2.1 Notations

We work over the field of complex numbers ${\mathbb C}$ .

Given a normal variety X, we denote by $\Omega ^1_X$ its sheaf of Kähler differentials and, by $T_X:=(\Omega ^1_X)^{*}$ its tangent sheaf. For any positive integer p, we denote . Let A be a ${\mathbb R}$ -Weil divisor on X and let D be a prime divisor. We denote by $\mu _DA$ the coefficient of D in A. A log pair $(X,\Delta )$ is a pair of a normal variety and a ${\mathbb Q}$ -divisor $\Delta $ such that $K_X+\Delta $ is ${\mathbb Q}$ -Cartier. We refer to [Reference Kollár and MoriKM98] for the classical definitions of singularities (e.g., klt, log canonical) appearing in the minimal model program, except for the fact that in our definitions we require the pairs to have effective boundaries. In addition, we say that a log pair $(X,\Delta )$ is sub log canonical, or sub lc, if $a(E,X,\Delta )\ge -1$ for any geometric valuation E over X. A fibration $f\colon X\to Y$ is a surjective morphism between normal varieties with connected fibres. We refer to [Reference Cascini and SpicerCS21, Section 2.6] for some of the basic notions, commonly used in the MMP.

A foliation of rank $\boldsymbol {r}$ on a normal variety X is a rank r coherent subsheaf ${\mathcal F}\subset T_X$ such that

1. 1. ${\mathcal F}$ is saturated in $T_X$ , and

2. 2. ${\mathcal F}$ is closed under Lie bracket.

Note that if $r=1$ then (2) is automatically satisfied. By (1), it follows that $T_X/{{\mathcal F}}$ is torsion free. We denote by $\mathcal N^{*}_{\mathcal F}:=(T_X/{{\mathcal F}})^{*}$ the conormal sheaf of ${\mathcal F}$ . The normal sheaf $\mathcal N_F$ of ${\mathcal F}$ is the dual of the conormal sheaf. The canonical divisor of ${\mathcal F}$ is a divisor $K_{\mathcal F}$ on X such that $\mathcal O_X(-K_{\mathcal F})\simeq \det T_{{\mathcal F}}$ . The foliation ${\mathcal F}$ is said to be Gorenstein (resp. ${\mathbb Q}$ -Gorenstein) if $K_{{\mathcal F}}$ is a Cartier (resp. ${\mathbb Q}$ -Cartier) divisor. More generally, a rank $\boldsymbol {r}$ foliated pair $({\mathcal F}, \Delta )$ is a pair of a foliation ${\mathcal F}$ of rank r and a ${\mathbb Q}$ -divisor $\Delta \ge 0$ such that $K_{\mathcal F}+\Delta $ is ${\mathbb Q}$ -Cartier.

Let X be a normal variety and let ${\mathcal F}$ be a rank r foliation on X. We can associate to ${\mathcal F}$ a morphism

$$\begin{align*}\phi\colon \Omega^{[r]}_X \rightarrow \mathcal O_X(K_{\mathcal F})\end{align*}$$

defined by taking the double dual of the r-wedge product of the map $\Omega ^{[1]}_X\to {\mathcal F}^{*}$ , induced by the inclusion ${\mathcal F} \subset T_X$ . We will call $\phi $ the Pfaff field associated to ${\mathcal F}$ . Following [Reference DruelDru21, Definition 5.4], we define the twisted Pfaff field as the induced map

$$\begin{align*}\phi'\colon (\Omega^{[r]}_X \otimes \mathcal O_X(-K_{\mathcal F}))^{**} \rightarrow \mathcal O_X\end{align*}$$

and we define the singular locus of ${\mathcal F}$ , denoted by ${\operatorname {Sing} {\mathcal F}}$ , to be the cosupport of the image of $\phi '$ . We say that ${\mathcal F}$ is smooth at a closed point $x\in X$ if $x\notin {\operatorname {Sing} {\mathcal F}}$ and we say that ${\mathcal F}$ is a smooth foliation if ${\operatorname {Sing} {\mathcal F}}$ is empty.

Let $\sigma \colon Y\dashrightarrow X$ be a dominant map between normal varieties and let ${\mathcal F}$ be a foliation of rank r on X. We denote by $\sigma ^{-1}{\mathcal F}$ the induced foliation on Y (e.g., see [Reference DruelDru21, Section 3.2]). If $\sigma \colon Y\to X$ is a morphism then the induced foliation $\sigma ^{-1}{\mathcal F}$ is called the pulled back foliation. If $f\colon X\dashrightarrow X'$ is a birational map, then the induced foliation on $X'$ by $f^{-1}$ is called the transformed foliation of ${\mathcal F}$ by f and we will denote it by $f_{*}{\mathcal F}$ . Moreover, if $q\colon X' \rightarrow X$ is a quasi-étale cover and ${\mathcal F}' = q^{-1}{\mathcal F}$ then $K_{{\mathcal F}'}=q^{*}K_{\mathcal F}$ and [Reference DruelDru21, Proposition 5.13] implies that ${\mathcal F}'$ is smooth if and only if ${\mathcal F}$ is.

2.2 Singularities in the sense of McQuillan

The definition of foliation singularities used in [Reference McQuillanMcQ04] is slightly different than the notion defined above. We recall McQuillan’s definition now.

Let X be a normal variety, let ${\mathcal F}$ be a rank one foliation on X such that $K_{\mathcal F}$ is ${\mathbb Q}$ -Cartier. Let $x \in X$ be a point and let U be an open neighbourhood of x. Up to replacing U by a smaller neighbourhood we may find an index one cover $\sigma \colon U' \rightarrow U$ associated to $K_{{\mathcal F}}$ and such that $\sigma ^{-1}{\mathcal F}$ is generated by a vector field $\partial $ .

We say that ${\mathcal F}$ is singular in the sense of McQuillan at $x \in X$ provided there exists an embedding $U' \rightarrow M$ where M is a smooth variety and a lift $\tilde {\partial }$ of $\partial $ to a vector field on M such that $\tilde {\partial }$ vanishes at $\sigma ^{-1}(x)$ . We denote by ${\operatorname {Sing}^{+}{\mathcal F}}$ the locus of points $x \in X$ where ${\mathcal F}$ is singular in the sense of McQuillan. Note that ${\operatorname {Sing}^{+}{\mathcal F}}$ does not depend on the choice of $U'$ and it is a closed subset of X.

We have the following inclusion of singular loci:

We will show later that the equality holds if X admits klt singularities (cf. Proposition 2.32).

2.3 Invariant subvarieties

Let X be a normal variety, and let $\partial \in H^0(X,T_X)$ be a vector field. We say that an ideal sheaf J of X is $\boldsymbol {\partial }$ -invariant if $\partial (J)\subset J$ . Let $S\subset X$ be a subvariety. Then S is said to be $\boldsymbol {\partial }$ -invariant, or invariant by $\partial $ if the ideal sheaf ${\mathcal I}_{S}$ of S is $\partial $ -invariant.

Let ${\mathcal F}$ be a foliation on X. Then S is said to be ${\boldsymbol {\mathcal F}}$ -invariant, or invariant by ${\mathcal F}$ , if, in a neighbourhood U of the generic point of S, $T_{{\mathcal F}}$ is locally free and for any section $\partial \in H^0(U,{\mathcal F})$ , we have that $S\cap U$ is $\partial $ -invariant. If $D\subset X$ is a prime divisor then we define $\epsilon (D) = 1$ if D is not ${\mathcal F}$ -invariant and $\epsilon (D) = 0$ if it is ${\mathcal F}$ -invariant.

We will need the following version of Riemann-Hurwitz formula for foliations (e.g., see [Reference DruelDru21, Lemma 3.4]):

Proposition 2.2. Let $\sigma \colon Y \rightarrow X$ be a finite surjective morphism between normal varieties, let ${\mathcal F}$ be a foliation on X and let $\mathcal G := \sigma ^{-1}{\mathcal F}$ .

