathematics, Pi , volume =. 2025 , pages =

    work page 2025

  52. [52]

    Higher Dimensional Algebraic Geometry: A Volume in Honor of V

    Paolo Cascini and Calum Spicer , title =. Higher Dimensional Algebraic Geometry: A Volume in Honor of V. V. Shokurov , publisher =. 2025 , pages =

    work page 2025

  53. [53]

    and McKernan, James , title =

    Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James , title =. J. Amer. Math. Soc. , volume =. 2010 , pages =

    work page 2010

  54. [54]

    and McKernan, James and Xu, Chenyang , title =

    Hacon, Christopher D. and McKernan, James and Xu, Chenyang , title =. Ann. of Math. (2) , volume =. 2014 , pages =

    work page 2014

  55. [55]

    and McKernan, James and Xu, Chenyang , title =

    Hacon, Christopher D. and McKernan, James and Xu, Chenyang , title =. Ann. of Math. (2) , volume =. 2013 , pages =

    work page 2013

  56. [56]

    Birkar, Caucher , title =. Ann. of Math. (2) , volume =. 2021 , pages =

    work page 2021

  57. [57]

    Birkar, Caucher , title =. Ann. of Math. (2) , volume =. 2019 , pages =

    work page 2019

  58. [58]

    , title =

    Berman, Robert J. , title =. Inventiones Mathematicae , volume =. 2016 , doi =

    work page 2016

  59. [59]

    Duke Mathematical Journal , volume =

    Chi Li , title =. Duke Mathematical Journal , volume =. 2017 , doi =

    work page 2017

  60. [60]

    arXiv preprint arXiv:2003.11858 , year =

    Kewei Zhang , title =. arXiv preprint arXiv:2003.11858 , year =. 2003.11858 , archivePrefix =

    work page arXiv 2003

  61. [61]

    2026 , eprint=

    Birational boundedness of stable families , author=. 2026 , eprint=

    work page 2026

  62. [62]

    Journal für die reine und angewandte Mathematik (Crelles Journal) , author=

    K-stability of cubic fourfolds , volume=. Journal für die reine und angewandte Mathematik (Crelles Journal) , author=. 2022 , pages=. doi:10.1515/crelle-2022-0002 , number=

    work page doi:10.1515/crelle-2022-0002 2022

  63. [63]

    Theodoros Papazachariou , title =

    work page