On a theorem of Campana and P\u{a}un

Let $X$ be a smooth projective variety over the complex numbers, and $\Delta \subseteq X$ a reduced divisor with normal crossings. We present a slightly simplified proof for the following theorem of Campana and P\u{a}un: If some tensor power of the bundle $\Omega_X^1(\log \Delta)$ contains a subsheaf with big determinant, then $(X, \Delta)$ is of log general type. This result is a key step in the recent proof of Viehweg's hyperbolicity conjecture.


Introduction
The purpose of this paper is to present a slightly simplified proof for the following result by Campana and Pȃun [CP15,Theorem 7.6]. It is a crucial step in the proof of Viehweg's hyperbolicity conjecture for families of canonically polarized manifolds [CP15,Theorem 7.13], and more generally, for smooth families of varieties of general type [PS17, Theorem A].
Theorem 1. Let X be a smooth projective variety, and ∆ ⊆ X a reduced divisor with at worst normal crossing singularities. If some tensor power of Ω 1 X (log ∆) contains a subsheaf with big determinant, then K X + ∆ is big.
The simplification is that I have substituted an inductive procedure for the arguments involving Campana's "orbifold cotangent bundle"; otherwise, the proof of Theorem 1 that I present here is essentially the same as in the one in [CP15]. My reason for writing this paper is that it gives me a chance to draw attention to some of the beautiful ideas involved in the proof by Campana and Pȃun: slope stability with respect to movable classes; a criterion for the leaves of a foliation to be algebraic subvarieties; and positivity results for relative canonical bundles.
Remark 2. The most recent arXiv version of the paper by Campana and Pȃun (from June 14, 2017) also contains a brief summary of our proof; see [CP15,Section 8.1].

Strategy of the proof
Let (X, ∆) be a pair, consisting of a smooth projective variety X and a reduced divisor ∆ ⊆ X with at worst normal crossing singularities. We denote the logarithmic cotangent bundle by the symbol Ω 1 X (log ∆), and its dual, the logarithmic tangent bundle, by the symbol T X (− log ∆). Recall that T X (− log ∆) is naturally a subsheaf of the tangent bundle T X , and that it is closed under the Lie bracket on T X . Indeed, suppose that ∆ is given, in suitable local coordinates x 1 , x 2 , . . . , x n , by the equation x 1 x 2 · · · x k = 0; then T X (− log ∆) is generated by the n commuting vector fields , . . . , ∂ ∂x n , and is therefore closed under the Lie bracket. Suppose that Ω 1 X (log ∆) ⊗N contains a subsheaf with big determinant, for some N ≥ 1. The following observation reduces the problem to the case of line bundles.

Lemma 3.
If Ω 1 X (log ∆) ⊗N contains a subsheaf of generic rank r ≥ 1 and with big determinant, then Ω 1 X (log ∆) ⊗N r contains a big line bundle.
Proof. Let B ⊆ Ω 1 X (log ∆) ⊗N be a subsheaf of generic rank r ≥ 1, with the property that det B is big. After replacing B by its saturation, whose determinant is of course still big, we may assume that the quotient sheaf Ω 1 X (log ∆) ⊗N B is torsion-free, hence locally free outside a closed subvariety Z ⊆ X of codimension ≥ 2. On X \ Z, we have an inclusion of locally free sheaves which remains valid on X by Hartog's theorem.
For the purpose of proving Theorem 1, we are therefore allowed to assume that Ω 1 X (log ∆) ⊗N contains a big line bundle L as a subsheaf. Let Q denote the quotient sheaf, and consider the resulting short exact sequence Since K X + ∆ represents the first Chern class of Ω 1 X (log ∆), we obtain in N 1 (X) R , the R-linear span of codimension-one cycles modulo numerical equivalence. By assumption, the class c 1 (L) is big; Theorem 1 will therefore be proved if we manage to show that the class c 1 (Q) is pseudo-effective. In fact, we are going to prove the following more general result, which is of course just a special case of [CP15, Theorem 7.6 and Theorem 1.2].
Theorem 4. Let X be a smooth projective variety, and ∆ ⊆ X a reduced divisor with at worst normal crossing singularities. Suppose that some tensor power of Ω 1 X (log ∆) contains a subsheaf with big determinant. Then the first Chern class of every quotient sheaf of every tensor power of Ω 1 X (log ∆) is pseudo-effective.

