2.1 Distributions

A distribution ${\mathcal D}$ on a smooth algebraic variety consists of a pair $(T_{\mathcal D}, N^*_{\mathcal D})$ of coherent subsheaves of $T_X$ and $\Omega ^1_X$ such that $T_{\mathcal D}$ is the annihilator of $N^*_{\mathcal D}$ and $N^*_{\mathcal D}$ is the annihilator of $T_{\mathcal D}$ . Explicitly,

$$ \begin{align*} T_{\mathcal D} &= \{ v \in T_X \, | \, \omega(v) = 0 \text{ for every } \omega \in N^*_{\mathcal D} \} \, \text{ and}\\ N^*_{\mathcal D} &= \{ \omega \in \Omega^1_X \, | \, \omega(v)=0 \text{ for every } v \in T_{\mathcal D} \} \,. \end{align*} $$

The sheaf $T_{\mathcal D}$ is the tangent sheaf of the distribution, and the sheaf $N^*_{\mathcal D}$ is the conormal sheaf of the distribution. It follows from the definition that both sheaves $T_{\mathcal D}$ and $N^*_{\mathcal D}$ are reflexive (i.e., canonically isomorphic to their bi-duals). The dimension of ${\mathcal D}$ is the generic rank of $T_{\mathcal D}$ , and the codimension of ${\mathcal D}$ is the generic rank of $N^*_{\mathcal D}$ .

The dual of $T_{\mathcal D}$ is the cotangent sheaf of ${\mathcal D}$ and will be denoted by $\Omega ^1_{\mathcal D}$ . Similarly, the dual of $N^*_{\mathcal D}$ is the normal sheaf of ${\mathcal D}$ and will be denoted by $N_{\mathcal D}$ . These sheaves fit into the following exact sequences.

For every i between $1$ and $\dim {\mathcal D}$ , we will define $\Omega ^i_{\mathcal D}$ as the dual of $\wedge ^i T_{\mathcal D}$ ; that is,

$$\begin{align*}\Omega^i_{\mathcal D} = \operatorname{\mathrm{Hom}}\left( \bigwedge^i T_{\mathcal D}, {\mathcal O}_X \right) \,. \end{align*}$$

Since the sheaves $\Omega ^i_{\mathcal D}$ are defined as duals, they are reflexive sheaves. It is convenient to set $\Omega ^0_{\mathcal D} = {\mathcal O}_{\mathcal D}$ and $\Omega ^j_{\mathcal D} = 0$ if $j \notin \{0, \ldots \dim {\mathcal D}\}$ .

We will denote the determinant of $\Omega ^1_{\mathcal D}$ by $\omega _{\mathcal D}$ ; that is,

$$\begin{align*}\omega_{\mathcal D} = \left(\bigwedge^{\dim {\mathcal D}} \Omega^1_{\mathcal D}\right)^{**} \,. \end{align*}$$

Since $T_{\mathcal D}$ is reflexive, [Reference Hartshorne26, Proposition 1.10] implies that

$$\begin{align*}\Omega^1_{\mathcal D} \simeq \left(\bigwedge^{\dim {\mathcal D} - 1} T_{\mathcal D}\right)^{**}\otimes \omega_{\mathcal D} \,. \end{align*}$$

The singular set of ${\mathcal D}$ is the locus where $T_X/T_{\mathcal D}$ is not locally free and is denoted by $\operatorname {\mathrm {sing}}(\mathcal D)$ . Since $T_X/T_{\mathcal D}$ is torsion-free, the codimension of $\operatorname {\mathrm {sing}}(\mathcal D)$ is at least two. Outside $\operatorname {\mathrm {sing}}(\mathcal D)$ , the rightmost arrows of both exact sequences above are surjective; see, for instance, [Reference Giraldo and Pan-Collantes22, Section 2]. Taking determinants one gets an isomorphism of line-bundles

(2.1) $$ \begin{align} \omega_{X} \simeq \det(\Omega^1_{\mathcal D}) \otimes \det(N^*_{\mathcal D}) \, , \end{align} $$

where $\omega _X$ denotes the canonical sheaf of X.

If ${\mathcal D}$ is a distribution of codimension q, then there exists $ \omega \in H^0(X, \Omega ^q_X \otimes \det (N_{\mathcal D})) \, , $ without codimension one zeros, such that $T_{\mathcal D}$ is the kernel of the morphism

$$\begin{align*}T_X \to \Omega^{q-1}_X \otimes \det(N_{\mathcal D}) \end{align*}$$

defined by contraction with $\omega $ .

2.2 Foliations

A foliation ${\mathcal F}$ on a smooth algebraic variety X is a distribution with tangent sheaf $T_{{\mathcal F}}$ closed under Lie brackets. Explicitly, the ${\mathcal O}_X$ -morphism (also called the O’Neill tensor)

$$ \begin{align*} \bigwedge^2 T_{{\mathcal F}} & \longrightarrow N_{{\mathcal F}} \\ v\wedge w &\mapsto [v,w] \quad\mod T_{{\mathcal F}} \end{align*} $$

is identically zero. Here, we used Remark 2.1 to replace $T_X/T_{{\mathcal F}}$ by $N_{{\mathcal F}}$ . Coherent subsheaves of $T_X$ which are closed under Lie brackets are called involutive subsheaves.

If X is a complex algebraic variety, then the vector fields given by Lemma 2.2 can be integrated, in an analytic or formal neighborhood of x, to a germ of action of an abelian Lie group of dimension r. The situation in positive characteristic is utterly different as will be discussed at length in Section 4.

A coherent subsheaf $\mathcal L$ of $\Omega ^1_X$ is called integrable if $d \mathcal L$ is contained in the saturation of $\mathcal L \wedge \Omega ^1_X$ in $\Omega ^2_X$ . Cartan’s formula for the exterior derivative (see [Reference Cartier10, Lemma 3] for its validity in positive characteristic) implies that the involutiveness of $T_{{\mathcal F}}$ is equivalent to the integrability of $N^*_{{\mathcal F}}$ .

The integrability of $N^*_{{\mathcal F}}$ and the reflexiveness of $\Omega ^{\bullet }_{{\mathcal F}}$ imply that the exterior derivative of differential forms on X descends to a ${\mathsf k}$ -linear morphism

$$\begin{align*}d_{{\mathcal F}} : \Omega^{\bullet}_{{\mathcal F}} \to \Omega^{\bullet+1}_{{\mathcal F}} \end{align*}$$

which, like the exterior derivative, satisfies Leibniz’s rule. Indeed, if $U \subset X$ is a sufficiently small open subset contained in $X- \operatorname {\mathrm {sing}}({\mathcal F})$ and $\omega $ belongs to $\Omega ^i_{{\mathcal F}}(U)$ , then we consider any lift $\hat \omega $ of $\omega $ to $\Omega ^i_{X}(U)$ and define $d_{{\mathcal F}} \omega $ as the restriction of $d \hat \omega $ to $\wedge ^{i+1} T_{{\mathcal F}}$ . If $\hat \omega '$ is any other lift, then $\hat \omega - \hat \omega '$ belongs to $\Omega ^{i-1}_X(U) \wedge N^*_{{\mathcal F}}(U)$ and the integrability condition implies that $d ( \hat \omega - \hat \omega ' )$ vanishes on $\wedge ^{i+1} T_{{\mathcal F}}$ . If $j : X-\operatorname {\mathrm {sing}}({\mathcal F}) \to X$ denotes the inclusion, then we have just shown how to define a ${\mathsf k}$ -linear morphism $d_{{\mathcal F}}$ on $j^*\Omega ^{\bullet }_{{\mathcal F}}$ . The reflexiveness of $\Omega ^{\bullet }_{{\mathcal F}}$ implies that $d_{{\mathcal F}}$ extends to a ${\mathsf k}$ -linear morphism defined on the whole X which inherits the Leibniz’ property from d.

Every smooth variety X has two trivial foliations: one foliation of dimension zero given by the pair $(0,\Omega ^1_X) \subset (T_X,\Omega ^1_X)$ , called the foliation by points, and one foliation of dimension $\dim X$ given by the pair $(T_X,0)$ , called the foliation with just one leaf.

Part 1. Geometry of foliations in positive characteristic

3.1 Absolute Frobenius

Let X be a scheme over an algebraically closed field ${\mathsf k}$ of characteristic $p>0$ . The absolute Frobenius morphism of X is the endomorphism $\operatorname {\mathrm {Frob}} = \operatorname {\mathrm {Frob}}_X : X \to X$ of the scheme X, which acts as the identity on the topological space X and which acts on the structural sheaf of X by raising elements to their p-th powers.

A $p^{-1}$ -linear morphism $\varphi : \mathcal E_1 \to \mathcal E_2$ is equivalent to an ${\mathcal O}_X$ -linear morphism $\varphi : \operatorname {\mathrm {Frob}}_* \mathcal E_1 \to \mathcal E_2$ ; see [Reference Blickle and Schwede5, Sections 2 and 3]. If $\mathcal E$ is a coherent ${\mathcal O}_X$ -module, then $\mathcal E$ and $\operatorname {\mathrm {Frob}}_* \mathcal E$ are isomorphic, but the ${\mathcal O}_X$ -module structure on $\operatorname {\mathrm {Frob}}_* \mathcal E$ is different. If we denote the action of ${\mathcal O}_X$ on $\operatorname {\mathrm {Frob}}_* \mathcal E$ by $\star $ , then

$$\begin{align*}f \star \sigma = f^p \sigma \end{align*}$$

for any $f \in {\mathcal O}_X$ and any $\sigma \in \operatorname {\mathrm {Frob}}_* \mathcal E$ .

If $\mathcal E$ still denotes a coherent sheaf of ${\mathcal O}_X$ -modules, then

$$ \begin{align*} \operatorname{\mathrm{Frob}}^* \mathcal E & = {\mathcal O}_X \otimes_{\operatorname{\mathrm{Frob}}^{-1} {\mathcal O}_X} \operatorname{\mathrm{Frob}}^{-1} \mathcal E && \text{(Definition of pull-back)} \\ & = {\mathcal O}_X \otimes_{\operatorname{\mathrm{Frob}}^{-1} {\mathcal O}_X} \mathcal E && (\operatorname{\mathrm{Frob}}\text{ is the identity on }|X|). \end{align*} $$

Consequently, in $\operatorname {\mathrm {Frob}}^* \mathcal E$ we have that

(3.1) $$ \begin{align} 1 \otimes f \sigma = f^p \otimes \sigma \end{align} $$

for any $f \in {\mathcal O}_X$ and any $\sigma \in \mathcal E$ . It follows that a p-linear morphism $\varphi : \mathcal E_1 \to \mathcal E_2$ is equivalent to an ${\mathcal O}_X$ -linear morphism $\varphi : \operatorname {\mathrm {Frob}}^* \mathcal E_1 \to \mathcal E_2$ .

The ${\mathcal O}_X$ -module $\operatorname {\mathrm {Frob}}^* \mathcal E$ comes endowed with a canonical connection

$$ \begin{align*} {\nabla^{\mathrm{can}}} : \operatorname{\mathrm{Frob}}^* \mathcal E & \to \Omega^1_X \otimes \operatorname{\mathrm{Frob}}^* \mathcal E \\ f \otimes \sigma & \mapsto df \otimes (1 \otimes \sigma) \,. \end{align*} $$

It follows from Equation (3.1) that the kernel of ${\nabla ^{\mathrm {can}}}$ can be identified, as sheaves of abelian groups, with $1 \otimes \mathcal E \subset \operatorname {\mathrm {Frob}}^* \mathcal E$ .

