Local and global applications of the Minimal Model Program for co-rank one foliations on threefolds

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.


Introduction
Our primary goal in this paper is to use techniques and ideas from the foliated minimal model program (MMP) to deduce some structural and dynamical results for foliation singularities. Along the way of proving these results we further develop the MMP and explore some applications of these developments to global properties of foliations.
Local results. To every foliation singularity the MMP associates a numerical invariant, the discrepancy, which measures how the canonical class of the foliation changes under blow ups.
Our local results explore to what extent this numerical invariant characterizes the structural and dynamical behavior of the foliation singularity.
Here we are mostly interested in three classes of foliation singularities which are defined according to the behavior of the discrepancy, namely, terminal, canonical and log canonical, see Definition 1.4. Terminal singularities can be viewed as the mildest class of singularities of the MMP whereas log canonical are the most severe. For a two quick illustrations of this principle, terminal foliations on smooth surfaces are smooth foliations, and simple singularities (see Definition 1.8) are canonical, we refer the reader to Section 1.9 for a further discussion on the relations and parallels between the classes of singularities of the MMP and other classes of singularities.
The singularities of the MMP are birational generalizations of nice classes of foliation singularities and are natural from the perspective of the geometry of foliations. For example, canonical foliation singularities play a central role in the study of hyperbolicity properties of surfaces by McQuillan, see [McQ98,McQ08], and the classification of foliations with trivial canonical bundle, see [LPT18,Dru18].
We remark that simple singularities (which are roughly analogous to the singularities of smooth normal crossings pairs) are not preserved by the operations of the MMP, since the underlying variety may become singular in the course of the MMP. In other words, if one seeks to improve the global geometry of the foliation (by making K F more positive) one loses some control on the local geometry of the foliation. As a result, canonical singularities should be viewed as a compromise. They are flexible enough to allow for the operations of the MMP, but mild enough to have many of the same desirable properties as simple singularities.
A fundamental result in the study of singular foliations on smooth varieties is a theorem of Malgrange [Mal76] asserting that the classical Frobenius integrability criterion holds even in the presence of foliated singularities, provided that the codimension of the singular locus of the foliation is at least 3. We prove a version of Malgrange's theorem on singular threefolds.
Theorem 0.1 (= Theorem 5.1). Let P ∈ X be a germ of an isolated (analytically) Q-factorial threefold singularity with a co-rank 1 foliation F. Suppose that F has an isolated canonical singularity at P .
Then F admits a holomorphic first integral.
The above statement is close to optimal, cf. [CLN08, Examples 1.1-1.3]. As a consequence of the above theorem, we obtain the following strong classification result.
Theorem 0.2 (= Theorem 5.20). Let P ∈ X be a germ of normal threefold with a co-rank 1 foliation F with terminal singularities. Then F admits a holomorphic first integral.
Moreover, up to a Z/nZ × Z/mZ-cover, F admits a holomorphic first integral φ : (P ∈ X) → (0 ∈ C), where φ −1 (0) is a Du Val surface singularity and φ −1 (t) is smooth for t = 0. In particular, X is terminal. Moreover, (X, F) fits into the finite list of families contained in Proposition 5.19.
We remark that in Theorem 0.2 we make no assumption on the singularities of the underlying space other than normality.
We also remark that Theorems 0.1 and 0.2 should be viewed as analogues of the classification of terminal and canonical singularities on threefolds.
These classification results have been crucial in understanding the global geometry of threefolds, as well as the moduli space of surfaces, and we expect that the above results to play a corresponding role in the study of the global geometry of foliations of threefolds and moduli of surface foliations.
We next prove a result on the existence of separatrices of log canonical foliation singularities. Loosely speaking a separatrix may be thought of as a local solution to the differential equation defining the foliation, see Definition 1.11 for a precise definition. It is an interesting and challenging problem to decide when a foliation singularity admits a separatrix. The existence of a (converging) separatrix is an essential element in the study of foliations singularities as it provides a way to "organize" the dynamics around the foliation singularity. While separatrices do not necessarily exist for a general foliation singularity, we prove their existence for log canonical singularities.
Theorem 0.3 (= Theorem 6.1). Let F be a germ of a log canonical foliation singularity on 0 ∈ C 3 . Then F admits a separatrix.
Our strategy of proof actually provides a more general version of this result allowing the underlying analytic germ to be singular.
In [CC92] it is shown that non-dicritical foliation singularities always admit separatrices, confirming a conjecture of R. Thom. Log canonical singularities are in general dicritical and so [CC92] does not apply to prove existence of separatrices for this class of singularities. Theorem 0.3 is also closely related to a local analogue of a conjecture of Brunella that has been formulated in [CRVS15] and explored in [CRV15].
We remark that results analogous to Theorem 0.1 and Theorem 0.3 have been shown in [CLN08] and [MS19], respectively, under differing assumptions on the singularities of the foliation and variety. An advantage of our statements is that they hold for very natural classes of singularities which are satisfied by a wide range of foliations.
Classically, the technique of inversion of adjunction has proven crucial for understanding singularities by providing a precise relation between the singularities of a variety and the singularities of a divisor in the variety. We prove a foliated analogue of this result which should prove equally useful in the study of foliation singularities.
Theorem 0.4 (= Theorem 3.12). Let X be a Q-factorial threefold and let F be a co-rank one foliation. Consider a prime divisor S and an effective Q-divisor ∆ on X which does not contain S in its support. Let ν : S ν → S be the normalization and let G be the restricted foliation to S ν and write ν * (K F + ∆) = K G + Θ. Suppose that • if S is transverse to F, then (G, Θ) is lc; • if S is F-invariant, then (S ν , Θ) is lc.
Then (F, ǫ(S)S + ∆) is lc in a neighborhood of S.
We refer the reader to §3.3 for a discussion of this result and its relationship to the adjunction formula for lc pairs, cf. §3.2.
We now take a moment to explain some of the key ideas of the proofs in the above statements. Indeed, our central innovation is the systematic use of F-dlt modifications to study foliation singularities, see Theorem 2.4 for a recollection on the definition and existence of F-dlt modifications which were first shown to exist in [CS21]. An F-dlt modification (which is a foliated analogue of a classical dlt modification) is a special kind of partial resolution which extracts divisors for which the global properties of the foliation restricted to these divisors strongly reflect the local properties of the foliation singularity. In particular, for dicritical singularities, an F-dlt modification will always extract one exceptional geometric valuation transverse to the foliation.
To prove Theorem 0.3 we extract, by way of an F-dlt modification, an exceptional divisor which is transverse to the foliation (in other words, a dicritical component of the singularity). Showing the existence of a separatrix is then reduced to producing a global invariant algebraic divisor for the restricted foliation on this exceptional divisor. An adjunction calculation shows that this restricted foliation has trivial first chern class, and so the existence of an invariant algebraic divisor is a consequence of the classification of foliations with trivial first chern class.
To prove Theorem 0.1 we provide a precise bound on the singularities of X (we show that X is klt) by controlling the geometry of the invariant divisors on an F-dlt modification. We then show that the singularities of X are mild enough to allow us to prove the existence of a holomorphic Godbillon-Vey sequence associated to the foliation, §5.2, and we may then conclude roughly along the lines of Malgrange's original proof.
Global results. In [Spi20] and [CS21] much the of minimal model program for rank 2 foliations on threefolds was completed, including a cone and contraction theorem, existence of flips and special termination. However, the termination of flips was not proven. In this paper, we termination of flips, thereby completing the statement of the MMP for F-dlt pairs. We refer the reader to Definition 1.13 for the definition of F-dlt singularities, but we emphasize here that they are a very large and natural class of foliated singularities; for example, they include pairs (X, F) such that X is smooth, F has simple singularities.
Theorem 0.5 (= Theorem 2.1). Let X be a Q-factorial quasi-projective threefold. Let (F, ∆) be an F-dlt pair. Then starting at (F, ∆) there is no infinite sequence of flips.
A direct consequence of termination and the work in [CS21] is the following non-vanishing theorem.
Theorem 0.6 (= Theorem 2.6). Let F be a co-rank one foliation on a normal projective Q-factorial threefold X. Let ∆ be a Q-divisor such that (F, ∆) is a F-dlt pair. Let A ≥ 0 and B ≥ 0 be Q-divisors such that ∆ = A + B and A is ample. Assume that K F + ∆ is pseudo-effective Then K F + ∆ ∼ Q D ≥ 0.
We then turn our attention to the study of the non-klt centres of foliations. One of our central results in this direction is the proof of a foliated analogue of the connectedness of non-klt centres.
Theorem 0.7 (= Theorem 3.1). Let X be a projective Q-factorial threefold and let F be a rank 2 foliation on X. Let (F, ∆) be an F -dlt pair on X. Assume that −(K F + ∆) is nef and big. Then Nklt(F, ∆) is connected.
One of the fundamental ideas of the foliated MMP is that the negativity of foliated log pairs (F, ∆) with mild singularities is governed by the presence of rational curves, see, for example, [Spi20]. As a final application we prove a foliated version of the main result of [Sva19] which relates the hyperbolicity of a foliation to an analysis of the log canonical singularities of a foliation. Given a foliated pair (F, ∆) and an lc center S we will denote byS ⊂ S the locally closed subvariety obtained by removing from S the lc centers of (F, ∆) strictly contained in S.
Theorem 0.8 (= Theorem 7.1, Foliated Mori hyperbolicity). Let (F, ∆) be a foliated log canonical pair on a normal projective threefold X. Assume that • X is klt, • there is no non-constant morphism f : A 1 → X \ Nklt(F, ∆) which is tangent to F, and • for any stratum S of Nklt(F, ∆) there is no non-constant morphism f : A 1 →S which is tangent to F. Then K F + ∆ is nef.
Finally, we remark that the central idea in the proof of our connectedness and hyperbolicty results is a refinement of the technique of F-dlt modifications, see Theorem 7.2, and a careful analysis of the properties of F-dlt modifications through adjunction.
Acknowledgements. We would like to thank Paolo Cascini, Michael Mc-Quillan and Jorge V. Pereira for many valuable conversations, suggestions and comments. We also thank the referees for many suggestions and improvements to the exposition of the paper. This project was started during a visit of CS and RS to FRIAS; we would like to thank FRIAS and Prof. Stefan Kebekus for their hospitality and the financial support during the course of our visit. Most of this work was completed during several visits of RS to Imperial College, London. RS would like to thank Imperial College for the hospitality and the nice working environment. He would also like to thank University of Cambridge and Churchill College, Cambridge where he was a research fellow when part of this work was completed. RS was partially supported from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No. 842071.

Preliminaries
Notations and conventions. By the term variety, we will always mean an integral separated scheme over an algebraically closed field k. Unless otherwise stated, it will be understood that k = C. Unless otherwise specified, we adopt the same notations and conventions as in [KM98].
A contraction is a projective morphism f : X → Y of quasi-projective varieties with f * O X = O Z . If X is normal, then so is Z and the fibers of f are connected. A proper birational map f : X Y of normal quasiprojective varieties is a birational contraction if f −1 does not contract any divisor.
Given a Weil R-divisor D and a prime divisor E on a normal variety X, we will denote by µ E D the coefficient of E in D. If D is an Weil R-divisor on X then for any c ∈ R we define where * is any of =, ≥, ≤, >, <. We define the round down ⌊D⌋ of D to The support Supp(D) of an R-divisor D is the union of the prime divisors appearing in D with non-zero coefficient, Supp(D) := µ E D =0 E.
1.1. Recollection on foliations. We refer the reader to [Bru00] for basic notions in foliation theory.
A foliation on a normal variety X is a coherent subsheaf F ⊂ T X such that (1) F is saturated, i.e. T X /F is torsion free, and (2) F is closed under Lie bracket.
The rank of F is its rank as a sheaf. Its co-rank is its co-rank as a subsheaf of T X .
Let X be a normal variety and let F be a rank r foliation on X. A canonical divisor of F is a divisor K F such that O X (−K F ) ∼ = det(F). We define the normal sheaf of F as N F := (T X /F) * * . The conormal sheaf N * F of F is the dual of N F . If F is a foliation of co-rank one then, by abuse of notation, we denote by N * F a divisor associated to N * F . We can associate to F a morphism φ : Ω defined by taking the double dual of the r-wedge product of the map Ω and we define the singular locus of F, denoted by sing(F), to be the cosupport of the image of φ ′ .
Given a dominant rational map f : Y X and a foliation F on X we may pullback F to a foliation on Y , denoted f −1 F. Remark 1.1. If q : X ′ → X is a quasi-étale cover and F ′ = q −1 F then K F ′ = q * K F and [Dru18, Proposition 5.13] implies that F ′ is non-singular if and only if F is. In particular, it is not the case that we always have sing(X) ⊂ sing(F).
1.2. Invariant subvarieties. Let X be a normal variety and let F be a rank r foliation on X. Let S ⊂ X be a subvariety. Then S is said to be F-invariant, or invariant by F, if for any open subset U ⊂ X and any section ∂ ∈ H 0 (U, F), we have that 1.3. Foliation singularities. Frequently in birational geometry it is useful to consider pairs (X, ∆) where X is a normal variety, and ∆ is a Q-Weil divisor such that K X + ∆ is Q-Cartier. We refer the reader to [KM98,§ 2] for the relevant definitions and notations on the singularities of pairs. We will use the following definition that we recall as it is non-standard.
Definition 1.2. A normal variety X is said to be potentially lc (resp. potentially klt) if there exists an effective R-divisor D on X such that the log pair (X, D) has lc (resp. klt singularities).
It is possible to define singularities for pairs also in the foliated world, in analogy with the classical case of pairs. Definition 1.3. A foliated pair (F, ∆) is a pair of a foliation and a Q-Weil Note also that we are typically interested only in the cases when ∆ ≥ 0, although it simplifies some computations to allow ∆ to have negative coefficients.
Given a birational morphism π : X → X and a foliated pair (F, ∆) on X let F be the pulled back foliation onX and π −1 * ∆ be the strict transform. We can write , respectively, where ǫ(D) = 0 if D is invariant and 1 otherwise and where π varies across all birational morphisms. If (F, ∆) is log terminal and ⌊∆⌋ = 0 we say that (F, ∆) is foliated klt.
Notice that these notions are well defined, i.e., ǫ(E) and a(E, F, ∆) are independent of π. We say a(E, F, ∆) is the discrepancy of E (with respect to (F, ∆)), or the foliated discrepancy.
