1. Introduction
In [Reference CampanaCam04], Campana introduced the class of special varieties as the compact Kähler varieties admitting no maps onto an orbifold of general type. Campana also established that a Kähler manifold 
$X$ is special if and only if it has no Bogomolov sheaf, which are rank-one coherent subsheaves of 
$\Omega _X^p$ having Kodaira dimension 
$p$ for some 
$p>0$. Campana conjectured that special varieties should have properties related to Lang's conjectures on the distribution of entire curves or rational points on projective manifolds. More precisely, he formulated the following conjecture.
Conjecture 1 (Campana [Reference CampanaCam04])
A compact Kähler manifold 
$X$ is special if and only if it contains a Zariski-dense entire curve 
$f:\mathbb {C} \to X$, i.e. the image of 
$X$ is not contained in any proper subvariety of 
$X$. Moreover, if 
$X$ is projective and defined over a number field 
$k$, then 
$X$ is special if and only if 
$X(k)$ is potentially dense, i.e. 
$X(k')$ is Zariski dense in 
$X$ for some finite extension 
$k' \supset k$.
Recently, Wu [Reference WuWu20] introduced the notion of numerical Bogomolov sheaves as rank-one coherent subsheaves 
$L$ of 
$\Omega _X^p$ having numerical dimension 
$p$ for some 
$p>0$, and a complex projective (or, more generally, compact Kähler) manifold 
$X$ is said to be numerically special if it has no numerical Bogomolov sheaves. Campana [Reference CampanaCam20, Remark 7.3] raises the question of whether specialness is equivalent to numerical specialness.
It is worth noting that the existence of 
$L$ as given previously determines a distribution, namely 
$\mathrm {Ker}\ L$ of codimension at least 
$p$, where equality holds if 
$p=1$, and that this distribution is actually integrable, according to a theorem of Demailly [Reference DemaillyDem02]. This explains why foliations enter into the picture (see, in particular, Theorem D and § 7).
In this note, we address this problem for subsheaves of 
$\Omega ^1_X$, proving the following result.
Theorem A Let 
$X$ be a compact Kähler manifold admitting a rank-one coherent subsheaf 
$L \subset \Omega ^1_X$ of numerical dimension one. Then 
$X$ is not special, i.e. it admits a rank-one coherent subsheaf of maximal Kodaira dimension 
$p$ in 
$\Omega _X^p$ for some 
$p>0$.
We also study the conjectural characterization of special manifolds following Conjecture 1. Concerning the analytic characterization using entire curves, we prove the following.
Theorem B Let 
$X$ be a compact Kähler manifold admitting a rank-one coherent subsheaf 
$L \subset \Omega ^1_X$ of numerical dimension one. Then 
$X$ has no Zariski-dense entire curves 
$f:\mathbb {C} \to X.$
On the arithmetic side, we are not able to deal with rational points but rather we study a function field version of Campana's conjecture recently introduced in [Reference Javanpeykar and RousseauJR22]. In this setting, the analogue of potential density is given by geometric specialness as follows.
Definition 1.1 (Geometrically special varieties)
A complex projective variety 
$X$ is geometrically special if, for every dense open subset 
$U\subset X$, there exists a smooth projective connected curve 
$C$, a point 
$c$ in 
$C$, a point 
$u$ in 
$U$, and a sequence of morphisms 
$f_i:C\to X$ with 
$f_i(c) = u$ for 
$i=1,2,\ldots$ such that 
$C\times X$ is covered by the graphs 
$\Gamma _{f_i}\subset C\times X$ of these maps, i.e. the closure of 
$\bigcup _{i=1}^{\infty } \Gamma _{f_i}$ equals 
$C\times X$.
Then the analogue of Campana's conjecture on potential density is formulated as follows.
Conjecture 2 [Reference Javanpeykar and RousseauJR22]
A complex projective variety 
$X$ is special if and only if it is geometrically special.
In this setting, we prove the following.
Theorem C Let 
$X$ be a complex projective manifold admitting a rank-one coherent subsheaf 
$L \subset \Omega ^1_X$ of numerical dimension one. Then 
$X$ is not geometrically special.
One of the main ingredients in the proof of the previous results is the following statement of independent interest which adapts to compact Kähler manifolds previous work of the third author on projective manifolds [Reference TouzetTou16].
Theorem D Let 
$(X,\mathcal {F})$ be a foliated Kähler manifold such that 
$\mathcal {F}$ is a holomorphic codimension-one transversely hyperbolic foliation with quotient singularities. Assume that 
$\mathcal {F}$ is not algebraically integrable. Then, up to replacing 
$X$ by a non-singular Kähler modification, there exists a morphism 
$\Psi : X\to \mathbb {D}^N/\Gamma$ whose image has dimension 
$p\geq 2$ such that 
$\mathcal {F}=\Psi ^*\mathcal {G}$ where 
$\mathcal {G}$ is one of the tautological foliation on 
$\mathbb {D}^N/\Gamma$. Moreover, the hyperbolic transverse structure of 
$\mathcal {F}$ agrees with that obtained from pull-back.
Towards a generalization of the preceding results to foliations with higher codimensions, we prove the following statement.
Theorem E Let 
$X$ be a compact Kähler manifold and 
$\mathcal {F}$ be a smooth foliation of codimension 
$p$. If 
$c_1 (N_\mathcal {F}^*)$ is represented by a semi-positive 
$(1,1)$-form 
$\eta$ of constant rank 
$p$, then 
$X$ is not special.
The paper is organized as follows. In § 2, we collect some preliminary definitions and properties of transversely hyperbolic foliations. In § 3, we state the main properties of the conormal bundle of tautological foliations on irreducible polydisk quotients. In § 4, we prove Theorem D and derive Theorem A from it. In § 5, we prove Theorem B on entire curves. In § 6, we prove Theorem C on non-potential density in the (split) function field setting. Finally, in § 7, we prove Theorem E.
2. Transversely hyperbolic foliations
In this section, we collect useful information about transversely hyperbolic foliations on complex manifolds. We follow the terminology of [Reference Lo Bianco, Pereira, Rousseau and TouzetLPRT20, §§ 3 and 5].
2.1 Transversely hyperbolic foliations
Let 
$\mathcal {F}$ be a codimension-one foliation on a complex manifold 
$X$. The foliation 
$\mathcal {F}$ is transversely hyperbolic if the sheaf of holomorphic first integrals 
$\mathcal {O}_{X/\mathcal {F}}$ admits a locally constant subsheaf of sets 
$\mathcal {I}$ (called the sheaf of distinguished first integrals) such that:
(1) every
$f \in \mathcal {I}$ is non-constant and has image contained in the unit disk $\mathbb {D}$;(2) for every non-empty, connected, and simply connected open subset
$U$, $\mathcal {I}(U)$ is non-empty and equal to $\operatorname {Aut}(\mathbb {D}) \cdot f$ for any $f \in \mathcal {I}(U)$;(3) if
$f \in \mathcal {I}(U)$, $g \in \mathcal {I}(V)$, and $U\cap V$ is a connected open set then there exists $\varphi \in \operatorname {Aut}(\mathbb {D})$ such that $\varphi \circ f = g$.
The pull-back of the Poincaré metric on the unit disk by any local distinguished first integral 
$f \in \mathcal {I}$ is a closed semi-positive smooth 
$(1,1)$-form
\[ \eta=f^*\bigg( \frac{i}{\pi} \frac{du\wedge d \overline{u}}{(1-|u|^2)^2} \bigg)= -\frac{i}{\pi} \partial\bar{\partial} ( \log (1-{ |f|}^2) ) \]
that does not depend on the choice of 
$f$. If 
$\omega$ is a local generator of 
$N^*_{\mathcal {F}}$ then the 
$(1,1)$ form
\[ \eta = \frac{i}{\pi} \exp( 2 \psi) \omega \wedge \overline{\omega} \]
defines by duality a (singular) metric on the conormal bundle 
$N_\mathcal {F}^*$ with a plurisubharmonic continuous local weight 
$\psi =-\log (1-{ |f|}^2) +\log ( |g|)$ where 
$g$ is a holomorphic function such that 
$df=g\omega$. Therefore, the curvature form of the induced metric on 
$N^*_{\mathcal {F}}$ is (in the sense of currents)
\begin{equation} T= \frac{i}{\pi} \partial \overline {\partial} \psi = \sum_{D\in \mathcal{P}} m_D [D] +\eta, \end{equation}
the (locally finite) sum being taken over the set 
$\mathcal {P}$ of prime divisors. Here, 
$m_D\ge 0$ denotes the ramification order of the local distinguished first integrals along 
$D$. In other words, a distinguished first integral at a neighborhood of a general point of 
$D$ is of the form 
$z^{1+m_D}$ where 
$\{z=0\}$ is a suitable local defining equation of 
$D$. In particular, 
$T=\eta$ if, and only if, the developing map 
$\tilde {\varphi }: \tilde X\to \mathbb {D}$ is a submersion in codimension one. The divisor 
$\sum _{D\in \mathcal {P}} m_D D$ is the ramification divisor of the transversely hyperbolic foliation 
$\mathcal {F}$. Note, in particular, that 
$D$ is 
$\mathcal {F}$-invariant whenever 
$m_D >0$. Note also that 
$T$ is a closed positive 
$(1,1)$-current, whence the following result.
Proposition 2.1 If 
$X$ is a complex compact manifold and 
$\mathcal {F}$ is a transversely hyperbolic foliation on 
$X$, then 
$c_1(N^*_{\mathcal {F}})$ is pseudo-effective.
2.2 Pull-back of transversely hyperbolic structure
Let 
$f:X\to Y$ be a holomorphic map between complex manifolds and let 
$\mathcal {F}$ be a transversely hyperbolic foliation on 
$Y$. One can define the pull-back foliation 
$f^*\mathcal {F}$ provided that the image of the differential 
$df$ is not tangent to 
$\mathcal {F}$. In this case, if 
$\mathcal {F}$ carries a transverse hyperbolic structure with a sheaf of distinguished first integrals 
$\mathcal {I}$, then 
$f^*\mathcal {F}$ carries a transverse hyperbolic structure defined by the sheaf of distinguished first integrals 
$f^*\mathcal {I}$.
2.3 Transversely hyperbolic foliations with quotient singularities
We say that a codimension-one foliation 
$\mathcal {F}$ is a transversely hyperbolic foliation with poles if there exists a hypersurface 
$H$ such that 
${\mathcal {F}}{|_{X-H}}$ is a transversely hyperbolic foliation as defined in § 2.1.