3.2 Vector fields and the p-curvature of connections

Given a vector field/derivation $v \in T_X(U)$ on an open subset $U \subset X$ of a variety defined over a field ${\mathsf k}$ of characteristic $p>0$ , the p-th iteration of v is also a derivation since Leibniz’s rule implies

$$\begin{align*}v^p(f \cdot g) = \sum_{i=0}^p \binom{p}{i} v^{i}(f) \cdot v^{p-i}(g) = f v^{p}(g) + v^p(f) g \end{align*}$$

for every $f,g \in {\mathcal O}_X(U)$ .

Thus, one can define a morphism of sheaves of sets $\times ^{p} : T_X \to T_X$ which takes a vector field and raises it to its p-th power. This is not a morphism of sheaves of abelian groups, but a result of Jacobson ([Reference Katz28, Equation (5.2.4)]) says that

(3.2) $$ \begin{align} (v + w)^p = v^p + w^p + Q_p(v,w), \end{align} $$

where $Q_p$ is a Lie polynomial (i.e., a polynomial on (iterated Lie) brackets of v and w). Moreover, if $f \in {\mathcal O}_X$ and $v \in T_X$ , then

(3.3) $$ \begin{align} (f v)^p = f^p v^p - f v^{p-1}(f^{p-1}) v \, , \end{align} $$

according to [Reference Katz28, Equation (5.4.0)] modulo a sign misprint.

Recall from [Reference Katz28, Section 5] that the p-curvature of a connection $\nabla : \mathcal E \to \Omega ^1_X \otimes \mathcal E$ is the morphism of ${\mathcal O}_X$ -modules

$$ \begin{align*} \psi : \operatorname{\mathrm{Frob}}^* T_X & \longrightarrow \operatorname{\mathrm{End}}_{{\mathcal O}_X}(\mathcal E) \\ v & \mapsto (\nabla_v)^p - \nabla_{v^p} \, , \end{align*} $$

where $\nabla _\bullet $ is the ${\mathsf k}$ -endomorphisms of $\mathcal E$ that send defined by

$$ \begin{align*} \mathcal E & \longrightarrow \mathcal E \\ \sigma & \mapsto (i_\bullet \otimes \operatorname{\mathrm{id}}) \nabla\sigma \, , \end{align*} $$

where $i_{\bullet }$ denotes the contraction of $1$ -forms with the vector field $\bullet $ .

3.3 Relative Frobenius, p-closed flat connections and the Cartier correspondence

The absolute Frobenius is not a morphism of ${\mathsf k}$ -schemes. Nevertheless, if we set $S= \operatorname {\mathrm {Spec}} {\mathsf k}$ and consider the fiber product $X^{(p)} = X \times _{S} S$ of the structure morphism of X with the absolute Frobenius of S, then the absolute Frobenius $\operatorname {\mathrm {Frob}}$ factors as in the following commutative diagram:

The morphism $\operatorname {\mathrm {Frob}}_{X/S} : X \to X^{(p)}$ is a morphism of ${\mathsf k}$ -schemes and is called the relative Frobenius morphism of X over S; see [Reference Bucur, Houzel, Illusie, Jouanolou and Serre1, Éxpose XV]. Notice that $W_{X/S} : X^{(p)} \to X$ is an isomorphism of schemes (but not of ${\mathsf k}$ -schemes) defined through the twisting of the coefficients by means of the field automorphism $x \mapsto x^p$ .

Given any coherent sheaf $\mathcal E$ of ${\mathcal O}_X$ -modules on X, we set $\mathcal E^{(p)} = W_{X/S}^* \mathcal E$ . In other words, $\mathcal E^{(p)}$ is the base change of $\mathcal E$ to $X^{(p)}$ . As such, $\mathcal E^{(p)}$ is an ${\mathcal O}_{X^{(p)}}$ -module such that $\operatorname {\mathrm {Frob}}^* \mathcal E = \operatorname {\mathrm {Frob}}_{X/S}^* \mathcal E^{(p)}$ . If we consider ${\mathcal O}_{X^{(p)}}$ as a subsheaf of ${\mathcal O}_X$ , then the kernel of the canonical connection ${\nabla ^{\mathrm {can}}}$ on $\operatorname {\mathrm {Frob}}_{X/S}^* \mathcal E^{(p)}$ is an ${\mathcal O}_{X^{(p)}}$ -module that can be identified with $\mathcal E^{(p)}$ . Reciprocally, if $\mathcal E$ is a coherent sheaf on X endowed with a flat connection $\nabla $ of p-curvature zero, then the kernel of $\nabla $ is, in a natural way, an ${\mathcal O}_{X^{(p)}}$ -module such that

$$\begin{align*}\operatorname{\mathrm{Frob}}_{X/S}^* (\ker \nabla) \simeq \mathcal E \,. \end{align*}$$

This is essentially the content of the Cartier correspondence: there exists an equivalence of categories between the category of quasi-coherent sheaves on $X^{(p)}$ and the category of quasi-coherent ${\mathcal O}_X$ -modules with flat connections of p-curvature zero given by

$$\begin{align*}\mathcal E \mapsto (\operatorname{\mathrm{Frob}}_{X/S}^* \mathcal E , {\nabla^{\mathrm{can}}}) \quad \text{ and } \quad (\mathcal E, \nabla) \mapsto \ker \nabla; \end{align*}$$

see, for instance, [Reference Katz28, Theorem 5.1].

Lemma 3.3. Let $\mathcal E$ be a coherent sheaf of ${\mathcal O}_X$ -modules. Let $\mathcal K \subset \operatorname {\mathrm {Frob}}^*\mathcal E$ be a coherent subsheaf. If the morphism of ${\mathcal O}_X$ -modules

$$ \begin{align*} \varphi: \mathcal K & \longrightarrow \Omega^1_X \otimes \frac{\operatorname{\mathrm{Frob}}^* \mathcal E}{\mathcal K} \end{align*} $$

induced by ${\nabla ^{\mathrm {can}}}$ is identically zero, then there exists a coherent subsheaf $\mathcal L \subset \mathcal E$ such that $\mathcal K = \operatorname {\mathrm {Frob}}^* \mathcal L$

4.1 The p-curvature of foliations

Let ${\mathcal F}$ be a foliation on X. It follows from Equations (3.2) and (3.3) that the map which sends $v \in T_{{\mathcal F}}$ to the class of $v^p$ in ${T_X}/{T_{{\mathcal F}}} \subset N_{{\mathcal F}}$ is p-linear and additive. We can thus define the morphism of ${\mathcal O}_X$ -modules

$$ \begin{align*} {\psi_{{\mathcal F}}} : \operatorname{\mathrm{Frob}}^* T_{{\mathcal F}} & \longrightarrow N_{{\mathcal F}} \\ \sum f_i \otimes v_i & \mapsto \sum f_i v_i^p \quad\mod T_{{\mathcal F}} \,. \end{align*} $$

The morphism ${\psi _{{\mathcal F}}}$ is called the p-curvature of ${\mathcal F}$ . If ${\psi _{{\mathcal F}}} =0$ , then we say that the foliation is p-closed.

The foliation ${\mathcal F}$ determines a subsheaf ${\mathcal O}_{X/{\mathcal F}} \subset {\mathcal O}_X$ consisting of functions that are annihilated by any section of $T_{{\mathcal F}}$ ; that is,

$$\begin{align*}{\mathcal O}_{X/{\mathcal F}} = \{ a \in {\mathcal O}_X \,; \, v(a) =0 \text{ for any } v \in T_{\mathcal F} \} \, .\ \end{align*}$$

4.2 p-dense foliations

Some authors use the term foliation only to refer to p-closed foliations. In this work, we will also consider foliations which are not p-closed.

Notice that a foliation is p-dense if and only if any rational first integral $f \in {\mathsf k}(X)$ of ${\mathcal F}$ is such that $df =0$ .

Proposition 4.4. If ${\mathcal F}$ is a p-dense codimension one foliation on a smooth algebraic variety X defined by $\omega \in H^0(X,\Omega ^1_X \otimes N_{{\mathcal F}})$ and v is a rational vector field tangent to ${\mathcal F}$ such that $\omega (v^p) \neq 0$ , then the rational $1$ -form

$$\begin{align*}\frac{\omega}{\omega(v^p)} \end{align*}$$

is closed.

The result above first appeared in [Reference Cerveau and Lins Neto11, Theorem 6.2]. Its proof can be generalized to p-dense foliations of arbitrary codimension as shown by the next proposition.

Proof. Let $r = \dim {\mathcal F}$ . If $x \notin \operatorname {\mathrm {sing}}({\mathcal F})$ , then there exist generators $v_1, \ldots , v_r$ of $T_{{\mathcal F}} \otimes {\mathcal O}_{X,x}$ such that $[v_i , v_j] =0$ for every $i, j \in \{ 1, \ldots , r\}$ ; see Lemma 2.2. Notice that the iterated p-th powers of the vector fields $v_i$ commute with $v_1, \ldots , v_r$ and also among themselves. Since ${\mathcal F}$ is p-dense, we can choose among the iterated p-powers of $v_1, \ldots , v_r$ , vector fields $v_{r+1}, \ldots , v_{r+q}$ , $r+q = \dim X$ , such that

$$\begin{align*}v_1 \wedge \cdots \wedge v_r \wedge v_{r+1} \wedge \cdots \wedge v_{r+q} \end{align*}$$

does not vanish identically. Therefore, we can interpret $v_1, \ldots , v_{r+q}$ as ${\mathsf k}(X)$ -linearly independent commuting rational vector fields on X.

Let $\alpha _1, \ldots , \alpha _{r+q}$ be rational $1$ -forms on X dual to $v_1, \ldots , v_{r+q}$ . In other words, the $1$ -forms $\alpha _i$ are characterized by $\alpha _i(v_j)= \delta _{ij}$ . We claim that the $1$ -forms $\alpha _i$ are closed. As $v_1, \ldots , v_{r+q}$ are ${\mathsf k}(X)$ -linearly independent rational vector fields, it suffices to check that $d\alpha _i(v_j,v_k) =0 $ for $i,j,k \in \{ 1, \ldots , r+q\}$ . Since $v_1, \ldots , v_{r+q}$ commute, Cartan’s formula for the exterior derivative implies that

$$\begin{align*}d \alpha_i(v_j,v_k) = v_j(\alpha_i(v_k)) - v_k(\alpha_i(v_j)) - \alpha_i([v_j,v_k]) = 0, \end{align*}$$

establishing our claim. By construction, $\omega _1 = \alpha _{r+1}, \ldots , \omega _q = \alpha _{r+q}$ vanish on $T_{{\mathcal F}}$ and satisfy $\omega _1 \wedge \cdots \wedge \omega _q \neq 0$ as wanted.

The following example shows that without p-denseness of ${\mathcal F}$ , Proposition 4.5 does not hold.

A simple adaptation of the arguments used in the proof of Proposition 4.5 gives the following generalization.

A ${\mathcal F}$ -partial connection on a coherent sheaf $\mathcal E$ is a ${\mathsf k}$ -linear morphism $\nabla : \mathcal E \to \Omega ^1_{{\mathcal F}} \otimes \mathcal E$ which, like usual connections, satisfies Leibniz’s rule $\nabla (f s) = f \nabla (s) + d_{{\mathcal F}} f \otimes s$ . Bott’s partial connection is the ${\mathcal F}$ -partial connection on the normal sheaf of ${\mathcal F}$ which sends the class of $v\ \ \mod T_{{\mathcal F}}$ to the $1$ -form (defined only along $T_{{\mathcal F}}$ ) that maps a local section w of $T_{{\mathcal F}}$ to $[v,w] \ \ \mod T_{{\mathcal F}}$ .