Observe that in the case where F = T X no exceptional divisor is invariant, i.e., ǫ(E) = 1, and so this definition recovers the usual definitions of (log) terminal, (log) canonical.
We remark that we will be using the terminal, etc. classification to refer to both the singularities of the foliation and the singularities of the underlying variety. If necessary we will use the term foliation terminal, etc. to emphasize the fact that we are talking about the singularities of the foliation rather than the variety. Definition 1.5. Let (F, ∆) be foliated log pair.
(1) We say that W ⊂ X is a log canonical centre (lc centre) of (F, ∆) provided (F, ∆) is log canonical at the generic point of W and there exists some divisor E of discrepancy −ǫ(E) on some birational model over X such that E dominates W . (2) The nonklt locus Nklt(F, ∆) of (F, ∆) is the union of the centres of all divisors divisorial valuations E of discrepancy ≤ −ε(E).
(1) If ǫ(E i ) = 0 for all exceptional divisors E i over a centre W ⊂ X, the notions of log canonical and canonical centre coincide for W . In this case, we will refer to canonical centres as log canonical centres.
(2) Any F-invariant divisor D is an lc centre of (F, ∆) since D shows up in ∆ with coefficient at least 0 = ǫ(D).
Moreover, a direct computation shows that any strata of a simple singularity is an lc centre.
We have the following nice characterization due to [McQ08, Corollary I.2.2.]: Proposition 1.7. Let 0 ∈ X be a normal surface germ with a terminal foliation F of rank one.
Then there exists a cyclic cover σ : Y → X such that Y is a smooth surface and σ −1 F is a smooth foliation.
We emphasize that the above shows that even if 0 ∈ sing(X) it may be the case that 0 / ∈ sing(F). We will also make use of the class of simple foliation singularities, see [Can04, Appendix: About simple singularities].
Definition 1.8. We say that p ∈ X with X smooth is a simple singularity for F provided in formal coordinates x 1 , ..., x n around p, N * F is generated by a 1-form which is in one of the following two forms, where 1 ≤ r ≤ n.
(1) There are λ i ∈ C * such that and if a i λ i = 0 for some non-negative integers a i then a i = 0 for all i.
(2) There is an integer k ≤ r such that where p i are positive integers, without a common factor, ψ(s) is a series which is not a unit, and λ i ∈ C and if a i λ i = 0 for some non-negative integers a i then a i = 0 for all i. By Cano, [Can04], every foliation on a smooth threefold admits a resolution by blow ups centred in the singular locus of the foliation such that the transformed foliation has only simple singularities.
We recall the definition of non-dicritical foliation singularities, see [CM92,§2]. Definition 1.9. Given a foliated pair (X, F) we say that F has dicritical singularities if for some P ∈ X there exists a germ of a surface P ∈ S such that the restricted foliation F| S has infinitely many invariant curves passing through sing(F) ∩ S.
Otherwise, we say that F has non-dicritical singularities.
We remark that the above definition is equivalent on threefolds to the characterization appearing in [CS21], thanks to the existence of resolution of singularities. Namely, F has non-dicritical singularities if for any sequence of blow ups π : (X ′ , F ′ ) → (X, F) and any closed q ∈ X we have π −1 (q) is tangent to the foliation.
Definition 1.11. Given a germ 0 ∈ X with a foliation F such that 0 is a singular point for F we call a (formal) hypersurface germ 0 ∈ S a (formal) separatrix if it is invariant under F.
Note that away from the singular locus of F a separatrix is in fact a leaf. Furthermore being non-dicritical implies that there are only finitely many separatrices through a singular point. The converse of this statement is false.
Definition 1.12. Given a normal variety X, a co-rank one foliation F and a foliated pair (F, ∆) we say that (F, ∆) is foliated log smooth provided the following hold: (1) (X, ∆) is log smooth.
(3) If S is the support of the non-invariant components of ∆ then for any p ∈ X if Σ 1 , ..., Σ k are the separatrices of F at p (formal or otherwise), then S ∪ Σ 1 ∪ ... ∪ Σ k is normal crossings at p.
Given a normal variety X, a co-rank one foliation F and a foliated pair (F, ∆) a foliated log resolution is a high enough model π : Such a resolution on threefolds is known to exist by [Can04] We recall the class of F-dlt singularities introduced in [CS21, Definition 3.6].
Definition 1.13. Let X be a normal variety and let F be a co-rank one foliation on X. Suppose that K F + ∆ is Q-Cartier.
We say (F, ∆) is foliated divisorial log terminal (F-dlt) if (1) each irreducible component of ∆ is transverse to F and has coefficient at most 1, and (2) there exists a foliated log resolution (Y, G) of (F, ∆) which only extracts divisors E of discrepancy > −ǫ(E).
In the case of surfaces F-dlt singularities have a particularly simple characterization.
Lemma 1.14. Let X be a normal surface and let F be a co-rank one foliation on X. Suppose that K F is Q-Cartier and that F is F-dlt. Then for all P ∈ X one of the following holds: (1) F is terminal at P ; or (2) X is smooth at P and F has a simple singularity at P . In particular, if K F is Cartier then X is smooth.
Proof. Our dichotomy is a direct consequence of [CS21, Lemma 3.8]. Our second claim follows from Proposition 1.7 by observing that if P ∈ X is terminal for F and K F is Cartier near P then X is smooth.
1.4. Pulling back 1-forms. In § 5, we will need the following result.
Proposition 1.15. Let P ∈ X be an isolated potentially klt singularity and let µ :X → X be a resolution of singularities of X and let E be an irreducible µ-exceptional divisor. Let ω ∈ Ω [1] X and let ω := d refl µ(ω) and let ω E be the restriction of ω to E. Then ω E = 0.
Proof. This is a straightforward consequence of the existence of pull-back for differential forms on potentially klt varieties, cf. [Keb13, Theorem 1.2] 1.5. Singularities of X vs. singularities of F. The following is [CS21, Theorem 11.3]. Because we will refer to it frequently we include it here.
Theorem 1.16. Let (F, ∆) be a foliated pair on a quasi-projective threefold X. Assume that either (1) (F, ∆) is F-dlt or (2) (F, ∆) is canonical. Then F has non-dicritical singularities. Furthermore, if (F, ∆) is F-dlt and K X is Q-Cartier then X is klt.
We also have the following comparison of singularities result, which is a slight modification of [CS21, Lemma 3.16].
Lemma 1.17. Let X be a Q-factorial threefold and let F be a co-rank one foliation. Suppose that (F, ∆) is F-dlt. Then (X, ∆) is dlt.
Proof. Let π : X ′ → X be a foliated log resolution of (F, ∆) which only extracts divisors of foliation discrepancy > −ǫ(E). Observe that a foliated log resolution π : X ′ → X of (F, ∆) is a log resolution of (X, ∆). By Theorem 1.16, F has non-dicritical singularities, thus, we may apply [Spi20, Lemma 8.14] to conclude that π only extracts divisors of discrepancy > −1 with respect to K X + ∆, as required.
1.6. Extending separatrices. We recall the following extension of separatrices result.
Lemma 1.18. Let X be a normal quasi-projective threefold. Let F be a co-rank one foliation on X with non-dicritical singularities. Let V ⊂ X be a subvariety tangent to F, let q ∈ V be any point and let S q be a separatrix at q.
Then there exists an analytic open neighborhood U of V in X and an analytic divisor S on U which contains S q near q.
Proof. This is proven in [CS21, Lemma 3.5] (see also [Spi20, §5.1]). We remark that this is a slight extension of the techniques and ideas utilized in [CC92,§IV].
1.7. Special termination. We recall the following theorem, [CS21, Theorem 7.1]: Theorem 1.19 (Special Termination). Let X be a Q-factorial quasi-projective threefold. Let (F, ∆) be an F-dlt pair. Suppose (F i , ∆ i ) is an infinite sequence of (K F i + ∆ i )-flips. Then after finitely many flips, the flipping and flipped locus is disjoint from the lc centres of (F i , ∆ i ). In particular, (F i , ∆ i ) is log terminal in a neighborhood of each flipping curve.
1.8. MMP with scaling. A version of the MMP with scaling was proven in [CS21,§10], however, for our purposes we will need the MMP with scaling in a slightly different form than presented there. Here we briefly explain the necessary adjustments.
We recall the following lemma proven in [CS21, Lemma 3.27] Lemma 1.20. Let X be a normal projective Q-factorial threefold and let F be a co-rank one foliation on X. Let ∆ = A + B be a Q-divisor such that (F, ∆) is a F-dlt pair, A ≥ 0 is an ample Q-divisor and B ≥ 0. Let ϕ : X X ′ be a sequence of steps of the (K F + ∆)-MMP and let F ′ be the induced foliation on X ′ .
Then, there exist Q-divisors A ′ ≥ 0 and C ′ ≥ 0 on X ′ such that 1.8.1. Running the MMP with scaling. Let X be a projective Q-factorial threefold and let F be a co-rank 1 foliation on X. Let ∆ = A + B be a Q-divisor where A ≥ 0 is ample and B ≥ 0 so that (F, ∆) is a F-dlt pair.
Let H be a divisor on X so that K F + ∆ + H is nef. In practice we will often take H to be some sufficiently ample divisor on X.
Setting X 0 := X, F 0 := F, ∆ 0 := ∆ and H 0 := H we may produce a sequence φ i : X i X i+1 of K F i + ∆ i divisorial contractions and flips contracting an extremal ray R i and rational numbers λ i such that Moreover we have that λ i ≥ λ i+1 , and that R i · H i > 0 for all i. Assuming the relevant termination of flips we see that this MMP terminates in either a Mori fibre space or a model where K F i + ∆ i is nef. We call this process the MMP of (F, ∆) (or K F + ∆) with scaling of H.
1.9. (Pre) simple vs. (Log) canonical. We now briefly discuss some relations and parallels between the classes of singularities defined by the MMP and some of the other classes of singularities described above.
Intuitively, for a given singularity, the smaller its discrepancy the more severe the singularity is. So terminal singularities are, in this sense, the mildest kind of singularities appearing in the MMP. Indeed, terminal singularities on smooth surfaces are in fact smooth foliated points, although this equivalence fails in higher dimensions, as we will see in Theorem 5.20.
We observe that simple singularities are both non-dicritical and canonical, however canonical singularities are in general not simple, as the following example shows.
Example 1.21. The foliation on C 2 defined by the vector field x ∂ ∂x + (x + y) ∂ ∂y has canonical singularities (which may be verified since a single blow up resolves the singularities of the foliation to simple singularities, and this blow up has discrepancy = 0). However, since both its eigenvalues are positive integers it is not a simple singularity.
We remark that the above example also shows that canonical singularities are not in general F-dlt singularities. On the other hand, Theorem 1.16 shows that canonical and F-dlt singularities are non-dicritical.
Consider a germ of a vector field ∂ on C 2 and suppose that ∂ is singular at 0 and let m be the maximal ideal ideal at 0. We get an induced linear map ∂ : m/m 2 → m/m 2 which is non-nilpotent if and only if the foliation generated by ∂ is log canonical, see [MP13,Fact I.ii.4]. To our knowledge there is no similar criterion for characterizing log canonical foliations of rank ≥ 2.
We refer to [Can04,Definition 3] for the definition of presimple singularities. The difference between simple and presimple is (roughly) the additional requirement of a non-resonance condition on the eigenvalues of the foliation. For instance, x ∂ ∂x + y ∂ ∂y defines a foliation on C 2 with a presimple but not simple singularity. A single blow up of this foliation has discrepancy = −1 and resolves this foliation to a smooth foliation which shows that this foliation has a log canonical but not canonical singularity.
With this example in mind it might be useful to view the relation between simple and presimple singularities as analogous to the relation between canonical and log canonical singularities. However, we do not know if every presimple singularity is log canonical. It also does not seem to be the case that a canonical singularity is a log canonical singularity which satisfies a particular resonance condition, in light of Example 1.21.
We observe that on a smooth threefold a log canonical singularity which is non-dicritical is necessarily canonical. Indeed, by [Can04] and our assumption of non-dicriticality we may find a resolution of singularities of F call it π : X ′ → X, which only extracts F ′ := π −1 F invariant divisors. If E is any π-exceptional divisor then a(E, F) ≥ −ǫ(E) by log canonicity and so a(E, F) ≥ 0 since E is invariant and we may conclude that F has in fact canonical singularities.
We recall that the class of simple singularities is stable under blow ups contained in strata of the singular locus, however, it is important to realize that a canonical singularity may not remain canonical after a blow up in the singular locus. In fact, it is a subtle problem to decide when the blow up of a canonical singularity remains canonical.
We emphasize that in contrast to (pre)simple singularities the notion of (log) canonical singularities makes sense on singular varieties. Take for instance the foliation on C 2 generated by the vector field ∂ = x ∂ ∂x −y ∂ ∂y . This defines a canonical foliation singularity. If we let X = C 2 /(x, y) ∼ (−x, −y) then X is singular and note that ∂ descends to a vector field on X which still defines a canonical singularity.

Termination
Our goal in this section is to show the following: Theorem 2.1 (Termination). Let X be a Q-factorial quasi-projective threefold and let (F, ∆) be a F-dlt pair. Then starting at (F, ∆) there is no infinite sequence of flips.
Together with the existence of flips, [CS21, Theorem 6.4], and divisorial contractions, [CS21, Theorem 6.7], this has the following immediate corollary (whose proof is identical to the proof for the corresponding statement for varieties).
Corollary 2.2. Let X be a projective Q-factorial threefold and let (F, ∆) be an F-dlt pair. Then there is birational contraction f : X Y (which may be factored as a sequence of flips and divisorial contractions) such that if G is the transformed foliation then either (1) K G + f * ∆ is nef; or (2) there is a fibration g : Y → Z such that −(K G + f * ∆) is g-ample and the fibres of g are tangent to G.
We call such a contraction a (F, ∆) or K F + ∆-MMP.
We will also frequently need to run the relative MMP. The relative MMP can be deduced from the absolute MMP via standard arguments, see for instance [KM98, §3.6-7].
Corollary 2.3. Let X be a Q-factorial quasi-projective threefold and let (F, ∆) be an F-dlt pair. Let p : X → S be a surjective projective morphism. Then there is a birational contraction f : X Y /S (which may be factored as a sequence of flips and divisorial contractions) such that if G is the transformed foliation and if q : Y → S is the structure map either (1) K G + f * ∆ is q-nef; or (2) there is a fibration g : Y → Z/S such that −(K G + f * ∆) is g-ample and the fibres of g are tangent to G.