Then we may write

$$\begin{align*}K_{\mathcal G} = \sigma^{*}K_{\mathcal F} + \sum \epsilon(\sigma(D))(r_D-1)D\end{align*}$$

where the sum runs over all the prime divisors on Y and $r_D$ is the ramification index of $\sigma $ along D. In particular, if every ramified divisor is $\mathcal G$ -invariant then $K_{\mathcal G} = \sigma ^{*}K_{\mathcal F}$ .

Proof. We may assume that X is affine and that $T_{\mathcal F}$ is generated by a vector field $\partial $ which lifts to a vector field $\tilde {\partial }$ on Y which generates $T_{\mathcal G}$ .

We first prove (1). Let J denote the ideal of $p(W)$ and let I denote the ideal sheaf of W. In particular, $p_{*}I$ is the sheaf associated to J. Let $f \in J$ and notice that $p^{*}\partial f = \tilde {\partial }(p^{*}f)$ . Since W is $\mathcal G$ -invariant and p is proper, we have

$$\begin{align*}p^{*}\partial f= \tilde{\partial}(p^{*}f) \in H^0(Y, I)=J. \end{align*}$$

Thus, J is $\partial $ -invariant and (1) follows.

We now prove (2). Let I denote the ideal sheaf of Z and let $f_1, ..., f_k$ be generators of I. Then $p^{*}f_1,\dots ,p^{*}f_k$ are generators of $p^{-1}I\mathcal O_Y$ , the ideal sheaf of the scheme-theoretic preimage $p^{-1}(Z)$ . Since $\partial (f_i) \in I$ we get that

$$\begin{align*}\tilde{\partial}(p^{*}f_i) = p^{*}\partial f_i \in p^{-1}I\mathcal O_Y \end{align*}$$

and so $p^{-1}I\mathcal O_Y$ is invariant under $\tilde {\partial }$ , as required.

2.4 Foliation singularities

Let X be a normal variety and let $({\mathcal F},\Delta )$ be a foliated pair on X.

Given a birational morphism $\pi \colon \widetilde {X} \rightarrow X$ , let $\widetilde {\mathcal F}$ be the pulled back foliation on $\tilde {X}$ and let $\Delta '$ be the strict transform of $\Delta $ in $\widetilde X$ . We may write

$$\begin{align*}K_{\widetilde{\mathcal F}}+\Delta'= \pi^{*}(K_{\mathcal F}+\Delta)+ \sum a(E, {\mathcal F}, \Delta)E\end{align*}$$

where the sum runs over all the prime $\pi $ -exceptional divisors of $\tilde X$ .

The rational number $a(E,{\mathcal F},\Delta )$ denotes the discrepancy of $({\mathcal F},\Delta )$ with respect to E. If $\Delta =0$ , then we will simply denote $a(E,{\mathcal F})=a(E,{\mathcal F},0)$ .

Note that these notions are well defined, that is, $\epsilon (E)$ and $a(E, {\mathcal F}, \Delta )$ are independent of $\pi $ . Observe also that in the case where ${\mathcal F} = T_X$ , no exceptional divisor E over X is invariant, that is, $\epsilon (E)=1$ for all E, and so this definition recovers the usual definitions of (log) terminal and (log) canonical.

Let $P \in X$ be a, not necessarily closed, point of X. We say that $({\mathcal F}, \Delta )$ is terminal (resp. canonical, log canonical) at $\boldsymbol {P}$ if for any birational morphism $\pi \colon \widetilde X\to X$ and for any $\pi $ -exceptional divisor E on $\widetilde X$ whose centre in X is the Zariski closure $\overline P$ of P, we have that the discrepancy of E is $>0$ (resp. $\geq 0$ , $\geq -\epsilon (E)$ ). Sometimes we will phrase this as P is a terminal (resp. canonical, log canonical) point for $({\mathcal F}, \Delta )$ . We say that ${\mathcal F}$ is terminal near $P \in X$ if there is a neighborhood U of P such that ${\mathcal F}\vert _U$ is terminal. We will see (cf. Lemma 2.9) that being terminal at a closed point P is equivalent to ${\mathcal F}$ being smooth at P.

Given an irreducible subvariety $W \subset X$ , we say that $({\mathcal F},\Delta )$ is terminal at the generic point of W if $({\mathcal F},\Delta )$ is terminal at the generic point $\eta _W$ of W. We say that $({\mathcal F},\Delta )$ is terminal at a general point of W if $({\mathcal F},\Delta )$ is terminal at a general closed point of W.

Note that if W is a canonical centre of $({\mathcal F}, \Delta )$ , then $({\mathcal F}, \Delta )$ is not terminal at the generic point of W. We also remark that if ${\mathcal F}$ is smooth and $C\subset X$ is an ${\mathcal F}$ -invariant curve then ${\mathcal F}$ is terminal at a general point of C, but is not terminal at the generic point of C.

Given a normal variety X and a foliation ${\mathcal F}$ of rank one on X, we say that ${\mathcal F}$ has dicritical singularities if there exists a birational morphism $\pi \colon X'\to X$ and a $\pi $ -exceptional divisor E which is not $\pi ^{-1}{\mathcal F}$ -invariant. We say that ${\mathcal F}$ is nondicritical, if it is not dicritical.

Note that if ${\mathcal F}$ is a nondicritical foliation then the notions of log canonical and canonical coincide. In this case we might still refer to canonical centres as log canonical centres. We also remark that any ${\mathcal F}$ -invariant divisor is an lc centre and a canonical centre of $({\mathcal F}, \Delta )$ .

We will make frequent use of the following consequence of the negativity lemma:

Lemma 2.7. Let $\phi \colon X\dashrightarrow X'$ be a birational map between normal varieties and let

be a commutative diagram, where Y is a normal variety and f and $f'$ are proper birational morphisms. Let $({\mathcal F},\Delta )$ be a foliated pair on X. Let ${\mathcal F}'=\phi _{*}{\mathcal F}$ and let $({\mathcal F}',\Delta ')$ be a foliated pair on $X'$ such that $f_{*}\Delta =f^{\prime }_{*}\Delta '$ . Assume that $-(K_{\mathcal F}+\Delta )$ is f-ample and $K_{{\mathcal F}'}+\Delta '$ is $f'$ -ample.

Then, for any valuation E on X, we have

$$\begin{align*}a(E,{\mathcal F},\Delta)\le a(E,{\mathcal F}',\Delta'). \end{align*}$$

Moreover, the strict inequality holds if f or $f'$ is not an isomorphism above the generic point of the centre of E in Y.

The following is essentially [Reference McQuillan and PanazzoloMP13, Corollary III.i.5]:

Proof. We follow the same methods as [Reference Kollár and MoriKM98, Proposition 5.20]. Let $\overline f\colon \overline Y\to \overline X$ be a proper birational morphism and let $\overline E$ be an $\overline f$ -exceptional divisor on $\overline Y$ whose centre in $\overline X$ is $\overline Z$ . Then, by [Reference KollárKol96, Theorem VI.1.3], after possibly replacing $\overline {Y}$ by an higher model, we may assume that there exists a commutative diagram

where $ f$ is birational and p is finite. In particular, if $E=p(\overline {E})$ then E is f-exceptional and Z is the centre of E in X.

Assume now that $f\colon Y\to X$ is a proper birational morphism and let E be an f-exceptional divisor on Y whose centre in X is Z. Let $\overline Y$ be a component of the normalisation of $\overline X\times _X Y$ which maps onto Y and let $\overline {f}\colon \overline Y\to \overline X$ and $p\colon \overline Y\to Y$ be the induced morphisms. Let $\overline E$ be a prime divisor such that $p(\overline E)=E$ .