According to [Reference Lo Bianco, Pereira, Rousseau and TouzetLPRT20, Corollary 5.3], any transversely hyperbolic foliation defined on 
$X-H$, where 
$H$ is an hypersurface, extends through 
$H$ as a foliation. Moreover, [Reference Lo Bianco, Pereira, Rousseau and TouzetLPRT20, Theorem 5.2] describes the degeneracies of the transverse hyperbolic structure along 
$H$.
In this work, we are interested in the following subclass of the class of transversely hyperbolic foliations with poles.
Definition 2.2 A transversely hyperbolic foliation with quotient singularities on a complex compact manifold 
$X$ consists of a reduced divisor 
$H = \sum { H_i}$ (the divisor of poles of the transverse structure) and a transversely hyperbolic foliation 
$\mathcal {F}$ on 
$X-H$ such that for any 
$x \in H$, there exists a neighborhood 
$U_x\subset X$ of 
$x$, such that the local monodromy of the transverse hyperbolic structure
\[ \pi_1(U_x-H) \to \operatorname{Aut}(\mathbb{D}) \]is non-trivial and has finite image.
Let 
$U_x$ be as above. Every finite subgroup of 
$\operatorname {Aut}(\mathbb {D})$ is cyclic and generated by an elliptic transformation. Therefore, there is no loss of generality in assuming that the image of the local monodromy representation takes values in 
$S^1$. Let 
$f$ be the corresponding multivalued distinguished first integral. Then 
$f^n: U_x-H\to \mathbb {D}$ is well defined and extends through 
$H$ by boundedness as a holomorphic function 
$g:U_x\to \mathbb {D}$. This implies that, on 
$U_x$, 
$f$ takes the form
\begin{equation} f=f_x f_1^{ \nu_1} \cdots f_r^{\nu_r} \end{equation}
where the 
$\nu _1, \ldots, \nu _r$ are positive non-integral rational numbers, 
$f_1 \cdots f_r=0$ is a local defining equation of 
$H$ and 
$f_x$ is a holomorphic function. One can moreover assume, up to adding a non-negative integer exponent to the 
$\nu _i$ that 
$f_x$ is not identically zero on each branch 
$H_i\cap U_x$. Remark also that 
$H$ is necessarily 
$\mathcal {F}$-invariant.
The local expression for 
$f$ makes clear that the pull-back of the Poincaré metric on 
$\mathbb {D}$ by any multivalued distinguished first integral for 
${\mathcal {F}}{|_{U_x-H}}$ extends through 
$H$ as the closed positive current
\[ \eta_x= -\frac{i}{\pi} \partial\bar{\partial} \left( \log (1-{ |f|}^2)\right). \]If one consider the (locally) well-defined logarithmic form
\[ \xi=\dfrac{df}{f}= \sum_i\nu_i \dfrac{df_i}{f_i} + \dfrac{df_x}{f_x}, \]one can rewrite
\[ \eta_x= \frac{i}{\pi} \left(\dfrac{{ |f|}^2} { { (1-{ |f|}^2)}^2}\right)\xi\wedge \bar{\xi}. \] Set 
$g=f_x\prod _i f_i$. The holomorphic form 
$\omega = g\xi$ has no zeros in codimension one (see the proof of Proposition 2.6), hence it is a local generator of 
$N^*_{\mathcal {F}}$ in a neighborhood of 
$x$. One can then readily check that
\[ \eta_x = \frac{i}{\pi} \exp( 2 \psi) \omega \wedge \overline \omega \]with
\[ \psi= -\log (1-{ |f|}^2) +\sum_{i=1}^r ( \nu_i-1) \log( |f_i|) +\sum_{j=1}^s(m_j-1)\log( |f_ { x,j}|), \]
where 
$f_x=\prod _{j=1}^{s} f_ { x,j}^{m_j}$ is the writing of 
$f_x$ as a product of irreducible factors. By construction, these local 
$(1,1)$-forms glue together with that defined on 
$X-H$ by (2.1) and then give rise by duality to a global singular metric on 
$N^*_{\mathcal {F}}$ with local weight 
$\psi$. Observe that 
$\eta$, considered as a closed positive current, has a 
$(1,1)$ continuous plurisubharmonic potential of the form 
$\varphi = -\log (1-{ |f|}^2)$ so that its Lelong numbers 
$\nu (\eta,x)=0$ at any 
$x\in X$. When 
$X$ is compact, Demailly's approximation Theorem [Reference DemaillyDem92, Theorem 4.1] implies that 
$\eta$ represents a 
${nef}$ class in 
$H_{\partial \bar {\partial }}^{1,1}(X,\mathbb {R})$. Observe also that 
$\eta$ is nothing but the unique positive current giving no mass to 
$H$ and extending the semi-positive form 
$\eta _{|X-H}$. In particular, 
$\eta$ coincides with its absolute continuous part with respect to the Lebesgue measure.
Let us make explicit the curvature current 
$T$ of the singular metric defined by 
$\eta$ on 
$N_\mathcal {F}^*$ in restriction to 
$U_x$. A straightforward calculation yields
\[ T_{ |U_x}=\frac{i}{\pi} \partial \overline{\partial} \psi = \eta + \sum_{i=1}^r (\nu_{i} - 1 ) [H_i]_{ | U_x}+ [D_x] , \]
where 
$D_x$ is an integral effective divisor whose support lies in the polar locus of the logarithmic derivative 
${df_x}/{f_x}$. More precisely, if 
$D=D_1+\cdots +D_s$ is the divisor of poles of 
${df_x}/{f_x}$ with corresponding residues 
$m_1,\ldots,m_s$, then 
$D_x=\sum _{i=1}^s (m_i-1) D_i$.
The previous discussion is summarized in the following result.
Proposition 2.3 Let 
$X$ be a complex manifold and let 
$\mathcal {F}$ be a transversely hyperbolic foliation with quotient singularities on 
$X$. Let 
$H$ be the divisor of poles of the transverse structure. Consider the (singular) metric on 
$N_\mathcal {F}^*$ defined by 
$\eta$, the trivial extension through 
$H$ of the pull-back of the Poincaré metric by local distinguished first integrals. Let 
$T$ be its curvature current (in particular 
$T$ represents 
$c_1(N^*_{\mathcal {F}})\in H_{\partial \bar {\partial }}^{1,1} (X,\mathbb {R}) )$, then
\begin{equation} T =\sum_{ D\in\mathcal{P}} r_D [D] +\eta, \end{equation}
where 
$D$ ranges over the set 
$\mathcal {P}$ of prime divisors, 
$r_D \in \mathbb {\mathbb {Q}}_{>-1}$ and the sum is locally finite. Moreover:
(1) the current
$\eta$ is a smooth semi-positive $(1,1)$-form in restriction to $X-H$ and when $X$ is compact, represents a nef class in $H_{\partial \bar {\partial }}^{1,1}(X,\mathbb {R})$; and(2) if
$r_D\not =0$, $\mathcal {F}$ admits, at a general point of $D$, a distinguished (maybe multivalued) first integral of the form $z^{r_D +1}$ where $z=0$ is a local defining equation of $D$; and(3) the set
$\{D\in \mathcal {P} \arrowvert r_D\notin \mathbb {N}\}$ coincides with the set $\{H_i, i\in I\}$.
Definition 2.4 The divisor 
$\sum _{ D\in \mathcal {P}} r_D D$ will be called the divisorial part of 
$\mathcal {F}$ (with respect to the given transversely hyperbolic structure).
Remark 2.5 The decomposition presented in (2.3) is compatible with restriction to open subsets 
$U$ (where the transverse hyperbolic structure of 
$ {\mathcal {F}}{|_{U}}$ is given by restriction of the sheaf 
$\mathcal {I}$). In particular, the divisorial part of the restricted transverse structure is just 
$\sum _{ D\in \mathcal {P}} r_D {D}{|_{U}}$.
2.4 Divisorial part along invariant hypersurfaces
Let 
$\mathcal {F}$ be a transversely hyperbolic foliation with quotient singularities on a complex manifold 
$X$. Let 
$H=\sum H_i$ be its divisor of poles. Let us denote by 
$\mathcal {I}_{d \log }$ the sheaf defined on 
$X-H$ by the collections of logarithmic differential 
$df/f$ where 
$f\in \mathcal {I}$, see also [Reference TouzetTou13, Définition 5.3].
Proposition 2.6 Let 
$K$ be a hypersurface of 
$X$. Assume that there exists a neighborhood 
$U$ of 
$K$ and a section of 
$\mathcal {I}_{d \log }$ on 
$U-K$ which extends through 
$K$ as a logarithmic one-form 
$\omega$ such that 
$K\subset { (\omega )}_\infty$. The following assertions hold true.
(1) The irreducible components of the hypersurface
$K$ are $\mathcal {F}$-invariant.(2) The germ of
$\omega$ along $K$ is unique.(3) If
$D$ is a prime divisor of $U$, then the residue $\lambda _D$ of $\omega$ along $D$ belongs to $\mathbb {Q}_{\geq 0}$.(4) Up to shrinking
$U$, $\omega$ has no zeros in codimension one and the divisorial part of $ {\mathcal {F}}{|_{U}}$ is $\sum _{ D\in \mathcal {P}} r_D D$ where $r_D= 0$ if $\lambda _D=0$, and $r_D= \lambda _D- 1$ otherwise.
Proof. Item (1) is obvious. Indeed, 
$K$ is a component of the polar locus of a closed meromorphic form defining the foliation on 
$U$.
Let 
$x\in K$. If 
$x\notin H$, then there exists in the neighborhood 
$U_x$ of 
$x$ and a section 
$f$ of 
$I$ over 
$U_x-K$ such that 
$\omega ={df}/{f}$. As noted previously, 
$f$ extends through 
$K$ as a section of 
$\mathcal {I}$ over 
$U$. As 
$K\subset { (\omega )}_\infty$, one necessarily has 
$f(x)=0$. Hence, this section is unique modulo multiplication by a complex number of modulus one. Consequently, 
$\omega$ is unique in restriction to 
$U-H$.
If 
$x\in H\cap K$, there exists a neighborhood 
$U_x$ of 
$x$ and a multivalued distinguished first integral 
$f$ on 
$U_x-K$ with finite and non-trivial multiplicative monodromy taking values in 
$S^1$. The uniqueness of 
$\omega ={df}/{f}$ follows from the observations already made in § 2.3. This establishes the uniqueness stated in item (2).