We call the contraction f : X Y /S constrcuted in the above corollary a (F, ∆)-MMP or (K F + ∆)-MMP over S. In case (1) of the above statement, we then say that Y (or, alternatively, (G, f * ∆)) is a minimal model of X (or, alternatively, (F, ∆)) over S; in case (2), instead, we say that that Y (or, alternatively, (G, f * ∆)) is a Mori fibre space for X (or, alternatively, (F, ∆)) over S.
Corollary 2.3 immediately implies the following extension of the existence of F-dlt modification to the relative setting, cf. [CS21, Theorem 8.1].
Theorem 2.4 (Existence of F-dlt modifications). Let F be a co-rank one foliation on a normal quasi-projective threefold X. Let (F, ∆) be foliated pair.
Then there exists a birational morphism π : We call such a modification an F-dlt modification.
Proof. The proof is analogous to that of [CS21, Theorem 8.1]. In particular, it suffices to consider a log resolution π 1 : Y 1 → X of (F, ∆) and then run the K F 1 + Γ-MMP relatively over X, where F 1 is the transform of F on Y 1 and Γ := π −1 1 * ∆ + ε(E)E, where the sum is taken over the prime π 1exceptional divisors. The relatively minimal model produced by this MMP is the desired modification (G, Γ).
Remark 2.5. We use the notation of Theorem 2.4.
(2) If the foliated log pair (F, ∆) is lc, then the previous part of the remark implies that F = 0 and K G + Γ = π * (K F + ∆). Moreover, property (1) in the statement of Theorem 2.4 implies that Nklt(G, Γ) is the union of all codimension 1 subvarieties contained in it. Hence, As a consequence of the existence of the MMP we have the following non-vanishing theorem.
Theorem 2.6. Let F be a co-rank one foliation on a normal projective Qfactorial threefold X. Let ∆ be a Q-divisor such that (F, ∆) is a F-dlt pair. Let A ≥ 0 and B ≥ 0 be Q-divisors such that ∆ = A + B and A is ample.
Proof. We run a K F +∆-MMP. By [CS21, Theorem 6.4 and Theorem 6.7] all the required divisorial contractions and flips exist. By Theorem 2.1 there is no infinite sequence of flips and so this MMP terminates, call it φ : X X ′ . Let F ′ be the transform of F. By Lemma 1.20 we may find an ample divisor Thus, we may apply [CS21,Theorem 9.4] to conclude that K F ′ + ∆ ′ is semi-ample and so there exists and our result follows.
2.1. Singular Bott partial connections. We recall Bott's partial connection. Let F be a smooth foliation on a complex manifold X. We can define a partial connection on N F locally by where w is any local lift of w to T X and ω i are local generators of Ω 1 F , ∂ i are dual generators of F and q : T X → N F is the quotient map. One can check that these local connections patch to give a global connection.
Lemma 2.7. Let F be a rank r foliation on a complex analytic variety X. Let S ⊂ X be a local complete intersection subvariety of X of dimension r and suppose that S is F-invariant.
Let Z = sing(X) ∪ sing(F). Suppose that Z ∩ S is codimension at least 2 in S. Then there is a connection S ⊗ N S/X by the push-pull formula, [Har77, Exercise II.5.1(d)]. Thus we get a map S ⊗ N S/X and by observing that i * ∇ • satisfies the Leibniz condition (since it does so away from a set of codimension at least 2) we see that this is our desired connection.

Proof of Theorem 2.1.
Lemma 2.8. Let X be a normal complex analytic threefold and let (F, ∆) be a log terminal co-rank 1 foliation on X. Let C ⊂ X be a compact curve tangent to F. Let S be a germ of an invariant surface containing C. Suppose that K X , K F and S are Q-Cartier. Then Proof. Let H ⊂ X be a sufficiently ample divisor meeting C transversely and choose H to be sufficiently general so that (F, ∆ + (1 − ǫ)H) is log terminal for all ǫ > 0.
We may then find a Galois cover π : X ′ → X ramified over H and sing(X) such that if we write S ′ = π −1 (S) and F ′ = π −1 F then S ′ and K F ′ are both Cartier.
Write ∆ ′ = π * ∆ and C ′ = π −1 (C). We claim that (F ′ , ∆ ′ ) is log terminal. Indeed, let r be the ramification index along H. We have by foliated By construction S ′ is Cartier and so we may apply Lemma 2.7 to produce a connection S ′ → Ω 1 B , and so we may pull back ∇ to get a connection by composing . In particular, since n * O(S ′ ) admits a holomorphic connection, it is flat which implies 0 = S ′ · n(B) = m(S · C).
Proof of Theorem 2.1. By Special Termination, [CS21, Theorem 7.1], it suffices to show that any sequence of log terminal flips terminates. Let be one such flip and let C ⊂ X i be an irreducible component of exc(φ). Let f : X i → Z denote the base of the flip. Since C is tangent to the foliation any divisor E dominating C on some By [CS21, Lemma 3.14] by taking U to be a sufficiently small analytic neighborhood of z = f (C) we may find a unique F i -invariant divisor on X i,U := f −1 (U ) containing C. Call this divisor S.
Since X i,U is klt and projective over U we may find a small Q-factorialization of X i,U denoted g : X i,U → X i,U . Let F i be the transformed foliation, write , let S be the strict transform of S and let C be the strict transform of C. Since g is small, we see that (F i , ∆ i ) is still log terminal.
Let P ∈ C be a point and let T be a germ of a F i -invariant divisor at P . We claim that T = S (as germs). Indeed, suppose otherwise. Since S is Q-Cartier we know that T ∩ S contains a 1-dimensional component Σ. Since Σ is the intersection of two invariant divisors we see that Σ ⊂ sing(F) and Σ is tangent to the foliation, in particular, we see that F i is terminal at the generic point of Σ. This, however, is a contradiction of Proposition 1.7 which implies that terminal foliation singularities are non-singular in codimension 2.
By Lemma 2.8 we see that On the other hand by [CS21, Corollary 3.20] and the observation in the previous paragraph we see that the collection of F i -invariant divisors meeting C is exactly S itself and so is log terminal and so our result follows by termination for threefold log terminal flips, see for example [KM98,Theorem 6.17].
To finish we present an example of a foliation flip, another example may be found in [Spi20, Example 9.1].
Example 2.9. Let b : Y → C 2 be the blow up at the origin with exceptional curve C and let p : X → Y be the total space of the line bundle O Y (C). Observe that X contains a single projective curve, which we will continue to denote by C. Let G be the foliation on Y given by the transform of the foliation generated by ∂ ∂x 1 on C 2 (with coordinates (x 1 , x 2 )) and let . It is straightforward to check that K G · C = 0 and that G is smooth at the generic point of C. Moreover, observe that S and D 2 are G-invariant whereas D 1 is not. Consider the map σ : C 2 → C 2 given by (x 1 , x 2 ) → (−x 1 , x 2 ) and observe that σ lifts to a map τ : X → X. Let X := X/ τ and observe that π : X → X is ramified to order 2 along S and D 1 . Observe moreover that τ preserves G and so it descends to a foliation F on X. A foliated Riemann-Hurwitz computation shows that K G = π * K F + D 1 . In particular, if we let Σ = π(C) we see that K F · Σ < 0 and so Σ is a K F -flipping curve.
Notice that Σ meets sing(F) at a single point which is a Z/2 quotient singularity.

Connectedness
3.1. Connectedness of the nonklt locus for foliated pairs. The aim of this section is to prove the following connectedness statement which constitutes one of the pillars in the analysis of the birational structure of foliated singularities. The analogue result in the non-foliated case has a long history and is rather classical; recently, [Bir20,FS20] fully settled the Connectedness Principle in full generality for pairs.
Theorem 3.1 immediately implies the following more general result.
Proof. It suffices to consider g : X ′ → X a F-dlt modification, K G + ∆ X ′ = g * (K F + ∆) and apply Theorem 3.1 to the pair (G, ∆ X ′ ) and the map f • g : X ′ → Y .
We will prove Theorem 3.1 in the course of this section by proving different cases that fit together to provide a argument for it.
Before proving the theorem we indicate a quick application of Theorem 3.1 to the geometry of (weak) Fano foliations, see also [AD13].
We will denote by sing * (F) the union of all codimension 2 components of sing(F).
Corollary 3.3. Let X be a smooth projective threefold and let F be a co-rank 1 foliation on X. Assume that −K F is big and nef. Then either (1) F has an algebraic leaf; or (2) sing * (F) is connected.
Proof. Let us observe that Nklt(F) = sing * (F) ∪ I ∪ Z where I is the union of all the F-invariant divisors and Z is a finite collection of points. We take an F-dlt modification µ : X → X of F and let F be the induced foliation on X, which exists by Theorem 2.4. Writing K F +∆ = µ * K F , then µ(Nklt(F , ∆)) = Nklt(F), see Remark 2.5. If F has no algebraic leaves then I = ∅. We conclude applying Theorem 3.1 that Nklt(F , ∆) is connected and our result follows.
Remark 3.4. The corresponding statement to Theorem 3.1 for rank 1 foliations is an essentially trivial consequence of the arguments in [BM16].
We now turn to the proof of Theorem 3.1. We will work in the following setting. We denote by f : X → Y a contraction of normal quasi projective varieties, with X a Q-factorial threefold.
Recall that f being a contraction means that f is surjective, projective with We assume the existence of a co-rank 1 foliation F on X and of a foliated log pair (F, ∆) with ∆ = a i D i . We will denote by We remark that we allow ∆ to have F-invariant components, however ∆ ′ will have no F-invariant components.
We start by addressing the birational case.
Lemma 3.5. With the notation above, we assume that f is birational, Suppose moreover that every lc centre of (F, ∆) is contained in a codimension 1 lc centre of (F, ∆).
Let us recall that, as observed in Remark 2.5, Proof. We assume that Nklt(F, ∆) is disconnected in a neighborhood of some fiber of f and we will show that such assumption leads to a contradiction. Let us observe that we may assume that ∆ ′′ ≥ 0 is f -ample, otherwise exc(f ) ⊂ Supp(∆ ′′ ) and there is nothing to prove. Thus −(K F + ∆ ′ ) = −(K F + ∆) + ∆ ′′ is f -ample. By [Spi20, Lemma 8.10] we see that f only contracts curves tangent to F. Case 1. The morphism f is a divisorial contraction. Suppose that f contracts a divisor E. Observe that since ρ(X/Y ) = 1 we have that E is irreducible. If E is invariant then it is an lc centre and so there is nothing to prove. Thus we may assume that E is not invariant. If Thus we may assume that f (E) = C is a curve in Y . We may find t ≥ 0 so that we may write , and so by assumption Nklt(G, Γ E ) contains at least two components meeting a fibre of f Let Σ 0 be an irreducible curve contracted by f . Since Σ 0 is tangent to G it is therefore a rational curve with K G · Σ 0 ≥ −2. Moreover, exactly one of the following two scenarios hold: (1) either Σ 0 meets two distinct components of Nklt(G, Γ E ); or, (2) the fibre containing Σ 0 is a union of two rational curves meeting at a point and, up to switching the two components of this fibre, we can assume that Σ 0 meets at least one connected component of Nklt(G, Γ E ).
where, in scenario (1), p 1 , p 2 are the intersections of Σ 0 with two distinct connected components of Nklt(F, ∆ ′ ) along Σ 0 , while in scenario (2), p 1 , is the intersection of Σ 0 with the other component of the fibre and p 2 is the intersection of Σ 0 with Nklt(F, ∆ ′ ). However, deg( The morphism f is a flipping contraction. We denote by the flip of f and by Σ a curve in the exceptional locus. Then there exists two divisorial components D 1 , D 2 of Nklt(F, ∆ ′ ) which intersect Σ, and do not contain it. But then on X + , the strict transforms D + i of the D i contain the exceptional locus of the map f + , hence this must be contained in the intersection of the D + i and as such it is a non-klt center. Since Σ is tangent to F we may assume that (F, ∆ ′ ) is terminal along Σ, as otherwise Σ would be an lc centre, see [CS21, Lemma 3.14]); on the other hand, the above observation implies that F + , the birational transform of F on X + , is canonical along the exceptional locus of f + . But this leads to a contradiction, as by the Negativity Lemma the discrepancies of (F, ∆ ′ ) along the f + -exceptional locus must decrease since −(K F + ∆ ′ ) is f -ample, see, for example, [CS21, Lemma 2.7].
Lemma 3.6. With the notation above, we assume that f is birational and (F, ∆ ′ ) is F-dlt. Suppose moreover that every lc centre of (F, ∆ ′ ) is contained in a codimension 1 lc centre of (F, ∆ ′ ). If −(K X + ∆) is f -ample then Nklt(F, ∆) is connected in a neighborhood of any fibre of f .
Proof. Let y ∈ Y be a point on Y and let X y denote the fiber of f over Y . We assume that Nklt(F, ∆) is disconnected in a neighborhood of X y and we will show that such assumption leads to a contradiction. As each lc center of (F, ∆ ′ ) is contained in a codimension 1 lc center and Nklt(F, ∆) is disconnected in a neighborhood of X y , there exist prime divisors E 1 , E 2 on E such that: As Then We can then run the (K F + ∆ ′ + G)-MMP with scaling of G over Y , see Section 1.8, (3.2) We quickly explain how to run such an MMP. As we are only interested in what happens over a neighborhood of y ∈ Y , we can assume that each step of (3.2) is non-trivial in a neighborhood of X i,y . As G is ample, then there exists 0 < η ≪ 1 such that In particular, we may observe moreover that at each step any lc centre of (F i , ∆ ′ i ) is contained in a codimension 1 lc centre of (F i , ∆ ′ i ). Indeed, by [CS21, Lemma 2.7] an lc centre cannot lie in exc(s −1 i ) and so if W is an lc centre of (F i , ∆ ′ i ) then each s j for j ≤ i must be an isomorphism at the generic point of W .