Lemma 2.3 easily implies that $\epsilon (E)=\epsilon (\overline E)$ . Let $r_E$ be the ramification index of p along E. Then, as in the proof of [Reference Kollár and MoriKM98, Proposition 5.20], Proposition 2.2 implies that

$$\begin{align*}a(\overline E,\overline{\mathcal F},\overline {\Delta })=r_Ea(E,{\mathcal F},\Delta)+\epsilon(E)(r_E-1). \end{align*}$$

It follows easily that $a(\overline E,\overline {\mathcal F},\overline {\Delta })> -\epsilon (\overline E)$ if and only if $ a(E,{\mathcal F},\Delta )>-\epsilon (E). $ Thus, the first claim follows.

Note that if q is a quasi-étale morphism and $\Delta =0$ then $\overline \Delta =0$ . Lemma 2.6 implies that if ${\mathcal F}$ (resp. $\overline {\mathcal F}$ ) is canonical, then $\epsilon (E)=0$ (resp. $\epsilon (\overline E)=0$ ). Thus, the second claim follows using the same arguments as above.

Let $f\colon X\to Y$ be a holomorphic morphism between analytic varieties. We say that f is a submersion if, for any point $x\in X$ , it induces a surjective morphism $df_x:T_xX\to T_{f(x)}Y$ .

Note that the above lemma implies the well-known fact that if X is a surface and ${\mathcal F}$ is a terminal rank one foliation on X then X has at worst quotient singularities. One can ask more generally if there is a similar way to control the singularities of the underlying variety in higher dimensions and higher ranks, and if such a bound holds if ${\mathcal F}$ has only canonical singularities. For foliations of co-rank one on a normal threefold, some of these questions were addressed in [Reference Cascini and SpicerCS21]. We will approach some cases of this problem in the rank one case in dimension three (cf. Section 4).

We remark that if ${\mathcal F}$ is log canonical then there is no bound on the singularities of the underlying variety, at least from the perspective of Mori theory, as the example in [Reference McQuillanMcQ08, Example I.2.5] shows.

We also remark that by Lemma 2.1 if ${\mathcal F}$ is a rank one foliation on a normal variety X such that ${\mathcal F}$ is terminal at a closed point $P\in X$ then $P \notin {\operatorname {Sing} {\mathcal F}}$ .

2.5 Foliations on a surface

The goal of this section is to present some results for foliations on a surface which will be used later on. To this end, we employ Mumford’s intersection theory for Weil divisors on a normal projective surface (e.g., see [Reference FultonFul84, Example 8.3.11]).

Lemma 2.12. Let X be a normal projective surface and let ${\mathcal F}$ be a rank one foliation on X which is algebraically integrable. Let $\Delta , \Theta \geq 0$ be ${\mathbb Q}$ -divisors on X such that

1. 1. $\mu _C \Theta \leq \mu _C\Delta $ for any curve C which is not ${\mathcal F}$ -invariant, and

2. 2. $(X, \Theta )$ is log canonical.

Then X is covered by ${\mathcal F}$ -invariant curves M such that

$$\begin{align*}(K_X+\Theta)\cdot M \leq (K_{\mathcal F}+\Delta)\cdot M. \end{align*}$$

Proof. We may assume, without loss of generality, that the coefficients of $\Delta $ are at most one. Let $p\colon X' \rightarrow X$ be an F-dlt modification of $({\mathcal F},\Delta )$ (cf. [Reference Cascini and SpicerCS21, Theorem 1.4]). Then we may write $K_{{\mathcal F}'}+p_{*}^{-1}\Delta +E = p^{*}(K_{\mathcal F}+\Delta )$ and $K_{X'}+p_{*}^{-1}\Theta +E' = p^{*}(K_X+\Theta )$ , where $E, E'$ are p-exceptional ${\mathbb Q}$ -divisors and the coefficients of E (resp. $E'$ ) are greater or equal (resp. less or equal) to one. Since ${\mathcal F}'$ is algebraically integrable and nondicritical, it follows that ${\mathcal F}'$ is induced by a fibration $\pi \colon X' \rightarrow B$ . Let F be a general fibre of $\pi $ and observe that

1. (i) $K_{{\mathcal F}'}\cdot F = K_{X'}\cdot F$ ,

2. (ii) $p_{*}^{-1}\Theta \cdot F \leq p_{*}^{-1}\Delta \cdot F$ , and

3. (iii) $E-E' \ge 0$ .

Thus, if $M=p(F)$ then

$$\begin{align*}\begin{aligned} (K_X+\Theta)\cdot M &=(K_{X'}+p_{*}^{-1}\Theta+E')\cdot F\\ &\le (K_{{\mathcal F}'}+p_{*}^{-1}\Delta+E)\cdot F = (K_{\mathcal F}+\Delta)\cdot M \end{aligned} \end{align*}$$

and the claim follows.

2.6 Adjunction

We now explain some generalities comparing foliation adjunction and classical adjunction on a threefold:

Proposition 2.14. Let X be a normal threefold and let ${\mathcal F}$ be a foliation of rank one on X with canonical singularities. Let $\Gamma \ge 0$ be a ${\mathbb Q}$ -divisor on X with ${\mathcal F}$ -invariant support and let $S\subset X$ be a reduced and irreducible ${\mathcal F}$ -invariant divisor such that $(X,\Gamma +S)$ is log canonical. Let $\nu \colon S^\nu \to S$ be its normalisation.

We may write

$$\begin{align*}K_{\mathcal F}|_{S^\nu} = K_{\mathcal G}+\Delta \quad\text{and}\quad (K_X+\Gamma+S)|_{S^\nu} = K_{S^\nu}+\Theta \end{align*}$$

where $\mathcal G$ is the induced foliation and $\Delta , \Theta \geq 0$ are ${\mathbb Q}$ -divisors on $S^{\nu }$ . Let $C\subset S^{\nu }$ be a curve.

Then the following hold:

1. 1. if $\nu (C)$ is contained in ${\operatorname {Sing} {\mathcal F}}$ then $\mu _C\Delta \geq 1$ and, in particular, $\mu _C\Delta \geq \mu _C\Theta $ ;

2. 2. if $\nu (C)$ is not contained in ${\operatorname {Sing} {\mathcal F}}$ and C is not $\mathcal G$ -invariant (i.e., ${\mathcal F}$ is terminal at the generic point of $\nu (C)$ ), then $\mu _C\Delta = \mu _C\Theta = \frac {n-1}{n}$ where n is the Cartier index of $K_{{\mathcal F}}$ at the generic point of C.

Note that, in the notations above, if C is $\mathcal G$ -invariant then there is in general no natural relation between $\mu _C\Delta $ and $\mu _C\Theta $ , as shown in the following example:

2.7 Jordan decomposition of a vector field

We follow the notation of [Reference MartinetMar81]. Let be the completion of ${\mathbb C}^m$ at the origin $0\in X$ and let $\partial $ be a vector field on X which leaves invariant. Let ${\mathfrak m}$ be the maximal ideal defining W and note that, by the Leibniz rule, the ideal ${\mathfrak m}^n$ is $\partial $ -invariant for all positive integer n. Thus, we get a linear map

$$\begin{align*}\partial_n\colon {\mathfrak m}/{\mathfrak m}^{n+1} \rightarrow {\mathfrak m}/{\mathfrak m}^{n+1}. \end{align*}$$

We may write $\partial _n = \partial _{S, n} +\partial _{N, n}$ as the Jordan decomposition of $\partial _n$ into its semisimple and nilpotent parts. This decomposition respects the exact sequences

$$\begin{align*}0 \rightarrow {\mathfrak m}^n/{\mathfrak m}^{n+1} \rightarrow {\mathbb C}[[X]]/{\mathfrak m}^{n+1} \rightarrow {\mathbb C}[[X]]/{\mathfrak m}^n \rightarrow 0\end{align*}$$

for each positive integer n and it yields a decomposition $\partial = \partial _S+\partial _N$ .

