Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree

Abstract

In this paper, we prove that in any projective manifold, the complements of general hypersurfaces of sufficiently large degree are Kobayashi hyperbolic. We also provide an effective lower bound on the degree. This confirms a conjecture by S. Kobayashi in 1970. Our proof, based on the theory of jet differentials, is obtained by reducing the problem to the construction of a particular example with strong hyperbolicity properties. This approach relies the construction of higher order logarithmic connections allowing us to construct logarithmic Wronskians. These logarithmic Wronskians are the building blocks of the more general logarithmic jet differentials we are able to construct.As a byproduct of our proof, we prove a more general result on the orbifold hyperbolicity for generic geometric orbifolds in the sense of Campana, with only one component and large multiplicities. We also establish a Second Main Theorem type result for holomorphic entire curves intersecting general hypersurfaces, and we prove the Kobayashi hyperbolicity of the cyclic cover of a general hypersurface, again with an explicit lower bound on the degree of all these hypersurfaces.I J := {I ∈ I | Supp(I) ⊂ {0, . . . , k} \ J}, and let us also consider the restricted universal complete intersection varietiesLet us denote by pr : Y → Gr k the canonical projection, and for any J ⊂ {0, . . . , k} we set pr J := pr ↾Y J : Y J → Gr k . Observe that the pr J is generically finite. This observation is crucial in the rest of the argument which highly rests on the understanding of the geometry of the non-finite locus of pr J :E J := y ∈ Y J | dim y pr −1 J (pr J (y)) > 0 , and its image in Gr k :

Degrees $d \geqslant \big( \sqrt{n}\, \log\, n\big)^n$ and $d \geqslant \big( n\, \log\, n\big)^n$ in the Conjectures of Green-Griffiths and of Kobayashi

“…Theorem 1.2 improves the degree bound d n 2n obtained in [12]. For standard presentations of the research field, and for up-to-date history, including degree bound comparisons, the reader is referred to the introductions of the articles [12,14,3,7,10,8,2,6,5,4,15,1,9], listed in chronological order of prepublication.…”

“…Our aim is to minorize this derivative: g ,ρ (θ) = sin θ + 2 ρ ( + 1) sin ( + 1)θ + ρ +1 − 4 sin θ − 4( + 1) sin ( + 1)θ + 4 sin θ + ρ 2 +1 8 sin θ + 2( + 1) sin ( + 1)θ − 8 sin θ + ρ 2 +2 − 2 sin θ + ρ 3 +1 − 4 sin θ , by a quantity which can be seen to be positive. However, we have to treat the special case = 1 separately, namely for all 0 < ρ 1 4 and for all 0 < θ π 4 , we first check that: g 1,ρ (θ) = sin θ + 4ρ sin 2θ + ρ 2 − 8 sin 2θ + ρ 3…”

On the set of divisors with zero geometric defect

“…We underline that the feature of Theorem 1.1 is the truncation 1 of the defect. By a recent result in [3], for every integer d greater than an explicit number depending on n and for a generic divisor D of L d , we have δ [1] f (D) ≤ 1 − 1/d; see also [12] for a weaker estimate. We also notice that Theorem 1.1 is sharp in the case where n = 1 by a result of Drasin [9].…”

On the existence of logarithmic and orbifold jet differentials

“…In the case of orbifold surfaces P 2 , 1 − 1 ρ C where C is a smooth curve of degree d, such existence results have been obtained in [CDR20] for k = 2, d 12 and ρ 5 depending on d. In [DR20], the existence of jet differentials is obtained for orbifolds P n , d i=1 1− 1 ρ H i in any dimension for k = 1, ρ 3 along an arrangement of hyperplanes of degree d 2n 2n ρ−2 + 1 . In [BD18], it is established that the orbifold P n , 1 − 1 d D , where D is a general smooth hypersurface of degree d, is hyperbolic i.e. there is no non-constant orbifold entire curve f : n+3 .…”