Our next result provides an interpretation of Propositions 4.5 and 4.7 in terms of Bott’s partial connection and the canonical connection on $\operatorname {\mathrm {Frob}}^* T_{{\mathcal F}}$ .

Proposition 4.8. The morphism ${\psi _{{\mathcal F}}}$ is such that

$$\begin{align*}(r \otimes {\psi_{{\mathcal F}}}) \circ {\nabla^{\mathrm{can}}} = {\nabla^{\mathcal F}} \circ {\psi_{{\mathcal F}}}, \end{align*}$$

where $r : \Omega ^1_{X} \to \Omega ^1_{{\mathcal F}}$ is the natural restriction morphism, ${\nabla ^{\mathrm {can}}}$ is the canonical connection on $\operatorname {\mathrm {Frob}}^*T_{{\mathcal F}}$ and ${\nabla ^{\mathcal F}}$ is Bott’s connection on $N_{{\mathcal F}}$ . In other words, ${\psi _{{\mathcal F}}}$ is a morphism of ${\mathcal F}$ -partial connections between and $(N_{{\mathcal F}}, {\nabla ^{\mathcal F}})$ .

4.3 The degeneracy divisor and the kernel of the p-curvature

Proposition 4.8 implies that the kernel of ${\psi _{{\mathcal F}}}$ is invariant by the partial connection . Therefore, the morphism

$$\begin{align*}\ker {\psi_{{\mathcal F}}} \longrightarrow \Omega^1_{\mathcal F} \otimes \frac{\operatorname{\mathrm{Frob}}^* T_{{\mathcal F}}}{\ker {\psi_{{\mathcal F}}}} \end{align*}$$

induced by is zero. This is not sufficient to apply Lemma 3.3 as it may happen that $\ker {\psi _{{\mathcal F}}}$ is not invariant by the connection ${\nabla ^{\mathrm {can}}}$ ; that is, the morphism of ${\mathcal O}_X$ -modules

(4.1) $$ \begin{align} \ker {\psi_{{\mathcal F}}} \longrightarrow \Omega^1_X \otimes \frac{\operatorname{\mathrm{Frob}}^* T_{{\mathcal F}}}{\ker {\psi_{{\mathcal F}}}} \end{align} $$

induced by ${\nabla ^{\mathrm {can}}}$ is not necessarily zero.

Lemma 4.9. If ${\mathcal F}$ is p-dense, then the morphism (4.1) is zero. In particular, there exists a distribution ${\mathscr V({{\mathcal F}})}$ contained in ${\mathcal F}$ such that

$$\begin{align*}\ker {\psi_{{\mathcal F}}} = \operatorname{\mathrm{Frob}}^* T_{{\mathscr V({{\mathcal F}})}} \,. \end{align*}$$

In this case, the distribution ${\mathscr V({{\mathcal F}})}$ will be called the kernel of the p-curvature of ${\mathcal F}$ .

Example 4.11. Let ${\mathcal F}$ be a codimension one foliation on $X= \mathbb A^n_{{\mathsf k}}$ with $N^*_{{\mathcal F}}$ generated by $1$ -form

$$\begin{align*}\omega = \left( \prod_{i=1}^r x_i \right) \left( \sum_{i=1}^r \lambda_i \frac{dx_i}{x_i} \right) \, , \end{align*}$$

where $1<r\le n$ and $\lambda _1, \ldots , \lambda _r \in {\mathsf k}^*$ . The tangent sheaf of ${\mathcal F}$ is locally free and generated by the vector fields $v_{j} = \lambda _j x_1 \frac {\partial }{\partial x_1} - \lambda _i x_j \frac {\partial }{\partial x_j}$ for $1< j\le r$ and $\frac {\partial }{\partial x_{r+1}}, \ldots , \frac {\partial }{\partial x_n}$ . Observe that $\frac {\partial ^p }{\partial x_{r+1}}=\ldots = \frac {\partial ^p }{\partial x_n}=0$ and that

$$\begin{align*}\omega(v_{j}^p) = \left( \prod_{i=1}^r x_i \right) ( \lambda_j^p \lambda_1 - \lambda_1^p \lambda_j ) \, \end{align*}$$

for any $1 < j \le r$ . Hence, ${\mathcal F}$ is p-closed if and only if $(\lambda _1: \ldots : \lambda _r) \in \mathbb P^{r-1}_{{\mathsf k}}(\mathbb F_p)$ . Furthermore, if ${\mathcal F}$ is not p-closed, then $\Delta _{{\mathcal F}}$ is the simple normal crossing divisor given by $\{ x_1 \cdots x_r =0\}$ ,

Proposition 4.12. Let ${\mathcal F}$ be a p-dense codimension one foliation on a smooth algebraic variety X of characteristic $p>0$ . If ${\mathscr V({{\mathcal F}})}$ denotes the kernel of the p-curvature of ${\mathcal F}$ , then the identity

$$\begin{align*}{\mathcal O}_X( {\Delta_{{\mathcal F}}}) = (\omega_{{\mathcal F}} \otimes \omega_{{\mathscr V({{\mathcal F}})}}^{*})^{\otimes p} \otimes \det N_{{\mathcal F}} \, \end{align*}$$

holds in the Picard group of X.

Proof. The definition of ${\Delta _{{\mathcal F}}}$ implies that the sequence

(4.2) $$ \begin{align} 0 \longrightarrow \operatorname{\mathrm{Frob}}^{*}T_{{\mathscr V({{\mathcal F}})}} \longrightarrow \operatorname{\mathrm{Frob}}^{*}T_{\mathcal{F}} \longrightarrow N_{\mathcal{F}}(-\Delta_{\mathcal{F}})\longrightarrow 0 \end{align} $$

is exact on the complement of a codimension two subset of X. The result follows by taking determinants.

Proof. Suppose first that H is $\mathcal {F}$ -invariant. Let $U \subset X$ be a sufficiently small open subset intersecting H and let $h \in {\mathcal O}_X(U)$ be a reduced equation for $H\cap U$ . Since H is ${\mathcal F}$ -invariant, for any local section $v \in T_{{\mathcal F}}(U)$ , we have that $v(h)$ is a multiple of h. Therefore, the same holds true for $v^p(h)$ . In other words, for any $v \in T_{{\mathcal F}}(U)$ , the restriction of v and $v^p$ to $H\cap U$ is contained in the kernel of . Hence, the support of ${\Delta _{{\mathcal F}}}$ contains H.

Suppose now that $\operatorname {\mathrm {ord}}_{H}(\Delta _{\mathcal {F}}) = \alpha \not \equiv 0 \ \ \mod p$ . In the notation above, after restricting U if necessary, we can chose commuting generators $v_1, \ldots , v_r$ of $T_{{\mathcal F}}(U)$ such that

$$\begin{align*}v_1\wedge \cdots \wedge v_r \wedge v_1^p = h^{\alpha} \theta, \end{align*}$$

where $r = \dim {\mathcal F}$ , and $\theta \in \bigwedge ^{r+1} T_X(U)$ does not vanish along H; see Lemma 2.2. Observe that

$$\begin{align*}[v_1, h^{\alpha} \theta] = 0 \implies \alpha v_1(h) h^{\alpha -1} \theta = - h^{\alpha} [v_1, \theta] \,. \end{align*}$$

Since $\alpha \not \equiv 0 \ \ \mod p$ , it follows that h divides $v_1(h)$ . This is sufficient to show that ${H = \{ h=0\}}$ is ${\mathcal F}$ -invariant.

4.4 The Cartier operator

Let X be a smooth algebraic variety defined over an algebraically closed field of characteristic $p>0$ .

The de Rham complex $(\Omega ^{\bullet }_X,d)$ of X is a complex of sheaves of abelian groups but is not a complex of ${\mathcal O}_X$ -modules. Nevertheless, for any $f \in {\mathcal O}_X$ and any $\omega \in \Omega ^i_X$ , we have that

$$\begin{align*}d (f^p \cdot \omega) = f^p \cdot \omega \,. \end{align*}$$

It follows that $(\operatorname {\mathrm {Frob}}_* \Omega ^{\bullet }_X, d)$ is a complex of ${\mathcal O}_X$ -modules. If we denote by $Z\Omega ^{\bullet }_X$ the sheaf of closed differential, and by $B \Omega ^{\bullet }_X$ the sheaf of locally exact differentials, then both $\operatorname {\mathrm {Frob}}_* Z \Omega ^{\bullet }_X$ and $\operatorname {\mathrm {Frob}}_* B \Omega ^{\bullet }_X$ are coherent ${\mathcal O}_X$ -modules. The sheaf cohomology groups of the complex $(\operatorname {\mathrm {Frob}}_* \Omega ^{\bullet }_X, d)$ are the ${\mathcal O}_X$ -modules

$$\begin{align*}{\mathscr{H}}^i \operatorname{\mathrm{Frob}}_* \Omega^{\bullet}_X = \frac{\operatorname{\mathrm{Frob}}_*Z \Omega^i_X }{\operatorname{\mathrm{Frob}}_*B \Omega^i_X } \,. \end{align*}$$

Cartier proved in [Reference Cartier9] the existence of a unique morphism $\mathbf {C}:\operatorname {\mathrm {Frob}}_*Z \Omega ^{\bullet }_X \to \Omega ^{\bullet }_X$ of sheaves of graded-commutative ${\mathcal O}_X$ -algebras such that

1. (1) $\ker \mathbf {C} = \operatorname {\mathrm {Frob}}_* B \Omega ^{\bullet }_X$ ; and

2. (2) $\mathbf {C}(f^{p-1} df) = df$ for every $f \in {\mathcal O}_X$ ; and

3. (3) $\mathbf {C}$ is surjective and, consequently, induces an isomorphism of graded-commutative ${\mathcal O}_X$ -algebras between ${\mathscr {H}}^{\bullet } \operatorname {\mathrm {Frob}}_* \Omega ^{\bullet }_X$ and $\Omega ^{\bullet }_X$ .

The morphism $\mathbf {C}$ is the (absolute) Cartier operator on X. If $\omega $ is a closed q-form, we will say that $\mathbf {C}(\omega )$ is the Cartier transform of $\omega $

Lemma 4.16. Let X be a smooth variety defined over a field ${\mathsf k}$ of characteristic $p>0$ . If $\omega \in \Omega ^1_X$ and $v \in T_X$ , then

$$\begin{align*}( i_v \mathbf{C}(\omega) ) ^p = i_{v^p} \omega - v^{p-1}(i_v \omega) \,. \end{align*}$$

More generally, if $\omega \in \Omega ^j_X$ and $v \in T_X$ , then

$$\begin{align*}i_v \mathbf{C}(\omega) = \mathbf{C}\left( i_{v^p} \omega - (L_v)^{p-1}(i_v \omega) \right) \,. \end{align*}$$

4.5 The Cartier transform of a p-dense foliation

Let ${\mathcal F}$ be a p-dense foliation of codimension q on a smooth algebraic variety X. Let $\omega _1, \ldots , \omega _q$ be closed rational $1$ -forms defining ${\mathcal F}$ . The existence of such $1$ -forms is guaranteed by Proposition 4.5. If $\omega $ is any closed $1$ -form vanishing on $T_{{\mathcal F}}$ , we can write

$$\begin{align*}\omega = \sum_{i=1}^q f_i \omega_i \end{align*}$$

for some rational functions $f_i \in {\mathsf k}(X)$ . If we differentiate the expression above and take the wedge product of the result with $\omega _1 \wedge \cdots \wedge \widehat {\omega _i} \wedge \cdots \wedge \omega _q$ , we see that $df_i \wedge \omega _1 \wedge \cdots \wedge \omega _q=0$ for any i. Since ${\mathcal F}$ is p-dense, it follows that $df_i =0$ for every i (i.e., there exists $g_i \in {\mathsf k}(X)$ such that $f_i = g_i^p$ ). Therefore, the identity

$$\begin{align*}\mathbf{C}(\omega ) = \sum_{i=1}^q g_i \mathbf{C}(\omega_i) \, \end{align*}$$

holds. It follows that the distribution defined by the Cartier transform of all closed rational $1$ -forms vanishing on $T_{{\mathcal F}}$ can be defined by the Cartier transform of only $q = \operatorname {\mathrm {codim}} {\mathcal F}$ closed rational $1$ -forms.