Let 
$F=f_1\cdots f_r \cdot f_{r+1}\cdots f_p=0$ be a local reduced equation for the polar locus of 
$\omega$ in a small neighborhood 
$U_x$ of 
$x\in K$, where 
$f_1\cdots f_r=0$ is a local equation for 
$H$. By construction, there exists 
$\nu _1,\ldots,\nu _r\in {\mathbb {Q}}_{>0}$, 
$m_{r+1},\ldots,m_{p}\in \mathbb {N}_{>0}$, such that
\[ \omega_{|U_x}= \sum_{i=1}^r \nu_i\frac{df_i}{f_i} +\sum_{i=r+1}^p {m_i}\frac{df_i}{f_i} +\omega_0 \]
where 
$\omega _0$ is some holomorphic one form. In particular, the property mentioned in item (3) is satisfied.
Equivalently, 
$\mathcal {F}$ admits on 
$U_x$ a multivalued distinguished first integral of the form 
$e^{\int \omega }=f=uf_1^{\nu _1}\cdots f_r^{\nu _r} f_{r+1}^{m_{r+1}}\cdots f_p^{m_p}$ where 
$u$ is a unit. As previously, this enables the computation of the divisorial part of 
$ {\mathcal {F}}{|_{U_x}}$, namely 
$\sum _{i=1}^r( \nu _i -1) D_i+\sum _{i=r+1}^p (m_i-1)D_i$ where 
$D_i=\{f_i=0\}$. This proves the second assertion of item (4).
By considering the well-defined real first integral 
$g=|f|$, one remark that there is no invariant hypersurface passing through 
$x$ except the poles 
$D_i$. In particular, the germ of 
$\omega$ along 
$K$ has no zeros in codimension one. This establishes the first point of item (4).
2.5 Behavior under pull-back by a surjective morphism
If 
$\varphi :X\to Y$ is a surjective morphism between complex compact manifolds and 
$\mathcal {F}$ is a transversely hyperbolic foliation with quotient singularities on 
$Y$, the pull-back foliation 
$\varphi ^*\mathcal {F}$ carries also a transversely hyperbolic structure with quotient singularities directly inherited from that of 
$\mathcal {F}$, that is induced on 
$X-\varphi ^{-1} (H)$ by the sheaf 
$\varphi ^*{\mathcal {I}}$. From Proposition 2.3, one obtains a decomposition of 
$N_\mathcal {F}^*$ which reads (in 
$\mathrm {Pic} (Y)\otimes \mathbb {Q}$) as
\[ N_\mathcal{F}^*= L+D, \]
where 
$D$ is the divisorial part of 
$\mathcal {F}$ (see Definition 2.4), 
$L$ is a nef 
$\mathbb {Q}$-line bundle whose Chern class is represented by 
$\eta$. A similar decomposition holds for the conormal sheaf of 
$\varphi ^* \mathcal {F}$. Both decompositions are indeed naturally related as shown by the next result.
Proposition 2.7 With assumptions and notation as previously,
\[ N_{\varphi^* \mathcal{F}}^*= \varphi^* (L) + D' \]
where 
$D'$ is the divisorial part of 
$\varphi ^* \mathcal {F}$.
Proof. Let 
$\mathcal {G}=\varphi ^*\mathcal {F}$. On 
$X$, we have a 
$\mathcal {G}$-invariant divisor 
$I$ which, roughly speaking, is the locus where 
$\varphi$ ramifies over the direction transverse to 
$\mathcal {F}$. More precisely, if 
$\omega$ is a generator of 
$N^*_{\mathcal {F}}$ on an open subset 
$U$, the restriction of 
$I$ to 
$\varphi ^{-1}(U)$ is the zeros divisor of 
$\varphi ^*\omega$. The line bundles 
$\varphi ^* N_\mathcal {F}^*$ and 
$N_\mathcal {G}^*$ are related by the equality 
$N_\mathcal {G}^*=\varphi ^* N_\mathcal {F}^* +I.$
Then, we have just to verify that 
$D'=\varphi ^*D +I$ where 
$D$ is the divisorial part of 
$\mathcal {F}$. It suffices to show that the equality
\begin{equation} D'_{|\varphi^{-1}(U)} =\varphi^*D_{|U} +I_{|\varphi^{-1}(U)} \end{equation}
holds for every member 
$U$ of an open cover 
${(U)}_{U\in \mathcal {U}}$ of 
$Y$. First, let 
$x\in Y-\mathrm {Supp}(D)$. In some neighborhood 
$U$ of 
$x$, 
$N_\mathcal {F}^*$ is generated by 
$df$ where 
$f\in \mathcal {I} (U)$ and (2.4) is obviously true.
If 
$x\in \operatorname {Supp}(D)$, let 
$D_1,\ldots,D_q$ be the components of 
$\operatorname {Supp}(D)$ such that 
$x\in D_i, i=1,\ldots,q$ are ordered such that 
$r_{D_i}\notin \mathbb {N}$ for 
$i=1,\ldots,p$, 
$r_{D_i}\in \mathbb {N}_{>0}$ for 
$i=p+1,\ldots,q$. According to § 2.3, 
$\mathcal {F}$ is defined in some small neighborhood 
$U$ of 
$x$ by a closed logarithmic form 
$\xi = \sum _{i=1}^q (r_{D_i}+1){df_i}/{f_i} + \sum _{i=q+1}^s {df_i}/{f_i}$ where 
$f_i=0$, 
$i\leq q$, is a local reduced equation of 
$D_i$ and 
$f_i=0$, 
$i>q$, are additional poles with residues equal to one. Moreover, 
$ {\xi }{|_{U-\bigcup _{i=1}^p D_i}}$ is a section of 
$\mathcal {I}_{d\log }$. Note that 
$\omega = \varphi ^*\xi$ is a closed logarithmic form on 
$\varphi ^{-1} (U)$ fulfilling the hypothesis of Proposition 2.6, with 
$K=\varphi ^{-1}( \bigcup D_i)$ (in restriction to 
$\varphi ^{-1} (U))$. Item (4) of Proposition 2.6 determines the divisorial part of 
$ {\mathcal {G}}{|_{\varphi ^{-1}(U)}}$. An elementary calculation yields
\[ D'_{\arrowvert\varphi^{-1} (U)}={\bigg(\sum_{i=1}^q r_{D_i} \varphi^*(D_i) +\sum_{i=1}^s (\varphi^* (D_i)-\varphi^* (D_i)_{\rm red}) \bigg)}{|_{\varphi^{-1}(U)}}. \] On the other hand, according to the first part of item (4) of Proposition 2.6, 
$f\omega$ is a local generator of 
$\mathcal {F}$ on 
$U$, where 
$f=\prod _i f_i$. This implies that
\[ {I}{|_{\varphi^{-1}(U)}}= \sum_{i=1}^s {( (\varphi^* (D_i)-\varphi^* (D_i)_{\rm red}))}{|_{\varphi^{-1}(U)}} , \]thus proving equality (2.4).
We also state the following two lemmas for further use.
Lemma 2.8 Let 
$\varphi :X\to Y$ be a surjective morphism with connected fibers between compact complex manifolds. Let 
$\mathcal {G}$ be a transversely hyperbolic foliation on 
$X$ with quotient singularities. Assume that there exists on 
$Y$ a codimension-one holomorphic foliation 
$\mathcal {F}$ such that 
$\mathcal {G}=\varphi ^*\mathcal {F}$. Then 
$\mathcal {F}$ carries a transversely hyperbolic with quotient singularities structure. Moreover, the pull-back of this structure via 
$\varphi$ coincides with that of 
$\mathcal {G}$ wherever defined.
Proof. Let 
$H$ be the divisor of poles of 
$\mathcal {G}$. As 
$H$ is necessarily 
$\mathcal {G}$ invariant, the restriction of 
$\varphi$ to 
$H$ is not surjective. Therefore, there exists a non-empty open Zariski subset 
$U$ of 
$Y$ such that 
$\mathcal {G}$ is transversely hyperbolic (without poles) in restriction to 
$V:=\varphi ^{-1} (U)$. In addition, one can suppose that 
$ {f}{|_{V}}$ is a smooth morphism onto 
$U$. Let 
$(W)$ be a covering of 
$U$ by open subsets (in the Euclidean topology) such that the sheaf 
$\mathcal {I}$ of distinguished first integrals of 
$\mathcal {G}$ is constant on 
$\varphi ^{-1}(W)$. The fibers being compact submanifolds, every global section of 
${ \mathcal {I}}_{\varphi ^{-1}(W)}$ descends to 
$W$. Consequently, 
$\mathcal {F}$ admits on 
$U$ a transversely hyperbolic structure defined by the locally constant sheaf 
$\mathcal {J}$ such that 
$\mathcal {I}=\varphi ^*\mathcal {J}$. Consider the analytic subset of 
$Y$ defined by 
$Z= Y-U$. The transverse hyperbolic structure defined by 
$\mathcal {J}$ extends through 
$Z-K$ where 
$K$ is the union of codimension-one components of 
$Z$ around which the local monodromy is non-trivial. Let 
$Z_0$ be a component of 
$Z$. Pick a general point 
$p$ of 
$Z_0$ and let 
$\gamma$ be a loop around 
$Z_0$ in some neighborhood 
$V_p$ of 
$p$. Let 
$q\in \varphi ^{-1} (p)$. Let 
$\mathbb {D}_q$ be a small disk centered at 
$q$ such that 
$\mathbb {D}_q-\{q\}$ is transverse to 
$\mathcal {G}$ and 
$\varphi ( \mathbb {D}_q )-\{p\}$ is transverse to 
$\mathcal {F}$. Let 
$\varepsilon :[0,1]\to \mathbb {D}_q -\{q\}$ be a small loop of index one around 
$q$. Obviously, 
$\varphi (\varepsilon )$ is a loop freely homotopic to a non-zero multiple of 
$\gamma$ in 
$V_p-Z_0$. Because 
$\mathcal {G}$ has quotients singularities, this implies that the local monodromy representation along 
$\gamma$ has finite image, whence the result.
