REVIEW 1 major objections 5 minor 34 references
Local and global applications of the Minimal Model Program for co-rank one foliations on threefolds
T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The minimal model program for threefold foliations converts discrepancy classes into local integrability: canonical singularities have first integrals, terminal singularities are explicit quotients, and log canonical singularities always…
desk verdict Completes the foliated MMP on threefolds with a local analytic gap in the separatrix theorem that a referee should press. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the F-dlt modification, a foliated analogue of a dlt modification: a birational morphism that extracts only divisors of foliation discrepancy equal to the transversality defect $-\epsilon(E)$, arranged so that the transformed pair is foliated dlt. Its work is to turn a local singularity into a global model: in the dicritical case it extracts a transverse exceptional divisor $E_0$, and the existence of a separatrix is reduced, via foliated adjunction, to finding an invariant algebraic curve on $E_0$ for a restricted foliation with trivial first Chern class, which is then guaranteed by the classification of rank-one foliations with trivial canonical class on surfaces. A secondary mechanism, used for the Malgrange-type results, is the holomorphic Godbillon-Vey sequence attached to the foliation's defining 1-form, whose existence follows from control of the ambient singularity and which produces the holomorphic first integral.
What would settle it
Construct a log canonical co-rank one foliation germ on an isolated klt threefold that has no formal or analytic separatrix; a concrete route is to take a separatrix-free Jouanolou-type 1-form on $\mathbb{C}^3$, perturb it to make the singular locus isolated, and compute foliated discrepancies to see whether any such germ is log canonical while remaining separatrix-free.
Extended reading notes
Core claim
The central claim, stated as Theorem 6.1, is that a germ of a log canonical co-rank one foliation singularity on an isolated klt threefold singularity always admits a separatrix: a formal or convergent invariant hypersurface through the singular point. Because log canonical singularities are generally dicritical, the classical non-dicritical separatrix theorem does not apply; the paper supplies a new proof by passing to an F-dlt modification and finding an invariant algebraic curve on a transverse exceptional divisor. The paper also proves a singular Malgrange theorem (Theorem 5.1): an isolated canonical foliation singularity on an isolated analytically Q-factorial threefold has a holomorphic first integral, and for terminal singularities (Theorem 5.20) no assumption on the ambient germ is needed and the germs are classified, up to a $\mathbb{Z}/n \times \mathbb{Z}/m$-cover, as smoothings of Du Val surface singularities. The global half of the paper establishes termination of flips for F-dlt pairs, connectedness of the non-klt locus, a non-vanishing theorem, and a foliated Mori hyperbolicity statement.
Load-bearing premise
The load-bearing premise is that the foliated minimal model program for co-rank one foliations on threefolds works as established in [CS21]; in particular, the separatrix proof for the dicritical log canonical case needs an F-dlt modification that extracts a transverse exceptional divisor, so a failure of that existence theorem would remove the support from Theorem 6.1.
Editorial extensions
If this is right
- Every log canonical co-rank one foliation singularity on an isolated klt threefold has at least one invariant analytic curve, so the local dynamical system defined by the foliation has a genuine solution curve through the singular point.
- An isolated canonical foliation singularity on a Q-factorial threefold is locally a pullback of a smooth foliation: it admits a holomorphic first integral, and consequently a separatrix.
- Terminal foliation singularities on threefolds have an explicit description: up to a $\mathbb{Z}/n \times \mathbb{Z}/m$-cover they are smoothings of Du Val surface singularities, and the underlying threefold is terminal.
- The foliated MMP terminates for F-dlt co-rank one pairs on Q-factorial threefolds, completing the program started in the companion paper; pseudo-effective $K_F + \Delta$ then has a nonzero section, and non-klt loci satisfy a connectedness principle.
- A foliated hyperbolicity criterion holds: under potential kltness and the absence of non-constant $\mathbb{A}^1$-curves tangent to the foliation, both in the complement of the non-klt locus and in its strata, $K_F + \Delta$ must be nef.