4.6 The Cartier transform and the kernel of the p-curvature

The proposition below gives an alternative description of the kernel of the p-curvature of a p-dense foliation ${\mathcal F}$ .

Proof. Lemma 4.9 implies the existence of a distribution ${\mathscr V({{\mathcal F}})}$ on X such that ${\ker {\psi _{{\mathcal F}}} = \operatorname {\mathrm {Frob}}^* T_{{\mathscr V({{\mathcal F}})}}}$ .

Let $v \in T_{{\mathcal F}}$ . For any closed rational $1$ -form vanishing on $T_{{\mathcal F}}$ , Lemma 4.16 implies that

$$\begin{align*}( i_v \mathbf{C}(\omega) ) ^p = i_{v^p} \omega - v^{p-1}(i_v \omega) = i_{v^p} \omega \,. \end{align*}$$

If $v \in T_{{\mathscr V({{\mathcal F}})}}$ , then, by the definition of ${\mathscr V({{\mathcal F}})}$ , $i_{v^p} \omega =0$ for any $\omega \in N^*_{{\mathcal F}}$ . Therefore, when $\omega $ is closed, $i_v \mathbf {C}(\omega ) =0$ . This shows that $T_{{\mathscr V({{\mathcal F}})}}$ is contained in the saturation of $T_{{\mathcal F}} \cap T_{{\mathscr {C}({\mathcal F})}}$ .

Reciprocally, if $v \in T_{{\mathcal F}} \cap T_{{\mathscr {C}({\mathcal F})}}$ , then, by definition, $i_{v} \mathbf {C}(\omega )=0$ for any closed rational rational $1$ -form $\omega $ vanishing on $T_{{\mathcal F}}$ . Consequently, $i_{v^p} \omega =0$ for any closed rational section of $N^*_{{\mathcal F}}$ . Since $N^*_{{\mathcal F}}$ is generically generated by closed rational $1$ -forms, as proved in Proposition 4.5, this is sufficient to show that $T_{{\mathcal F}} \cap T_{{\mathscr {C}({\mathcal F})}}$ is contained in $T_{{\mathscr V({{\mathcal F}})}}$ .

4.7 Global expression for the kernel of the p-curvature

Assume that ${\mathcal F}$ is a p-dense foliation of codimension one on a smooth algebraic variety X. For later use, we will present a twisted section of $\Omega ^1_{{\mathcal F}}$ defining the distribution ${\mathscr V({{\mathcal F}})}$ .

Let $U $ be the open set $X - ( \operatorname {\mathrm {sing}}({\mathcal F}) \cup \operatorname {\mathrm {sing}}({\mathscr V({{\mathcal F}})}) )$ and consider a sufficiently fine open covering $U_i$ of U. Choose for each $U_i$ a nowhere vanishing section $\omega _i$ of $N^*_{{\mathcal F}}(U_i)$ and a vector field $v_i \in T_{{\mathcal F}}(U_i) \subset T_X(U_i)$ everywhere transverse to . Therefore, $f_i = \omega _i(v_i^p)$ generates ${\mathcal O}_{U_i}(-{\Delta _{{\mathcal F}}})$ , and the rational $1$ -form $\omega _i/f_i$ is closed.

Let $\eta _i$ be the restriction of the rational $1$ -form $\mathbf {C}(\omega _i/f_i)$ to . We claim that $\eta _i$ is a regular section of without codimension one zeros. Indeed, if $v \in T_{{\mathcal F}}(U_i)$ , then, using Lemma 4.16, we can write

$$\begin{align*}\omega_i(v^p) = f_i \left( \frac{\omega_i}{f_i} \right) (v^p) = f_i \left(\mathbf{C}\left(\frac{\omega_i}{f_i} \right) (v) \right)^p \,. \end{align*}$$

Since $\omega _i(v^p) \in {\mathcal O}_X(-{\Delta _{{\mathcal F}}})(U_i) = f_i {\mathcal O}_X(U_i)$ , we deduce that $\eta _i (v) = \mathbf {C}\left (\frac {\omega _i}{f_i} \right )(v) \in {\mathcal O}_X(U_i)$ for no matter which $v \in T_{{\mathcal F}}(U_i)$ . Moreover, $\eta _i$ has no zeros since $\eta _i(v_i) =1$ .

Taking into account Proposition 4.12, we deduce that the collection $\{ \eta _i \}$ defines a regular section $\eta $ of the sheaf . Since $\Omega ^1_{{\mathcal F}} \otimes {\omega _{{\mathscr V({{\mathcal F}})}}} \otimes \omega ^*_{{\mathcal F}}$ is reflexive, $\eta $ extends to an element of

$$\begin{align*}H^0(X, \Omega^1_{{\mathcal F}} \otimes {\omega_{{\mathscr V({{\mathcal F}})}}} \otimes \omega^*_{{\mathcal F}}) = H^0(X, \Omega^1_{{\mathcal F}} \otimes \det{N_{{\mathscr V({{\mathcal F}})}}} \otimes N^*_{{\mathcal F}}) \, , \end{align*}$$

without codimension one zeros, defining the distribution ${\mathscr V({{\mathcal F}})} \subset {\mathcal F}$ .

5.1 Equidimensional and dominant rational maps

Let X and Y be smooth varieties of dimensions n and m, respectively, defined over an algebraically closed field ${\mathsf k}$ and let $\varphi :X \dashrightarrow Y$ be a dominant rational map. Throughout this section, we will make use of the following assumption on $\varphi $ .

When ${\mathsf k}$ has characterisitic zero, Item (3) follows from Item (2), but this is no longer true when ${\mathsf k}$ has positive characteristic.

Define the ramification divisor $\operatorname {\mathrm {Ram}}(\varphi ^{\circ }) \in \operatorname {\mathrm {Div}}(X^{\circ })$ of $\varphi ^{\circ }$ as

(5.1) $$ \begin{align} \operatorname{\mathrm{Ram}}(\varphi^{\circ}) = \sum_{H} (\varphi^{\circ})^* H - ((\varphi^{\circ})^* H)_{\mathrm{red} } \, , \end{align} $$

where the sum runs over all irreducible hypersurfaces of $Y^{\circ }$ . Assumption 5.1, Item (3), guarantees that this is a finite sum. Moreover, each irreducible component of the support of $\operatorname {\mathrm {Ram}}(\varphi ^{\circ })$ is mapped (dominantly) to an irreducible hypersurface of $Y^{\circ }$ . Define the branch divisor of $\varphi ^{\circ }$ , $\operatorname {\mathrm {Branch}}({\varphi ^{\circ }})$ , as the reduced divisor with support equal to the image of $\operatorname {\mathrm {Ram}}(\varphi ^{\circ })$ . Set $\operatorname {\mathrm {Ram}}(\varphi )$ as the divisor on X given by the closure of $\operatorname {\mathrm {Ram}}(\varphi ^{\circ })$ .

If $\mathcal L$ is a line-bundle on Y, then we will abuse the notation and write $\varphi ^* \mathcal L$ for the unique line-bundle on X such that . Since X is smooth and $\operatorname {\mathrm {codim}} X- X^{\circ } \ge 2$ , such a line-bundle always exists.

The next proposition, as well as Proposition 5.2 below, are well-known for equidimensional morphisms between normal complex varieties; see, for instance, [Reference Druel20, Subsection 2.9].

Proposition 5.1. Let $\varphi :X \dashrightarrow Y$ be a rational map satisfying Assumption 5.1. If $\mathcal V(\varphi )$ is the foliation on X defined by $\varphi $ , then

$$\begin{align*}\omega_{\mathcal V(\varphi)} = \omega_X \otimes \varphi^* \omega_Y^* \otimes {\mathcal O}_X(- \operatorname{\mathrm{Ram}}(\varphi)) \,. \end{align*}$$

Proof. Further restricting $X^{\circ }$ and $Y^{\circ }$ (keeping the assumption $\operatorname {\mathrm {codim}} X- X^{\circ } \ge 2$ ), we can, and will, assume that all irreducible components of $\operatorname {\mathrm {Ram}}(\varphi ^\circ )$ and $\operatorname {\mathrm {Branch}}(\varphi ^\circ )$ are smooth, pairwise disjoint and that the rank of the differential of $\varphi ^{\circ }$ is equal to $m-1=\dim Y-1$ on the support of $\operatorname {\mathrm {Ram}}(\varphi ^{\circ })$ and equal to m everywhere else.

Set $B= \operatorname {\mathrm {Branch}}(\varphi ^{\circ })$ and $R = ((\varphi ^{\circ })^*(B))_{red}$ . Notice that $\operatorname {\mathrm {Ram}}(\varphi ^{\circ }) = (\varphi ^{\circ })^*B - R$ . As usual, we will write $T_{X^\circ }(-\log R)$ for the subsheaf of $T_{X^\circ }$ formed by vector fields tangent to R and similarly for $T_{Y^\circ }(-\log B)$ .

Let $\mathcal V(\varphi )$ be the foliation defined by $\varphi $ . Assumption 5.1, Item (3), (or rather the stronger version of it made on the first paragraph of this proof) implies that the tangent sheaf of

fits into the exact sequence

where the morphism $T_{X^{\circ }}(-\log R) \longrightarrow (\varphi ^{\circ })^* (T_{Y^{\circ }}(- \log B))$ is induced the differential $d\varphi ^{\circ }$ of $\varphi ^{\circ }$ . It suffices to take the determinant of this exact sequence to conclude.

5.2 Pull-backs of foliations under dominant rational maps

Before continuing the discussion, we introduce two notations:

1. (1) If D is an arbitrary divisor on smooth variety carrying a foliation ${\mathcal F}$ , we will write $D = D_{{\mathcal F}} + D_{{\mathcal F}^{\perp }}$ for the unique decomposition of D as the sum of divisors where all the irreducible components of the support of $D_{{\mathcal F}}$ are ${\mathcal F}$ -invariant and all irreducible components of the support of $D_{{\mathcal F}^{\perp }}$ are not ${\mathcal F}$ -invariant.

2. (2) We will write $T_{{\mathcal F}}(-\log D)$ for the subsheaf of $T_{{\mathcal F}}$ formed by vector fields tangent to ${\mathcal F}$ and D. This definition implies that $T_{{\mathcal F}}(-\log D)= T_{{\mathcal F}}(-\log D_{{\mathcal F}^{\perp }})$ and $T_{{\mathcal F}}(-\log D_{{\mathcal F}})= T_{{\mathcal F}}$ . Note that $\det T_{{\mathcal F}}(-\log D) = \det T_{{\mathcal F}} \otimes {\mathcal O}_X(-D_{{\mathcal F}^{\perp }})$ .