We summarise briefly some of the key properties of this decomposition:

1. 1. $[\partial _S, \partial _N] = 0$ ;

2. 2. we may find coordinates $y_1, ..., y_m$ on $\widehat {{\mathbb C}^m}$ and $\lambda _1,\dots ,\lambda _m\in {\mathbb C}$ so that $\partial _S = \sum _i \lambda _i y_i \partial _{y_i}$ ; and

3. 3. if $Z \subset \widehat {{\mathbb C}^m}$ is $\partial $ -invariant then Z is both $\partial _S$ and $\partial _N$ -invariant.

We briefly explain (3). Let $I_Z\subset {\mathbb C}[[X]]$ be the ideal of Z and let $I_{Z, n}$ denote its restriction to ${\mathfrak m}/{\mathfrak m}^{n+1}$ , for each positive integer n. Then $I_{Z, n} \subset {\mathfrak m}/{\mathfrak m}^{n+1}$ is a $\partial _n$ -invariant subspace and, in particular, it is both $\partial _{S,n}$ and $\partial _{N,n}$ -invariant. Thus, (3) follows.

More generally, we can define the Jordan decomposition for any vector field $\partial $ on the completion of a variety X at a point $P\in X$ . Indeed, consider an embedding $\iota \colon Z \hookrightarrow {\mathbb C}^m$ and a lift $\widetilde {\partial }$ of $\partial $ to a vector field on ${\mathbb C}^m$ . We can define $\widetilde {\partial }_S$ and $\widetilde {\partial }_N$ as above. Then $\widetilde {\partial }_S$ and $\widetilde {\partial }_N$ leave Z invariant and, therefore, they restrict to vector fields $\partial _S$ and $\partial _N$ on Z. Thus, $\partial = \partial _S+\partial _N$ and this decomposition has all the properties of the Jordan decomposition, as described above.

2.8 Characterising log canonical vector fields

Let X be a normal variety and let $\partial $ be a vector field which defines a foliated pair $({\mathcal F},D)$ such that $K_{\mathcal F}+D$ is Cartier. Then we say that $\partial $ is terminal (resp. canonical, log canonical) if the foliated pair $({\mathcal F},D)$ is such.

Let $P \in Z$ be a germ of a normal variety and let $\partial \in H^0(Z, T_Z)$ be a vector field which leaves P invariant. By Lemma 2.9, $\partial $ is singular at P. Let where ${\mathfrak m}$ is the maximal ideal at P and observe that $\partial $ induces a linear map $\partial _0\colon V \rightarrow V$ . Let ${\mathcal F}$ be the foliation defined by $\partial $ so that $\partial $ is a section of ${\mathcal F}(-D)$ for some divisor $D \geq 0$ . We assume that D is reduced.

We recall the following results:

We will also need the following:

Proof. We may freely replace $\partial $ by its semisimple part, and so we may assume that $\partial $ is semisimple. In suitable coordinates and after possibly rescaling by a unit, we may write

$$\begin{align*}\partial = -x_1\frac{\partial}{\partial x_1}+a_2x_2\frac{\partial}{\partial x_2}+ a_3x_3\frac{\partial}{\partial x_3}\end{align*}$$

and $C = \{x_2 = x_3 = 0\}$

Fix $i\in \{1,\dots ,q\}$ . By (2), it follows that $f_i \in (x_2, x_3)$ , and we may write

$$\begin{align*}f_i = \sum_{k,l,m\ge 0} a_{klm}^ix_1^kx_2^lx_3^m\end{align*}$$

for some $a_{klm}^i\in {\mathbb C}$ such that $a^i_{k00} = 0$ for all $k\ge 0$ . We have

$$\begin{align*}\partial f_i = \sum a^i_{klm}(-k+a_2l+a_3m)x_1^kx_2^lx_3^m. \end{align*}$$

Thus, (1) implies that

$$\begin{align*}\lambda_i = -k+a_2l+a_3m \end{align*}$$

for all non-negative integers $k,l,m$ such that $a^i_{klm}\neq 0$ .

If $a^i_{kl0}$ (resp. $a^i_{k0m}$ ) is nonzero for some $i, k, l$ (resp. $i,k,m$ ) it follows immediately that $a_2$ (resp. $a_3$ ) is a positive rational number.

Assume that $a^i_{k0m} = 0$ for all $i,k, m$ . Then it follows that

$$\begin{align*}\{x_2 = 0\} \subset \{f_1 = ... = f_k = 0\} \end{align*}$$

contradicting the fact that $\{x_2 = x_3 = 0\}$ is an irreducible component of the latter scheme. A similar contradiction holds if $a^i_{km0} = 0$ for all $i, k, m$ .

2.9 Canonical bundle formula

We recall some results on the canonical bundle formula which will be used later (see [Reference AmbroAmb04] for more details).

Let $(X,\Delta )$ be a sub log canonical pair and let $f\colon X\to Y$ be a fibration. Assume that the horizontal part $\Delta ^h$ of $\Delta $ is effective and that there exists a ${\mathbb Q}$ -Cartier ${\mathbb Q}$ -divisor D on Y such that

$$ \begin{align*}K_X+\Delta\sim_{{\mathbb Q}} f^{*}D.\end{align*} $$

If P is a prime divisor on Y, we denote by $\eta _P$ its generic point and we define the log canonical threshold of $f^{*}P$ with respect to $(X,\Delta )$ to be

Let . Then we define the discriminant of f with respect to $\Delta $ as , where the sum runs over all the prime divisors P in Y. Let r be the smallest positive integer such that there exists a rational function $\phi $ on X satisfying

$$\begin{align*}K_X+\Delta+\frac 1 r (\phi)=f^{*}D. \end{align*}$$

Then there exists a ${\mathbb Q}$ -divisor $M_Y$ such that

$$ \begin{align*}K_X+\Delta+\frac 1 r(\phi)=f^{*}(K_Y+B_Y+ M_Y).\end{align*} $$

$M_Y$ is called the moduli part of f with respect to $\Delta $ .

Lemma 2.19. Let $(X,\Delta )$ be a two dimensional log canonical pair, let $f\colon X\to Y$ be a fibration onto a curve Y and let D be a ${\mathbb Q}$ -divisor on Y such that $K_X+\Delta \sim _{{\mathbb Q}}f^{*}D$ . Let $y\in Y$ be a closed point and assume that there exists an open neighbourhood U of y such that, if we denote

then $(X_U,\Delta |_{X_U})$ is log smooth and there exists an isomorphism

$$\begin{align*}\phi_u\colon X_u \to X_y\qquad\text{such that} \qquad \phi^{*}_u(\Delta|_{X_y})=\Delta|_{X_u} \qquad\text{for all }u\in U. \end{align*}$$

Then the moduli part of f with respect to $\Delta $ is trivial, that is, $M_Y\sim _{{\mathbb Q}}0$ .

2.10 A recollection on approximation theorems

We recall some approximation results proven in [Reference Cascini and SpicerCS21, Section 4].

We consider the following set up. Let $\tilde {X} = {\text {Spec}\, {\tilde {A}}}$ be an affine variety where $\tilde {A}$ is a henselian local ring with maximal ideal ${\mathfrak m}$ and let $W \subset \tilde {X}$ be a closed subscheme, defined by an ideal $\tilde {I}\subset \tilde A$ . Let where $\widehat {A}$ is the completion of $\tilde {A}$ along $\tilde {I}$ and let $\widehat {D}$ be a divisor on $\widehat {X}$ . Equivalently, $\widehat {D}$ is given by a reflexive sheaf $\widehat {M}$ on $\widehat {X}$ and a choice of a section $\hat {s} \in \widehat {M}$ .

The following is a slight generalisation of Artin-Elkik approximation theorem:

Theorem 2.20. Set up as above. Let m be a positive integer such that $m\widehat {D}$ is Cartier on $\widehat {X} \setminus W$ .