Reading between the lines
- If Theorem 6.1 extends beyond dimension three, it would settle the log canonical case of the local separatrix problem for foliations in all dimensions; the paper itself raises exactly this as an open question.
- The F-dlt modification construction suggests a practical separatrix search: extract a transverse exceptional divisor, compute the restricted foliation, and test for invariant algebraic curves; this could be implemented on explicit 1-forms to detect separatrices computationally.
- The terminal classification should constrain deformation theory and moduli of threefold foliations, since every terminal germ must sit inside one of the six Du Val smoothing families before a finite quotient is taken.
- The connectedness and hyperbolicity results point toward a log canonical version of the foliated MMP; the paper explicitly asks whether log canonical flips exist, and its separatrix theorem removes a known obstruction to that program.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops local and global applications of the recently developed minimal model program for co-rank one foliations on threefolds. Locally, it proves a singular analogue of Malgrange's theorem (Theorem 5.1), a classification of terminal foliation singularities as cyclic quotients of smoothings of Du Val singularities (Theorem 5.20), and the existence of separatrices for log canonical foliation singularities on isolated klt bases (Theorem 6.1). Globally, it proves termination of flips (Theorem 2.1), a non-vanishing theorem (Theorem 2.6), connectedness of non-klt loci (Theorem 3.1), inversion of adjunction (Theorem 3.12), a foliated relative vanishing theorem (Theorem 4.3), and a foliated Mori hyperbolicity statement (Theorem 7.1). The proofs systematically use F-dlt modifications and the foliated MMP, and the paper explicitly relies on [CS21] and [Spi20] for existence of flips, contractions, special termination, and F-dlt modifications.
Significance. If the results hold, this is a substantial advance: it completes the termination part of the foliated MMP on threefolds and gives a general separatrix theorem for log canonical foliation singularities, converting a discrepancy condition into dynamical information. The paper is carefully structured, with the main reductions clearly identified: foliation flips are reduced to usual threefold log terminal flips in Section 2.2, and lc separatrices are constructed through F-dlt modifications in Section 6. The statements are sharp, as Example 6.3 shows that the isolated klt hypothesis on the base is necessary for the stated level of generality, and Jouanolou's example shows that log canonicity is close to optimal. The proofs are detailed and I found no post-hoc selection or circular reasoning in the central arguments; the main concern is an unresolved gap between the stated quasi-projective MMP statements and the local analytic setting used in one load-bearing lemma.
major comments (1)
- [§6, Lemma 6.8] The proof of Lemma 6.8 invokes Corollary 2.3 to run a (K_F + Σ ε(E_i')E_i' − δE_0')-MMP over X, where X is a germ of a normal threefold and not assumed to be quasi-projective. Corollary 2.3 is stated only for Q-factorial quasi-projective threefolds equipped with a projective morphism to an algebraic base; no algebraization of the analytic germ or local analytic version of the foliated MMP is supplied. This is load-bearing: Lemma 6.8 produces the transverse exceptional divisor E0 with π^{-1}(P) ⊂ E0, and Lemma 6.9 and the curve case of Theorem 6.1 use this divisor to construct the separatrix in the log canonical but not canonical case. If the needed local statement is already contained in [CS21], the paper should give a precise citation; otherwise the proof of Theorem 6.1 currently covers only germs that are restrictions of quasi-projective models.
minor comments (5)
- [Abstract] The abstract contains the typo 'non-vanshing'; it should read 'non-vanishing'.
- [§2, Corollary 2.3] The phrase 'constrcuted' should be 'constructed'.
- [§4, proof of Lemma 4.2] There is a typo 'Lemm 4.1' in the sentence referring to Lemma 4.1; it should read 'Lemma 4.1'.
- [Throughout] Several displayed diagrams and formulas contain OCR artifacts such as '/d47/d47', '/d31/d31', and 'Tr eor em` e'; these should be cleaned in the final version so that the arrows and foreign-language citations are readable.