Lemma 2.9 Let 
$\varphi :X\to Y$ be a surjective morphism with connected fibers between compact complex manifolds. Let 
$\mathcal {G}$ be a transversely hyperbolic foliation on 
$X$. Denote by 
$\rho$ its monodromy representation. Assume that there exists a representation 
$\rho ': \pi _1(Y)\to \operatorname {Aut} (\mathbb {D})$ such that 
$\rho =\varphi ^* \rho '$. Then there exists on 
$Y$ a transversely hyperbolic foliation 
$\mathcal {F}$ such that 
$\mathcal {G}=\varphi ^*\mathcal {F}$ and whose monodromy representation is 
$\rho '$.
Proof. Let 
$U\subset Y$ be a non-empty open Zariski subset such that 
$\varphi$ restricts to a smooth morphism on 
$\varphi ^{-1} (U)$. By assumption on the representation 
$\rho$, the sheaf 
$\mathcal {I}$ of distinguished first integrals of 
$\mathcal {G}$ is globally constant over 
$W$, where 
$W$ is any simply connected open subset of 
$U$. Like before, this implies that any section 
$s\in \mathcal {I} (\varphi ^{-1}(U))$ is constant on the fibers of 
$\varphi$. Consequently, there exists on 
$U$ a transversely hyperbolic foliation 
$ {\mathcal {F}}{|_{U}}$, which then extends as a foliation 
$\mathcal {F}$ on the whole 
$Y$ (as recalled in § 2.3) and whose monodromy representation is given by the composition morphism:
\[ \pi_1 (U)\to \pi_1(Y)\xrightarrow{\rho'} \operatorname{Aut}(\mathbb{D}) . \]Since the first arrow is surjective, the result follows.
Remark 2.10 Let 
$\mathcal {F}$ be a transversely hyperbolic foliation with quotient singularities on a compact complex manifold 
$X$. Then there exists a smooth modification 
$\pi :\hat X \to X$ obtained by a sequence of successive blow-ups with smooth centers such that the divisorial part of 
$\pi ^*\mathcal {F}$ is supported on an invariant normal crossing divisor 
$D=D_1+\cdots + D_r$. In particular there exists a 
$r$-tuple of rational numbers 
$(\lambda _1,\ldots,\lambda _r)\in \mathbb {Q}_{<1}^r$ such that 
$E:=N_ { \pi ^*\mathcal {F}}+\sum _i \lambda _i D_i$ is a pseudoeffective 
$\mathbb {Q}$-line bundle whose Chern class is represented by a non-trivial positive 
$(1,1)$-form 
$\eta$ with 
$L_{\rm loc}^1$ coefficient (and actually smooth on a Zariski dense open subset). When 
$X$ is compact Kähler, 
$\pi ^*\mathcal {F}$ is a particular case of a KLT foliation in the terminology of [Reference TouzetTou16, § 8.1]. Note also that the existence of 
$\eta$ guarantees that the positive part of 
$L$ in its Zariski decomposition is non-trivial.
2.6 Uniqueness of the transverse structure
The following result is established in [Reference Lo Bianco, Pereira, Rousseau and TouzetLPRT20, Corollary 5.6].
Proposition 2.11 Let 
$\mathcal {F}$ be a transversely hyperbolic foliation (with quotient singularities) on a projective manifold. Assume that 
$\mathcal {F}$ is not algebraically integrable. Then, the hyperbolic transverse structure is unique, i.e. any transverse hyperbolic structure for 
$\mathcal {F}$ on a dense Zariski subset is defined by the same sheaf of distinguished first integrals.
2.7 Relationship with numerical properties of the conormal bundle
The following theorem is essentially proved in [Reference TouzetTou13] and describes the interplay between the existence of a transverse hyperbolic structure and positivity properties of the conormal bundle of a foliation. The following is essentially a reformulation of some results established in [Reference TouzetTou13] (to which we will precisely refer in the proof, see also [Reference TouzetTou16, § 3.2]) where it is recalled that the coefficients 
$r_D$ appearing in (2.3) coincide with the coefficients of the divisorial Zariski decomposition of 
$c_1(N^*_{\mathcal {F}})$ and must be, therefore, non-negative.
Theorem 2.12 Let 
$\mathcal {F}$ be a codimension-one foliation on a compact Kähler manifold 
$X$ equipped with a Kähler form 
$\Theta$. Assume that 
$N_\mathcal {F}^*$ is pseudo-effective with numerical dimension one. Let 
$N=\sum _{i=1}^r \lambda _i N_i$ be the negative part in the Zariski decomposition of 
$c_1(N_\mathcal {F} ^*)$. Then:
(1) the coefficients
$\lambda _i$ are positive rational numbers;(2) the intersection matrix
$m_{ij}=N_i\cdot N_j \cdot \Theta ^ {n-2}$ is negative definite;(3)
$\mathcal {F}$ admits a transverse hyperbolic structure with quotient singularities on $X$ such that
(a) the divisor of poles is
$H= \sum \beta _i N_i$, where $\beta _i=0$ if $\lambda _i\in \mathbb {N}$, $\beta _i=1$ otherwise, and(b) the divisorial part of
$\mathcal {F}$ with respect to the given transversely hyperbolic structure is $N$.
Proof. The first item has already been established when 
$X$ is projective in [Reference TouzetTou13, Proposition 2.14(vi)].
Item (2) is a consequence of [Reference TouzetTou13, Corollaire 2.15], taking into account that the family 
$\{ N_1,\ldots,N_r\}$ is exceptional in the sense recalled in [Reference TouzetTou13, Définition 2.6 and Théorème 2.7].
Assume first that (1) holds in general, as claimed in the statement. Then items (3a) and (3b) can be derived directly from [Reference TouzetTou13, Théorème 1, with 
$\varepsilon =1$] and [Reference TouzetTou13, Proposition 5.1]. Indeed, the hyperbolic transverse structure is there defined by the equality 
$T=\eta$ of Théorème 1, valid outside the support of 
$N$ and which provides the collection of distinguished first integrals on 
$X-\mathrm {Supp}(N)$ (see [Reference TouzetTou13, Lemme 5.1]). Moreover, according to Proposition 5.1, the degeneracy of the hyperbolic structure along 
$\mathrm {Supp}(N)$ is explicitly described in terms of local multivaluate first integrals of the form 
$f\prod _i f_i^{\lambda _i +1}$, where 
$f_i=0$ is a local reduced equation of 
$N_i$ and 
$f$ is holomorphic with a divisor of zeroes either empty, either reduced and not contained in 
$\mathrm {Supp}(N)$.
It then remains to prove the rationality of the coefficients 
$\lambda _i$ in full generality, including the case where 
$X$ is Kähler and possibly non-projective. Still according to [Reference TouzetTou13, Proposition 5.1], there exists in a small neighborhood 
$U$ of 
$\mathrm {Supp}(N)$ a closed logarithmic form 
$\omega$ defining the foliation on 
$U$ (which restricts to a section of 
${\mathcal {I}}_{d\log }$ on 
$U-\mathrm {Supp}(N)$) with the following additional properties.
(1) The divisor of poles of
$\omega$ has the following form:
where\[ {(\omega)}_\infty=\sum N_i +A \]$A$ is a hypersurface of $U$ intersecting $\mathrm {Supp}(N)$ along a codimension-two subset.(2) We have
$\mathrm {Res}_{N_i} \omega =\lambda _i +1$, $\mathrm {Res}_{A} \omega =1$.(3) The generator
$\omega$ has no zeros in codimension one.
As an immediate consequence, the real Chern classes class of 
$N^*_{\mathcal {F}}$ and of 
$N$ coincide in 
$H^2(U,\mathbb {R})$. On the other hand, the class of 
$N^*_{\mathcal {F}}$ lies in 
$H^2(U,\mathbb {Q})\subset H^2(U,\mathbb {R})$ as the class of any line bundle.
Suppose by contradiction that at least one of the coefficients 
$\lambda _i$ lies in 
$\mathbb {R}-\mathbb {Q}$. By rationality of 
$c_1(N_i)$, one promptly deduces that there exists 
$(\nu _1,\ldots,\nu _r)\in \mathbb {R}^r-\{0\}$ such that 
$\sum _i \nu _i c_1 (N_i)=0$ in 
$H^ 2(U,\mathbb {R})$. Now, by de Rham's isomorphism, 
$c_1(N_i)$ can be represented on 
$U$ by a real closed two form 
$\theta _i$ and the linear dependance relation above is equivalent to the fact that 
$\theta :=\sum \nu _i \theta _i$ is exact on 
$U$. Let us evaluate the intersection product 
$I=\big ( \sum \nu _i c_1( N_i)\big )^2 \Theta ^{n-2}$. By item (2) of the theorem, it is a negative real number but one can alternatively compute this intersection as 
$I=\sum \nu _i\int _{N_i}\theta \wedge \Theta ^{n-2} =0$ by exactness of 
$\theta$, whence the contradiction.
3. Pull-backs of tautological foliations on irreducible polydisk quotients
3.1 Irreducible polydisk quotients
Let 
$N\ge 2$ be an integer. A discrete subgroup 
$\Gamma \subset \operatorname {Aut}(\mathbb {D})^N$ is a lattice if the quotient 
$\mathbb {D}^N/\Gamma$ has finite volume. A lattice 
$\Gamma \subset \operatorname {Aut}(\mathbb {D})^N$ is irreducible if it is not commensurable to a product of 
$\Gamma _1 \times \Gamma _2 \subset \operatorname {Aut}(\mathbb {D})^{N_1} \times \operatorname {Aut}(\mathbb {D})^{N_2}$ with 
$N_1, N_2 \ge 1$, 
$N_1+N_2=N$.
If 
$\Gamma \subset \operatorname {Aut}(\mathbb {D})^N$ is an irreducible lattice, then the quotient 
$\mathbb {D}^{N}/\Gamma$ is a singular variety with finitely many cyclic quotient singularities according to [Reference ShimizuShi63, Theorem 2].
The quotient 
$\mathbb {D}^{N}/\Gamma$ carries 
$N$ distinct codimension-one tautological foliations 
$\mathcal {G}_1, \ldots, \mathcal {G}_N$, defined on 
$\mathbb {D}^N$ by one of the natural projections to 
$\mathbb {D}$. The foliations 
$\mathcal {G}_i$ are transversely hyperbolic foliations on the complement of the singular points of 
$\mathbb {D}^{N}/\Gamma$.