Lemma 3.5 shows that the number of connected components of Nklt(F i , ∆ i ) in a neighborhood of X i,y cannot decrease with i. Assume at the i-th step X i−1 X i of (3.2) the strict transform of one of the E j , say E 1 , gets contracted. Denoting by R i−1 the generator of the extremal ray of N E(X i−1 /Y ) contracted at this step and by E 1,i−1 the strict transform of E 1 on X i−1 , then R i−1 · E 1,i−1 < 0. Lemma 3.5 implies that E 1,i−1 ∩ E 2,i−1 = ∅ in a neighborhood of X i−1,y , as otherwise the number of connected components would have decreased at some point of the MMP. This observation implies that and we can repeat the argument just illustrated as E 2,i ∩ D 1,i = ∅ around X i,y and they belong to different connected components of Nklt it must contain the whole fiber X ′ y , which leads to a contradiction. Proposition 3.7. With the above notation, we assume that f is birational Proof. First, observe that we may freely replace (F, ∆ ′ ) by a higher model so that every lc centre of (F, ∆ ′ ) is contained in a codimension 1 lc centre. Indeed, by Theorem 2.4 we may find a modification µ : every lc centre is contained in a codimension 1 lc centre and Nklt We next reduce the general case to the case of ample H, which then follows from Lemma 3.6.
We claim that for δ sufficiently small then Indeed, let r Z : Z → X be a foliated log resolution of (F, ∆ ′ + δB 2 ). We denote by G the strict transform of F on Z. Thus, For fixed sufficiently small δ > 0 satisfying (3.3), let π : Y → X be a F-dlt modification of (F, ∆ ′ + δB 2 ). By Theorem 2.4, writing • C ≥ 0, and • the support of C is contained in Nklt(G, Γ) and is π-exceptional. Moreover, the Q-factoriality of X implies that there exists an effective π- For ǫ > 0 sufficiently small we know that (Γ + Θ) ′ = Γ (in the notation at the beginning of the section) and −(K G + Γ + Θ) = π * (H − δB) − ǫG is f -ample and this concludes the proof.
Proof of Theorem 3.1. In view of Lemma 3.5 and 3.6, we are only left to prove the case where f is a non-birational contraction. Hence, we assume that Nklt(F, ∆) is disconnected in a neighborhood of some fiber X y , y ∈ Y of f with dim X > dim Y and we derive a contradiction.
Step 1. In this step, we assume H to be f -ample.
As H is f -ample, there exists 0 < ǫ ≪ 1 such that G = H − ǫF is f -ample. We can then run the (K F + ∆ ′ )-MMP with scaling of G over Y , see Section 1.8, (3.4) Proof of Claim 3.8. By the definition of the MMP with scaling, at each step of (3.4) there exists a positive real number λ i such that For any i, λ i > 1: in fact, assuming λ i ≤ 1 we reach an immediate contradiction since would then be non-pseudoeffective over Y -this holds true in view of the fact that dim X i > dim Y . By (3.5), and the condition λ i > 1 together with (3.6) imply that (ǫF i + ∆ ′′ i ) · R i > 0, which proves the claim.
We now prove the second part of the statement. If s i+1 : X i → X i+1 is a divisorial contraction, let E be the prime divisor contracted by s i+1 . Since F i · R i > 0 or ∆ ′′ i · R i > 0 it follows that the image of the exceptional locus of s i+1 is contained in Nklt(F i+1 , ∆ ′ i+1 ). But then Lemma 3.5 implies that the number of connected components of Nklt(F i+1 , ∆ ′ i+1 ) in a neighborhood of X i+1,y must be the same as that of Nklt is a flip, let z − i : X i → Z i be the associated flipping contraction and z + i+1 : X i+1 → Z i the other small map involved in the flip. By the first part of the proof, we know that Nklt i are all small maps. Hence it suffices to prove that the number of connected components of Nklt ))) around Z i,y . Lemma 3.5 implies that the number of connected components of Nklt(F i , ∆ ′ i ) in a neighborhood of X i,y must be the same as that of ) around Z i,y , which concludes the proof.
By Special Termination, [CS21, Theorem 7.1], and Claim 3.8, the run of the MMP in (3.4) must terminate and, since K F + ∆ ′ is non-pseudoeffective over Y , the final step will be a Mori fibre space By Claim 3.9 it suffices to prove that Nklt(F n , ∆ n ) is connected in a neighborhood of X n,y . On X n , Nklt(F n , ∆ n ) = Supp(F n + I n ). As I n is F ninvariant every component of I n must be vertical over Z. As F n · R n > 0 or ∆ ′′ n · R n > 0, there exists at least one component of F n which dominates Z and contains only one horizontal component. Let z ∈ Z be a point and observe that dim(g −1 (z)) ≤ 2.
If dim(g −1 (z)) = 2 for all z ∈ Z, then since ρ(X/Z) = 1, it follows that every horizontal component of F n is g-ample; hence, any 2 horizontal components of F n intersect along any fibre of g. If dim(g −1 (z)) = 1 for some (equivalently any) z then since −(K Fn + ∆ n ) is g-ample we see that F n contains at most 1 horizontal component. Thus, Nklt(F n , ∆ n ) must be connected a neighborhood of X n,y . But this gives a contradiction.
Step 2. In this step we reduce the general case to the case of f -ample H.
Here it suffices to copy the proof of Proposition 3.7 verbatim.
3.2. Adjunction for foliated pairs. The goal of this section is to illustrate adjunction theory for foliated threefolds. Let us highlight the fact that in [CS21] a Q-factorial threefold X is simply an analytic variety which is (globally) Q-factorial. We will work in this set-up throughout § 3.2-3.3; the reader should keep this observation in mind when encountering foliated adjunction throughout the paper.
Let us recall the following adjunction for foliations with non-dicritical singularities.
Lemma 3.10 (Adjunction). [CS21, Lemma 3.18] Let X be a Q-factorial threefold, let F be a co-rank one foliation with non-dicritical singularities. Suppose that (F, ǫ(S)S + ∆) is lc (resp. lt, resp. F-dlt) for a prime divisor S and a Q-divisor ∆ ≥ 0 on X which does not contain S in its support. Let ν : S ν → S be the normalization and let G be the restricted foliation to S ν . Then, there exists Θ ≥ 0 on S ν such that (3.7) Moreover, we have: • Suppose ǫ(S) = 1. Then (G, Θ) is lc (resp. lt, resp. F-dlt).
We wish to generalize this result to an adjunction formula which holds in full generality.
Lemma 3.11 (General Adjunction). Let X be a threefold and let F be a co-rank one foliation on X. Suppose that (F, ǫ(S)S + ∆) is a foliated log pair for a prime divisor S and a Q-divisor ∆ ≥ 0 on X which does not contain S in its support. Let ν : S ν → S be the normalization and let G be the restricted foliation to S ν . Then, there exists Θ ≥ 0 on S ν such that (3.8) In the hypotheses of Lemma 3.11, we will refer to Θ as the different Diff S ∆ of ∆ on S.
Proof. Let π : Y → X be a F-dlt modification for (F, ǫ(S) + ∆) and let S ′ be the strict transform of S on Y . Writing is F-dlt, Lemma 3.10 implies that there exists Θ 1 such that Hence, it suffices to take Θ : The two equations (3.7), (3.8) represent the adjunction formula for foliations, where (3.8) is a generalized version of the one proven in [CS21]. On the other hand, in the more general framework of Lemma 3.11, it is not possible control the singularities of the restriction of the pair (F, ∆) to a codimension one log canonical center.
3.3. Inversion of adjunction. We are now ready to prove inversion of adjunction for foliated pairs.
Theorem 3.12. Let X be a Q-factorial threefold and let F be a co-rank one foliation. Consider a prime divisor S and an effective Q-divisor ∆ on X which does not contain S in its support. Let ν : S ν → S be the normalization and let G be the restricted foliation on S ν and Θ be the foliation different Proof. Let π : Y → X be an F-dlt modification for the pair (F, ǫ(S)S + ∆) and let S ′ be the strict transform of S on Y . Writing When ǫ(S ′ ) = 1, we will denote by G ′ the restriction of F Y to the normalization ν 1 : S ′ν → S ′ of S ′ and let Ξ ′ be the different given by adjunction of (F Y , ε(S ′ )S ′ + ∆ Y ).
Step 1. In this step we prove that (F, Step 2 In this step we prove that if ǫ(S ′ ) = 1, then (G ′ , Ξ ′ ) is lc, then we deduce that (F Y , S ′ + ∆ Y ) is lc in a neighborhood of S ′ . By Lemma 1.17, S ′ is normal. Hence, S ′ν = S ′ and G ′ = G| S ′ .
Hence, considering the birational morphism ψ : S ′ → S ν , then As (G, Θ) is lc, the same holds for (G ′ , Ξ ′ ). As shown in Step 1, we need Let G be any prime component of E ∩ S ′ . Claim 3.13. G is an lc center of (F Y , ∆ ′ Y ) Proof of Claim 3.13. This fact is an immediate consequence of Lemma 1.17 and its proof if ε(E) = 1, while, if ε(E) = 0, then [CS21, Lemma 3.16] implies that Y has quotient singularities at the generic point of G at which point the conclusion follows from a local computation on foliated surfaces, upon localizing at the generic point of G.
By [CS21, Lemma 3.8] and Claim 3.13, (F Y , ∆ ′ Y ) is log smooth at the generic point of G: in particular, Y is smooth at the generic point of G and E meets S ′ generically transversely along G. Hence, taking Step 3 In this step we prove that if ǫ(S ′ ) = 0 and if (S ν , Θ) is lc, then (S ′ν , Ξ ′ ) is lc, then we prove that (F Y , ∆ Y ) is lc in a neighborhood of S ′ . We have . Considering the birational morphism ψ : S ′ν → S ν , then As (S ν , Θ) is lc, the same holds for (S ′ν , Ξ ′ ). We need to show that Supp(∆ ′′ Y )∩ S ′ = ∅. Seeking a contradiction, let E be a prime component of Supp This is the adaptation to foliations of the classic statement of inversion of adjunction for log pairs, cf. [KM98, Theorem 5.50]. Nonetheless, it is not the most general form of inversion of adjunction that one could hope for. In fact, if we look at the statement of Lemma 3.10, we see that the natural divisor to look at when ǫ(S) = 0 would be, in the notation of the lemma, the divisor Θ ′ rather than the foliated different Θ -let us recall that Θ ′ := ⌊Θ⌋ red + {Θ}. As, by definition Θ ′ ≤ Θ it follows immediately that if (S, Θ) is lc, so is (S, Θ ′ ), but it would be even more interesting to have a statement of inversion of adjunction that only assumes (S, Θ ′ ) is lc.

A vanishing result
In this section we prove a relative vanishing theorem for foliations. We make the following easy observation whose proof we leave to the reader.
Then R i f * L = 0. Lemma 4.2. Let f : Y → X be a surjective birational projective morphism of normal varieties of dimension at most 3 and let (F, ∆) be an F-dlt foliated pair on Y with ⌊∆⌋ = 0. Suppose that Y is Q-factorial and that every fibre of f is tangent to F.
Let P ∈ X be a closed point. Then there exists anétale neighborhood σ : X ′ → X of P , a small Q-factorialization µ : W → Y ′ := Y × X X ′ and a reduced divisor T i on W such that writing ∆ W = µ * ∆ and F W = µ −1 F and f ′ : W → X ′ for the induced map we have Proof. First, notice that since Y is Q-factorial we may apply [CS21,Theorem 11.3] to see that Y is klt. Since (F, ∆) is F-dlt we also know that F is nondicritical by Theorem 1.16.
Let {S 1 , ..., S N } be the collection of all separatrices of F meeting f −1 (P ), formal or otherwise. Fix n ≫ 0 sufficiently large. By [CS21, §4, 5] there is anétale cover σ : is as well and we may find an F-dlt modification π : Z → Y ′ such that µ is small. Observe that Z is Q-factorial and so R ′ i := π −1 * R i and S i ′ := π −1 * S i are Q-Cartier. Set G = π −1 F and we may write K G + Γ = π * (K F ′ + ∆ ′ ). We also remark that (G, Γ) is necessarily terminal at the generic point of a curve C ⊂ exc(π). Indeed, otherwise C would be a lc centre of (G, Γ) since it is tangent to G which by [CS21, Lemma 3.8] would imply that (F ′ , ∆ ′ ) is log smooth at π(C), a contradiction.
Note that S i ′ are all the separatrices (formal or otherwise) which meet π −1 (g −1 (p)) and that R ′ i still approximate all the S i ′ . Note that in particular they have the same intersection numbers with all curves contained in π −1 (g −1 (p)).
Since Y is klt, the same is true of Z and so we may run a K Z + δ R ′ i -MMP over Y ′ for some δ > 0 sufficiently small. Denote this MMP φ : Z W and set T i := φ * R ′ i and S i = φ * S ′ i . Set F W = φ * G and ∆ W = φ * Γ. Observe that each step of this MMP is K G + Γ trivial and that (F W , ∆ W ) is F-dlt. We claim that W satisfies all our required properties.
Item 1 holds since K Y is Q-Cartier and so K Z (and hence K W ) is trivial over Y ′ . Thus K W + δ T i being nef over Y ′ implies that T i is nef over Y ′ .
To prove (2), let C ⊂ π −1 (P ). Note that by non-dicriticality of (F, ∆) and our assumptions on f we have that C is tangent to F W . Moreover, if (F W , ∆ W ) is canonical at the generic point of C then (F W , ∆ W ) is log smooth at a general point of C, [CS21, Lemma 3.8]. So, up to relabelling the S i we may assume that C ⊂ S 1 and that S 1 gives a strong separatrix at a general point of C if (F W , ∆ W ) is canonical at the generic point of C and (F W , ∆ W ) has a saddle node at the general point of C, [Bru00, pg. 3] for a recollection on saddle nodes and weak separatrices on surfaces, but which works equally in the current setting.
By [CS21,Corollary 3.20] we may write where Θ ≥ Θ ′ and the coefficient of C in both these divisors is the same. It follows that and since {T 1 , ..., T N } approximate the S 1 we have Since C was arbitrary we get our claimed nefness. Next, observe that each step of the MMP φ : Z Y is K G + Γ trivial so we still have that (F W , ∆ W ) is F-dlt and so F W is non-dicritical by Theorem 1.16 and all the log canonical centres of (F W , ∆ W ) are contained in Supp( T i ). So we may apply [CS21, Lemma 3.16] to see that (W, ∆ W + (1 − ǫ)( T i )) is klt for all ǫ > 0. This gives us item 3. Suppose moreover that either where A is f -ample then replacing ∆ by ∆ − δA for δ > 0 small we may freely assume that L − (K F + ∆) is f -ample. Moreover, perhaps replacing ∆ by ∆ − ǫ⌊∆⌋ for some ǫ > 0 sufficiently small we may assume that ⌊∆⌋ = 0. As in Lemma 4.2 we see that Y is klt and so it has rational singularities. By Lemm 4.1 and the fact that Y has rational singularities we see that R i f * L = 0 provided R i f ′ * L ′ = 0 where L ′ = µ * τ * L where µ and τ are as in Lemma 4.2 (and its proof).