Then, for all positive integer n, there exists a divisor $D^n$ on $\tilde {X}$ such that

$$\begin{align*}D^n = \widehat{D}\quad \mod \tilde{I}^n\qquad\text{and}\qquad \mathcal O_{\tilde{X}}(mD^n) \otimes \widehat{A} \cong \mathcal O_{\tilde{X}}(m\widehat{D}). \end{align*}$$

We will use this theorem under the following additional constraints. Let $X = {\text {Spec}\, A}$ be an affine variety and let $P \in X$ be closed point and suppose $\tilde {A}$ is the henselisation of A at P.

In our applications here we will always take $W = P$ and so the additional hypotheses of the corollary are always satisfied.

We also recall the following:

2.11 Resolution of singularities of threefold vector fields

We recall the following example from [Reference McQuillan and PanazzoloMP13].

Example 2.23. [Reference McQuillan and PanazzoloMP13, Example III.iii.3] Consider the ${\mathbb Z}/2{\mathbb Z}$ -action on ${\mathbb C}^3$ given by $(x, y, z) \mapsto (y, x, -z)$ . Let X denote the quotient of ${\mathbb C}^3$ by this action.

Consider the vector field on ${\mathbb C}^3$ given by

$$\begin{align*}\partial_S := x \frac{\partial}{\partial x} - y \frac{\partial}{\partial y}\end{align*}$$

and

$$\begin{align*}\partial_N := a(xy, z)x\frac{\partial}{\partial x} - a(xy, -z)y\frac{\partial}{\partial y} + c(xy, z)\frac{\partial}{\partial z}\end{align*}$$

where $a,c$ are formal functions in two variables such that c is not a unit and it satisfies $c(xy, z) = c(xy, -z)$ . Let $\partial := \partial _S+\partial _N$ . Note that $\partial \mapsto -\partial $ under the group action. Thus, $\partial $ induces a foliation ${\mathcal F}$ on X with an isolated canonical singularity and such that $2K_{\mathcal F}$ is Cartier, but $K_{\mathcal F}$ is not Cartier.

By [Reference McQuillan and PanazzoloMP13, Possibility III.iii.3.bis], there does not exist a birational morphism $f\colon Y\to X$ such that the induced foliation $f^{-1}{\mathcal F}$ is both Gorenstein and canonical. Moreover, by [Reference McQuillan and PanazzoloMP13, III.iii.3.bis], we also have that the curve $\{x= y = 0\}$ is not algebraic, nor analytically convergent.

2.12 Nakamaye’s theorem and the structure of extremal rays

Let X be a normal projective variety and let M be a ${\mathbb Q}$ -Cartier divisor on X. We define the exceptional locus of M to be

where the union runs over all the subvarieties $V\subset X$ of positive dimension such that $M|_V$ is not big. We denote by ${\mathbb B}(M)$ the stable base locus of M,

where the intersection runs over all the sufficiently divisible positive integers m. Finally, given a ray R in the cone of curves $\overline {\mathrm {NE}}(X)$ , we define the locus of $\boldsymbol {R}$ to be the subset

$$\begin{align*}{\operatorname{loc} \, R}:=\bigcup_{[C]\in R}C. \end{align*}$$

We recall the following result originally due to Nakamaye, in the case of smooth varieties.

Proof. Let A be an ample divisor. Consider the rational map $\phi \colon X \dashrightarrow {\mathbb P}^N$ defined by the linear system $|m(M-\epsilon A)|$ where $\epsilon>0$ is a sufficiently small rational number and m is a sufficiently divisible and large positive integer and note that $\phi $ is birational onto the closure of its image $Y\subset {\mathbb P}^N$ . Let $p\colon \overline X\to X$ and $q\colon \overline X\to Y$ be birational morphisms which resolve the indeterminancy locus of $\phi $ .

By Lemma 2.28, it follows that $p({\operatorname {Exc} \, q}) = W$ . We may write

$$\begin{align*}p^{*}(m(M-\epsilon A)) = H +F \end{align*}$$

where $F \geq 0$ is q-exceptional and $H=q^{*}L$ for some very ample Cartier divisor L on Y. Since X is ${\mathbb Q}$ -factorial we may choose $G \geq 0$ to be p-exceptional so that $-G$ is p-ample. Choose $\delta>0$ sufficiently small so that is ample.

We therefore have $F+\delta G = p^{*}(mM) - A'-H$ and $F+\delta G$ is a ${\mathbb Q}$ -Cartier q-exceptional divisor. Since $p({\operatorname {Exc} \, q}) = W$ , it follows that $p^{*}M$ restricted to ${\operatorname {Exc} \, q}$ is numerically trivial. Thus, if k is a sufficiently divisible positive integer so that $k(F+\delta G)$ is a Cartier divisor, then

$$\begin{align*}-k(F+\delta G)\vert_{k(F+\delta G)} \equiv k(A'+H)\vert_{k(F+\delta G)}. \end{align*}$$

Since ampleness of a line bundle on a scheme is equivalent to ampleness of the line bundle restricted to the reduction and normalisation, and since $A'+H$ restricted to the reduction and normalisation of each component of ${\operatorname {Exc} \, q}$ is ample, we see that $-k(F+\delta G)\vert _{k(F+\delta G)}$ is ample.

We may therefore apply Artin’s Theorem [Reference ArtinArt70, Theorem 6.2] to produce a morphism of algebraic spaces $\overline {X} \rightarrow Z$ which contracts $F+\delta G$ to a point. By the rigidity lemma this contraction factors through $\overline {X} \rightarrow X$ giving our desired birational contraction $\phi \colon X \rightarrow Z$ .

2.13 Cone theorem

The cone theorem for rank one foliations was initially proven in [Reference Bogomolov and McQuillanBM16, Corollary IV.2.1] when ${\mathcal F}$ is Gorenstein and in [Reference McQuillanMcQ04] when ${\mathcal F}$ is ${\mathbb Q}$ -Gorenstein. A more general version is proven in [Reference Cascini and SpicerCS25a], which we recall here.

Theorem 2.30. Let X be a normal projective ${\mathbb Q}$ -factorial variety and let $({\mathcal F}, \Delta )$ be a rank one foliated pair on X.

Then there are ${\mathcal F}$ -invariant rational curves $C_1,C_2,\dots $ not contained in ${\operatorname {Sing}{\mathcal F}} $ such that

$$\begin{align*}0<-(K_{\mathcal F}+\Delta)\cdot C_i\le 2\dim X\end{align*}$$

and

$$ \begin{align*}\overline{\mathrm{NE}}(X)=\overline{\mathrm{NE}}(X)_{K_{\mathcal F}+\Delta\ge 0}+Z_{-\infty}+ \sum_i \mathbb R_+[C_i]\end{align*} $$

where $Z_{-\infty }\subset \overline {\mathrm {NE}}(X)$ is a subset contained in the span of the images of $\overline {\mathrm {NE}}(W) \rightarrow \overline {\mathrm {NE}}(X)$ where $W \subset X$ are the non-log canonical centres of $({\mathcal F}, \Delta )$ .

2.14 A remark on the different notions of singularity

The following proposition is not needed in this paper, but we believe it is of independent interest as it clarifies the relation between different notions of foliation singularities appearing in the existing literature.