Slopes and foliations
To simplify the presentation, we will prove Theorem 4 by contradiction. Suppose then that, for some integer N ≥ 1, and for some quotient sheaf Q of Ω 1 X (log ∆) ⊗N , the class c 1 (Q) was not pseudoeffective. Let Q tor ⊆ Q denote the torsion subsheaf. Since and since c 1 (Q tor ) is effective, we may replace Q by Q/Q tor , and assume without any loss of generality that Q is torsion-free (and nonzero).
By the characterization of the pseudo-effective cone in [BDPP13, Theorem 2.2], there is a movable class α ∈ N 1 (X) R such that c 1 (Q) · α < 0. As shown in [CP11,GKP16], there is a good theory of α-semistability for torsion-free sheaves, with almost all the properties that are familiar from the case of complete intersection curves. We use this theory freely in what follows. By assumption, and so Q is a torsion-free quotient sheaf of Ω 1 X (log ∆) ⊗N with negative α-slope. The dual sheaf Q * is therefore a saturated subsheaf of T X (− log ∆) ⊗N with positive α-slope. At this point, we recall the following result about tensor products.
Theorem 5. Let α ∈ N 1 (X) R be a movable class. If F and G are torsion-free and α-semistable coherent sheaves on X, then their tensor product Proof. For the reflexive hull of the tensor product, this is proved in [GKP16, Theorem 4.2 and Proposition 4.4], based on analytic results by Toma [CP11,Appendix]. Since F⊗G and its reflexive hull are isomorphic outside a closed subvariety of codimension ≥ 2, the assertion follows. (The formula for the α-slope of F⊗G is of course valid for arbitrary nonzero torsion-free coherent sheaves F and G .) Similarly, the fact that T X (− log ∆) ⊗N has a subsheaf with positive α-slope implies, again by [GKP16, Theorem 4.2 and Proposition 4.4], that T X (− log ∆) must also contain a subsheaf with Proof. This is clear from the construction of the maximal destabilizing subsheaf in [GKP16, Corollary 2.4]. Note that F ∆ is the first step in the Harder-Narasimhan filtration of T X (− log ∆), see [GKP16,Corollary 2.26].
Recall that we have an inclusion T X (− log ∆) ⊆ T X . We define another coherent subsheaf F ⊆ T X as the saturation of F ∆ in T X ; then T X /F is torsion-free, and (3.2) We will see in a moment that F is actually a (typically, singular) foliation on X. Recall that, in general, a foliation on a smooth projective variety is a saturated subsheaf F ⊆ T X that is closed under the Lie bracket on T X . From the Lie bracket, one constructs an O X -linear mapping Proof. The Lie bracket of two sections of T X (− log ∆) is a section of T X (− log ∆), and so we get a logarithmic O'Neil tensor The key point is that N ∆ = 0. Indeed, by Theorem 5, the tensor product F ∆⊗ F ∆ , modulo torsion, is again α-semistable of slope which is strictly greater than the slope of any nonzero subsheaf of T X (− log ∆)/F ∆ by Lemma 6. This inequality among slopes implies that N ∆ = 0, see for instance [GKP16, Proposition 2.16 and Corollary 2.17].
The O'Neil tensor N and the logarithmic O'Neil tensor N ∆ are both induced by the Lie bracket on T X , and so we have the following commutative diagram: The vertical arrow on the right is injective by (3.2). Now N ∆ = 0 implies that N factors through the cokernel of the vertical arrow on the left; but the cokernel is a torsion sheaf, whereas T X /F is torsion-free. The conclusion is that N = 0.
The next step in the proof is to show that the foliation F is actually algebraic. This is a simple consequence of the powerful algebraicity theorem of Campana  Theorem 8. Let X be a smooth projective variety over the complex numbers, and let F ⊆ T X be a foliation. Suppose that there exists a movable class α ∈ N 1 (X) R , such that every nonzero quotient sheaf of F has positive α-slope. Then F is an algebraic foliation, and its leaves are rationally connected.
To apply this in our setting, we observe that every quotient sheaf of F is, at least over the open subset X \ ∆, also a quotient sheaf of F ∆ , because F and F ∆ agree outside the divisor ∆. As F ∆ is α-semistable with µ α (F ) > 0, it follows easily that every quotient sheaf of F has positive α-slope. We can now invoke Theorem 8 and conclude that the foliation F is algebraic. In other words [CP15,§4], there exists a dominant rational mapping p : X Z to a smooth projective variety Z, such that F = ker dp : T X → p * T Z outside a subset of codimension ≥ 2. More precisely, let us follow [CKT16, Construction 2.29] and denote by the symbol T X/Z the unique reflexive sheaf on X that agrees with ker dp : T X → p * T Z on the big open subset where p is a morphism. Using this notation, the algebraicity of F may be expressed as indeed, F is reflexive, due to the fact that T X /F is torsion-free.
Remark 9. Theorem 8 also says that the fibers of p are rationally connected, but we are not going to make any use of this extra information. This means that the proof of Theorem 4 only uses characteristic zero methods.