Next, observe that L ′ − (K F W + ∆ W ) is f ′ -big and nef and is strictly positive on any curve not contracted by µ. Thus we see by Lemma 4.2 Item 2 that is f ′ -big and nef and is strictly positive on any curve not contracted by µ. So for ǫ > 0 sufficiently small since T i is µ-nef by Lemma 4.2 Item 1 is f ′ -big and nef. Thus we may apply relative Kawamata-Viehweg vanishing to conclude that R i f ′ * L ′ = 0 for i > 0.

Malgrange's theorem
In this section we prove a version of Malgrange's theorem on singular threefolds. A weaker version of this statement was proven in [Spi20]. Results in this direction were achieved in [CLN08] and some of our ideas have been inspired by their approach.
Let P ∈ X be a germ of a threefold and let F be a co-rank 1 foliation on X defined by a holomorphic 1-form ω. We say that f ∈ O X,p is a first integral for F if df ∧ ω = 0.
Theorem 5.1. Let P ∈ X be a germ of an isolated (analytically) Q-factorial threefold singularity with a co-rank 1 foliation F. Suppose that F has an isolated canonical singularity at P .
Then F admits a holomorphic first integral.
It would be ideal to drop the Q-factoriality assumption in the theorem. We are able to do this when F is terminal, see Corollary 5.15.
Theorem 5.1 has the following immediate consequence.
Corollary 5.2. Let P ∈ X be a germ of an isolated threefold singularity with a co-rank 1 foliation F. Suppose that X is Q-factorial and that F is canonical. Then F has a separatrix at P .
Proof. If F is smooth outside of P then this follows directly from Theorem 5.1. Otherwise let Z ⊂ sing(F) be a curve and note that Z is tangent to F. Observe that there is a germ of a separatrix for all Q ∈ Z − P . By Theorem 1.16 F is non-dicritical and so by Lemma 1.18 we may extend S Q to a neighborhood of Z, which in turn gives separatrix at P .
Recall that in general even if F is non-dicritical, if P ∈ X is a singular point then there may be no separatrices at P . See [Cam88] results in this direction on surfaces.
5.1. Controlling the singularities of X and F. The goal of this subsection is to show that under the hypotheses of Theorem 5.1 we have that X has log terminal singularities.
We will need the following version of the classical Camacho-Sad formula for F-dlt foliations. It follows as a special case of the Camacho-Sad formula for foliations on varieties with quotient singularities proven in [DO19, Proposition 3.12]. We refer to [DO19, Definition 3.10] for the definition of the Camacho-Sad index.
Lemma 5.3. Let X be a normal surface and F an F-dlt foliation. Let C be a compact F-invariant curve. Then Lemma 5.4. Let P ∈ X be a germ of an isolated threefold singularity and let F be a co-rank 1 foliation with canonical singularities such that F is smooth away from P . Suppose that K X is Q-Cartier. Then X is log terminal.
Proof. If F is terminal then the result follows from Theorem 1.16. So suppose that F has canonical but not terminal singularities.
Let µ : (X, F) → (X, F) be an F-dlt modification. Let E = E i = exc(µ). Since F is canonical we have µ * K F = K F . Moreover, we may freely assume that µ is not the identity. By Theorem 1.16 F is non-dicritical and so E is F-invariant. Let Z be a 1-dimensional component of sing(F ) ∩ E. By [CS21, Lemma 3.14] either F is terminal at the generic point of Z or X is smooth at the generic point of Z, and at a general point of Z F has simple singularities and there there are two separatrices (possibly formal) containing Z. However, Proposition 1.7 applied to a general hyperplane passing through Z and the restricted foliation on this hyperplane implies that F cannot be terminal at the generic point of Z.
Write K X + E i = π * (K X )+ a i E i . By [Spi20, Lemma 8.9] we see that K F −(K X + E i ) = − a i E i is π-nef away from finitely many curves which implies by the Negativity Lemma, [Wan19, Lemma 1.3] , that a i E i ≥ 0 and since Supp( E i ) = π −1 (P ), then either (1) a i > 0 for all i; or (2) a i = 0 for all i.
By [CS21, Lemma 3.16], we see that that (X, (1 − ǫ) E i ) is klt and so if we are in Case 1 then we see immediately that X is klt.
So suppose for sake of contradiction that we are in Case 2, i.e., a i = 0 for all i and so K X + E ∼ Q 0.
We first claim that if Z ⊂ sing(F ) ∩ E is a 1-dimensional component admitting two separatrices contained in E then Z is not a saddle node. Indeed, suppose for sake of contradiction that there exists Z ⊂ E i such that Z is a saddle node and E i is the weak separatrix of the saddle node, see [Bru00,pg. 3] for a recollection on saddle nodes and weak separatrices on surfaces, but which works equally in the current setting. Write By Lemma 3.10 we know that Θ ≥ Θ ′ . Since E i is the weak separatrix of a saddle node along Z in appropriate (formal) local coordinates around a general point of Z we see that F is generated by a 1-form ω of the form z(1 + νw k )dw + w k dz where E i = {z = 0}, ν ∈ C and k ≥ 2. The coefficient of Z in Θ is the order of vanishing of ω| E i along Z, which in turn is exactly k ≥ 2. On the other hand, since (X, E j ) is log canonical we see that the coefficient of Z in Θ ′ is at most 1. However, this implies that K F − (K X + E j ) cannot be π-trivial, a contradiction. A similar argument shows that for each 1-dimensional component Z ⊂ sing(F ) ∩ E that Z admits two separatrices, both of which are contained in E. In particular, each 1-dimensional component Z ⊂ sing(F ) ∩ E admits 2 non-zero eigenvalues.
The rest of the argument proceeds in an essentially identical fashion to the proof of the first part of [McQ08, Theorem IV.2.2]. We will explain the rest of his argument for the reader's convenience. Since K F ∼ Q 0 and K X + E ∼ Q 0 we see that N * F + E ∼ Q 0. Let H ⊂ X be a general ample divisor, and let G be the restricted foliation on H and let E ∩ H = ∪C i = C. Set S = sing(G) ∩ C and notice that If H is general enough we see that (F , H) is F-dlt and so G is F-dlt. Even better, if H is general enough we see that N * F | H = N * G and so N * G + C i ∼ Q 0. Observe that since G is F-dlt we see that C is a nodal curve.
For q ∈ S let ∂ q be a vector field generating G near q.
Claim 5.5. The ratio λ q of the eigenvalues of ∂ q is a root of unity.
Proof of claim. We may check this after taking a cover ramified along a general ample divisor A, and so after taking the index one cover associated to K G on H \ A we may freely assume that K G is Cartier. By Lemma 1.14 it follows that H is smooth. For any p ∈ C \ S set U p to be small open set so that G is defined by a 1-form ω p = dz p where {z p = 0} = C ∩ U p . For any q ∈ S set U q to be a small open subset so that G is defined by a 1-form ω q = x q a q dy q + y q b q dx q where {x q y q = 0} = C ∩ U q and where a q (q), b q (q) = 0. Let If p, p ′ ∈ C \ S so that U pp ′ = ∅ then dz p = h pp ′ dz p ′ and so g pp ′ = h pp ′ when restricted to C. If p ∈ C \ S and q ∈ S so that U pq = ∅ we have that z p = h pq (x q y q ) and so dz p = h pq d(x q y q ), when restricted to C, which in turn gives that dz p = h pq a −1 q ω q or = h pq b −1 q ω q depending on whether p ∈ {x q = 0} or p ∈ {y q = 0}. In particular, we see that after restricting to C we have an equality We may therefore find invertible functions f p on C p so that f p /f q = (h pq g qp ) m ). Without loss of generality we may assume that for p ∈ C \ S that f p = 1. From our previous calculations we see that for q ∈ S that f q = a m q = b m q (where we consider a q , b q as functions restricted to C). In particular, 1 = f q /f q = (a q (q)/b q (q)) m = λ m q as required. Since C is contractible we see that C 2 < 0 which implies On the other hand, Lemma 5.3 gives us which in turn gives us However, each λ p is a root of unity and so the modulus of p∈S λ p + 1 λp = p∈S λ p +λ p is bounded by 2#S. This is our sought after contradiction. 5.2. Holomorphic Godbillon-Vey sequences. We say that a 1-form ω is integrable provided ω ∧ dω = 0.
Definition 5.6. Let M be a complex manifold of dimension ≥ 2 and let ω be an integrable holomorphic 1-form on M . A holomorphic Godbillon-Vey sequence for ω is a sequence of holomorphic 1-forms (ω k ) on M such that ω 0 = ω and the formal 1-form Lemma 5.7. Let P ∈ X be an analytic germ of an isolated Q-factorial klt singularity with dim(X) ≥ 3. Then Proof. Notice that since X is klt it is also a rational singularity. Consider the long exact sequence coming from the exponential exact sequence . By [Fle81, Lemma 6.2] we know that im(a) = 0, in particular b is injective.
On the other hand, we see that we have an injection [Siu69] i * L is a coherent reflexive sheaf on X. By assumption Cl(P ∈ X) is torsion and so the same is true of The following result is proven in [CLN08, Lemma 2.1.1].
Lemma 5.8. Let M be a complex manifold of dimension ≥ 3 and let ω be a holomorphic 1-form on M . Assume that the codimension of sing(ω) is at least 3 and H 1 (M, O M ) = 0. Then ω admits a holomorphic Godbillon-Vey sequence.
Corollary 5.9. Let P ∈ X be a germ of an isolated analytically Q-factorial klt 3-fold singularity. Let ω be an integrable 1-form on X − P such that sing(ω) has codimension at least 3 in X − P . Then ω admits a holomorphic Godbillon-Vey sequence.
Proof. By Lemma 5.7 we have H 1 (X − P, O X−P ) = 0 in which case we may apply Lemma 5.8 to conclude.

A few technical lemmas.
Lemma 5.10. Let P ∈ X be an analytic germ of a Q-factorial and klt singularity with dim(X) ≥ 3. Let π : (Q ∈ Y ) → (P ∈ X) be a quasi-étale morphism of germs. Then Q ∈ Y is Q-factorial.
Proof. Let π : Y → X be the Galois closure of π. Observe that π is quasietale and if Y is Q-factorial then Y is Q-factorial, [KM98,Lemma 5.16]. Thus we may freely replace Y by Y and so may assume that π is Galois with Galois group G. Suppose for sake of contradiction that Y is not Q-factorial. Since π is quasietale we see that Y is klt and therefore Y admits a small Q-factorialization f : Y ′ → Y such that (1) G acts on Y ′ ; (2) f is G equivariant; and (3) f is not the identity. Indeed, such a Y ′ can be found by taking a G-equivariant resolution µ : W → X and running a G-equivariant K W +(1−ǫ) E i -MMP over X where E i is the union of the µ-exceptional divisors and ǫ > 0 is sufficiently small. Let X ′ = Y ′ /G and observe that we have a birational morphism g : X ′ → X. Moreover, we see that g : X ′ → X is small, a contradiction of the fact that X is Q-factorial.
Lemma 5.11. Let π : Y → X be a finite morphism of complex varieties and let F be a co-rank one foliation on X. Then F admits a holomorphic (resp. meromorphic) first integral if and only if π −1 F does.
Proof. We may assume without loss of generality that π : Y → X is Galois, in which case the claim is easy.
We say that f ∈ C[[x 1 , ..., x n ]] is a power if there exists g ∈ C[[x 1 , ..., x n ]] and an integer m ≥ 2 such that g m = f . Observe that if f is a first integral of ω and g m = f then g is also a first integral of ω. Let ∆ denote the formal completion of C at the origin.
Lemma 5.12. Consider C 3 × C with coordinates (z 1 , z 2 , z 3 , t) and let Ω = dt + t i ω i be a formal 1-form where ω i ∈ H 0 (U, Ω 1 U ) is a holomorphic 1form on 0 ∈ U ⊂ C 3 . Suppose that Ω is integrable. Let 0 ∈ D ⊂ U be a normal crossings divisor such that ω i is zero when restricted to D for all i. Let X be the formal completion of U × C along D × 0.
Then Ω admits a first integral in H 0 ( X, O X ).
Remark 5.13. A priori, the formal Frobenius theorem only guarantees that Ω admits a first integral in H 0 ( C 4 , O C 4 ), with C 4 the completion of C 4 at the origin.
Proof. Following a change of coordinates and for ease of notation we will assume that D = {z 1 z 2 z 3 = 0} (the cases where D has 1 or 2 components are simpler).
Since ω i vanishes when restricted to D for j = 1, 2, 3 we may write where f i j and θ i j are holomorphic. It follows that we may write Ω = dt + F j dz j +z j Θ j where F j (z 1 , z 2 , z 3 , t) ∈ H 0 (X j , O X j ), Θ j = H j i (z 1 , z 2 , z 3 , t)dz i ∈ H 0 (X j , Ω 1 X j ) and where X j is the formal completion of U × C along {t = z j = 0}.
We may then apply [CLN08, Lemma 3.1.1.] (or, more precisely, its proof) to find a first integral G j ∈ H 0 (X j , O X j ) of Ω. Moreover, if we write G j = m,n t m z n j g j mn where g j mn is a convergent power series in the set of variables {z 1 , z 2 , z 3 } \ z j then we may choose G j so that g j 00 = 0. In particular, observe that this implies that if φ ∈ Aut( ∆), then φ • G j is still an element of H 0 (X j , O X j ). Indeed, if we write φ • G j = mn t m z n j g ′ mn then g ′ mn = P (g j lp ) l≤m,p≤n where P is some polynomial depending on φ, in particular, g ′ mn is convergent provided all the g j lp are. Without loss of generality we may also assume that G j is not a power.
By considering G 1 , G 2 , G 3 as elements in H 0 ( C 4 , O C 4 ), with C 4 the completion of C 4 at the origin, we may apply [MM80, Théoremè de factorisation] to find φ ij ∈ Aut( ∆) so that G i = φ ij • G j . Thus, perhaps replacing G j by φ ij • G j we may assume that G 1 , G 2 , G 3 all give the same element in G is in fact an element of H 0 ( X, O X ) and we are done.