- [§5.4] The references to 'Theor´ em` e A' and 'factorisation' contain encoding errors and should be typeset correctly.
Circularity Check
No circularity: the derivation chain rests on prior MMP results as external inputs, not on the paper's own conclusions.
full rationale
Walking the derivation chains, I find no step in which a target conclusion is used as an input, nor any fitted or normalized quantity renamed as a prediction. The main local theorems (5.1 and 6.1) and global theorems (2.1, 3.1, 4.3, 7.1) are proved from the discrepancy definitions, foliated adjunction, and MMP statements imported from [CS21] and [Spi20]. Those imports are prior results by overlapping authors, but they are cited as established black boxes with proofs elsewhere, not as consequences of the present claims. In particular, neither [CS21] nor [Spi20] contains the separatrix existence or first-integral conclusions proved here, so the central claims are not assumed. The load-bearing step for the non-canonical case of Theorem 6.1 is Lemma 6.8, which invokes Corollary 2.3 to run a foliated MMP over an analytic germ; the paper does not spell out a local analytic algebraization or a quasi-projective compactification covering that germ. That is a domain-of-applicability gap and a correctness risk, not a circular reduction: the MMP input does not already contain the separatrix output. No equation in the paper reduces by construction to its own input, and the self-citations to [CS21] and [Spi20] are external support under the stated standard, since they are prior theorems with independent proofs rather than re-statements of the present results. I therefore find no significant circularity.
Assumptions & free parameters
assumptions (10)
- domain assumption All spaces are over C; local statements concern complex analytic germs while global statements concern normal quasi-projective varieties (Section 1, Notations).
- standard math Foliated log resolutions exist for threefold foliations after blow ups centered in the singular locus ([Can04]).
- standard math Foliated MMP existence results: flips, divisorial contractions, cone theorem, and related structural theorems ([CS21, Theorems 6.4, 6.7]).
- standard math Existence of F-dlt modifications with Q-factorial klt output ([CS21, Theorem 8.1]; extended here as Theorem 2.4).
- standard math Special Termination for F-dlt pairs ([CS21, Theorem 7.1]).
- standard math Termination of threefold log terminal flips for usual pairs ([KM98, Theorem 6.17]).
- standard math McQuillan's classification of rank one canonical foliations on surfaces with c1(K_F) = 0 ([McQ08, Theorem IV.3.6]).
- standard math Finiteness of local fundamental groups and extension of finite quasi-etale covers ([TX17, Corollary 1.4]; [GKP16, Proposition 3.13]).
- standard math Reflexive differential form pullback and extension theorems ([Keb13, Theorem 1.2]; [GKKP11, Theorem 4.3]).
- standard math Formal first integral factorization and convergence criteria ([MM80]; [KKMSD73, Chapter II]).
Cite this review
Pith. "Pith review of Local and global applications of the Minimal Model Program for co-rank one foliations on threefolds." pith.science (2026). https://pith.science/paper/HHIJQDR2
@misc{pith2026190805037,
author = {Pith},
title = {Pith review of: Local and global applications of the Minimal Model Program for co-rank one foliations on threefolds},
year = {2026},
howpublished = {\url{https://pith.science/paper/HHIJQDR2}},
note = {Machine review of arXiv:1908.05037}
}
read the original abstract
We provide several applications of the minimal model program to the local and global study of co-rank one foliations on threefolds. Locally, we prove a singular variant of Malgrange's theorem, a classification of terminal foliation singularities and the existence of separatrices for log canonical singularities. Globally, we prove termination of flips, a connectedness theorem on lc centres, a non-vanshing theorem and some hyperbolicity properties of foliations.