Proposition 5.2. Let $\varphi :X \dashrightarrow Y$ be a rational map satisfying Assumption 5.1 and let $\mathcal G$ be a foliation on Y. If ${\mathcal F}$ is the pull-back of $\mathcal G$ under $\varphi $ , then

$$ \begin{align*} \det( N^*_{{\mathcal F}} ) & = \varphi^* \det(N^*_{\mathcal G}) \otimes {\mathcal O}_{X}\left( \operatorname{\mathrm{Ram}}(\varphi)_{{\mathcal F}} \right), \text{ and } \\ \omega_{{\mathcal F}} &= \varphi^* \omega_{\mathcal G} \otimes \omega_{\mathcal V(\varphi)} \otimes {\mathcal O}_{X}\left( \operatorname{\mathrm{Ram}}(\varphi)_{{\mathcal F}^{\perp}}\right). \end{align*} $$

Proof. The restriction of the tangent sheaf of ${\mathcal F}$ to $X^{\circ }$ is defined by the exact sequence

where the rightmost arrow is given by the composition of the differential of $\varphi ^{\circ }$ with the natural projection

.

As in the proof of Proposition 5.1, set $B= \operatorname {\mathrm {Branch}}(\varphi ^{\circ })$ and $R = ((\varphi ^{\circ })^*(B))_{red}$ . Assumption 5.1, Item (3), gives an exact sequence

with the rightmost morphism surjective in codimension one. Taking determinants, we obtain that

$$\begin{align*}\omega_{{\mathcal F}} = \varphi^* \omega_{\mathcal G} \otimes \omega_{\mathcal V(\varphi)} \otimes {\mathcal O}_{X}\left( \operatorname{\mathrm{Ram}}(\varphi)_{{\mathcal F}^{\perp}} \right)\, , \end{align*}$$

as claimed. The formula for the conormal bundle of ${\mathcal F}$ follows from Equation (2.1) (adjunction for foliations) combined with Proposition 5.1.

5.3 Degeneracy divisor of the pull-back under a dominant rational map

Throughout this subsection, ${\mathsf k}$ is an algebraically closed field of characteristic $p>0$ . We will determine a formula comparing the degeneracy divisor of the p-curvature of a p-dense codimension one foliation $\mathcal G$ on a smooth projective variety with the degeneracy divisor of its pull-back under a dominant rational map $\varphi :X \dashrightarrow Y$ satisfying Assumption 5.1.

Proposition 5.3. Let $\varphi :X \dashrightarrow Y$ be a rational map and let $\mathcal G$ be a p-dense codimension one foliation on Y. If $\varphi $ satisfies Assumption 5.1 and ${\mathcal F} = \varphi ^* \mathcal G$ , then ${\Delta _{{\mathcal F}}}$ is linearly equivalent to

$$\begin{align*}\varphi^* {\Delta_{\mathcal G}} - \operatorname{\mathrm{Ram}}(\varphi)_{{\mathcal F}} + p \left( \operatorname{\mathrm{Ram}}(\varphi)_{{{\mathcal F}^{\perp}}} - \operatorname{\mathrm{Ram}}(\varphi)_{{{\mathscr V({{\mathcal F}})}}^{\perp}} \right) \,. \end{align*}$$

Proof. The proof builds on a combination of Proposition 4.12 with Proposition 5.2.

First, observe that ${\mathscr V({{\mathcal F}})}$ , the kernel of p-curvature of ${\mathcal F} = \varphi ^* \mathcal G$ , coincides with $\varphi ^* {\mathscr V({\mathcal G})}$ . Therefore, Proposition 5.2 implies that

$$ \begin{align*} N_{{\mathcal F}} &= \varphi^* N_{\mathcal G} \otimes {\mathcal O}_X\left( - \operatorname{\mathrm{Ram}}(\varphi)_{{\mathcal F}} \right) \\ \omega_{{\mathcal F}} &= \varphi^* \omega_{\mathcal G } \otimes \omega_{\mathcal V(\varphi)} \otimes {\mathcal O}_{X}\left( \operatorname{\mathrm{Ram}}(\varphi)_{{\mathcal F}^{\perp}} \right) \\ \omega_{{\mathscr V({{\mathcal F}})}} &= \varphi^* \omega_{{\mathscr V({\mathcal G})} } \otimes \omega_{\mathcal V(\varphi)} \otimes {\mathcal O}_{X}\left( \operatorname{\mathrm{Ram}}(\varphi)_{{\mathscr V({{\mathcal F}})}^{\perp}} \right) \,. \end{align*} $$

Hence, we deduce from Proposition 4.12 that ${\mathcal O}_X({\Delta _{{\mathcal F}}})$ is isomorphic to

$$\begin{align*}\varphi^* {\mathcal O}_Y({\Delta_{\mathcal G}}) \otimes {\mathcal O}_X \left( - \operatorname{\mathrm{Ram}}(\varphi)_{{\mathcal F}} + p\left( \operatorname{\mathrm{Ram}}(\varphi)_{{{\mathscr V({{\mathcal F}})}}^{\perp}} - \operatorname{\mathrm{Ram}}(\varphi)_{{{\mathcal F}^{\perp}}}\right) \right) \, \end{align*}$$

in the Picard group of X.

Corollary 5.4. Assumptions as in Proposition 5.3. Assume also that X has no non-zero effective divisor linearly equivalent to zero. If $\operatorname {\mathrm {Ram}}(\varphi )_{{{\mathscr V({{\mathcal F}})}}^{\perp }} =\operatorname {\mathrm {Ram}}(\varphi )_{{{\mathcal F}^{\perp }}}=0$ and all the coefficients of ${\Delta _{{\mathcal F}}}$ are strictly smaller than the characteristic of ${\mathsf k}$ , then

$$\begin{align*}{\Delta_{{\mathcal F}}} = \varphi^* {\Delta_{\mathcal G}} - \operatorname{\mathrm{Ram}}(\varphi)_{{\mathcal F}} \,. \end{align*}$$

Proof. According to Proposition 4.4, $\mathcal G$ is defined by a closed rational $1$ -form $\eta $ . Since $\varphi ^* \eta $ is a closed $1$ -form defining ${\mathcal F}$ , Proposition 4.14 implies that

$$ \begin{align*} {\Delta_{\mathcal G}} &= (\eta)_{\infty} - (\eta)_0 \quad\mod p \, , \\ {\Delta_{{\mathcal F}}} &= (\varphi^* \eta)_{\infty} - (\varphi^* \eta)_{0} \quad\mod p. \end{align*} $$

A local computation gives that $(\varphi ^*\eta )_{\infty } - (\varphi ^*\eta )_{0} = \varphi ^*\left ( (\eta )_{\infty } - (\eta )_{0} \right ) - \operatorname {\mathrm {Ram}}(\varphi )_{{\mathcal F}}$ in accordance with Proposition 5.2. Consequently, we can write

$$\begin{align*}{\Delta_{{\mathcal F}}} = \varphi^* {\Delta_{\mathcal G}} - \operatorname{\mathrm{Ram}}(\varphi)_{{\mathcal F}} \quad\mod p \,. \end{align*}$$

Our assumption $\operatorname {\mathrm {Ram}}(\varphi )_{{{\mathscr V({{\mathcal F}})}}^{\perp }} =\operatorname {\mathrm {Ram}}(\varphi )_{{{\mathcal F}^{\perp }}}=0$ combined with Proposition 5.3 implies that ${\Delta _{{\mathcal F}}}$ and $\varphi ^* {\Delta _{\mathcal G}} - \operatorname {\mathrm {Ram}}(\varphi )_{{\mathcal F}}$ are linearly equivalent. Since $\mathcal G$ -invariant hypersurfaces are contained ${\Delta _{\mathcal G}}$ according to Proposition 4.13, the divisor $\varphi ^* {\Delta _{\mathcal G}} - \operatorname {\mathrm {Ram}}(\varphi )_{{\mathcal F}}$ is effective. Therefore, ${\Delta _{{\mathcal F}}}$ and $\varphi ^* {\Delta _{\mathcal G}} - \operatorname {\mathrm {Ram}}(\varphi )_{{\mathcal F}}$ are effective divisors, with the same reduction modulo p, and the coefficients of ${\Delta _{{\mathcal F}}}$ are smaller than p; it follows from our assumptions on X that they must be same divisor.

Example 5.5. Let $n\ge 2$ and $\ell>0$ be integers such that p does not divide $\ell $ . The table below, where $H= \{ x_n=0\}$ , describes the behaviour of the degeneracy divisor of the p-curvature of the pull-back ${\mathcal F} = \varphi ^* \mathcal G$ of a foliation $\mathcal G$ on $\mathbb A^n_{{\mathsf k}}$ under the tame morphism

$$ \begin{align*} \varphi : \mathbb A^n_{{\mathsf k}} \to \mathbb A^n_{{\mathsf k}} \\ (x_1, \ldots, x_{n-1} ,x_n) & \mapsto (x_1, \ldots, x_{n-1}, x_n^{\ell}) \,. \end{align*} $$

5.4 Non-invariant hypersurfaces and differents

Let ${\mathcal F}$ be a distribution on a smooth algebraic variety X and let $H \subset X$ be a smooth hypersurface. Let $\delta $ be the natural composition

If H is not ${\mathcal F}$ -invariant, then the image of $\delta $ is equal to $N_H \otimes \mathcal I$ for some ideal sheaf $\mathcal I \subset {\mathcal O}_H$ . The effective divisor defined by the codimension one part of $\mathcal I$ will be called the different of ${\mathcal F}$ and H and will be denoted by $\mathrm {diff}_{H}({{\mathcal F}})$ . Notice that the kernel of $\delta $ is the tangent sheaf of the distribution

.

Note that when X is a surface and H is a curve, $\mathrm {diff}_{H}({{\mathcal F}})$ is nothing but the tangency divisor of ${\mathcal F}$ and H as defined by Brunella [Reference Brunella8, Chapter 3]. The terminology adopted here is in accordance with [Reference Spicer42, Section 3] and is justified by the analogy with the homonymous concept from birational geometry (see, for instance, [Reference Kollár29, Section 4.1]) suggested by the following observation.

Lemma 5.6. Let ${\mathcal F}$ be a distribution of arbitrary dimension on a smooth algebraic variety X and let $H \subset X$ be a smooth hypersurface. If H is not ${\mathcal F}$ -invariant and

is the restriction of ${\mathcal F}$ to H, then

where we adopt the convention that when $\mathcal H$ is the foliation/distribution by points, then $\omega _{\mathcal H} = {\mathcal O}_H$ and $N_{\mathcal H} = T_H$ .

5.5 Differents and non-invariant subvarieties of higher codimension

A variant of the discussion about differents carried out in Subsection 5.4 can be made for the restriction of a distribution ${\mathcal F}$ to a smooth subvariety Y of codimension greater than one. One extra difficulty comes from the fact that the natural morphism

can have any rank between $0$ and $\operatorname {\mathrm {codim}} Y$ . When it has rank zero, the subvariety Y is ${\mathcal F}$ -invariant. If the rank of $\delta $ is equal to $q=\operatorname {\mathrm {codim}} Y$ , then we will say that Y is strongly transverse to ${\mathcal F}$ . In this case, the image of $\wedge ^q \delta $ is of the form $\det N_Y \otimes \mathcal I$ for some ideal sheaf $\mathcal I \subset {\mathcal O}_Y$ , and we define the different $\mathrm {diff}_{Y}({{\mathcal F}})$ as the effective divisor on Y defined by the codimension one part of $\mathcal I$ .