3.2 Morphisms to irreducible polydisk quotients
Let 
$\Gamma \subset \operatorname {Aut}(\mathbb {D})^N$ be an irreducible lattice and let 
$X$ be a complex compact manifold. Assume there exists a morphism 
$\rho : X \to \mathbb {D}^N/\Gamma$ with image of positive dimension 
$p$. Let 
$\mathcal {F}_1 = \rho ^* \mathcal {G}_1, \ldots, \mathcal {F}_p= \rho ^* \mathcal {G}_p$ be the pull-back to 
$X$ of 
$p$ among the 
$N$ tautological foliations on 
$\mathbb {D}^N/\Gamma$ such that the foliations are in general position, i.e. if 
$\omega _p$ is a local generator of 
$N_{ \mathcal {F}_i }^*$, 
$i=1,\ldots, p$, then 
$\omega _1\wedge \cdots \wedge \omega _p$ does not vanish identically.
The foliations 
$\mathcal {F}_i$ are all transversely hyperbolic foliations with finite quotient singularities. The transverse hyperbolic structure is not necessarily defined over codimension-one components of the fibers of 
$\rho$ over 
$\operatorname {Sing}(\mathbb {D}^N/\Gamma )$.
Let 
$D_j$ be the divisorial part of the foliation 
$\mathcal {F}_j$ and let 
$H_1,\ldots,H_k$ be some pairwise distinct prime divisors such that 
$D_j= \sum _{i=1}^k r_{ij} H_i$ where 
$r_{ij}\in \mathbb {Q}_{>-1}$. Let 
$L_j$ be the nef 
$\mathbb {Q}$-line bundle provided by Proposition 2.3 such that
\[ N_{ \mathcal{F}_j }^*=L_j + D_j= L_j+ \sum_{i=1}^k r_{ij} H_i . \]Lemma 3.1 The following assertions hold true:
(1) for any
$j \in \{1, \ldots, p\}$
where\[ N_{ \mathcal{F}_j }^*=L_j + \sum_{i=1}^k r_{ij} H_i, \]$L_j$ is a nef $\mathbb {Q}$-line bundle with $\nu (L_j)\geq 1$;(2) the hypersurface
$H_i$ is $\mathcal {F}_j$ invariant whenever $r_{ij} \neq 0$;(3) the monodromies of the transversely hyperbolic structures of
$\mathcal {F}_1, \ldots, \mathcal {F}_p$ around $H_i$ all have the same order $n_i=\mathrm {Min}\{m\in \mathbb {N}_{>0}|mr_{ij}\in \mathbb {Z}\}$; in particular, $r_{ij} \in \mathbb {N}$ for some $j \in \{1, \ldots, p\}$ if, and only if $r_{ij} \in \mathbb {N}$ for all $j \in \{ 1, \ldots, p\}$.
Proof. Each of the foliations 
$\mathcal {F}_j$ is transversely hyperbolic outside its polar locus 
$H_j$. Fix some 
$j$. From [Reference ShimizuShi63, Theorem 2], the local monodromies of the transversely hyperbolic structures of 
$\mathcal {F}_1, \ldots, \mathcal {F}_p$ around 
$H_j$ have all the same order, which is non-trivial. It follows that the 
$p$ foliations 
$\mathcal {F}_1,\ldots,\mathcal {F}_p$ share the same polar locus. Once we have made this observation, the lemma directly follows from Proposition 2.3.
3.3 A big divisor
In our next statement, we keep the notation used in Lemma 3.1.
Lemma 3.2 Assume that 
$X$ is a projective manifold such that the morphism 
$\rho$ is generically finite (i.e. 
$p= \mathrm {dim}\ X$). Then, the divisor 
$L = \sum _{j=1}^p L_j$ is big.
Proof. The nef 
$\mathbb {Q}$-divisors 
$L_j$ have Chern–Hodge classes in 
$H^1(X,\Omega ^1_X)$ represented by semi-positive 
$(1,1)$-form 
$\eta _j$ obtained by pull-back of the Poincaré metric under distinguished first integrals. Recall that the 
$(1,1)$-forms 
$\eta _i$ are smooth outside the (common) polar locus of the transverse structures, and have 
$L_{\rm loc}^1$ coefficients on 
$X$.
Consequently, the Chern class of 
$\mathbb {Q}$-line bundle 
$L=\sum _{j=1}^n L_j$ is represented by a semi-positive 
$(1,1)$-form
\[ \eta = \sum_{i=1}^p \eta_i \]
with the same type of regularity and we can take its 
$p$th power 
$\eta ^p$ which is meant pointwise.
Recall that on a 
$p$-dimensional complex compact manifold, the volume of a line bundle 
$E$ is defined as
\[ v(E)=\limsup _{k\to \infty}\frac{p!}{k^n}h^0(X,kE) . \]
The line bundle is big, i.e. has maximal Kodaira–Itaka dimension 
$\kappa (E)=p$ exactly when 
$v(E)>0$ and, in that case, the lim sup is actually a genuine limit. More generally, one can define the volume of a 
$\mathbb {Q}$-line bundle 
$E$ as 
$v(L):= l^{-p}v(lE)$ where 
$l$ is a positive integer such that 
$lE$ is a line bundle. It is easily seen that this definition of volume does not depend on the choice of 
$E$.
According to [Reference BoucksomBou02, Theorem 1.2], 
$v(L)\geq \int _X \eta ^p$. One concludes by noting that the right-hand side is positive. Indeed, 
$\eta ^p$ restricts on 
$X-H$ to the smooth and non-trivial semi-positive form 
$(p!) \eta _1 \wedge \cdots \wedge \eta _p$.
4. Numerical specialness for rank-one subsheaves of $\Omega ^1$
Proposition 4.1 Let 
$\Gamma \subset \operatorname {Aut}(\mathbb {D})^N$ be an irreducible lattice and let 
$X$ be a compact complex manifold. Assume that there exists a morphism 
$\Psi : X \to \mathbb {D}^N/\Gamma$ with image of positive dimension 
$p$ such that the pull-back 
$\mathcal {F}= \Psi ^*\mathcal {G}$ of one of the tautological foliation 
$\mathcal {G}$, equipped with the pull-back transverse hyperbolic structure, has a 
$\mathbb {Q}$-effective divisorial part. Then there exists an invertible subsheaf 
$L\subset \Omega _X^p$ whose Kodaira dimension satisfies 
$\kappa (L)= p$.
Remark 4.2 According to a classical result of Bogomolov, known as Bogomolov–Castelnuovo–de Franchis inequality, the Kodaira dimension of rank-one subsheaves of 
$\Omega _X^p$, 
$X$ compact Kähler, is bounded from above by 
$p$. In this sense, the subsheaf 
$L\subset \Omega _X^p$ exhibited in the previous proposition has maximal Kodaira dimension.
Proof. Let 
$V$ be a smooth projective model of the image of 
$\Psi$ determined by some birational morphism 
$\rho : V\to \mathrm {Im}\, \Psi \subset \mathbb {D}^N/\Gamma$. Retaining the notation of § 3.2, we have on 
$V$, 
$p=\mathrm {dim}\ V$ foliations in general position: 
$\mathcal {H}_j=\rho ^*\mathcal {G}_j$. One can moreover assume that 
$\mathcal {G}_1=\mathcal {G}$.
The conormal bundle of each of these foliations splits as
\[ N_{ \mathcal{H}_j }^*=L_j' + D_j' \]
such that 
$D_j'$ is the divisorial part of 
$\mathcal {H}_j$. We also have from Lemma 3.2 that 
$\sum L_j'$ is a big 
$\mathbb {Q}$-divisor.
Note also, using for instance Proposition 2.7, that the divisorial part of 
$\mathcal {F}$ under pull-back by any birational morphism remains effective. Then, up to performing some blow-ups on 
$X$, one can suppose that 
$\rho$ factors through a dominant morphism 
$\varphi : X\to V$.
Set 
$\mathcal {F}_j=\varphi ^*\mathcal {H}_j$, 
$j=1,\ldots,p=\mathrm {dim}\ V$, so that 
$\mathcal {F}=\mathcal {F}_1$. Consider the decomposition
\[ N_{ \mathcal{F}_j }^*=L_j + D_j= L_j+ \sum_{i=1}^k r_{ij} H_i \]
as given in § 3.2. In particular, the coefficients 
$r_{ij}$ are rational and greater than 
$-1$.
After renumbering the hypersurfaces 
$H_i$, one can assume by Lemma 3.1, the existence of an integer 
$k' \le k$ such that 
$r_{ij} \in \mathbb {Q} - \mathbb {Z}$ for any 
$j \in \{ 1, \ldots, p\}$ if and only if 
$i\leq k'$. We can thus write, for any 
$j$,
\begin{equation} \sum_{i=1}^k r_{ij} H_i = \sum_{i=1}^{k'} r_{ij} H_i + R_j, \end{equation}
where 
$R_j$ is an effective divisor. By assumption on the divisorial part of 
$\mathcal {F}=\mathcal {F}_1$, we have that 
$r_{i1}>0$ for every 
$i \in \{ 1, \ldots, k\}$. Therefore, we can write
\begin{equation} \sum_{i=1}^{k'} \bigg( p-1 + \sum_{j=1}^p r_{ij} \bigg) H_i = \sum_{i=1}^{k'} \bigg( r_{i1} + \sum_{j=2}^p (1+ r_{ij}) \bigg) H_i \ge 0 \end{equation}
thanks to the lower bound 
$r_{ij}>-1$.
Consider the morphism
\begin{align*} \sigma : N^*_{\mathcal{F}_1} \otimes \cdots \otimes N^*_{\mathcal{F}_p} &\longrightarrow \Omega^p_X \\ \omega_1 \otimes \cdots \otimes \omega_n & \mapsto \omega_1 \wedge \cdots \wedge \omega_p. \end{align*}
The saturation of the image of 
$\sigma$ is an invertible subsheaf 
$\mathcal {O}_X(F) \subset \Omega ^p_X$ isomorphic to
\[ N^*_{\mathcal{F}_1} \otimes \cdots \otimes N^*_{\mathcal{F}_p} \otimes \mathcal{O}_X(\operatorname{tang}(\mathcal{F}_1, \ldots, \mathcal{F}_p)) , \]
where 
$\operatorname {tang}(\mathcal {F}_1, \ldots, \mathcal {F}_p)$ is the tangency divisor of the foliations 
$\mathcal {F}_1, \ldots, \mathcal {F}_p$. As the hypersurfaces 
$H_1, \ldots, H_{k'}$ are invariant by all the foliations 
$\mathcal {F}_j$, it follows that
\begin{equation} \operatorname{tang}(\mathcal{F}_1, \ldots, \mathcal{F}_p) \ge (p-1) \sum_{i=1}^{k'} H_i. \end{equation}From (4.1) and (4.3), one can infer
\[ F \ge \sum_{j=1}^{p} \sum_{i=1}^{ k'} r_{ij}H_i + (p-1) \sum_{i=1}^{k'} H_i + \sum_{j=1}^p L_j =\sum_{i=1}^{k'} \bigg( p-1 + \sum_{j=1}^p r_{ij} \bigg) H_i+ \sum_{j=1}^p L_j, \]where the last equality is obtained by reversing the order of summation.
