Foliations with positive slopes and birational stability of orbifold cotangent bundles

Abstract

Let X be a smooth connected projective manifold, together with an snc orbifold divisor ∆, such that the pair (X, ∆) is log-canonical. If K X + ∆ is not pseudo-effective, we show, among other things, that any quotient of its orbifold cotangent bundle has a pseudo-effective determinant. This improves considerably our previous result [18], where generic positivity instead of pseudoeffectivity was obtained. One of the new ingredients in the proof is a version of the Bogomolov-McQuillan algebraicity criterion for holomorphic foliations whose minimal slope with respect to a movable class (instead of an ample complete intersection class) is positive.

Characterization of generic projective space bundles and algebraicity of foliations

“…We end this subsection with a consequence of [Hör12, Lemma 2.14] (see also [CP15b], [CP15a] and [Dru17b] for related results). Recall that a Weil Q-divisor D on a normal projective variety X is said to be pseudo-effective if, for any ample divisor L on X and any rational number ε > 0, there exists an effective…”

Brody hyperbolicity of base spaces of certain families of varieties

“…(2) For the next step, in the case of Viehweg's hyperbolicity conjecture the point was to apply a powerful criterion detecting the log general type property, due to Campana-Pȃun [CP15]. In the present case of Brody hyperbolicity, this step is by contrast of an analytic, and in some sense more elementary flavor.…”

On Generalised Abundance, II

“…Let (X, ∆) be a klt pair such that K X + ∆ is nef and X is not uniruled, and let L be a nef Q-divisor on X. Our main technical result, Theorem 4.1 below, gives a general criterion for the existence of sections of some multiple of K X + ∆ + L. The criterion generalises the main results of [LP18a]: the proof uses the birational stability of the cotangent bundle from [CP15] (this is where the non-uniruledness of X is necessary) together with the very carefully chosen MMP techniques in Theorem 4.2 (this is where the nefness of K X + ∆ is used).…”