Lemma 5.14. Let X be a normal complex variety, let D ⊂ X be a compact subvariety and let X be the completion of X along D. Let F be a co-rank 1 formal foliation on X and suppose that that D is tangent to F.
Suppose that the following hold: (1) for all p ∈ D there exists an open neighborhood p ∈ U p ⊂ X and F p ∈ H 0 ( U p , O Up ) with F p a first integral of F and where U p is the formal completion of U p along D; (2) for any p, q ∈ D we have sing(X) ∩ U p ∩ U q = ∅; and (3) for any p, q ∈ D if U p ∩ U q = ∅ then F p | Up∩ Uq is not a power.
Then we may produce a representation ρ : π 1 (D) → Aut( ∆) such that if this representation if trivial then F admits a first integral F ∈ H 0 ( X, O X ). Moreover, if the F p can be taken to be convergent, then F may be taken to be convergent as well.
Proof. Without loss of generality we may assume that F p | D = 0 for all p.
If p = q is such that U p ∩ U q = ∅ choose some z ∈ U p ∩ U q ∩ D. By considering F p , F q as elements in the completion O X,z we may apply [MM80, Théoremè de factorisation] to find a φ p,q ∈ Aut( ∆) such that F p = φ p,q • F q .
If ρ(γ) = 1, then for 1 ≤ j ≤ n − 1 we may replace F p j by (φ p j ,p j−1 • ... • φ p 1 ,p 0 ) −1 • F p j and so we may freely assume that F p j = F p 0 for all j. Thus, if the image of ρ is {1} it follows that all the F p glue to give a section F ∈ H 0 ( X, O X ).
Our claim about convergence follows by observing that if the F p are all convergent then the φ p,q may be taken to be convergent as well.

Proof of Theorem 5.1.
Proof of Theorem 5.1. First, by Lemma 5.4 we know that X is klt.
Next, by Lemma 5.10 we may replace X by a quasi-étale cyclic cover and so may assume that N

[ * ]
F is Cartier and so F |X\P is defined by an integrable 1-form ω which is non-vanishing on X \ P . By Corollary 5.9 we have that that ω admits a holomorphic Godbillon-Vey sequence (ω k ).
Let L X be the link of X. By [TX17, Corollary 1.4] we know that π 1 (L X ) is finite and let L → L X be the universal cover. We may find a Galoisétale morphism of complex spaces Y ′ → X \ P corresponding to this cover and by [GKP16,Proposition 3.13] this cover extends to a Galois quasi-étale cover π : Y → X. So by replacing X by Y we may assume that π 1 (L X ) = {1}.
Let µ : Y → X be a log resolution of X and let E = E k be the sum of the µ-exceptional divisors. Let Y * := Y \ µ −1 (P ) ∼ = X \ P . By [GKKP11,Theorem 4.3] we see that ω i | X\P extends to a holomorphic 1-form ω i on Y .
There exist maps where a is a surjective and b is an isomorphism, since Y deformation retracts onto E. This implies that π 1 (E) is trivial. Define and recall that by definition Ω is an non-singular integrable 1-form defined on Y × C, the completion of Y ×C along E ×0, and where t is a local coordinate on C. We then have that Ω defines a smooth foliation G on Y × C. Observe that by construction By Corollary 1.15 we have ω k vanishes when restricted to E. Thus we may apply Lemma 5.12 to Ω to find for all p ∈ E a neighborhood p ∈ U p ⊂ Y and first integral of Ω denoted Since G is smooth without loss of generality we may assume that F p is not a power on U p × C ∩ U q × C for any p, q.
We may therefore apply Lemma 5.14 and since π 1 (E) = {1} we produce a formal first integralF ∈ H 0 ( Y × C, O Y ×C ). RestrictingF to Y × 0 we see that ω 0 admits a first integralf ∈ H 0 ( Y , O Y ). We now show that we can take this first integral to be convergent.
Chapter II] we may find dominant proper generically finite morphism W σ − → Y such that the central fibre of (f • σ) is reduced and σ is ramified only over foliation invariant divisors. Write E = σ −1 (E), W the completion of W along E andf =f • σ.
From the above construction we see that we may writef =f r such that for all p ∈ E we havef is not a power in O W ,p . Thus we may apply [MM80, Theorémè A] to find a φ p ∈ Aut( ∆) so that φ p •f is convergent on a neighborhood U p of p. We may apply Lemma 5.14 by taking F p = φ p •f to produce a representation ρ : π 1 (W ) → Aut( ∆) which vanishes when σ −1 µ −1 F admits a convergent first integral.
By taking the Stein factorization of W → X we produce a birational morphism W → X ′ contracting E to a point, and so that r : X ′ → X is branched only over the separatrices of F. We claim that X ′ is klt. Indeed, we see that K r −1 F = r * K F and so r −1 F has canonical singularities. Let S be a separatrix of r −1 F at r −1 (P ) (which exists since we know that r −1 F admits a formal first integral). By [CS21, Lemma 3.16] we know that (X ′ , S) is log canonical and since S is Q-Cartier it follows that X ′ is in fact klt.
Thus, perhaps passing to a higher quasi-étale cover we may freely assume that π 1 (W ) = 0. Thus σ −1 µ −1 F admits a convergent first integral. By Lemma 5.11 this implies that µ −1 F, and hence F, admits a convergent first integral. 5.5. Classification of terminal foliation singularities. We will need the following which is a direct generalization of [Spi20, Lemma 9.7] Corollary 5.15. Let P ∈ X be a normal threefold germ and let F be a terminal co-rank 1 foliation. Then F admits a holomorphic first integral. In particular K X is Q-Cartier.
Remark 5.16. A priori we only know that K F is Q-Cartier.
Proof. After replacing P ∈ X by a finite cover we may assume that K F is Cartier. Since F is terminal and K F is Cartier this implies that P ∈ X is in fact an isolated singularity. Moreover, perhaps shrinking about P we may assume that Cl(P ∈ X) is generated by the classes of divisors D 1 , ..., D N on X.
By Theorem 2.4 we may take an F-dlt modification of F. Since F is terminal we see that µ is small, i.e., µ −1 (P ) is a union of curves. Observe that Y is Q-factorial. In particular, D ′ i := µ −1 * D i is Q-Cartier and so if P ∈ U ⊂ X is a smaller germ then µ : µ −1 (U ) → U is also an F-dlt modification of F| U . Indeed, to see this it suffices to show that µ −1 (U ) is globally Q-factorial. If D is any global divisor on U then observe that µ * D ∼ a i D i by assumption and so D ∼ a i D ′ i and hence is Q-Cartier. Thus we may freely replace X by a smaller germ about P at any point should we need to do so. We claim Claim 5.17. For all Q ∈ µ −1 (P ) ⊂ Y we have that Y is analytically Qfactorial about Q.
Claim 5.18. Y is simply connected.
Proof of Claim 5.18. Let T be a germ of a G-invariant surface containing µ −1 (P ). Since G is terminal and µ −1 (P ) is connected we see that T is irreducible. Let S = µ * T and observe by the proper mapping theorem that S is a divisor on X.
Observe that since F, and hence G, is terminal and Gorenstein (i.e., K F is Cartier) we have that (X, F) and (Y, G) are both smooth in codimension 2 and so K G | T = K T and K F | S = K S and so we see that µ * K S = K T . By [CS21, Lemma 3.16] we see that T is a log terminal surface and so P ∈ S is a germ of a log terminal singularity. Thus we see that exc(T → S) = exc(µ) is a tree of rational curves and therefore µ −1 (P ) is simply connected. Notice that Y deformation retracts onto µ −1 (P ) and so Y is simply connected.
Assuming Claim 5.17 we complete the proof as follows. Observe that µ −1 F is terminal and so for all q ∈ µ −1 (P ) by Theorem 5.1 there exists a holomorphic first integral F q defined on a neighborhood U q of q so that F q | T = 0.
Let s : Y ′ → Y be an index 1 cover associated to T ramified only over T , see [KM98, Definition 2.52, Lemma 2.53], and let µ ′ : Y ′ → X ′ be the Stein factorization of Y ′ → X. Notice that r : X ′ → X is ramified only along invariant divisors so K r −1 F = r * K F , in particular r −1 F is still terminal. Replacing X by X ′ we may freely assume that T is Cartier.
In particular, for any q, up to taking a root, we may assume that (F q = 0) = T ∩ U q , i.e., (F q = 0) is reduced. Thus for any q and q ′ so that U q ∩ U q ′ = ∅, we see that F q is not a power on U q ∩ U q ′ . Moreover, since Y is smooth in codimension 2 we see that µ −1 (P ) ∩ sing(X) consists of a finite collection of points, and so by shrinking the U q if necessary we may also assume that U q ∩ U q ′ ∩ sing(X) = ∅. We may then apply Lemma 5.14 to produce representation ρ : π 1 (Y ) → Aut( ∆). Since π 1 (Y ) is trivial we see that ρ is trivial and so we get global first integral on Y , which descends to X.
To show that K X is Q-Cartier, let φ : (P ∈ X) → (0 ∈ C) be a holomorphic first integral for F where 0 ∈ C is a (germ of a) curve. Let F = φ −1 (0) and observe that K F = K X/C (−mF ) where K X/C = K X − φ * K C and where m + 1 is the multiplicity of the fibre over 0. By assumption K F is Q-Cartier, φ * K C is Cartier since C is a smooth curve and F = 1 m+1 φ * 0 is Q-Cartier and so K X is Q-Cartier as claimed, thus completing the proof.
We now prove Claim 5.17, Proof of Claim 5.17. Let Q ∈ µ −1 (P ) ⊂ Y be any point. We make the following preliminary observation. Let D be any divisor defined in an (analytic) neighborhood U of Q and suppose that D ∩ µ −1 (P ) = Q. Then, perhaps shrinking X to a smaller neighborhood of P , we may extend D to a divisor on all of Y . Indeed, for any Q ′ ∈ µ −1 (P )\(µ −1 (P )∩U ) we may find an open set V Q ′ ⊂ Y such that V Q ′ ∩D = ∅. By compactness of µ −1 (P )\(µ −1 (P )∩U ) we may find Q 1 , ..., Q n such that µ −1 (P ) ⊂ U ′ := U ∪ V Q 1 ∪ · · · ∪ V Qn . By construction we see that D is an analytic divisor defined on all of U ′ , by setting D ∩ V Q i = ∅. We may then find an open subset W of P in X such that µ −1 (W ) ⊂ U ′ . Replacing X by W we see that our observation follows. So, suppose that Q ∈ µ −1 (P ) and and suppose for sake of contradiction that Y is not analytically Q-factorial about Q and let D be a local divisor defined on a neighborhood V of Q which is not Q-Cartier. A priori, it is possible D ∩ µ −1 (P ) is 1-dimensional and so it is not clear if we can extend D to a divisor on all of Y .
Since Y is klt this implies there exists a small Q-factorialization about Q. Let f : Z → (Q ∈ Y ) be this Q-factorialization and let D ′ be the strict transform of D and let f −1 (Q) = i C i be a decomposition into irreducible components.
Observe that for all i we may find an irreducible effective Cartier divisor S i defined on Z such that S i · C j = δ ij and such that S i ∩ f −1 (Q) is a single point.
By choosing a i ∈ Q appropriately we may assume that D ′ + a i S i is numerically trivial over Y . Since f is small we see that (D ′ + a i S i ) − K Z is nef and big over Y and therefore by the relative basepoint free theorem, [KM98, Theorem 3.24] for n > 0 sufficiently divisible we have that n(D ′ + a i S i ) ∼ f 0. In particular, if we let T i = f * S i we see that D + a i T i is Q-Cartier near Q.
Since T i ∩ µ −1 (P ) is a point, by our observation at the beginning of this proof we may extend T i to a divisor on all of Y , in particular, it follows that T i is Q-Cartier. This in turn implies that D is in fact Q-Cartier, proving our claim.
We can now provide a classification of terminal foliation singularities.
Proposition 5.19. Let (P ∈ X) be a normal threefold germ and let F be a co-rank one foliation on (P ∈ X). Suppose K X and K F are Cartier and suppose that F is terminal. Then F is given by the smoothing of a Du Val surface singularity, i.e., F admits a first integral φ : (P ∈ X) → (0 ∈ C) where φ −1 (0) is a Du Val surface singularity and φ −1 (t) is smooth for t = 0. In particular, X is terminal.
Moreover, it is possible to write down a list of all such smoothings. In an appropriate choice of coordinates we have that and that F is defined by the 1-form dt, i.e., our first integral is just (x, y, z, t) → t and where ψ(x, y, z) is one of the following, see [KM98,Theorem 4.20]: (1) x 2 + y 2 + z n+1 with n ≥ 0; (2) x 2 + zy 2 + z n−1 with n ≥ 4; (3) x 2 + y 3 + z 4 ; (4) x 2 + y 3 + yz 3 ; (5) x 2 + y 3 + z 5 ; (6) x. Conversely, if g(x, y, z, t) is such that X has at worst an isolated singularity at P and F is defined by dt then F has a terminal singularity at P .
Theorem 5.20. Let P ∈ X be a threefold germ and let F be a co-rank one foliation on X and suppose that F is terminal. Then P ∈ X is a quotient of one of the foliations 1-6 in the above list by G = Z/m × Z/n.
Proof. By Corollary 5.15 we see that K F and K X are both Q-Cartier so we may find a Galois cover π : (X ′ , F ′ ) → (X, F) with Galois group Z/n × Z/m so that K F ′ and K X ′ are both Cartier. Indeed, Let X 1 → X and X 2 → X be the index one coves associated to K F and K X , with Galois groups Z/m 1 and Z/m 2 respectively. Then if X ′ is the normalization of a component of X 1 × X X 2 dominating X then X ′ → X is Galois and its Galois group is a subgroup of Z/m 1 × Z/m 2 as required. By Proposition 5.19 we see that (X ′ , F ′ ) is one of the foliations 1-6 and we can conclude.
Corollary 5.21. Let p ∈ X be a germ of a normal threefold and let F be a co-rank one foliation on X and suppose that F is terminal. Then X and F admit a Q-smoothing, i.e., there exists a family of foliated threefold germs X t and F t such that (X 0 , F 0 ) = (X, F) and such that for t = 0 we have that (X t , F t ) is a quotient of a smooth foliation on a smooth variety.