Another important point is that 
$L_j=\varphi ^* (L_j')$ according to Proposition 2.7. From Lemma 3.2 and because 
$\varphi$ is dominant, one deduces that the Kodaira dimension of 
$L=\sum L_j$ is at least 
$p$. Therefore,
\[ \displaystyle{ F \quad \ge \quad \underbrace{\sum_{i=1}^{k'} \bigg( p-1 + \sum_{j=1}^p r_{ij} \bigg) H_i}_{\text{ $\mathbb{Q}$-effective according to inequality (4.2)}} \qquad + \underbrace{\sum_{j=1}^p L_j \, ,}_{\text{$\kappa\big( \sum_{j=1}^p L_j \big)\ge p $}} \, } \]
and 
$F$ can be written as the sum of a nef 
$\mathbb {Q}$-line bundle 
$L$ of Kodaira dimension at least 
$p$ and an effective 
$\mathbb {Q}$-divisor. It follows that 
$\kappa (F)\geq p$.
Let 
$X$ be a 
$n$-dimensional compact Kähler manifold equipped with a codimension-one foliation 
$\mathcal {F}$ whose conormal sheaf 
$N_\mathcal {F}^*$ is pseudo-effective. Denote by 
$\nu$ (respectively, 
$\kappa$) the numerical (respectively, Kodaira) dimension of 
$N_\mathcal {F}^*$. Theorem 2.12 guarantees that the Zariski decomposition of 
$c_1(N_\mathcal {F}^*)$ reads as
\[ c_1(N_\mathcal{F}^*)=N +Z, \]
where 
$N$ is a 
$\mathbb {Q}$-effective divisor such the intersection matrix 
$(N_i \cdot N_j \cdot \Theta ^{n-2})$ is negative definite, where the 
$N_i$'s are the irreducible components of 
$\operatorname {Supp}(N)$, 
$Z$ is a nef class and 
$\Theta$ any Kähler class.
According to [Reference TouzetTou16, Theorem 4], there are three possible cases according to the value of 
$\nu$ (defined by 
$Z^\nu \not =0, Z^{ \nu +1}=0$):
(1)
$\nu =0=\kappa$;(2)
$\nu =1=\kappa$;(3)
$\nu =1$, $\kappa =-\infty$ (the non-abundant case).
The two last cases also strongly differ from the dynamical viewpoint (see [Reference TouzetTou16, Theorem 4]): in case (2), 
$\mathcal {F}$ is algebraically integrable; in case (3), the foliation is quasi-minimal (all leaves, except finitely many, are dense for the Euclidean topology). We are interested in the last situation where abundance does not hold.
4.1 Proof of Theorem D
Assume for a while that 
$X$ is projective. By [Reference TouzetTou16, Theorem 6] and also taking into account Remark 2.10, there exists a morphism 
$\Psi : X\to \mathbb {D}^N/\Gamma$ whose image has dimension at least two such that 
$\mathcal {F}=\Psi ^*\mathcal {G}$ where 
$\mathcal {G}$ is one of the tautological foliations on 
$\mathbb {D}^N/\Gamma$. Moreover, the transverse hyperbolic structure of 
$\mathcal {F}$ as described in Theorem 2.12 is obtained by pulling-back via 
$\Psi$ the natural transverse hyperbolic structure of 
$\mathcal {G}$. This is implicitly proved in [Reference TouzetTou16, § 6] (especially pp. 22–23) but one can also invoke the uniqueness property stated in Proposition 2.11.
Let us now consider the general case of compact Kähler manifolds. Up to renumbering the components 
$N_i$ of the negative part 
$N$, one can assume that for some 
$q\in \mathbb {N}$, 
$\lambda _i\in \mathbb {Q}-\mathbb {N}$ for 
$i=1, \ldots,q$, 
$\lambda _i\in \mathbb {N}_{>0}$ for 
$i>q$. Set 
$N'= \sum _{i=1}^q \lambda _i N_i$. Let 
$\rho : \pi _1 (X-\mathrm {Supp}(N'))\to \operatorname {Aut} (\mathbb {D})$ be the monodromy representation of the transverse hyperbolic structure. According to [Reference TouzetTou16, Proposition 4.6], the image of 
$\rho$ is Zariski dense (actually dense in the Euclidean topology). By Selberg's lemma, there exists a finite index torsion-free normal subgroup in the image of 
$\rho$. This enables to construct a finite Galois cover 
$R: \hat X\to X$ with branch locus 
$\mathrm {Supp}(N')$ such that the pull-back representation 
$R^*\rho$ is actually well defined as a morphism 
$\pi _1(\hat X)\to \operatorname {Aut} (\mathbb {D})$ with torsion-free image. According to [Reference VâjâituVâj96, Theorem 1], 
$\hat X$ is a Kählerian analytic space and then, by Hironaka [Reference HironakaHir77], admits a resolution of singularities which is a compact Kähler manifold 
$Y$. We have thus constructed a surjective morphism with generically finite fibers 
$\psi :Y\to X$ between compact Kähler manifolds such the pull-back foliation 
$\mathcal {F}_Y:=\psi ^*\mathcal {F}$ is transversely hyperbolic without poles. The associated monodromy representation is nothing but 
$\rho _Y:=\psi ^*\rho : \pi _1(Y)\to \operatorname {Aut}(\mathbb {D})$ with dense and torsion-free image. Note that one can prove than 
$\rho _Y$ (and, equivalently, 
$\rho$) has Zariski-dense image in 
$\operatorname {Aut}(\mathbb {D})$ without resorting to [Reference TouzetTou16, Proposition 4.6]. Indeed, let 
$\tilde Y$ be the universal covering of 
$Y$. Denote by 
$f: \tilde Y\to \mathbb {D}$ be the 
$\rho$-equivariant holomorphic obtained by developing the transverse hyperbolic structure of 
$\mathcal {F}_Y$. Because 
$Y$ is Kähler, 
$f$ is also harmonic and the Zariski density of the image of 
$\rho$ follows from [Reference CorletteCor88, Reference LabourieLab91] (regarding 
$\mathbb {D}$ as a symmetric space of the non-compact type).
By [Reference ZuoZuo96, Reference Campana, Claudon and EyssidieuxCCE15] and up to taking a bimeromorphic smooth model of 
$Y$, 
$\rho _Y$ factors through 
$\rho _V:\pi _1(V)\to \operatorname {Aut} (\mathbb {D})$ via a surjective morphism 
$e:Y\to V$ with connected fibers, where 
$V$ is a projective manifold of the general type.
In particular, 
$e$ factors through the algebraic reduction map 
${\rm red}_Y : Y\to {\operatorname {Red}} (Y)$ (here and, henceforth, we assume, up to taking appropriate smooth models, that all algebraic reduction spaces are projective manifolds and all reduction maps are morphisms). By Lemma 2.9, there exists on 
${\operatorname {Red}} (Y)$ a transversely hyperbolic foliation 
$\mathcal {F}_1$ such that 
$\mathcal {F}_Y= {\rm red}_Y^*\mathcal {F}_1$, and such that the monodromy representations factor accordingly.
Observe also that the group of deck transformations of the Galois cover 
$\hat X\to X$ induces on 
$Y$ a finite group of bimeromorphic transformations 
$G$ preserving the foliation 
$\mathcal {F}_Y$. Consider the action 
$G\times \mathbb {C} (Y)\to \mathbb {C} (Y)$ defined by 
$g \cdot f=f\circ g^{-1}$ and denote by 
$K$ the kernel of this action. By the very definition of 
${\operatorname {Red}} (Y)$, 
$G$ acts on 
$Y$ by preserving the fibration of the reduction map and it induces a faithful action of 
$G_1:=G/K$ on 
${\operatorname {Red}}(Y)$ by birational transformations also preserving the foliation 
$\mathcal {F}_1$. By construction, observe also that the field of rational functions of 
${\operatorname {Red}}(X)$ is precisely 
$\mathbb {C}(X)= { \mathbb {C} (Y)}^G$. This implies that there exists on 
${\operatorname {Red}} (X)$ a foliation, 
$\mathcal {F}_2$ such that 
$\mathcal {F}_1=r_{Y,X}^*\mathcal {F}_2$ where 
$r_{Y,X}:{\operatorname {Red}} (Y)\to {\operatorname {Red}} (X)$ is the rational map induced by the inclusion 
$\mathbb {C}(X)\subset \mathbb {C} (Y)$ and then makes the following diagram commutative, up to replacing 
$Y$, 
$\operatorname {Red}(Y)$, 
$\operatorname {Red}(X)$, and 
$V$ by suitable non-singular models.
In particular, we have that 
$\mathcal {F}= {\rm red}_X^*\mathcal {F}_2$. According to Lemma 2.8, 
$\mathcal {F}_2$ admits a transversely hyperbolic structure with quotient singularities compatible with that of 
$\mathcal {F}$. Applying Theorem D in the projective case, one deduces that 
$\mathcal {F}$ is obtained by pull-back of a tautological foliation on a polydisk quotient also compatible with the transverse hyperbolic structure (Proposition 2.11). The proof is thus complete.
Theorem 4.3 Let 
$(X,\mathcal {F})$ be a foliated compact Kähler manifold with 
$\mathrm {codim}(\mathcal {F})=1$. Assume that 
$N_\mathcal {F}^*$ is pseudo-effective and not abundant in the sense defined above. Then there exists an invertible subsheaf 
$L\subset \Omega _X^p$ for some 
$p\geq 2$ such that 
$\kappa (L)=p$.
Proof. The decomposition provided by Proposition 2.3 and Theorem 2.12 takes the form
\begin{equation} N_\mathcal{F}^*= P+N, \end{equation}
where the negative part 
$N$ is effective. This property is clearly invariant under pull-back by bimeromorphic morphisms, so that one can apply Theorem D. From Proposition 4.1, one derives the existence of an invertible subsheaf 
$L\subset \Omega _X^p$ for some 
$p\geq 2$ such that 
$\kappa (L)\geq p$ and hence equal to 
$p$ by Bogomolov's upper bound.