Pseudo-effectivity
Let us first convince ourselves that Z cannot be a point. This will later allow us to argue by induction on the dimension, because the general fiber of p has dimension less than dim X. Lemma 10. With notation as above, we must have dim Z ≥ 1.
Proof. If dim Z = 0, then F = T X and F ∆ = T X (− log ∆), and consequently, the logarithmic tangent bundle T X (− log ∆) is α-semistable of positive slope. Since the tensor product of αsemistable sheaves remains α-semistable [GKP16, Proposition 4.4], this means that any tensor power of Ω 1 X (log ∆) is α-semistable of negative slope. But that contradicts the hypothesis of Theorem 4, namely that some tensor power of Ω 1 X (log ∆) contains a subsheaf with big determinant, because the α-slope of such a subsheaf is obviously positive.
The only properties of F ∆ that we are still going to use in the proof of Theorem 4 are the identity in (3.2), and the fact that c 1 (F ∆ ) · α > 0 for a movable class α ∈ N 1 (X) R . In return, we are allowed to assume that p : X → Z is a morphism.
Lemma 11. Without loss of generality, p : X → Z is a morphism.
Proof. Choose a birational morphism f :X → X, for example by resolving the singularities of the closure of the graph of p : X Z inside X × Z, with the following properties: the rational mapping p • f extends to a morphismp :X → Z; both KX /X andp * ∆ are normal crossing divisors; and f is an isomorphism over the open subset where p is already a morphism.
Let∆ be the reduced normal crossing divisor whose support is equal to the preimage of ∆ inX. Then Ω 1X (log∆) ∼ =p * Ω 1 X (log ∆), and since the pullback of a big line bundle byp stays big, it is still true that some tensor power of Ω 1X (log∆) contains a big line bundle as a subsheaf. Now definẽ which is a saturated subsheaf of TX . The intersectioñ is a saturated (and hence reflexive) subsheaf of TX (− log∆), whose pushforward to X is isomorphic to F ∆ , by (3.2) and the fact that F ∆ is reflexive. Consequently, where the classα =p * α ∈ N 1 (X) R is of course still movable. Nothing essential is therefore changed if we replace the rational mapping p : X Z by the morphismp :X → Z; the divisor ∆ ⊆ X bỹ ∆ ⊆X; the sheaf F ∆ by the intersection TX /Z ∩ TX (− log∆) ⊆ TX and the movable class α ∈ N 1 (X) R by its pullbackα =p * α.
Let R(p) denote the ramification divisor of the morphism p : X → Z; see [CKT16, Definition 2.16] for the precise definition. Recall from [CKT16, Lemma 2.31] the following formula for the first Chern class of our foliation F ⊆ T X , in N 1 (X) R : Computing the first Chern class of F ∆ is a little tricky [CP15, Proposition 5.1], but at least we can use the fact that F = T X/Z to estimate the difference Recall that the horizontal part ∆ hor ⊆ ∆ is the union of all irreducible components of ∆ that map onto Z; evidently, ∆ hor is again a reduced divisor on X with at worst normal crossing singularities.
Proof. It is easy to see from (3.2) that we have an inclusion of sheaves The sheaf on the right-hand side is supported on the divisor ∆, and a brief computation shows that is isomorphic to the direct sum of the normal bundles of the irreducible components of ∆. The rank of F /F ∆ at the generic point of D is thus either 0 or 1, and where a D = 0 if F = F ∆ at the generic point of D, and a D = 1 otherwise. To prove that c 1 (F /F ∆ )− ∆ hor is effective, we only have to argue that F = F ∆ at the generic point of each irreducible component of ∆ hor . This is a consequence of the fact that F = T X/Z , as we now explain. Fix an irreducible component D of the horizontal part ∆ hor . At the generic point of D, the morphism p : X → Z is smooth. After choosing suitable local coordinates x 1 , . . . , x n in a neighborhood of a sufficiently general point of D, we may therefore assume that p is locally given by where d = dim Z, and that the divisor ∆ is defined by the equation x n = 0. In these local coordinates, F = T X/Z is the subbundle of T X spanned by On the other hand, the subsheaf T X (− log ∆) is spanned by the vector fields , . . . , ∂ ∂x 1 , and so it is clear from (3.2) that F = F ∆ in a neighborhood of the given point.
From Lemma 12, we draw the conclusion that where α ∈ N 1 (X) R is the movable class from above. We will therefore reach the desired contradiction if we manage to prove that the divisor class K X/Z + ∆ hor − R(p) is pseudo-effective. According to [CP15,Theorem 3.3] or to [CKT16,Theorem 7.1], it is actually enough to check that K F + ∆ F is pseudo-effective for a general fiber F of the morphism p; and we can prove, by induction on the dimension, that K F + ∆ F is not only pseudo-effective, but even big. The results that we use here are slight improvements of [Cam04,Theorem 4.13], which is itself a generalization of Viehweg's weak positivity theorem.