Proof. This is a direct consequence of the classification in Proposition 5.19. Indeed, in each case we may explicitly construct a smoothing of X and F by perturbing the defining equations of X and F. 5.6. Structure of terminal flips. We finish by providing a rough structural statement for terminal foliated flips.
Theorem 5.22. Let X be a Q-factorial threefold and let F be a co-rank 1 foliation on X with terminal singularities. Let φ : X → Z be a K F -flipping contraction and let C = Exc(φ). Then there exists an analytic open neighborhood C ⊂ U and a holomorphic first integral F : U → C of F.
Proof. By Theorem 4.3 we have that R 1 f * O X = 0, and so C is in fact a tree of rational curves, in particular it is simply connected. For all p ∈ C by Theorem 5.1, we may find a holomorphic first integral of F near p. However, since C is simply connected by arguing as in the proof of Corollary 5.15 we may produce a first integral in a neighborhood of C.

Existence of separatrices for log canonical foliation singularities
The goal of this section is to prove the following.
Theorem 6.1. Let P ∈ X be an isolated klt singularity. Let F be a germ of a log canonical co-rank 1 foliation singularity on P ∈ X. Then F admits a separatrix.
Recall that log canonical foliation singularities which are not canonical are always dicritical and in general dicritical singularities do not admit separatrices as the following classical example due to Jouanolou shows.
Example 6.2. The foliation on 0 ∈ C 3 defined by (x m z − y m+1 )dx + (y m x − z m+1 )dy + (z m y − x m+1 )dz has no separatrices at the origin for m ≥ 2. The blow up of this foliation at 0 has discrepancy = −m, and therefore is not log canonical for m ≥ 2.
As the next example shows a log canonical singularity may not admit a separatrix if no assumption is made on the base space.
Example 6.3. Let A be an abelian surface that admits an automorphism τ so that X := A/ τ is a rational surface and A → X isétale in codimension 1. We may find a linear foliation on A which admits no algebraic leaves and is τ -invariant and so descends to a foliation F without algebraic leaves on X.
Let P ∈ Y be the cone over X with vertex P and let G be the cone over F. It is easy to check that G is log canonical and admits no separatrices at P . However, observe that P ∈ Y is log canonical and not klt.
We also have the following interesting corollary.
Corollary 6.4. Let F be a germ of a foliation 0 ∈ C 3 and let i : (0 ∈ S) → (0 ∈ C 3 ) be a germ of a surface transverse to F such that i −1 F is log canonical, e.g., is a radial singularity. Then F admits a separatrix.
Proof. This follows by combining Theorems 6.1 and 3.12 We now proceed with the proof of Theorem 6.1. We will first need the following generalization of Lemma 1.18.
Lemma 6.5. Let X be a complex threefold with a co-rank 1 foliation F with non-dicritical singularities. Let D ⊂ X be a compact subvariety. and let V ⊂ D be a closed proper subvariety of D tangent to F with the following property: Let q ∈ V be any point, let U q be a neighborhood of q and let S q ⊂ U q be a separatrix at q. Then there exists an analytic open neighborhood U of D and an invariant subvariety S ⊂ U such that S ∩ U q = S q .
Proof. Let π : X → X be a resolution of singularities of X and F and so that π −1 (V ) is an invariant divisor.
Observe that Condition (⋆) still holds for π −1 (V ) and π −1 (D). Moreover, if q ∈ V is some point and π −1 (S q ) admits an extension, S, to a neighborhood U of π −1 (D) then since π is proper, π(S) ⊂ π(U ) is an extension of S q to a neighborhood of D.
Thus, without loss of generality we may assume that X is smooth, F has simple singularities and that V is a divisor.
Let q ∈ V be a point and let S q be any separatrix at q. By Lemma 1.18 we may find a neighborhood U ′ of V and an invariant divisor S ′ which agrees with S q near q. Let D ′ = D ∩ U ′ . By (⋆) we see that S ′ ∩ (D ′ − V ) = ∅. Thus, perhaps shrinking U ′ if necessary, for all p ∈ D − V there exists a neighborhood U p of p such that U p ∩ S ′ = ∅.
Taking U = U ′ ∪ p∈D−V U p we see that S ′ extends to a subvariety of U and we are done.
We recall the following classification result due to [McQ08].
Theorem 6.6. Let X be a normal projective surface and let L be a rank one foliation on X with canonical foliation singularities. Suppose c 1 (K L ) = 0.
Then there exists a birational morphism µ : X → X ′ contracting only rational curves tangent to L and a cyclic cover, τ : Y → X ′ ,étale in codimension one such that one of the following holds where G = τ −1 µ * L: (1) µ is an isomorphism, X = C × E/G where g(E) = 1, C is a smooth projective curve, G is a finite group acting on C × E and G is the foliation induced by the G-invariant fibration C × E → C; (2) µ is an isomorphism and G is a linear foliation on the abelian surface Y ; (3) µ is an isomorphism, Y is a P 1 -bundle over an elliptic curve and G is transverse to the bundle structure and leaves at least one section invariant; or (4) Up to blowing up Y at P ∈ sing(L) we have Y is a compactification of G m × G a and L restricted to this open subset is generated by a G m × G a invariant vector field; or (5) Up to blowing up Y at P ∈ sing(L) we have that Y is a compactification of G m × G m and L restricted to this open subset is generated by a G m × G m invariant vector field.
Proof. This follows directly from [McQ08,Theorem IV.3.6] except for the claim in items 1 -3 that µ is an isomorphism. In each of these cases we claim that µ * L is terminal. This follows because for all P ∈ X ′ there exists a cyclic cover (namely τ ) such that τ −1 µ * L is smooth in a neighborhood of τ −1 (P ) and so we may apply Proposition 1.7 to conclude.
Since µ * L is terminal and c 1 (K L ) = 0 this implies that µ is an isomorphism.
Lemma 6.7. Let S be a surface and let L be a co-rank 1 foliation on S. Suppose that c 1 (K L ) = 0 and that L has canonical singularities.
Then the following hold.
(1) For all p ∈ sing(L) each separatrix at p is algebraic. In particular, the union of all such separatrices is an algebraic subvariety of S. Proof. To prove item 1 observe that in order to check if each separatrix at a singular point is algebraic we may freely contract curves tangent to the foliation, as well as replacing by a finite cover. Thus, it suffices to check the claim for each of the 5 types of foliation listed in the statement of Theorem 6.6. In cases 1 -3 the foliation is smooth and so there is nothing to prove. Thus it remains to consider cases 4 and 5.
In this case, we see that the vector field generating L on G m ×G a or G m × G m , respectively, is smooth. Hence sing(L) is contained in the boundary of the compactification. Moreover, since L is invariant under the action of G m × G a or G m × G m we see that every separatrix of p ∈ sing(L) must be contained in the boundary.
To prove item 2 again we may freely contract curves tangent to L and replace by a finite cover. Thus we may assume that (S, L) is one of the foliations listed in Theorem 6.6. We argue based on the case.
If we are in case 5 or 4 then sing(L) is non-empty and so by item 1 as proven above we may take V to be the union of all separatrices at sing(L).
If we are in case 1 then L is algebraically integrable and we may take V to be the closure of a general leaf.
If we are in case 3 let Σ be the invariant section. We claim that L is smooth along Σ. Indeed, on one hand K L · Σ = K Σ + ∆ where ∆ ≥ 0 is supported on sing(L) ∩ Σ. On the other hand by assumption K L · Σ = 0 and since Σ is an elliptic curve we have K Σ = 0 and so ∆ = 0. This gives us sing(L) ∩ Σ = ∅ and so we may take V = Σ.
Otherwise S is an abelian variety and there is nothing more to prove.
Lemma 6.8. Let P ∈ X be a germ of a normal threefold and let F be a corank one foliation on X. Suppose that F is log canonical but not canonical.
Proof. Let µ : (X, F ) → (X, F) be an F-dlt modification of (X, F) and Observe that X is Q-factorial and klt and F has non-dicritical singularities.
Since F is not canonical it must be the case that µ extracts some divisor transverse to the foliation. We may therefore assume, after relabeling, that ǫ(E ′ 0 ) = 1 and is F-dlt and so by Corollary 2.3 we may run a Since the MMP preserves Q-factoriality and klt singularities and the output of the MMP has non-dicritical singularities we see that items 6 and 3 are satisfied. Item 4 follows by construction and item 5 follows since the MMP preserves F-dlt singularities. Since we see that each ray R contracted by this MMP has positive intersection with the strict transform of E ′ 0 , in particular E ′ 0 is not contracted by this MMP. Set E 0 = φ * E ′ 0 . Since E ′ 0 is transverse to the foliation E 0 is as well proving item 1. Moreover we have that By the Negativity Lemma, [Wan19, Lemma 1.3], for all x ∈ X either π −1 (x) is disjoint from E 0 or π −1 (x) is contained in E 0 . By our choice of E 0 we have E 0 ∩ π −1 (P ) = ∅ which proves item 2.
Lemma 6.9. Let P ∈ X be a germ of a klt singularity with a co-rank one foliation F with log canonical but not canonical singularities. Let π : (Y, G) → (X, F) be a birational morphism as in Lemma 6.8 above.
We will find a closed subset V ⊂ E 0 satisfying the hypotheses of Lemma 6.5 in order to produce a G-invariant divisor in a neighborhood of E 0 whose pushforward will be our desired separatrix.
Let {E i } denote the collection of π-exceptional divisors so that we have K G + E 0 = π * K F and (G, E 0 ) is log canonical and where E i is G-invariant for i = 0. Note that since E i is invariant we see that C i := π(E i ) is a curve tangent to F passing through P .
Since G is non-dicritical and E 0 is the only π-exceptional divisor which is not G-invariant it follows that F restricted to X \ P is non-dicritical.
By foliation adjunction, Lemma 3.10, we know that where ∆ 0 ≥ 0 and where n : E n 0 → E 0 is the normalization. Next, since X is klt we may write K Y + E 0 + B = π * K X + aE 0 where a > 0 and B is not necessarily effective, but is supported on the G-invariant π-exceptional divisors. Write n * (K Y + E 0 + B) = K E n 0 + Θ 0 . We claim that −E 0 | E 0 is big. Let A be an ample divisor on Y . We may find a divisor D ≥ 0 on X so that D + π * A is Q-Cartier. We may then write π * (D + π * A) First we handle the case ∆ 0 = 0. In this case K G 0 is not psef, hence G 0 is algebraically integrable, by [BM16,Main Theorem]. Take V to be the closure of general leaf of G 0 . Observe that G 0 is non-dicritical since π −1 F is, and so V is disjoint from the closure of any other leaf of G 0 . Moreover, in this case we see that E n 0 is a P 1 -fibration over a curve and that V is a general fibre in this fibration. In particular, notice that K E n 0 · V = −2. We claim that n −1 (n(V )) = V . Indeed, if not then E 0 would not be normal in a neighborhood of some point of n(V ). Let W ⊂ sing(E 0 ) be a one dimensional component meeting n(V ). Observe that since V is general that W is transverse to the foliation. Since (G, (1 − ǫ)E 0 ) is F-dlt for all 1 > ǫ > 0 it follows from [Spi20, Lemma 3.11] that (Y, (1−ǫ)E 0 ) is dlt at the generic point of W . It follows by [KM98,Corollary 5.55] in a neighborhood of a general point of W we have that E 0 consists of two smooth components meeting transversely.
Since V is general, it follows that in an (analytic) neighborhood of V that n −1 (W ) consists of two components transverse to G 0 . A straightforward calculation shows that the coefficient of each of these components in ∆ 0 and Θ 0 is = 1. Notice that V · Θ − 0 = 0 and so (K E n 0 + Θ 0 ) · V ≥ 0. However, V is a movable curve and this contradicts the fact that −(K E n 0 + Θ 0 ) is big. Thus for all q ∈ n(V ) if S q is a separatrix of π −1 F at q we see that S q ∩ E 0 ⊂ n(V ) and so we may apply Lemma 6.5 to produce an extension T of S q to a neighborhood U of E 0 . Perhaps shrinking U we may assume that U = π −1 (W ) for some neighborhood W of P . Notice also that since V was chosen to be general we may assume that T is not contained in the union of the π-exceptional divisors. Since U → W is proper we see that S = π * T ⊂ V is a divisor and is invariant under F, and hence is our desired separatrix. Now we handle the case ∆ 0 = 0. First observe that ∆ 0 = 0 implies that E 0 is normal. By foliation adjunction, Lemma 3.10, G 0 is log canonical and since G 0 is non-dicritical we see that G 0 is in fact canonical.
First, suppose that there exists a quasi-étale cover r : Y → E 0 be a quasietale cover of E 0 such that Y is an abelian variety. We claim that G 0 is algebraically integrable in this case (in which case we are done by arguing as above). If Θ − 0 = 0 then G 0 , and hence r −1 G 0 , admits an invariant algebraic curve and by Theorem 6.6 we see that G 0 is algebraically integrable. So suppose for sake of contradiction that Θ − 0 = 0. In this case,−K E 0 is big and so we have that −K Y = −r * K E 0 is big, contradicting K Y ∼ 0.
Next, suppose that there is no such cover. We may apply Lemma 6.7 to produce V ⊂ E 0 such that each component of V is tangent to G 0 and each separatrix of G 0 meeting V is contained in V . Thus, we may apply Lemma 6.5 to produce an invariant divisor T in a neighborhood of E 0 and which contains V . We claim that T is not contained in the union of the π-exceptional divisors. Supposing the claim we see that S = π * T is our desired separatrix.
We now prove the claim. First if E 0 is the only exceptional divisor there is nothing to show. So suppose that there is some other π-exceptional divisor E i . Notice that if Q ∈ C i \ P is a general point then there exists a separatrix of F at Q, call it S Q (recall C i = π(E i ) is tangent to F). This follows because in a neighborhood of Q we know that X has quotient singularities, since X is klt and klt singularities are quotient singularities outside a subset of codimension ≥ 3. Thus (up to replacing X by a cover) we may assume that X is smooth at Q. Next, we know that F is non-dicritical in a neighborhood of X and so we may apply [CC92, Existence of Separatrix Theorem] to produce S Q .