Remark 4.4 In general, one cannot deduce non-specialness of 
$X$ just from the existence of a proper morphism to an irreducible quotient of a polydisk. Indeed in [Reference GranathGra02, Theorem 12.1] Granath produces examples of rational and 
$K3$ surfaces obtained as minimal resolutions of singular compact quotients 
$\mathbb {D}^2/\Gamma$.
Remark 4.5 Another way to interpret the preceding proof is the following. Considering as before 
$V$ a smooth projective model of the image of 
$\Psi$, 
$V$ is equipped with a natural divisor 
$\Delta =\sum _i (1-{1}/{m_i})D_i$ with normal crossing support over 
$\operatorname {Sing} (\mathbb {D}^N/\Gamma )$. It follows from [Reference Cadorel, Diverio and GuenanciaCDG20] that 
$K_V+\Delta$ is big. As above, up to performing some blow-ups on 
$X$, we have a dominant morphism 
$\varphi : X\to V$. Then it follows from the proof of Theorem 4.3 that 
$\varphi : X\to (V, \Delta )$ is an orbifold morphism in the sense of Campana i.e. 
$\varphi$ ramifies over 
$D_i$ with multiplicity at least 
$m_i$ [Reference CampanaCam11, Definition 2.3]. This implies that 
$\varphi ^*(K_V+\Delta ) \subset \Omega _X^p$ [Reference CampanaCam11, Proposition 2.11] is a Bogomolov sheaf where 
$p:=\dim V$.
5. Entire curves
First, remark that in the case 
$\nu =1=\kappa$, we have a Bogomolov sheaf 
$L\subset \Omega _X$ which corresponds (see [Reference CampanaCam04]) to a fibration of general type 
$F: X \to C$ onto a curve. This means that the orbifold base of the fibration 
$(C, \Delta )$ is of general type. It is now a classical fact [Reference NevanlinnaNev70] that in this setting, for any entire curve 
$f:\mathbb {C} \to X$, 
$F\circ f: \mathbb {C} \to C$ has to be constant. Therefore, 
$f:\mathbb {C} \to X$ cannot be Zariski dense.
Thus, we now deal with the non-abundant case 
$\nu =1$, 
$\kappa =-\infty$.
Theorem 5.1 Let 
$(X,\mathcal {F})$ be a foliated compact Kähler manifold with 
$\mathrm {codim}(\mathcal {F})=1$. Assume that 
$N_\mathcal {F}^*$ is pseudo-effective and not abundant. Then any entire curve 
$f: \mathbb {C} \to X$ is algebraically degenerate, i.e. 
$f(\mathbb {C})$ is not Zariski dense.
We start with a lemma.
Lemma 5.2 Let 
$(X,\mathcal {F})$ be a foliated compact Kähler manifold with 
$\mathrm {codim}(\mathcal {F})=1$. Assume that 
$N_\mathcal {F}^*$ is pseudo-effective and not abundant. Then any entire curve 
$f: \mathbb {C} \to X$ is tangent to 
$\mathcal {F}$.
Proof. From [Reference TouzetTou16, Theorem 3.1] (see also Theorem 2.12), we have a singular transverse metric 
$h$ with curvature current 
$\Theta _h=-(h+[N])$. It is a smooth transverse metric of constant curvature 
$-1$ on 
$X\setminus (\operatorname {Sing}(\mathcal {F}) \cup \operatorname {Supp} N)$. Suppose 
$f: \mathbb {C} \to X$ is not tangent to 
$\mathcal {F}$. In particular, 
$f(\mathbb {C}) \not \subset \operatorname {Sing}(\mathcal {F}) \cup \operatorname {Supp} N$. Therefore, 
$f^*h$ induces a non-zero singular metric 
$\gamma (t)=\gamma _0(t)i\,dt\wedge d\overline {t}$ on 
$\mathbb {C}$ where 
$-\operatorname {Ric} \gamma \geq \gamma$ in the sense of currents. However, the Ahlfors–Schwarz lemma (see [Reference DemaillyDem97, Theorem 3.2]) implies that 
$\gamma \equiv 0$, a contradiction.
5.1 Proof of Theorems B and 5.1
The subsheaf 
$L$ determines a foliation 
$\mathcal {F}$ whose conormal bundle 
$N_\mathcal {F}^*$ has numerical dimension 
$\nu =1$. According to the previous discussion, it suffices to consider the case 
$\kappa =\kappa (N_\mathcal {F}^*)=-\infty$. From the preceding lemma, we can suppose that 
$f: \mathbb {C} \to X$ is tangent to 
$\mathcal {F}$. By Theorem D (up to replacing 
$X$ by a smooth model) there exists a morphism 
$\Psi : X\to \mathfrak H:=\mathbb {D}^N/\Gamma$ such that 
$\mathcal {F}=\Psi ^*\mathcal {G}$ where 
$\mathcal {G}$ is one of the tautological foliation on 
$X$. Therefore, 
$\Psi (f): \mathbb {C} \to \mathfrak H$ is tangent to 
$\mathcal {G}$ and is constant thanks to the hyperbolicity of the leaves on 
$\mathfrak H$ by [Reference Rousseau and TouzetRT18, Proposition 3.1]. This concludes the proof.
6. Geometric specialness
In this section, we prove Theorem C. We start by proving in our setting a particular case of Lang–Vojta's conjecture.
Conjecture 3 (Lang–Vojta)
Let 
$X$ be a projective variety of general type and 
$L$ an ample line bundle. Then there is a proper algebraic subset 
$Z \not \subset X$ and a constant 
$\alpha$, such that for every smooth projective connected curve 
$C$ and every morphism 
$f:C\to X$ with 
$f(C) \not \subset Z$, one has
\[ \deg f^*L \leq \alpha (2g(C)-2). \]Here, we prove the following particular case.
Proposition 6.1 Let 
$\Gamma \subset \operatorname {Aut}(\mathbb {D})^N$ be an irreducible lattice and let 
$X$ be a complex projective manifold. If there exists a generically finite morphism 
$\Psi : X \to \mathbb {D}^N/\Gamma$ such that the pull-back of one of the tautological foliations (equipped with the pull-back transverse hyperbolic structure) has a 
$\mathbb {Q}$-effective divisorial part, then there exists a big line bundle 
$L$ on 
$X$, a proper algebraic subset 
$Z \not \subset X$ and a constant 
$\alpha$, such that for every smooth projective connected curve 
$C$ and every morphism 
$f:C\to X$ with 
$f(C) \not \subset Z$, one has
\[ \deg f^*L \leq \alpha (2g(C)-2). \]Proof. Using the same notation as in § 3, we consider the decomposition
\[ N_{ \mathcal{F}_j }^*=L_j + D_j= L_j+ \sum_{i=1}^k r_{ij} H_i. \]
The morphism 
$f: C \to X$ induces a morphism 
$f': C \to \mathbb {P}(T_X)$ which implies the algebraic tautological inequality
\[ \deg(f'^*(\mathcal{O}(1))) \leq 2g(C)-2. \]
From [Reference Rousseau and TouzetRT18], we know that if 
$\Psi (f(C))$ is not constant, then 
$f(C)$ is not contained in a leaf of any foliation 
$\mathcal {F}_j$. We take 
$Z$ to be the union of the positive dimensional fibers of 
$\Psi$ and the 
$H_i$ and assume that 
$f(C)$ is not contained in 
$Z$.
Let 
$\pi : \mathbb {P}(T_X) \to X$ be the natural projection. To the foliation 
$\mathcal {F}_i$ is associated a divisor 
$Z_i \subset \mathbb {P}(T_X),$ linearly equivalent to 
$\mathcal {O}(1)+\pi ^*{N}_{\mathcal {F}_i}.$ Then the algebraic tautological inequality gives
\[ \deg(f^*({N}^*_{\mathcal{F}_i})) \leq \deg(f'^*(Z_i))+ \deg(f^*({N}^*_{\mathcal{F}_i})) \leq 2g(C)-2. \]
The first inequality comes from the non-tangency of the algebraic curve with the foliation which implies 
$0 \leq \deg (f'^*(Z_i))$. By assumption on the divisorial part of 
$\mathcal {F}_1$, we have that 
$r_{i1}>0$ for every 
$i \in \{ 1, \ldots, k'\}$. Therefore,
\[ \deg f^*L_1+ \sum_{i=1}^{k'} r_{i1} \deg f^*H_i \leq 2g(C)-2. \]
As 
$L_1$ is nef, we obtain 
$\deg f^*H_i \leq {1}/{r_{i1}}(2g(C)-2)$ for all 
$i \in \{ 1, \ldots, k'\}$. Let 
$L:=\sum _j L_j$, then 
$L$ is big according to Lemma 3.2 and the previous inequalities give
\[ \deg f^*L\leq \sum_j \deg f^* N_{ \mathcal{F}_j }^*-\sum_j \sum_{i=1}^{k'} r_{ij} \deg f^* H_i \leq (2g(C)-2)\bigg(p+\sum_j \sum_{i=1}^{k'} \frac{|r_{ij}|} {r_{i1}}\bigg). \]Corollary 6.2 Under the same assumptions, 
$X$ is not geometrically special.
Proof. Suppose 
$X$ is geometrically special. Consider the Zariski open set 
$U:=X \setminus Z$. Then there exists a smooth projective connected curve 
$C$, a point 
$c$ in 
$C$, a point 
$u$ in 
$U$, and a sequence of morphisms 
$f_i:C\to X$ with 
$f_i(c) = u$ for 
$i=1,2,\ldots$ such that 
$C\times X$ is covered by the graphs 
$\Gamma _{f_i}\subset C\times X$ of these maps. From the previous Proposition 6.1, all these pointed maps have bounded degree. Therefore by Bend-and-Break [Reference DebarreDeb01, Chapter 3], we obtain a rational curve passing through 
$u$. Such a curve has to be tangent to the foliation 
$\mathcal {F}_1$ by Lemma 5.2. This gives a contradiction.
6.1 Proof of Theorem C
As in the proof of Theorem B, consider the foliation 
$\mathcal {F}$ associated with 
$L$. In the case 
$\nu =1=\kappa$, we have a Bogomolov sheaf 
$L\subset \Omega _X$ which corresponds (see [Reference CampanaCam04]) to a fibration of general type 
$F: X \to D$ onto a curve. This means that the orbifold base of the fibration 
$(D, \Delta )$ is of general type. This implies finiteness of orbifold morphisms 
$f: C \to (D, \Delta )$ [Reference CampanaCam05, Theorem 3.8] and therefore 
$X$ cannot be geometrically special in this case.