Induction on the dimension
In this section, we use induction on the dimension to finish the proof of Theorem 4 and Theorem 1.
Proposition 13. Suppose that Theorem 1 is true in dimension less than dim X. If some tensor power of Ω 1 X (log ∆) contains a subsheaf with big determinant, then K X/Z + ∆ hor is pseudo-effective.
Proof. Let F be a general fiber of the morphism p : X → Z; since dim Z ≥ 1, we have dim F ≤ dim X − 1. Denote by ∆ F the restriction of ∆; since F is a general fiber, ∆ F is still a normal crossing divisor. Clearly (K X/Z + ∆ hor ) F = K F + ∆ F , and according to [CKT16,Theorem 7.3], the pseudo-effectivity of K X/Z +∆ hor will follow if we manage to show that K F + ∆ F is pseudo-effective. By hypothesis and by Lemma 3, there is a nonzero morphism from a big line bundle L to some tensor power of Ω 1 X (log ∆). Since F is a general fiber of p : X → Z, we can restrict this morphism to F to obtain a nonzero morphism Here L F denotes the restriction of L to the fiber; since L is big, L F is also big. The inclusion of F into X gives rise to a short exact sequence which induces a filtration on the k-th tensor power of the locally free sheaf in the middle. Since the normal bundle N F |X is trivial of rank dim Z, we find, by looking at the subquotients of this filtration, that there is a nonzero morphism L F → Ω 1 F (log ∆ F ) ⊗j for some 0 ≤ j ≤ k. Because L F is big, we actually have 1 ≤ j ≤ k. Since we are assuming that Theorem 1 is true for the pair (F, ∆ F ), the class K F + ∆ F is big on F , hence pseudo-effective. Appealing to [CKT16,Theorem 7.3], we deduce that the class K X/Z +∆ hor is pseudo-effective on X.
By induction on the dimension, the two assumptions of Proposition 13 are met in our case, and the class K X/Z + ∆ hor is therefore pseudo-effective. According to [CKT16,Theorem 7.1], this implies that K X/Z + ∆ hor − R(p) is also pseudo-effective. 1 Going back to the inequality in (4.5), we find that 0 ≥ − K X/Z + ∆ hor − R(p) · α ≥ c 1 (F ∆ ) · α > 0, and so we have reached the desired contradiction. The conclusion is that c 1 (Q) is indeed pseudoeffective, and so Theorem 4 and Theorem 1 are proved.
Remark 14. Most of the argument, for example the proof of Lemma 10, goes through when some tensor power of Ω 1 X (log ∆) contains a subsheaf with pseudo-effective determinant. But Theorem 4 is obviously not true under this weaker hypothesis: for example, on the product E × P 1 of an elliptic curve and P 1 , there are nontrivial one-forms, yet the canonical bundle is not pseudo-effective. What happens is that the last step in the proof of Proposition 13 breaks down: when L is not big, it may be that j = 0 (and L F is then trivial).