Proof. By Lemma 2.1 we have the inclusion ${\operatorname {Sing} {\mathcal F}} \subset {\operatorname {Sing}^{+}{\mathcal F}} $ , so suppose for the sake of contradiction that there exists a closed point $x \in {\operatorname {Sing}^{+}{\mathcal F}} \setminus {\operatorname {Sing} {\mathcal F}}$ . We may freely replace X by a neighbourhood of $x \in X$ and we may also freely replace X be the index one cover associated to $K_{{\mathcal F}}$ . Thus, we may assume that ${\mathcal F}$ is defined by a vector field $\partial $ . Since $x \not \in {\operatorname {Sing} {\mathcal F}}$ the morphism $\Omega ^{[1]}_X \rightarrow \mathcal O_X$ induced by pairing with $\partial $ is surjective, and so there exists a section $\omega \in \Omega ^{[1]}_X$ such that $\partial (\omega ) =1$ . Let $p\colon X' \rightarrow X$ be a functorial resolution of X, cf. [Reference KollárKol07b, Theorems 3.35 and 3.45]. By [Reference Greb, Kebekus and KovácsGKK10, Corollary 4.7] there exists a vector field $\partial '$ on $X'$ such that $p_{*}\partial ' = \partial $ . Since X is klt, [Reference Greb, Kebekus, Kovács and PeternellGKKP11, Theorem 1.4] implies that is a holomorphic 1-form on $X'$ . Note that we still have $\partial '(\omega ') = 1$ , in particular, $\partial '$ defines a smooth foliation ${\mathcal F}'$ on $X'$ .

Since $x \in {\operatorname {Sing}^{+}{\mathcal F}} $ , it follows that x is invariant by $\partial $ , and so $p^{-1}(x)$ is invariant by $\partial '$ . Perhaps passing to a higher functorial resolution we may assume that $p^{-1}(x)$ is a divisor and that there exists an exceptional Cartier divisor G such that $-G$ is p-ample. Since G is supported on p-exceptional divisors and the p-exceptional locus is ${\mathcal F}'$ -invariant we have a partial connection $\nabla \colon \mathcal O_{X'}(G) \rightarrow \mathcal O_{X'}(G) \otimes \mathcal O_{X'}(K_{{\mathcal F}'})$ . Let E be an irreducible component of $p^{-1}(x)$ , and let ${\mathcal F}^{\prime }_E$ be the restricted foliation. We may restrict the partial connection $\nabla $ to a partial connection

$$\begin{align*}\nabla_E \colon \mathcal O_E(G\vert_E) \rightarrow \mathcal O_E(G\vert_E)\otimes \mathcal O_E(K_{{\mathcal F}^{\prime}_E}). \end{align*}$$

Since ${\mathcal F}^{\prime }_E$ is smooth, we may apply Bott vanishing to conclude that $G\vert _E^{\dim E} \equiv 0$ , cf. [Reference Carrell and LiebermanCL77, Proposition 5.1], which contradicts the fact that $-G\vert _E$ is ample.

In light of this Proposition we ask the following:

3.1 A version of Reeb stability

Our goal is to generalise Reeb stability theorem to foliations defined over singular varieties.

More specifically, let X be a normal variety and let ${\mathcal F}$ be a rank one foliation on X which is terminal at all closed points. Let $C \subset X$ be a compact ${\mathcal F}$ -invariant curve and let $\Sigma \subset {\operatorname {Sing} X }$ be the locus where ${\mathcal F}$ is not Gorenstein. By definition of invariance, the set $\{c_1, ..., c_N\}=C\cap \Sigma $ is finite. Let $C^\circ = C \setminus \{c_1, ..., c_N\}$ and let $n_k$ be the Cartier index of $K_{\mathcal F}$ at $c_k$ for each $k=1,\dots ,N$ . We now define the holonomy of ${\mathcal F}$ along $C^\circ $ .

Since C is compact, by Lemma 2.9, we may find open sets $U_1,\dots ,U_\ell $ in X such that C is contained in the union $\cup U_i$ and for each $i=1,\dots ,\ell $ , there exists a finite morphism $q_i\colon V_i\to U_i$ and a fibration $f_i\colon V_i\to T_i$ such that is the foliation induced by $f_i$ , $q_i$ is unramified outside $\Sigma $ , and if $c_k\in U_i$ for some $k=1,\dots ,N$ then the ramification index of $q_i$ at $c_k$ is $n_k$ . In particular, the preimage of the curve C in $V_i$ is mapped to a point $z_i\in T_i$ .

Pick distinct $i,j$ such that $U_{i,j}:=U_i\cap U_j$ is not empty and it intersects C. After possibly shrinking $U_i$ or $U_j$ , we may assume that $U_{i,j}$ does not intersect $\Sigma $ . Let $V^j_i:=q_i^{-1}(U_{i,j})$ and let $V_{i,j}=V_i^j\times _{U_{i,j}} V_j^i$ . Note that the induced morphism $q_{i,j}\colon V_{i,j}\to U_{i,j}$ is unramified and there exists a morphism $f_{i,j}\colon V_{i,j}\to T_{i,j}$ such the pulled back foliation ${\mathcal F}_{i,j}$ on $V_{i,j}$ is induced by $f_{i,j}$ . Indeed, $f_{i,j}$ is the Stein factorisation of the morphism $V_{i,j}\to T_i$ . Let $\sigma _{i,j}\colon T_{i,j}\to T_i$ be the induced morphism. Note that the preimage of C in $V_{i,j}$ is mapped to a point $z_{i,j}\in T_{i,j}$ such that $\sigma _{i,j}(z_{i,j})=z_i$ . After possibly shrinking $U_i$ and $U_j$ , we may assume that $\sigma _{i,j}$ is surjective. It follows that $\sigma _{i,j}$ is étale. Thus, after replacing $V_i$ by $V_{i}\times _{T_{i}}T_{i,j}$ we may assume that $T_i=T_j$ . After repeating this process, finitely many times, we may assume that and do not depend on $i=1,\dots ,k$ . Note that, by the construction above, the choice of the germ $(T,z)$ is uniquely determined by ${\mathcal F}$ and C.

Pick $c\in C^\circ $ . Let $\gamma _1,\dots ,\gamma _N$ be loops based at c around $c_1,\dots ,c_N$ , respectively. The orbifold fundamental group $\pi (C^\circ ,c;n_1,\dots ,n_N)$ of $C^\circ $ with weight $n_k$ at $c_k$ is defined as the quotient of $\pi (C^\circ ,c)$ by the normal subgroup generated by $\gamma _1^{n_1},\dots ,\gamma _N^{n_N}$ . We now want to define the holonomy map

$$ \begin{align*}\rho\colon \pi(C^\circ,c;n_1,\dots,n_N)\to \operatorname{\mathrm{Aut}}(T,z),\end{align*} $$

where $\operatorname {\mathrm {Aut}}(T,z)$ denotes the group of biholomorphic automorphisms on the germ $(T,z)$ . Let $\gamma \colon [0,1]\to C^\circ $ be a continuous path which is contained in $U_i$ for some $i=1,\dots ,\ell $ . Then, since $q_i\colon V_i\to U_i$ is unramified outside $\Sigma $ , there exists a lifting $\tilde \gamma \colon [0,1]\to V_i$ of $\gamma $ in $V_i$ . Note that $f_i$ maps the image of $\tilde \gamma $ to the point $z\in T$ . Proceeding as in the construction of the classic holonomy map (e.g., see [Reference Camacho and NetoCN85]), we can define a homomorphism

$$ \begin{align*}\rho'\colon \pi(C^0,c)\to \operatorname{\mathrm{Aut}} (T,z).\end{align*} $$

Note that if $c_k\in U_i$ for some i and k, then the ramification index of $q_i$ at any point in $q_i^{-1}(c_k)$ is equal to $n_k$ . Thus, it follows that $\rho '(\gamma _k^{n_k})$ is the identity automorphism of T for any $k=1,\dots ,N$ and, in particular, the holonomy map

$$ \begin{align*}\rho\colon \pi(C^\circ,c;n_1,\dots,n_N)\to \operatorname{\mathrm{Aut}}(T,z)\end{align*} $$

is well defined.