Let S ′ Q := π −1 * S Q . By Lemma 1.18 we may extend S ′ Q to an invariant divisor in an (analytic) open neighborhood of m i=1 E i . Call this extension H and by construction H is not contained in m i=0 E i . Let Σ = H ∩ m i=1 E i and notice that Σ is a closed analytic subset of E i . Let Σ 0 = Σ ∩ E 0 and let x ∈ Σ 0 be a point. We know that m i=1 E i ∩ E 0 ⊂ V by construction. However, H ∩E 0 is a separatrix of G 0 at x intersecting V : in fact, H ∩E 0 ⊂ V . Then, in a neighborhood of E 0 , we have H ⊂ T , in particular, T is not contained in the union of the π-exceptional divisors.
We are now ready to prove the main theorem of this section.
Proof of Theorem 6.1. Suppose first that F has canonical singularities. If X is Q-factorial then we may apply Corollary 5.2 to produce a separatrix. Otherwise, since X is klt, it admits a small Q-factorialization µ : X ′ → X. Since F is non-dicritical we know that µ −1 (P ) is tangent to the foliation and is therefore contained in a germ of an invariant surface, S. We may then take µ * S as our desired separatrix.
So we may assume that F is not canonical and let π : (Y, G) → (X, F) be a modification as in Lemma 6.8 and let E 0 be a divisor as in the statement of the Lemma.
There are two cases, either π −1 (P ) is of dimension 2 or it is of dimension 1. Notice moreover, that if there exists some π-exceptional divisor E transverse to G such that E is centred over a curve in X then by choosing E = E 0 in the proof of Lemma 6.8 we have that π −1 (P ) is of dimension 1.
If π −1 (P ) is of dimension 2 we may therefore freely assume that the only π-exceptional divisor transverse to G is π −1 (P ). We may apply Lemma 6.9 to conclude.
Otherwise C := π −1 (P ) ⊂ E 0 is a curve. Let G 0 be the induced foliation on E 0 . Suppose first that some component C 0 ⊂ C is transverse to G 0 . Then we may apply Lemma 6.5 with D = π −1 (P ) and V a general point in C 0 to produce an invariant divisor S in a neighborhood of π −1 (P ). In this case π * S will be our desired separatrix. Now suppose that each component of C is invariant by G 0 . In this case, perhaps shrinking X, we may assume that the union of all convergent separatrices meeting C is an analytic subset of E 0 , call it C. In this case we may apply Lemma 6.5 with D = E 0 and V = C to produce a separatrix S in a neighborhood of E 0 . Again, π * S is our desired separatrix.
Remark 6.10. In fact, the arguments above prove a slightly stronger claim which may be of interest. In the set up as above, if we let C ⊂ sing(F) be a curve of singularities passing through P then C is contained in a separatrix.

Foliations and hyperbolicity
The goal of this section is to prove the following foliated version of [Sva19, Theorem 1.1]. Given a foliated pair (F, ∆) and an lc center S we will denote byS ⊂ S the locally closed subvariety obtained by removing from S the lc centers of (F, ∆) strictly contained in S.
Theorem 7.1. Let (F, ∆) be a foliated log canonical pair on a normal projective variety X. Assume that • X is potentially klt, • there is no non-constant morphism f : A 1 → X \ Nklt(F, ∆) tangent to F, and • for any stratum S of Nklt(F, ∆) there is no non-constant morphism f : A 1 →S which is tangent to F. Then K F + ∆ is nef.
The notions of potentially kltness and potentially lcness have been defined in Definition 1.2 7.1. A special version of dlt modifications. We prove a refinement of Theorem 2.4, which will be useful in the proof of the main result of this section.
Theorem 7.2 (Existence of special F-dlt modifications). Let F be a corank one foliation on a normal projective variety X of dimension at most 3. Let (F, ∆ = a i D i ) be a foliated pair. We will denote by ∆ ′ := a i <ǫ(D i ) a i D i + a j ≥ǫ(D j ) ǫ(D j )D j . Then there exists a birational morphism π : Y → X which extracts divisors E of foliation discrepancy ≤ −ǫ(E) such that if we write K G +Γ = π * (K F +∆) then (G, Γ ′ := π −1 * ∆ ′ + E i π−exc. ǫ(E i )E i ) is F-dlt. Furthermore, we may choose (Y, G) so that (1) if W is a non-klt centre of (G, Γ) then W is contained in a codimension one lc centre of (G, Γ ′ ), Proof. For the proof of (1), (2), and (3) one can refer to [CS21, Theorem 8.1]. Let π Z : Z → X be a modification of (F, ∆) satisfying these three properties. Let us denote by (H, Θ, Θ ′ ) the triple given by the birational transform of F on Z, By these inequalities K H + Θ ′<1 ∼ R,X −Θ ′′ , where Θ ′′ := Θ − Θ ′<1 . As K H + Θ ′<1 is big/X, there exists A ample/X and an effective divisor G such that K H + Θ ′<1 ∼ R,X A + G. We can decompose G as where G 1 is the part of F supported on π Z -exceptional divisors or Hinvariant divisors, G 2 is the part of G whose components are not H-invariant but contain an H-invariant lc center for (H, Θ), and G 3 : Choosing an effective divisor L whose support coincides with the divisorial part of exc(π Z ) such that A − L is ample, then Let us choose a sufficiently general effective A ′ ∼ R A − L and define G ′ := Claim 7.3. For ǫ ′ ≪ 1, there exists an F-dlt modificationr :Z → Z of (H, Θ ′<1 +ǫ ′ G ′ ) such that for anyr-exceptional prime divisor E, a(E; H, Θ ′ ) = −ǫ(E).
. Then, exactly one of the following two possibilities holds: • K X + ∆ is nef, or • X \ Nklt(∆) contains an algebraic curve whose normalization is A 1 .
In the case of a general foliated log pair, using dlt modifications we get the following criterion, which will be fundamental in the proof of Theorem 7.1.
Corollary 7.5. Let X be a normal, projective, Q-factorial threefold. Let (F, ∆ = i b i D i ≥ 0) be a foliated log pair such that (F, Assume that X \ Nklt(F, ∆) does not contain algebraic curves tangent to F whose normalization is A 1 . Then K F + ∆ is nef if and only if K F + ∆ is nef when restricted to Nklt(F, ∆).
Proof. If K F + ∆ is nef, then, a fortiori, it is nef when restricted to any subvariety of X. We now assume that K F + ∆ is nef when restricted to Nklt(∆). As (F, ∆ ′ ) is F-dlt, it follows that Nklt(F, ∆) = µ D i ∆≥ǫ(D i ) D i = Nklt(F, ∆ ′ ), by definition of ∆ ′ . Now, let us suppose that K F + ∆ is not nef. Then there exists a negative extremal ray R ⊂ N E(X). Since K F + ∆ is nef when restricted along Nklt(F, ∆), it follows that R·D i ≥ 0 for any D i with µ D i ∆ ≥ ǫ(D i ). Hence, R is a negative extremal ray also for K F + ∆ ′ . As (F, ∆ ′ ) is an F-dlt pair, it is non-dicritical by Theorem 1.16; in particular, [CS21,Lemma 3.30] implies that any curve C ⊂ X satisfying [C] ∈ R is tangent to F. Moreover, [CS21, Theorem 6.7] implies that there exists a contraction φ : X → Y within the category of projective varieties which only contracts curves in X whose numerical class belongs to R. In particular as K F + ∆ is nef along Nklt(F, ∆), it follows that each fiber of φ intersects Nklt(F, ∆) in at most finitely many points. As X is Q-factorial, it follows that each fiber of φ intersecting Nklt(F, ∆) must have dimension at most 1; otherwise, if X y , y ∈ Y , were a 2-dimensional fiber, no component of ∆ ′ could intersect X y , as this intersection would contain a (K F + ∆)-negative curve contained in Nklt(F, ∆), hence there would be a rational curve C ⊂ X \ Nklt(F, ∆), thus leading to a contradiction. Let Σ ⊂ X be an irreducible curve contracted by φ. We claim that Σ is a rational curve. Indeed, Σ is tangent to F, thus we may find a germ of an invariant surface, call it S, containing Σ. If Σ ⊂ sing(F), then S is simply a leaf containing Σ, while if Σ ⊂ sing(F), then we may take S to be a strong separatrix at a general point of Σ. As (F, ∆ ′ ) is F-dlt, then we can apply Lemma 3.10 to write ν * (K F + ∆ ′ ) = K S ν + ∆ ′ S ν , where ν : S ν → S is the normalization of S, and (S ν , ∆ ′ S ν ) is lc. Taking T to be the normalization of φ(S) then the strict transform of Σ on S ν is a (K S ν + ∆ ′ S ν )-negative curve contracted by the morphism S ν → T and is therefore (by classical adjunction) necessarily a rational curve. The Q-factoriality of X implies that we are in either of the following two cases: 1) φ is a Mori fibre space and all the fibres are one dimensional; 2) φ is birational and the exceptional locus intersects Nklt(∆).
We claim that in both cases R 1 φ * O X = 0. In fact, in case 1) as all fibers are rational curves, we have that φ must be a K X -negative contraction, while in case 2) the conclusion can be reached by direct application of Theorem 4.3. At this point, Theorem 3.1 implies that Nklt(F, ∆) is connected in a neighborhood of every fibre of φ. In case 1), the generic fibre of µ is a smooth projective rational curve. Theorem 3.1 implies that the generic fibre intersects Nklt(∆) in at most one point. This concludes the proof in case 1). In case 2), the positive dimensional fibres are chains of rational curves and by the vanishing R 1 φ * O X = 0, the generic fibre has to be a tree of smooth rational curves. By Theorem 3.1, Nklt(F, ∆) intersects this chain in at most one point. In particular, there exists a complete rational curve C such that C ∩ (X \ Nklt(F, ∆)) = f (A 1 ), where f is a non-constant morphism, which provides the sought contradiction.
Proof of Theorem 7.1. We divide the proof into two distinct cases.
Case 1: We first prove the theorem under the assumption that (F, ∆) is F-dlt. If K F + ∆ is nef along Nklt(F, ∆) the conclusion follows from Corollary 7.5. Hence, we can assume that there exists a positive dimensional lc center W for (F, ∆) and K F + ∆ is not nef along W . By induction on the dimension, we can consider W to be a minimal (with respect to inclusion) lc center satisfying such property, so that (K F + ∆)| W is nef when restricted to the lc centers of (F, ∆) strictly contained in W . Clearly, dim W > 0 and [Spi20,Theorem 4.5] implies that if dim W = 1, then W is tangent to F. As (F, ∆) is F-dlt, it follows that either one of the following conditions hold: a) W is a component of ∆ of coefficient 1; b) W is an invariant divisor; c) W ⊂ Sing(F), dim W = 1, and F is canonical along W by [CS21, Lemma 3.12]; or d) dim W = 1 and it is tangent to F, W ⊂ Sing(F), but W ⊂ D, where D is a component of ∆ with µ D ∆ = 1.
Case 1.a. If W is a component of ∆ of coefficient 1, then we can apply the adjunction formula along the normalization ν : W ν → W : where G is the restriction of F to W ν and Θ is the different as defined in Lemma 3.10. The adjunction formula guarantees that (G, Θ) is F-dlt, see Lemma 3.10, that ν −1 (Z) = Nklt(G, Θ), where Z is the union of all lc centers of (F, ∆) strictly contained in W . This follows from [CS21, Lemma 3.8] as • for any stratum S of Nklt(F, ∆) there is no non-constant morphism f : A 1 →S. Then K F + ∆ is nef.
Proof. Assume for sake of contradiction that K F + ∆ is not nef and (F, ∆) satisfies all the hypotheses in the statement of the proposition. We divide the proof into two distinct cases.
Case 1: We assume that (F, ∆) is F-dlt and we show that the above hypothesis leads to a contradiction. By [CS21, Theorem 3.31], since K F + ∆ is not nef, there exists a rational curve C ⊂ X with (K F + ∆) · C < 0 and C is tangent to F. As C is F-invariant we see that C cannot be contained in Supp(∆). Thus, where ν : C ν → C is the normalization, and Supp(⌊∆ C ν ⌋) ⊃ ν −1 (sing(F) ∪ ⌊∆⌋). Finally, observe Nklt(F, ∆) and all its strata are supported on sing(F)∪⌊∆⌋ to conclude that the normalization of C − Z ′ is P 1 or A 1 where Z ′ are all the strata of Nklt(F, ∆) meeting C. This is our desired contradiction.
Case 2: We assume that (F, ∆) is lc and we reduce the proof to Case 1. Let π : Y → X be an F-dlt modification for the pair (F, ∆), Hence, also K F Y + Γ is not nef and by Case 1 there is is a rational curve C ⊂ Y tangent to F Y such that C · (K F Y + Γ) < 0; moreover, the normalization morphism C ν → Y induces either a non-constant morphism f : A 1 → Y \ Nklt(F Y , Γ) or a non-constant morphism f : A 1 →S, for some stratum S of Nklt(F Y , Γ). The curve π(C) is tangent to F, thus, it is F-invariant, since F Y has rank 1. If C ∩ (Y \ Nklt(F Y , Γ)) = ∅, it follows from Theorem 7.2 and adjunction that π • f : A 1 → X \ Nklt(F, ∆) is a well-defined morphism. Hence we can assume that C is an lc center of (F Y , Γ) and thatC is a copy of A 1 embedded in Y . But then, again, the adjunction formula and Theorem 7.2 imply that π(C) is also a copy of A 1 embedded in X, thus proving the proposition.

Some questions
The proof of Theorem 6.1 and its generalizations and possible applications raise several questions.
Question 8.1. Let 0 ∈ X be a germ of a klt singularity and F a log canonical co-rank one foliation on X. Does F admit a separatrix at 0? Question 8.2. Let F be a co-rank 1 foliation on a klt variety (X, ∆) with c 1 (K F ) = 0 and −(K X + ∆) big.
(3) For p ∈ sing(F) is every separatrix at p algebraic?
More generally, one may wonder if log canonical singularities of foliations all dimension admit separatices. By examples of Gomez-Mont and Luengo [GML92] it is known that a vector field on C 3 does not always admit a separatrix, however the examples given there are not log canonical.
Question 8.3. Let F be a foliation of any rank on C m . Let 0 be an log canonical singularity of F. Does F admit a separatrix at 0?
In the proof of existence of flips given in [CS21] the existence of separatrices played a central role, and thus the methods given there do not immediately imply the existence of log canonical flips. With Theorem 6.1 in mind we ask the following. This extension seems to be important to apply the methods of the foliated MMP to several classes of folations of interest: Fano foliations, for instance, have worse than canonical singularities.