Reference graph
Works this paper leans on
-
[1]
C. Araujo and S. Druel, On F ano foliations , Adv. Math. 238 (2013), 70--118. 3033631
work page 2013
-
[2]
F. Bogomolov and M. McQuillan, Rational curves on foliated varieties, Foliation theory in algebraic geometry, Simons Symp., Springer, Cham, 2016, pp. 21--51. 3644242
work page 2016
-
[3]
Brunella, Birational geometry of foliations, Monograf\' as de Matem\' a tica
M. Brunella, Birational geometry of foliations, Monograf\' as de Matem\' a tica. [Mathematical Monographs], Instituto de Matem\' a tica Pura e Aplicada (IMPA), Rio de Janeiro, 2000, Available electronically at http://www.impa.br/Publicacoes/Monografias/Abstracts/brunella.ps. 1948251
work page 2000
-
[4]
Camacho, Quadratic forms and holomorphic foliations on singular surfaces, Math
C. Camacho, Quadratic forms and holomorphic foliations on singular surfaces, Math. Ann. 282 (1988), no. 2, 177--184. 963011
work page 1988
-
[5]
Cano, Reduction of the singularities of codimension one singular foliations in dimension three, Ann
F. Cano, Reduction of the singularities of codimension one singular foliations in dimension three, Ann. of Math. (2) 160 (2004), no. 3, 907--1011. 2144971 (2006f:32041)
work page 2004
-
[6]
F. Cano and D. Cerveau, Desingularization of nondicritical holomorphic foliations and existence of separatrices, Acta Math. 169 (1992), no. 1-2, 1--103. 1179013
work page 1992
-
[7]
D. Cerveau and A. Lins Neto, Frobenius theorem for foliations on singular varieties, Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 3, 447--469. 2473858
work page 2008
-
[8]
F. Cano and M. Ravara-Vago, Local B runella's alternative II . P artial separatrices , Int. Math. Res. Not. IMRN (2015), no. 23, 12840--12876. 3431638
work page 2015
Show all 34 references
-
[9]
F. Cano, M. Ravara-Vago, and M. Soares, Local B runella's alternative I . RICH foliations , Int. Math. Res. Not. IMRN (2015), no. 9, 2525--2575. 3344680
2015
-
[10]
Cascini and C
P. Cascini and C. Spicer, Mmp for co-rank one foliations on threefolds, 2018, ArXiv e-print, arXiv:1808.02711 https://arxiv.org/abs/1808.02711
2018 arXiv
-
[11]
Druel, Codimension one foliations with numerically trivial canonical class on singular spaces, 2018, ArXiv e-print, arXiv:1809.06905 https://arxiv.org/abs/1809.06905
S. Druel, Codimension one foliations with numerically trivial canonical class on singular spaces, 2018, ArXiv e-print, arXiv:1809.06905 https://arxiv.org/abs/1809.06905
2018 arXiv
-
[12]
Flenner, Divisorenklassengruppen quasihomogener S ingularit\" a ten , J
H. Flenner, Divisorenklassengruppen quasihomogener S ingularit\" a ten , J. Reine Angew. Math. 328 (1981), 128--160. 636200
1981
-
[13]
D. Greb, S. Kebekus, S. J. Kov \'a cs, and T. Peternell, Differential forms on log canonical spaces, Publ. Math. Inst. Hautes \'Etudes Sci. (2011), no. 114, 87--169. 2854859
2011
-
[14]
D. Greb, S. Kebekus, and T. Peternell, \'etale fundamental groups of K awamata log terminal spaces, flat sheaves, and quotients of abelian varieties , Duke Math. J. 165 (2016), no. 10, 1965--2004. 3522654
2016
-
[15]
G\' o mez-Mont and I
X. G\' o mez-Mont and I. Luengo, Germs of holomorphic vector fields in C^3 without a separatrix , Invent. Math. 109 (1992), no. 2, 211--219. 1172688
1992
-
[16]
M. L. Green, The hyperbolicity of the complement of 2n+1 hyperplanes in general position in P_ n and related results , Proc. Amer. Math. Soc. 66 (1977), no. 1, 109--113. 457790
1977
-
[17]
Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, Graduate Texts in Mathematics, No
R. Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, Graduate Texts in Mathematics, No. 52. 0463157
1977
-
[18]
Kebekus, Pull-back morphisms for reflexive differential forms, Adv
S. Kebekus, Pull-back morphisms for reflexive differential forms, Adv. Math. 245 (2013), 78--112. 3084424
2013
-
[19]
Kempf, F
G. Kempf, F. F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I , Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. 0335518
1973
-
[20]
Koll\' a r and S
J. Koll\' a r and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. 1658959
1998
-
[21]
Loray, J
F. Loray, J. V. Pereira, and F. Touzet, Singular foliations with trivial canonical class, Invent. Math. 213 (2018), no. 3, 1327--1380. 3842065
2018
-
[22]
Malgrange, Frobenius avec singularit\' e s
B. Malgrange, Frobenius avec singularit\' e s. I . C odimension un , Inst. Hautes \' E tudes Sci. Publ. Math. (1976), no. 46, 163--173. 0508169
1976
-
[23]
McQuillan, Diophantine approximations and foliations, Inst
M. McQuillan, Diophantine approximations and foliations, Inst. Hautes \' E tudes Sci. Publ. Math. (1998), no. 87, 121--174. 1659270
1998
-
[24]
, Canonical models of foliations, Pure Appl. Math. Q. 4 (2008), no. 3, part 2, 877--1012
2008
-
[25]
L. G. Mendes, Kodaira dimension of holomorphic singular foliations, Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), no. 2, 127--143. 1785264
2000
-
[26]
Mattei and R
J.-F. Mattei and R. Moussu, Holonomie et int\'egrales premi\`eres, Ann. Sci. \'Ecole Norm. Sup. (4) 13 (1980), no. 4, 469--523. 608290
1980
-
[27]
McQuillan and G
M. McQuillan and G. Pacienza, Remarks about bubbles, 2012, ArXiv e-print, arXiv:1211.0203 https://arxiv.org/abs/1211.0203
2012 arXiv
-
[28]
Molina-Samper, Invariant surfaces for toric type foliations in dimension three, 2019, ArXiv e-print, arXiv:1905.00836 https://arxiv.org/abs/1905.00836
B. Molina-Samper, Invariant surfaces for toric type foliations in dimension three, 2019, ArXiv e-print, arXiv:1905.00836 https://arxiv.org/abs/1905.00836
2019 arXiv
-
[29]
J. V. Pereira and R. Svaldi, Effective algebraic integration in bounded genus, Algebr. Geom. 6 (2019), no. 4, 454--485. 3957403
2019
-
[30]
Reid, Young person's guide to canonical singularities, Algebraic Geometry Bowdoin 1985 (Proc
M. Reid, Young person's guide to canonical singularities, Algebraic Geometry Bowdoin 1985 (Proc. Symp. Pure Math., ed.), 1987, pp. 345--416
1985
-
[31]
Siu, Extending coherent analytic sheaves, Ann
Y.-T. Siu, Extending coherent analytic sheaves, Ann. of Math. (2) 90 (1969), 108--143. 0245837
1969
-
[32]
Spicer, Higher dimensional foliated M ori theory , 2017, ArXiv e-print, arXiv:1709.06850 https://arxiv.org/abs/1709.06850
C. Spicer, Higher dimensional foliated M ori theory , 2017, ArXiv e-print, arXiv:1709.06850 https://arxiv.org/abs/1709.06850. Accepted for publication in Compositio Mathematica
2017 arXiv
-
[33]
Svaldi, Hyperbolicity for log canonical pairs and the cone theorem, 2014, ArXiv e-print, arXiv:1410.2529v2 https://arxiv.org/abs/1410.2529
R. Svaldi, Hyperbolicity for log canonical pairs and the cone theorem, 2014, ArXiv e-print, arXiv:1410.2529v2 https://arxiv.org/abs/1410.2529
2014 arXiv
-
[34]
Tian and C
Z. Tian and C. Xu, Finiteness of fundamental groups, Compos. Math. 153 (2017), no. 2, 257--273. 3604863
2017
Reviewed August 14, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.