We are now ready to state our singular version of Reeb stability theorem:

As a direct application of Reeb stability theorem, we get the following result (see also [Reference McQuillanMcQ04, II.d.5]):

Proof. By definition of invariance, ${\mathcal F}$ is Gorenstein at the generic point of C. Let $c_1, ..., c_N\in C$ be the non-Gorenstein points of ${\mathcal F}$ and let $n_k$ denote the Cartier index of $K_{\mathcal F}$ at $c_k$ , for $k=1,\dots ,N$ . Let $C^\circ =C\setminus \{c_1,\dots ,c_N\}$ .

It follows from foliation adjunction (cf. Proposition 2.13), that C is a rational curve and

$$ \begin{align*}K_{\mathcal F}\cdot C = -2+\sum_{k=1}^N \frac{n_k-1}{n_k}.\end{align*} $$

In particular, since $K_{\mathcal F}\cdot C<0$ it follows that the orbifold fundamental group $\pi _1(C^\circ ,c;n_1,\dots ,n_N)$ is finite. Thus, Theorem 3.2 implies the claim.

4.1 Dlt modification

Let X be a normal threefold singularity and let ${\mathcal F}$ be a rank one foliation on X with canonical singularities. Let $S_1,\dots ,S_k$ be prime ${\mathcal F}$ -invariant divisors. Our goal here is to control the singularities of the pair , where $a_1,\dots ,a_k\in (0,1]\cap {\mathbb Q}$ , in terms of the singularities of ${\mathcal F}$ . As the following example shows, a canonical foliation singularity will in general have worse than quotient singularities on the ambient variety (in contrast to the surface case):

Proof. Observe that our hypotheses are preserved by shrinking X and by taking quasi-étale covers. Thus, we may assume without loss of generality that $K_{\mathcal F}$ is Cartier.

Suppose for the sake of contradiction that $(X, \Gamma )$ has a worse than log canonical singularity at 0. We may find $0<\lambda <1$ , sufficiently close to $1$ so that $(X, \lambda \Gamma )$ is not log canonical at $0$ and is klt away from $0$ . Thus, after replacing $\Gamma $ by $\lambda \Gamma $ , we may assume that $(X,\Gamma )$ is klt away from $0$ .

Let $\mu \colon \overline {X} \rightarrow X$ be a dlt modification of $(X, \Gamma )$ with respect to ${\mathcal F}$ , whose existence is guaranteed by Lemma 4.2. Let . Then, since ${\mathcal F}$ is Gorenstein, we have that $K_{\overline {\mathcal F}}=\mu ^{*}K_{\mathcal F}$ and since $(X,\Gamma )$ is klt away from $0$ , we have that every $\mu $ -exceptional divisor is centred in $0$ . Let $E=\sum _{i=1}^q E_i$ be the sum of the $\mu $ -exceptional divisors and let $\overline {\Gamma }$ be the strict transform of $\Gamma $ in $\overline X$ . Lemma 2.6 implies that E is $\overline {\mathcal F}$ -invariant.

By classical adjunction and by Proposition 2.13, for each $i=1,\dots ,q$ , we may write

$$ \begin{align*}(K_{\overline X}+\overline{\Gamma}+E)|_{E_i}=K_{E_i}+\Theta_i\qquad\text{and}\qquad K_{\overline{\mathcal F}}\vert_{E_i} = K_{\mathcal G_i}+\Delta_i \end{align*} $$

for some ${\mathbb Q}$ -divisors $\Delta _i, \Theta _i\ge 0$ on $E_i$ and where $\mathcal G_i$ is the induced foliation on $E_i$ . In particular, $(E_i,\Theta _i)$ is log canonical for all $i=1,\dots ,q$ .

We first prove the following:

Claim 4.4. For any $i=1,\dots ,q$ , the surface $E_i$ is covered by curves M such that $(K_{E_i}+\Theta _i)\cdot M \leq 0$ .

Proof of the Claim.

Note that $K_{\mathcal G_i}+\Delta _i\equiv 0$ . Suppose first that $\mathcal G_i$ is not algebraically integrable. If $\Delta _i\neq 0$ , as in the proof of Lemma 2.11 it follows that $\mathcal G_i$ is uniruled, a contradiction. Thus, we may assume that $\Delta _i = 0$ , and so, by Proposition 2.14, $\Theta _i$ only consists of $\mathcal G_i$ -invariant components. Thus, since $(E_i, \Theta _i)$ is log canonical, we have that $\Theta _i \leq \sum C_j$ where the sum runs over all the $\mathcal G_i$ -invariant divisors, and so we may apply Lemma 2.11 to conclude.

Now suppose that $\mathcal G_i$ is algebraically integrable. Again, by Proposition 2.14 and since $K_{\mathcal G_i}+\Delta _i\equiv 0$ , we may apply Lemma 2.12 to conclude. Thus, the claim follows.

Let $c\colon \overline {X} \dashrightarrow X_{can}$ be the log canonical model of $(\overline {X}, \overline {\Gamma }+E)$ over X, let and let $m\colon X_{can} \rightarrow X$ be the induced morphism.

By (2) of Lemma 4.2, we have that $K_{\overline {X}}+\overline {\Gamma }+E$ is nef over X. Thus, the inequality of the Claim is in fact an equality and as such, each such curve is contracted by c. This implies that $X_{can} \rightarrow X$ is a small contraction. In particular, $m^{*}(K_X+\Gamma ) =K_{X_{can}}+\Gamma _{can}$ . Our result follows, since $(X_{can}, \Gamma _{can})$ has log canonical singularities.

We also need the following:

Proof. The following proof relies on similar, and at the same time easier, ideas as in Theorem 4.3. Thus, we only sketch its main steps.

Observe that our hypotheses are preserved by shrinking X and by taking quasi-étale covers. Thus, we may assume without loss of generality that $K_{\mathcal F}$ is Cartier.

Let $\mu \colon \overline {X} \rightarrow X$ be a dlt modification of $(X, S)$ with respect to ${\mathcal F}$ , whose existence is guaranteed by Lemma 4.2. Let . Then, since ${\mathcal F}$ is Gorenstein, we have that $K_{\overline {\mathcal F}}=\mu ^{*}K_{\mathcal F}$ . After possibly replacing X by a neighbourhood of the generic point of C, we may assume that every $\mu $ -exceptional divisor is centred in C. Let $E=\sum _{i=1}^q E_i$ be the sum of the $\mu $ -exceptional divisors and let $\overline {S}$ be the strict transform of S in $\overline X$ .

By classical adjunction and by Proposition 2.13, for each $i=1,\dots ,q$ , we may write

$$ \begin{align*}(K_{\overline X}+\overline{S}+E)|_{E_i}=K_{E_i}+\Theta_i\qquad\text{and}\qquad K_{\overline{\mathcal F}}\vert_{E_i} = K_{\mathcal G_i}+\Delta_i \end{align*} $$

for some ${\mathbb Q}$ -divisors $\Delta _i, \Theta _i\ge 0$ on $E_i$ and where $\mathcal G_i$ is the induced foliation on $E_i$ . In particular, $(E_i,\Theta _i)$ is log canonical, for all $i=1,\dots ,q$ .

Fix $i=1,\dots , q$ and consider the induced morphism $p\colon E_i\to C$ . Let $\Sigma $ be the general fibre of p and let $\Sigma ^{\nu }\to \Sigma $ be its normalisation. Since $C\subset {\operatorname {Sing} {\mathcal F}}$ , it follows that a general closed point of C is ${\mathcal F}$ -invariant. Thus, Lemma 2.3 implies that $\Sigma $ is $\overline {\mathcal F}$ -invariant. By classical adjunction and by Proposition 2.13, there exist ${\mathbb Q}$ -divisors $\Gamma _i, \Delta '\ge 0$ on $\Sigma ^{\nu }$ such that

$$\begin{align*}(K_{E_i}+\Theta_i)|_{\Sigma^\nu} = K_{\Sigma^\nu}+\Gamma_i \qquad \text{and}\qquad 0\equiv K_{\overline{\mathcal F}}|_{\Sigma^\nu}=K_{\Sigma^\nu}+\Delta'. \end{align*}$$

By Proposition 2.14, it follows that the support of $\Gamma _i$ is contained in the support of $\Delta '$ and since $\Delta '$ is integral, whilst $(\Sigma ^{\nu },\Gamma _i)$ is log canonical, it follows that $\deg (K_{\Sigma ^\nu }+\Gamma _i)\le 0$ . Thus, our results follow as in the proof of Theorem 4.3.