In the non-abundant case 
$\nu =1$, 
$\kappa =-\infty$, there exists a morphism 
$\Psi : X\to \mathfrak H:=\mathbb {D}^N/\Gamma$ such that 
$\mathcal {F}=\Psi ^*\mathcal {G}$ where 
$\mathcal {G}$ is one of the tautological foliation on 
$X$. Consider 
$V$ a smooth projective model of the image of 
$\Psi$ given by a morphism birational onto its image 
$V \to \mathbb {D}^N/\Gamma$. Let 
$\phi : X \to V$ be the induced dominant morphism (modulo some blow-ups on 
$X$). Let 
$f: C \to X$ be a morphism. Following the same proof as in Proposition 6.1, we obtain 
$\deg (\phi \circ f)^*L'= \deg f^*L \leq \alpha (2g-2)$. Since 
$L'$ is a big line bundle on 
$V$, the same proof as in Corollary 6.2 applied to the sequence 
$\phi \circ f_i$ gives that 
$X$ is not geometrically special.
Remark 6.3 In [Reference Javanpeykar and RousseauJR22], it is proved that if there exists a Zariski-dense representation 
$\rho : \pi _1(X) \to G(\mathbb {C})$ (
$G$ an almost simple algebraic group), then 
$X$ is not geometrically special. We cannot apply this result here because the monodromy representation 
$\rho : \pi _1(X_0) \to \operatorname {Aut}(\mathbb {D})$ is a priori defined only on 
$X_0:= X \setminus \operatorname {Supp} N$ where 
$N$ is the negative part of 
$c_1(N_\mathcal {F} ^*)$.
7. Higher codimensions
As recalled in the introductory part, the existence of a rank-one coherent subsheaf 
$L$ of 
$\Omega _X^p$ having numerical dimension 
$p$ and such that 
$\mathrm {codim} (\mathrm {Ker}\ L)=p$ on a compact Kähler manifold 
$X$ can be translated into the existence of a codimension 
$p$ foliation 
$\mathcal {F}$ on 
$X$ such that 
$\mathrm {det}\ N_\mathcal {F}^*$ (which is somehow the canonical sheaf of the ‘space of leaves’ 
$X/\mathcal {F}$) has numerical dimension 
$p$. We do not know how to generalize Theorem A to codimension 
$p>1$, in particular because we do not have at our disposal sufficiently precise structure results for this category of foliations, unlike in the case 
$p=1$. However, it remains possible to reach the same conclusion under strong assumptions on the subsheaf of 
$\Omega ^p_X$. For instance, as stated in Theorem E in the introduction, this is the case if we assume that the subsheaf of 
$\Omega ^p_X$ defines a smooth foliation with conormal bundle having Chern class represented by a smooth 
$(1,1)$-form with semi-positive curvature of constant rank 
$p$.
Proof of Theorem E. We first assert that 
$\eta$ is basic for 
$\mathcal {F}$, i.e. descends on the local space of leaves (see, for instance, [Reference MolinoMol88, § 2.3] for the precise definition). More generally, one has the more general phenomenon, as proved by Demailly [Reference DemaillyDem02]: if 
$X$ is compact Kähler 
$\omega \in H^0(X, \Omega ^p \otimes L)$ such that 
$L$ is a line bundle whose dual 
$L^*$ is pseudo-effective, then 
$\Theta \wedge \omega =0$ where 
$\Theta$ is any closed positive current representing 
$c_1(L^*)$.
In our situation this implies that 
$\eta$ defines a holonomy transverse invariant Kähler metric for 
$\mathcal {F}$. Equivalently, the kernel of 
$\eta$ is exactly the tangent bundle to 
$\mathcal {F}$. One thus inherits another real basic 
$(1,1)$-form, namely the transverse Ricci form 
$r=-\mathrm {Ricci}(\eta )$. In some holomorphic coordinates 
$(z_1,\ldots,z_p)$ parameterizing the local space of leaves, it reads as
\[ \eta= {\sqrt{-1}} \sum_{i,j}g_{ij}\,dz_i \wedge d\bar{z_j} \]
where 
$g_{ij}$ depend only of the transverse variables 
$(z_1,\ldots,z_p)$ and
\[ r= -\frac{\sqrt{ -1}}{\pi}\partial\bar{\partial}\log \bigg(\frac{\eta^p}{{|dz_1\wedge\cdots \wedge dz_p|}^2}\bigg). \]
Note that 
$-r$ also represents 
$c_1(N_\mathcal {F}^*)$, so that there exists, by the 
$dd^c$ lemma, a smooth function 
$f:X\to \mathbb {R}$ such that 
$-r=\eta +dd^c f$. Note that 
$dd^c f$ is basic as it is a sum of two basic forms. This implies that 
$f$ is pluriharmonic along the leaves of 
$\mathcal {F}$. It turns out that 
$f$ is basic: indeed, let 
$\mathcal {L}$ be a leaf and 
$\overline {\mathcal {L}}$ its topological closure. Let 
$x\in \overline {\mathcal {L}}$ be such that 
$ {f}{|_{\overline {\mathcal {L}}}}$ reaches its maximum at 
$x$ and let 
${\mathcal {L}}_x$ be the leaf passing through 
$x$. By the maximum principle for pluriharmonic functions 
$f$ is constant on 
${\mathcal {L}}_x$, hence on 
$\overline {\mathcal {L}_x}$. On the other hand, the leaves closure form a partition of 
$X$, a common feature for Riemannian foliations [Reference MolinoMol88, Theorem 5.1]. In particular, 
$\overline {\mathcal {L}}= \overline {{\mathcal {L}}_x}$. As the original leaf 
$\mathcal {L}$ has been chosen arbitrarily, this enables us to conclude that 
$f$ is leafwise constant, as wanted. Then, 
$r$ and 
$\eta$ are not only cohomologous in the ordinary 
$\partial \bar {\partial }$ cohomology, but also in the basic 
$\partial \bar {\partial }$ cohomology. By the foliated version of Yau's solution to Calabi conjecture, due to El Kacimi [Reference El Kacimi-AlaouiEKA90, § 3.5], there exists for 
$\mathcal {F}$ an invariant transverse Kähler metric whose Ricci form is equal to 
$-\eta$.
If the leaves of 
$\mathcal {F}$ are closed, the leaf space 
$X/\mathcal {F}$ is naturally equipped with a structure of a Kähler orbifold (in the usual sense). The 
$(1,1)$-form 
$\eta$ descends on 
$X/\mathcal {F}$ as a positive 
$(1,1)$-form representing the Chern class of the (orbifold) canonical bundle 
$K_{X/\mathcal {F}}$. The orbifold analogue of Kodaira embedding theorem [Reference BailyBai57, § 7] enables to exhibit a Bogomolov sheaf on 
$X$. To wit, take 
$p^* K_{X/\mathcal {F}}$ with 
$p:X \to X/\mathcal {F}$ the natural quotient morphism. It follows that 
$X$ is not special.
Otherwise, if the leaves of 
$\mathcal {F}$ are not closed, the strategy consists in producing a representation 
$\pi _1(X)\to G$ with dense image, where 
$G$ is a real semi-simple algebraic group and apply Zuo's theorem [Reference ZuoZuo96] or [Reference Campana, Claudon and EyssidieuxCCE15, Theorem 1] to conclude: if there exists such a representation, up to replacing 
$X$ by a finite étale cover 
$\tilde X$, there exists a meromorphic fibration 
$f: \tilde X\to V$ where 
$V$ is a projective manifold of general type (through which the representation factorizes).
We now explain how to produce such a representation. Under our assumptions, 
$\mathcal {F}$ is a transversely Kähler foliation 
$\mathcal {F}$ whose leaves are not closed and whose transverse Ricci curvature is negative, that is 
$\eta$ is semi-negative with constant rank equal to 
$p$ the complex codimension of 
$\mathcal {F}$. This corresponds to the monodromy of the so-called commuting sheaf 
$\mathcal {C}$. This sheaf is a locally constant sheaf of Lie algebras 
$\mathfrak {g}$ of basic Killing vector fields which encodes the dynamic of 
$\mathcal {F}$ and defined in the general setting of Riemannian foliations (see [Reference MolinoMol88, § 5.3]). Under our negativity assumption, one can prove [Reference TouzetTou10, Théorème 1.1], that 
$\mathfrak {g}$ is semi-simple. This implies that the image of the representation 
$\alpha :\pi _1(X)\to \mathrm {Aut}(\mathfrak {g})$ (
$=\text {real semi-simple algebraic group}$) intersects 
$\mathrm {Aut}^0(\mathfrak {g})$ as a dense subgroup. Indeed, the closure of the image of 
$\alpha$ contains the image 
$\mathrm {Ad}\ g$ of the exponential of the adjoint representation of 
$\mathfrak {g}$ according to [Reference MolinoMol88, Appendix D, Proposition 3.7]. We can thus apply Zuo's theorem to produce the sought fibration 
$\tilde {X}\mapsto V$, thus proving that 
$X$ is not special.
Remark 7.1 In [Reference MokMok00], Mok considered compact Kähler manifolds 
$X$ equipped with a 
$d$-closed holomorphic one form twisted by a locally constant bundle of Hilbert spaces 
$E_\Phi$. This defines on 
$X$ a foliation which is transversely Riemannian on a dense open subset of 
$X$ and whose transverse metric (semi-Kähler structure in the language of [Reference MokMok00]) is cooked up from an orthonormal basis on the typical fiber of 
$E_\Phi$ and whose ‘Ricci curvature’ carries some negativity properties. Under some circumstances, Mok shows the existence of fibration of (an étale cover of) 
$X$ onto varieties of general type. It could be tempting to relate the existence of a numerical Bogomolov sheaf to the existence of a semi-Kähler structure arising from twisted forms. For instance, in codimension one, transversely hyperbolic foliations can be regarded as foliations defined by the kernel of a 
$E_\Phi$ valued closed holomorphic one form where 
$E_\Phi \to X$ is the locally constant bundle of Hilbert spaces arising from a unitary representation 
$\pi _1(X) \to U(H)$, where 
$H$ is the space of square integrable antiholomorphic forms on the disk [Reference MokMok97, § 4].