The definition implies that

for any smooth subvariety $Y \subset X$ strongly transverse to ${\mathcal F}$ . Likewise, we also have a natural analogue of Lemma 5.7.

5.6 The degeneracy divisor of the p-curvature for the restriction to strongly transverse subvarieties

We are now ready to state and prove a result that offers a comparison between the degeneracy divisor of the p-curvature of a foliation and its restriction to a smooth subvariety.

Proposition 5.9. Let ${\mathcal F}$ be a codimension one foliation on a smooth algebraic variety X defined over a field ${\mathsf k}$ of characteristic $p>0$ . If $Y\subset X$ is a smooth subvariety strongly transverse to by ${\mathscr V({{\mathcal F}})}$ (consequently, strongly transverse to ${\mathcal F}$ ) and $\mathcal Y$ is the restriction of ${\mathcal F}$ to Y, then ${\Delta _{\mathcal Y}}$ , the degeneracy divisor of the p-curvature of $\mathcal Y$ , is equal to

Proof. Let $\omega \in H^0(X, \Omega ^1_X\otimes N_{{\mathcal F}})$ be a twisted $1$ -form defining ${\mathcal F}$ . According to Lemma 5.7 the zero divisor $i^* \omega $ is $\mathrm {diff}_{Y}({{\mathcal F}})$ . We can thus write $i^* \omega = \delta \cdot \alpha $ , where $\delta $ belongs to $H^0(Y,{\mathcal O}_Y(\mathrm {diff}_{Y}({{\mathcal F}}))$ and $\alpha \in H^0(Y,\Omega ^1_Y \otimes N_{\mathcal Y})$ is a twisted $1$ -form defining .

If $f \in H^0(X,{\mathcal O}_X({\Delta _{{\mathcal F}}}))$ defines the degeneracy divisor of the p-curvature of ${\mathcal F}$ and $\eta \in H^0(X, \Omega ^1_{{\mathcal F}} \otimes \det N_{{\mathscr V({{\mathcal F}})}} \otimes N^*_{{\mathcal F}})$ is the element exhibited in Subsection 4.7 that defines ${\mathscr V({{\mathcal F}})}$ , then, using Lemma 4.16, we can write

$$\begin{align*}\alpha (v^p ) = \frac{i^* \omega(v^p)}{\delta} = \frac{i^*f}{\delta} \left( i^* \eta (v) \right)^p \end{align*}$$

for every $v \in T_{\mathcal Y}$ . Since the degeneracy divisor of the p-curvature of $\mathcal Y$ is the largest effective divisor ${\Delta _{\mathcal Y}}$ such that ${\Delta _{\mathcal Y}} \le (\alpha (v^p))_0$ for every $v \in T_{\mathcal Y}$ , we deduce that

The result follows from Item (2) of Lemma 5.8 .

5.7 The degeneracy divisor of the p-curvature on $\mathbb P^2_{{\mathsf k}}$ and $\mathbb P^1_{{\mathsf k}} \times \mathbb P^1_{{\mathsf k}}$

In this subsection, we review a result on the structure of the degeneracy divisor of the p-curvature of sufficiently general foliations on $\mathbb P^2_{{\mathsf k}}$ and on Hirzebruch surfaces, and sketch its proof, obtained by the first author in [Reference Mendson34]; see also [Reference Mendson33]. We refer the reader to these works for details.

Proof Sketch of the proof.

Let us first consider the case of $X = \mathbb P^2_{{\mathsf k}}$ . If ${\mathcal F}$ is a foliation of $\mathbb P^2_{{\mathsf k}}$ , then the degree of ${\mathcal F}$ , $\deg ({\mathcal F})$ , is defined by the number of tangencies between of ${\mathcal F}$ a non-invariant line $\ell $ , counted with multiplicities. By definition, $\deg ({\mathcal F}) \ge 0$ and a simple computation shows that $N_{{\mathcal F}} = {\mathcal O}_{\mathbb P^2}(\deg ({\mathcal F}) + 2 )$ and $\omega _{{\mathcal F}} = {\mathcal O}_{\mathbb P^2_{{\mathsf k}}}(\deg ({\mathcal F}) - 1))$ .

If $\deg ({\mathcal F})=0$ , then ${\mathcal F}$ is always p-closed. Indeed, if ${\mathcal F}$ is p-dense, then the degeneracy divisor of the p-curvature would be a section of $N_{{\mathcal F}} \otimes \omega _{{\mathcal F}}^{\otimes p} = {\mathcal O}_{\mathbb P^2_{{\mathsf k}}}(2 - p)$ containing the singular set of ${\mathcal F}$ . A clear absurdity for $p>2$ . In the case $p=2$ , we also reach a contradiction since $\operatorname {\mathrm {sing}}({\mathcal F})$ is non-empty because $c_2(\Omega ^1_{\mathbb P^2_{{\mathsf k}}}(2)) = 1$ .

If $\deg ({\mathcal F})=1$ , then ${\mathcal F}$ is defined by a regular vector field v (since $\omega _{{\mathcal F}} = {\mathcal O}_{\mathbb P^2_{{\mathsf k}}}$ ). In suitable affine coordinates, where $0$ is a singularity of v and the line at infinity is invariant, a general v takes the form

$$\begin{align*}v = \alpha x \frac{\partial}{\partial x} + \beta y \frac{\partial}{\partial y} \,. \end{align*}$$

If $\alpha /\beta \notin \mathbb F_p$ , then ${\mathcal F}$ is p-dense with ${\Delta _{{\mathcal F}}}$ equal to the sum of the lines $\{x=0\}$ , $\{y=0\}$ and the line at infinity.

If $\deg ({\mathcal F})=2$ , then we consider the foliation on $\mathbb P^2_{{\mathsf k}}$ defined, in affine coordinates, by

$$\begin{align*}\omega = (xdy -y dx) + \omega_2, \end{align*}$$

where $\omega _2$ is a general homogeneous $1$ -form with polynomial coefficients of degree two. The foliation defined by $\omega $ leaves the line at infinity invariant and three lines through $0 \in \mathbb A^2_{{\mathsf k}}$ defined by the vanishing of $\omega (R) = \omega _2(R)$ , where $R= x \frac {\partial }{\partial x} + y \frac {\partial }{\partial y}$ is the radial (or Euler) vector field. It follows that the degeneracy divisor is supported on the union of four lines above and a curve C of degree at most p, since the degeneracy divisor of the p-curvature of ${\mathcal F}$ has degree $p+4$ . Blowing up the origin, one sees that the strict transform of C must intersect the strict transform of a line through $0$ in exactly one point, and this implies, after some work, that for $\omega _2$ generic, C is an irreducible and reduced curve of degree p invariant by ${\mathcal F}$ .

For $\deg ({\mathcal F})>2$ , we start with a foliation of $\mathcal G$ on $\mathbb P^2_{{\mathsf k}}$ of degree two, with reduced ${\Delta _{\mathcal G}}$ and leaving invariant three lines in general position, say $\{xyz=0\}$ . We consider the endomorphism for every positive integer $\ell \ge 2$ not divisible by p; one considers the endomorphism of $\mathbb P^2_{{\mathsf k}}$ defined by $\varphi _{\ell }(x:y:z)= (x^{\ell }: y^{\ell }:z^{\ell })$ . The pull-back foliation ${{\mathcal F}}_{\ell } = \varphi _{\ell }^* \mathcal G$ is a foliation with $\omega _{{{\mathcal F}}_{\ell }} = {\mathcal O}_{\mathbb P^2_{{\mathsf k}}}(\ell )$ , hence of degree $\ell +1$ . Moreover, since $\varphi _{\ell }$ is étale on the complement of the invariant divisor $\{ xyz=0\}$ , Proposition 5.3 implies that ${\Delta _{{{\mathcal F}}_{\ell }}}$ is a reduced divisor. This shows that when Assertions (1), (2) and (3) are not valid, then Assertion (4) holds true.

To prove the result on Hirzebruch surfaces, one constructs examples of p-dense foliations with p-curvature having reduced degeneracy divisor by applying a series of birational transformations and ramifications as above starting from examples on $\mathbb P^2_{{\mathsf k}}$ . We refer to [Reference Mendson34] for details.

6.1 Families of foliations

Let T be an irreducible algebraic variety. For us, a family of codimension q foliations parametrized by T consists of an algebraic variety $\mathscr X$ , a smooth and projective morphism $\pi : \mathscr X \to T$ , a line bundle $N_{\mathscr F}$ on $\mathscr X$ , and a foliation $\mathscr F$ on $\mathscr X$ of codimension $q + \dim T$ defined by a relative q-form $\omega \in H^0(\mathscr X, \Omega ^q_{\mathscr X/T} \otimes N_{\mathscr F})$ with codimension two singular set over any point $t \in T$ such that the kernel of

$$ \begin{align*} T_{\mathscr X/T} & \longrightarrow \Omega^{q-1}_{\mathscr X/T} \otimes N_{\mathscr F} \\ v & \mapsto i_v \omega \end{align*} $$

is an involutive subsheaf of $T_{\mathscr X}$ of rank $\dim \mathscr X - \dim T - q$ .

When $\mathscr X = X \times T$ , we will say that $\mathscr F$ is a family of foliations on X.

6.2 Behavior of the kernel of the p-curvature under deformations

If X is a polarized smooth and connected projective variety of dimension n, with polarization given by an ample divisor H, and $\mathcal E$ is a reflexive sheaf on X, we set the degree of $\mathcal E$ with respect to the polarization H equal to $\det (\mathcal E) \cdot H^{n-1}$ .

Recall from the statement of Lemma 4.9 that ${\mathscr V({{\mathcal F}})}$ is the distribution with tangent sheaf determined by the kernel of the p-curvature of a p-dense foliation ${\mathcal F}$ ; that is,

$$\begin{align*}\operatorname{\mathrm{Frob}}^* T_{{\mathscr V({{\mathcal F}})}} = \ker \psi_{{\mathcal F}} \,. \end{align*}$$

Naturally, $\omega _{{\mathscr V({{\mathcal F}})}}= \det (T_{{\mathscr V({{\mathcal F}})}})^*$ is the canonical bundle of ${\mathscr V({{\mathcal F}})}$ .

Lemma 6.2. Let $\mathscr F$ be a family of codimension one foliations on a smooth projective variety X parameterized by an algebraic variety T. If $0 \in T$ is a closed point such that $\mathscr F_0$ is p-dense, then there exists an open neighborhood $U \subset T$ of $0$ such that for every closed point $t \in U$ , the foliation $\mathscr F_t$ is p-dense and the inequality

$$\begin{align*}\deg_H \omega_{{\mathscr V({\mathscr F_t})}} \le \deg_H \omega_{{\mathscr V({\mathscr F_0})}} \end{align*}$$

holds true for any polarization H.

Proof. Consider the relative p-curvature morphism

$$ \begin{align*} {\psi_{\mathscr F/T}} : \operatorname{\mathrm{Frob}}^* T_{\mathscr F/ T} & \longrightarrow N_{\mathscr F} \\ v&\mapsto \omega(v^p) \, , \end{align*} $$

where $\operatorname {\mathrm {Frob}} : \mathscr X \to \mathscr X$ is the absolute Frobenius.