Note that it is easy to produce examples of a canonical foliation of rank one on a normal variety and a collection of invariant divisors $\sum S_i$ so that $(X, \sum S_i)$ has zero-dimensional non-log canonical singularities, as shown in the following example:

4.2 Subadjuntion

We work in the following set up. Let X be a ${\mathbb Q}$ -factorial threefold with klt singularities, let ${\mathcal F}$ be a rank one foliation on X and let $\Gamma = \sum a_iS_i$ be a ${\mathbb Q}$ -divisor where $S_1,\dots ,S_k$ are ${\mathcal F}$ -invariant prime divisors and $a_1,\dots ,a_k\in (0,1)$ . Let $C \subset X$ be a ${\mathcal F}$ -invariant projective curve which is a log canonical centre of $(X, \Gamma )$ and suppose that there are no one-dimensional non-log canonical centres. Suppose moreover that ${\mathcal F}$ has canonical singularities and that ${\mathcal F}$ is terminal at a general point of C. Theorem 4.3 implies that $(X, \Gamma )$ is log canonical.

By subadjunction for varieties, cf. [Reference KollárKol07a, Theorem 8.6.1], we may write

$$\begin{align*}(K_X+\Gamma)\vert_{C^\nu} = K_{C^\nu} +\Theta\end{align*}$$

where $\nu \colon C^\nu \rightarrow C$ is the normalisation and $\Theta \ge 0$ is a ${\mathbb Q}$ -divisor.

Theorem 4.9. Set up as above. Then

1. 1. $(C^\nu , \Theta )$ is log canonical;

2. 2. $\lfloor \Theta \rfloor $ is supported on the preimage of centres of canonical singularities of ${\mathcal F}$ ;

3. 3. if ${\mathcal F}$ is terminal at $\nu (Q) \in C$ for some $Q \in C^\nu $ then $\mu _Q\Theta = \frac {n-1}{n}$ where n is the Cartier index of $K_{\mathcal F}$ at $\nu (Q)$ .

In particular, we have

$$\begin{align*}(K_X+\Gamma)\cdot C\le K_{\mathcal F}\cdot C. \end{align*}$$

Proof. Let $p\colon \overline {X} \rightarrow X$ be a dlt modification of $(X, \Gamma )$ and let $\overline {\Gamma }$ be the strict transform of $\Gamma $ in $\overline X$ . Since $(X,\Gamma )$ is log canonical, we may write

$$\begin{align*}K_{\overline{X}}+\overline{\Gamma}+E = p^{*}(K_X+\Gamma) \end{align*}$$

where E is the sum of all the prime exceptional divisors of p. Lemma 2.6 implies that E is $\overline {\mathcal F}$ -invariant. Since C is a log canonical centre of $(X, \Gamma )$ , after possibly going to a higher model we may assume that there exists an irreducible component $E_0$ of E dominating C. Set $E_1=E-E_0$ . By adjunction we may write

$$\begin{align*}(K_{\overline{X}}+E+\overline{\Gamma})\vert_{E_0} = K_{E_0}+\Theta_0 \end{align*}$$

where $\Theta _0 \geq 0$ .

Let be the restriction morphism. Then $K_{E_0}+\Theta _0$ is f-trivial and we may write $K_{E_0}+\Theta _0 = f^{*}(K_{C^\nu }+M+B)$ where is the moduli part of f and is the discrepancy part of f, as in Section 2.9. In particular, we have $\Theta =M+B$ . Note that M depends only on $(X, \Gamma )$ in a neighbourhood of the generic point of C. Moreover, for any $P \in C^\nu $ , $\mu _PB$ depends only on the germ of $(X, \Gamma )$ at $\nu (P)$ .

Since $(E_0,\Theta _0)$ is dlt, it follows that $(C^\nu , B)$ is log canonical. Moreover, (3) implies (2). Thus, it is enough to prove:

1. (a) $M = 0$ ;

2. (b) for any closed point $P \in C$ such that ${\mathcal F}$ is terminal at P, if n is the Cartier index of $K_{\mathcal F}$ at P, then $\mu _P \Theta = \frac {n-1}{n}$ .

We first prove (a). Since ${\mathcal F}$ is Gorenstein at the general point $P\in C$ and the support of $\Gamma $ is ${\mathcal F}$ -invariant, by Lemma 2.9 there exists an analytic neighbourhood U of P and an isomorphism

$$\begin{align*}c\colon U \to S \times \mathbb D \end{align*}$$

where S is an analytic surface and $\mathbb D \subset {\mathbb C}$ is a disc such that ${\mathcal F}|_U$ is induced by the natural submersion $F\colon U\to S$ and $\Gamma =F^{*}\Gamma _S$ for some ${\mathbb Q}$ -divisor $\Gamma _S\ge 0$ on S. Thus, we may assume that $p^{-1}(U)$ is isomorphic to $\overline S\times \mathbb D$ where $\overline S$ is an analytic surface and that $\Gamma +E=\overline F^{*}D$ for some ${\mathbb Q}$ -divisor D on $\overline S$ , where $\overline F\colon p^{-1}(U)\to \overline S$ is the natural morphism. It follows that for any two general points $P, Q \in C$ we have an isomorphism $(f^{-1}(P), \Theta _0\vert _{f^{-1}(P)}) \cong (f^{-1}(Q), \Theta _0\vert _{f^{-1}(Q)})$ . Lemma 2.19 implies that $M = 0$ and (a) follows.

We now prove (b). Let $P\in C$ be a closed point such that ${\mathcal F}$ is terminal at P. By Lemma 2.9 there exists an analytic neighborhood U of P in X and a quasi-étale cover $q\colon V \rightarrow U$ such that $q^{*}K_{\mathcal F}$ is Cartier and a holomorphic submersion $F\colon V \rightarrow B$ which induces ${\mathcal F}' = q^{-1}{\mathcal F}$ .

Let $C' = q^{-1}(C)$ and note that is ramified to order n at . We also have that $C'$ is ${\mathcal F}'$ -invariant. Since F is a submersion, it follows that $C'$ is smooth at $P'$ .

We may write

$$\begin{align*}K_V+\Gamma_V=q^{*}(K_X+\Gamma). \end{align*}$$

Note that $C'$ is a log canonical centre for $(V,\Gamma _V)$ and, therefore, by subadjunction for varieties, we may also write

$$\begin{align*}(K_V+\Gamma_V)|_{C'}=K_{C'}+\Theta', \end{align*}$$

so that $K_{C'}+\Theta ' = q_C^{*}(K_C+\Theta )$ .

Since $\Gamma _V$ is ${\mathcal F}'$ -invariant, after replacing U by a smaller analytic neighbourhood of $P'$ , we have that the submersion F defines an isomorphism

$$\begin{align*}c\colon V \to S \times \mathbb D \end{align*}$$

where $S\subset B$ is an analytic open set, $\mathbb D \subset {\mathbb C}$ is a disc and $\Gamma _V=F^{*}\Gamma _S$ for some ${\mathbb Q}$ -divisor $\Gamma _S\ge 0$ on S. It follows that $\mu _{P'}\Theta '=0$ and, therefore, by Riemann-Hurwitz we have that $\mu _P\Theta = \frac {n-1}n$ , as claimed. This concludes the proof of (b). Thus, (1), (2) and (3) follow.

Our final claim follows immediately from the results above and Proposition 2.13.