Let $\mathcal C$ be the support of $\operatorname {\mathrm {coker}} {\psi _{\mathscr F/T}}$ and fix an embedding of X in some $\mathbb P^N$ . According to [Reference Hartshorne25, Chapter III, Theorem 9.9], for every $m \gg 1$ , the Hilbert function $h_{\mathcal C}(m,t)= \chi (X, {\mathcal O}_{\mathcal C_t}(m))$ is lower semi-continuous, where ${\mathcal O}_{\mathcal C_t}(m) = {\mathcal O}_{\mathcal C_t} \otimes {\mathcal O}_{\mathbb P^N}(m)$ as usual. Let Z be the subset of T such that for every $z \in Z$ , corresponding to the p-closed foliations in the family. For every t in the complement $U= T-Z$ , the subscheme $\mathcal C_t$ of X is different from X and has divisorial part equal to ${\Delta _{\mathscr F_t}}$ . There exists a positive constant c such that $c \cdot \deg ({\Delta _{\mathscr F_t}})$ is the leading coefficient of the Hilbert function $h_{\mathcal C}(m,t)$ .

Since $\det ( \ker {\psi _{\mathscr F_t}} )^*$ , the Frobenius pull-back of the canonical bundle of the kernel of the p-curvature of $\mathscr F_t$ , is isomorphic to $\det ( \operatorname {\mathrm {Frob}}^* T_{\mathscr X_t})^* \otimes \det ( N_{\mathscr F_t} \otimes \mathcal I_{{\mathcal C}_t} )$ , we deduce from the lower semi-continuity of $h_{\mathscr C}$ , the upper-semicontinuity of the function $t \mapsto \deg (\det ( \ker {\psi _{\mathscr F_t}} ))$ on U as wanted.

If one further assumes that the degeneracy divisor of a p-dense codimension one foliation is free from p-th powers, then one gets the reverse inequality for the degree of the kernel of the p-curvature.

Lemma 6.3. Let $\mathscr F$ be a family of codimension one foliations on a smooth projective variety X parameterized by an algebraic variety T. If $0 \in T$ is a closed point such that $\mathscr F_0$ is p-dense and the degeneracy divisor of its p-curvature is free from p-th powers, then there exists a non-empty open subset $V \subset T$ such that for every closed point $t \in U$ , the foliation $\mathscr F_t$ is p-dense and the equality

$$\begin{align*}\deg_H \omega_{{\mathscr V({\mathscr F_t})}} = \deg_H \omega_{{\mathscr V({\mathscr F_0})}} \end{align*}$$

holds true for any polarization H.

Proof. Thanks to Lemma 6.2, it suffices to show the existence of an open neighborhood U of $ 0 \in T$ such that

$$\begin{align*}\deg_H \omega_{{\mathscr V({\mathscr F_t})}} \ge \deg_H \omega_{{\mathscr V({\mathscr F_0})}} \end{align*}$$

for any $t \in U$ . For that, let $\mathcal L$ be a very-ample line bundle on $\mathscr X = X \times T$ such that $T_{\mathscr F} \otimes \mathcal L$ is generated by global sections. Choose $v \in H^0(\mathscr X, T_{\mathscr F} \otimes \mathcal L)$ such that $v_0$ , the restriction of v to $X \times \{0 \}$ , is generically transverse to ${\mathscr V({\mathscr F_0})}$ . If $\omega \in H^0(\mathscr X, \Omega ^1_{\mathscr X/T} \otimes N_{\mathscr F})$ is a twisted $1$ -form defining the family of foliations $\mathscr F$ , then the section $\sigma = \omega ( v^p) \in H^0(\mathscr X, N_{\mathscr F} \otimes \mathcal L^{\otimes p})$ defines a divisor $\mathscr D$ on $\mathscr X$ . The choice of v guarantees that the support of $\mathscr D$ does not contain $X_0$ . Consequently, there exists an open neighborhood of $0 \in T$ such that the support of $\mathscr D$ does not contain $\mathscr X_t$ . For any $t \in U$ , is, by construction, equal to ${\Delta _{\mathscr F_t}} + p R_t$ for some effective divisor $R_t$ . Let $\mathscr R \in \operatorname {\mathrm {Div}}(\mathscr X)$ be the maximal effective divisor such that $p \mathscr R \le \mathscr D$ . Semi-continuity implies that is equal to $R_t$ for every t in an non-empty open subset V of T (not necessarily containing $0$ ).

Since we are assuming that ${\Delta _{\mathscr F_0}} = \mathscr D_0 - p R_0$ is free from p-th powers, we have that $\mathscr R_0 \le R_0$ . Using that the degrees $\deg _H \mathscr D_t$ and $\deg _H \mathscr R_t$ do not depend on t, we obtain that

$$ \begin{align*} \deg_H {\Delta_{\mathscr F_0}} & \le \deg_H \left( {\Delta_{\mathscr F_0}} + p (R_0 - \mathscr R_0) \right) \\ & = \deg_H \left( \mathscr D_0 \right) - \deg_H\left( p \mathscr R_0 \right) \\ & = \deg_H \left( \mathscr D_t \right) - \deg_H\left( p \mathscr R_t \right) = \deg_H {\Delta_{\mathscr F_t}} \end{align*} $$

for every $t \in V$ . Proposition 4.12 implies the result.

7.1 Lifts of foliations modulo $p^2$

Let ${\mathsf k}$ be an algebraically closed field of characteristic $p>0$ , let $W_{2}{({\mathsf k})}$ be the ring of Witt vectors of length $2$ with values in ${\mathsf k}$ , and set $S= \operatorname {\mathrm {Spec}}(W_{2}{({\mathsf k})})$ . A lift of a smooth algebraic variety X module $p^2$ is a flat smooth S-scheme $X'$ such that $X = X' \otimes {\mathsf k}$ . Likewise, a lift modulo $p^2$ of a foliation on X consists of a lift $X'$ of X modulo $p^2$ and subsheaf $T_{{\mathcal F}'} \subset T_{X'/S}$ , flat over S, such that

1. (1) the sheaf $T_{{\mathcal F}'}$ is a lift of $T_{{\mathcal F}}$ , i.e.,, $T_{{\mathcal F}} = T_{{\mathcal F}'} \otimes {\mathsf k}$ ; and

2. (2) the sheaf $T_{{\mathcal F}'}$ is involutive; that is, the morphism

   $$\begin{align*}\bigwedge^2 T_{{\mathcal F}'} \to \frac{T_{X'/S}}{T_{{\mathcal F}'}} \end{align*}$$
   induced by Lie brackets is identically zero.

As already noted by Miyaoka in [Reference Miyaoka35, Example 4.2], not every foliation admits a lift modulo $p^2$ . As it will be essential in what follows, we present below a variant of Miyaoka’s example.

Proof. First, observe that

$$\begin{align*}\omega \wedge d \omega = p(xyz)^{p-1} dx \wedge dy \wedge dz, \end{align*}$$

and, therefore, the reduction modulo p of $\omega $ defines an involutive subsheaf of $T_{\mathbb A^3_{{\mathsf k}}}$ . Let ${\mathcal F}$ be the corresponding foliation on $\mathbb A^3_{{\mathsf k}}$ . If ${\mathcal F}$ admits a lift modulo $p^2$ and we denote by $\overline \omega $ the reduction of $\omega $ modulo $p^2$ , then such lift is defined by the $1$ -form $\overline \omega + p \eta $ for a suitable $1$ -form $\eta \in \Omega ^1_{\mathbb A^3_{{\mathsf k}}}$ which satisfies

(7.1) $$ \begin{align} (\overline{\omega} + p \eta) \wedge d (\overline{\omega} + p \eta) = 0 \quad \text{in} \quad \Omega^1_{\mathbb A^3_{W_{2}{({\mathsf k})}}}. \end{align} $$

Expanding the left-hand side of Equation (7.1), we get

$$\begin{align*}p\left(z^py^{p-1}dy\wedge d\eta +(xyz)^{p-1}dx\wedge dy \wedge dz +x^{p-1}dx\wedge d\eta\right) = 0. \end{align*}$$

Since $d \eta $ has no terms that are scalar multiples of $(xy)^{p-1}$ , this expression is always non-zero. It follows that ${\mathcal F}$ does not admit a lift modulo $p^2$ .

7.2 Integrability of the Cartier transform of liftable foliations

Our next result is an amplification of Miyaoka’s Example 7.1.

Proof. We will prove the contrapositive, (i.e., if the Cartier transform of a p-dense codimension one foliation ${\mathcal F}$ is not a foliation, then ${\mathcal F}$ does not lift modulo $p^2$ ).

Proposition 4.5 guarantees the existence of a closed rational $1$ -form $\omega $ defining ${\mathcal F}$ . The Cartier transform of ${\mathcal F}$ is defined by the rational $1$ -form $\theta = \mathbf {C}(\omega )$ . By assumption, the $3$ -form $\theta \wedge d \theta $ does not vanish identically. In particular, $\dim X \ge 3$ .

Assume first that $\dim X=3$ . Let $a \in X$ be a closed point such that $\theta $ is regular (no poles) at x and $\theta \wedge d\theta (x) \neq 0$ . Denote by $\mathfrak m \subset {\mathcal O}_{X,a}$ the corresponding maximal ideal. Let $x,y \in \mathfrak m$ be such that $d\theta = dx\wedge dy \ \ \mod \mathfrak m \Omega ^2_{X,a}$ . We can integrate the $2$ -form $d\theta $ modulo $\mathfrak m^2$ and write

$$\begin{align*}\theta = x dy + dz + w^{p-1} dw \quad\mod \mathfrak m^2 \Omega^1_{X,a} \end{align*}$$

for some $z,w \in \mathfrak m$ . If $p>2$ , then we can assume that w is equal to zero as $w^{p-1} \in \mathfrak m^2$ . The non-vanishing of $\theta \wedge d \theta $ at a implies that $x,y,z$ generate $\mathfrak m$ .

Since $\theta = \mathbf {C}(\omega )$ , we can write

$$\begin{align*}\omega = x^p y^{p-1} dy + z^{p-1} dz + df \quad\mod \mathfrak m^2 \Omega^1_{X,a} \end{align*}$$

for some $f \in \mathfrak m$ .

Let $X'$ be a lift of X, let $a'$ be a lift of a and let $\omega '$ be a $1$ -form in $\Omega ^1_{X',a'}$ that defines a lift ${\mathcal F}'$ of ${\mathcal F}$ to $X'$ and with reduction modulo p equal to $\omega $ . We can write

$$\begin{align*}\omega' = (x^py^{p-1}dy + z^{p-1}dz + df') + p \eta \quad\mod \left< p^2, \mathfrak m^{2p} \Omega^1_{X',a'} \right> \, \end{align*}$$

for any lift $f'$ of f and a certain $\eta \in \Omega ^1_{X,a}$ . Expanding $\omega ' \wedge d\omega '$ , we get

$$ \begin{align*} & p \left( ( x^p y^{p-1} dy + z^{p-1} dz + df') \wedge d\eta + (xyz)^{p-1} dx \wedge dy \wedge dz \right) + \\ &+ p \left( (xy)^{p-1} df'\wedge dx \wedge dy \right) \end{align*} $$

modulo $\left < p^2, \mathfrak m^{3p-2} \Omega ^1_{X',a'} \right>$ . As in Example 7.1, for no matter which choice of $f'$ and $\eta $ , the term $(xyz)^{p-1} dx \wedge dy \wedge dz$ will not be cancelled. This shows that no lift of $\omega $ defines a foliation when $\dim X=3$ .

When $\dim X>3$ , we reduce to the previous case by restricting to the intersection of $\dim X - 3$ sufficiently general hyperplane sections.

As pointed out by the referee, thanks to Chow’s Lemma [43, Tag 0200], the result above also holds for proper smooth varieties.

8.1 Reduction modulo p of holomorphic foliations on projective manifolds

Let ${\mathcal F}$ be a singular holomorphic foliation defined on a complex projective manifold X. The variety X and the subsheaf $T_{\mathcal F} \subset T_X$ can be viewed as objects defined over a finitely generated $\mathbb Z$ -algebra $R \subset \mathbb C$ . More precisely, there exists a projective scheme $\mathscr X$ flat over $\operatorname {\mathrm {Spec}}(R)$ and subsheaf $T_{\mathscr F} \subset T_{\mathscr X/R}$ flat over $\operatorname {\mathrm {Spec}}(R)$ such that X is isomorphic to the base change $\mathscr X \otimes _{R} \mathbb C$ and ${\mathcal F}$ is isomorphic to $\mathscr F \otimes _R \mathbb C$ .

The pair $(\mathscr X, \mathscr F)$ is called an integral model for $(X,{\mathcal F})$ and is not uniquely determined by $(X,{\mathcal F})$ . Roughly speaking, it depends on the choice of generators for the ideal of X, on a finite presentation of $T_{{\mathcal F}}$ , as well as on the choice of an open subset of the affine scheme determined by the coefficients used to define the ideal and the presentation.

If ${\mathfrak p} \subset R$ is a maximal ideal, then the residue field ${\mathsf k}({\mathfrak p}) = R/{\mathfrak p}$ is a finite field of characteristic $p>0$ with algebraic closure ${\mathsf k} = \overline {{\mathsf k}({\mathfrak p})} \simeq \overline {\mathbb F_p}$ . The reduction of ${\mathcal F}$ modulo ${\mathfrak p}$ is the foliation ${{\mathcal F}}_{{\mathfrak p}}$ of $X_{{\mathfrak p}} = \mathscr X \otimes _R {\mathsf k}$ determined by $T_{{{\mathcal F}}_{{\mathfrak p}}} = T_{\mathscr F} \otimes _R {\mathsf k}$ .

Example 8.1. Let ${\mathcal F}$ be the codimension one foliation on $\mathbb A^3_{\mathbb C}$ defined by the $1$ -form

$$\begin{align*}\omega = \sqrt{-1} \cdot \frac{dx}{x} + \frac{dy}{y} + \frac{dz}{z} \,. \end{align*}$$

We can take the $\mathbb Z$ -algebra $R =\mathbb Z[\sqrt {-1}] \simeq \mathbb Z[\alpha ]/(\alpha ^2 +1)$ as the ring of definition of ${\mathcal F}$ . The ring R has the following (maximal) prime ideals:

1. (1) only one prime ideal over $2$ , namely $(1+\sqrt {-1})R=(1-\sqrt {-1})R$ ;

2. (2) for each rational prime $p=a^2 + b^2$ congruous to $1$ modulo $4$ , two distinct prime ideals, namely $(a \pm \sqrt {-1}b) R$ ; and

3. (3) for each rational prime p congruous to $3$ modulo $4$ , the prime ideal $pR$ .

If we reduce ${\mathcal F}$ modulo $(1+\sqrt {-1})R$ , we get a p-closed foliation over $\overline {\mathbb F}_2$ defined by $d \log (xyz)$ . Likewise, if we reduce ${\mathcal F}$ modulo $(a\pm b \sqrt {-1})R$ , $p=a^2+b^2$ being a prime congruous to $1$ modulo $4$ , we get a p-closed foliation over $\overline {\mathbb F}_p$ defined by $d \log (xyx)$ . In contrast, if we reduce ${\mathcal F}$ modulo ${\mathfrak p} = pR$ , where p is a rational prime congruous to $3$ modulo $4$ , we get a p-dense foliation with ${\mathscr V({\mathscr F_{{\mathfrak p}}})}$ defined by the kernel of the $2$ -form

$$\begin{align*}\omega_{{\mathfrak p}} \wedge \mathbf{C}( \omega_{{\mathfrak p}} ) = (\alpha_{{\mathfrak p}} \cdot \frac{dx}{x} + \frac{dy}{y} + \frac{dz}{z}) \wedge ( \alpha_{{\mathfrak p}}^{1/p} \cdot \frac{dx}{x} + \frac{dy}{y} + \frac{dz}{z}) \, , \end{align*}$$

where $\alpha _{{\mathfrak p}}$ stands for the class of $\alpha $ in the algebraic closure of $R/{\mathfrak p} \simeq \mathbb F_{p^2}$ .

Example 8.2. Now let ${\mathcal F}$ be the codimension one foliation on $\mathbb A^3_{\mathbb C}$ defined by the $1$ -form

$$\begin{align*}\omega = \alpha \cdot \frac{dx}{x} + \frac{dy}{y} + \frac{dz}{z} \, , \end{align*}$$

with $\alpha $ equal to an arbitrary transcendental number. We can take the $\mathbb Z$ -algebra $R=\mathbb Z[\alpha ]$ as the ring of definition of ${\mathcal F}$ . The maximal ideals of the ring R are of the form $(p,f(\alpha )) R $ , where p is a rational prime and $f \in \mathbb Z[\alpha ]$ is a monic polynomial with irreducible reduction modulo p. The reduction of ${\mathcal F}$ modulo ${\mathfrak p} = (p,f(\alpha )) R$ is p-closed if the degree of f is one; otherwise, $\mathscr F_{{\mathfrak p}}$ is p-dense.

8.2 A conjecture by Ekedahl, Shepherd-Barron and Taylor

In general, it is not easy to draw conclusions about the geometry of the holomorphic foliation ${\mathcal F}$ from the behavior of the reduction modulo ${\mathfrak p}$ for different primes ${\mathfrak p}$ .

Despite the important particular cases settled by Bost in [Reference Bost6], Conjecture 8.3 is wide open. It must be mentioned that Conjecture 8.3 generalizes a conjecture by Grothendieck and Katz, which is also wide open despite recent advances [Reference Shankar41, Reference Patel, Shankar and Whang37].

Under the assumptions of Conjecture 8.3, Theorem 4.1 implies the existence of algebraic subvarieties invariant by $\mathscr F_{{\mathfrak p}}$ that cover $X_{{\mathfrak p}}$ for a Zariski open and dense set of maximal primes ${\mathfrak p} \subset \operatorname {\mathrm {Spec}}(R)$ . This is not enough to guarantee the existence of algebraic subvarieties invariant by ${\mathcal F}$ covering X as the degrees of the subvarieties of $X_{{\mathfrak p}}$ invariant by $\mathscr F_{{\mathfrak p}}$ may be unbounded as a function of ${\mathfrak p}$ .

8.3 Lifting subvarieties/subdistributions of bounded degree

If one can guarantee the existence of subvarieties/subdistributions for the reduction of a foliation modulo a Zariski dense set of maximal primes with degree uniformly bounded, then the lemma below implies the existence of the same type of objects in characteristic zero respecting the same bounds.

To obtain a priori upper bounds for the degree of invariant subvarieties of the reduction to positive characteristic does not seem more manageable than obtaining the same upper bounds in characteristic zero. In contrast, if ${\mathcal F}$ is a holomorphic foliation such that the reduction modulo ${\mathfrak p}$ is not p-closed for a Zariski dense set of primes ${\mathfrak p} \subset \operatorname {\mathrm {Spec}}(R)$ , then the degree of the canonical bundle of ${\mathscr V({\mathscr F_{{\mathfrak p}}})}$ is uniformly bounded as a function of ${\mathfrak p}$ . Therefore, one gets from the lemma above the existence of certain subdistributions in characteristic zero. In Section 9, we will explore this fact to obtain new results about the irreducible components of the space of holomorphic codimension one foliations on projective spaces.

8.4 A cautionary remark

For foliations of codimension one, Conjecture 8.3 can be rephrased as follows:

By dropping the codimension one assumption, one obtains a strengthening of Conjecture 8.3, which turns out to be false as we now proceed to show.

Proof. By semi-continuity, there exists an open and dense subset $U \subset \operatorname {\mathrm {Spec}}(R)$ such that $h^0(\mathscr X_{{\mathfrak p}}, \Omega ^2_{\mathscr X_{{\mathfrak p}}})=1$ for every maximal prime ${\mathfrak p} \in U$ . For ${\mathfrak p} \in U$ , let $\omega _{{\mathfrak p}}$ be the generator of $H^0(\mathscr X_{{\mathfrak p}}, \Omega ^2_{\mathscr X_{{\mathfrak p}}})$ or, equivalently, the reduction modulo ${\mathfrak p}$ of a generator of $H^0(\mathscr X, \Omega ^2_{\mathscr X})$ . Replacing U by an open subset, we can assume that $\omega _{{\mathfrak p}}^n\neq 0$ for every ${\mathfrak p} \in U$ .

Fix an arbitray maximal prime ${\mathfrak p} \in U$ . Since $\omega _{{\mathfrak p}}^n\neq 0$ , the kernel of the morphism from $T_{X_{{\mathfrak p}}}$ to $\Omega ^1_{X_{{\mathfrak p}}}$ defined by contraction with $\omega _{{\mathfrak p}}$ is the tangent sheaf of a foliation ${{\mathcal F}}_{{\mathfrak p}}$ of dimension one. Let v be any rational section of $T_{{{\mathcal F}}_{{\mathfrak p}}}$ . Since $h^0(X,\Omega ^2_{X_{{\mathfrak p}}})=1$ , $\mathbf {C}(\omega _{{\mathfrak p}})$ is a multiple of $\omega _{{\mathfrak p}}$ . Consequently, $i_v \mathbf {C}(\omega _{{\mathfrak p}}) = 0$ . According to Lemma 4.16,

$$\begin{align*}0 = i_v \mathbf{C}(\omega_{{\mathfrak p}}) = \mathbf{C}(i_{v^p} \omega_{{\mathfrak p}} - (L_v)^{p-1}(i_v\omega_{{\mathfrak p}})) = \mathbf{C}(i_{v^p} \omega_{{\mathfrak p}}). \end{align*}$$

The vanishing of $ \mathbf {C}(i_{v^p} \omega )$ implies the existence of a rational function f on $X_{{\mathfrak p}}$ such that $i_{v^p} \omega _{{\mathfrak p}} = df$ (i.e., $i_{v^p} \omega _{{\mathfrak p}}$ is exact). If $df=0$ , then ${{\mathcal F}}_{{\mathfrak p}}$ is p-closed; if not, then f is a non-trivial rational first integral for ${{\mathcal F}}_{{\mathfrak p}}$ . In both cases, ${{\mathcal F}}_{{\mathfrak p}}$ is not p-dense as claimed.

If X is a complex projective symplectic manifold of dimension $2n$ with symplectic form $\omega $ and $Y\subset X$ is a smooth hypersurface with big normal bundle, then the characteristic foliation of Y (i.e., the one-dimensional foliation on Y induced by the pull-back of $\omega ^{n-1}$ to Y) has Zariski dense general leaf [Reference Abugaliev3, Theorem 1.7]. In contrast, Proposition 8.5 above implies that any integral model for ${\mathcal F}$ is not p-dense for almost every prime ${\mathfrak p}$ .