Related papers: Vanishing of Invariant 2-Jet Differentials and Imp…
The main goal of this work is to prove that a very generic surface of degree at least 21 in complex projective 3-dimensional space is hyperbolic in the sense of Kobayashi. This means that every entire holomorphic map $f:{\Bbb C} \to X$ to…
We prove that the complement of a very generic curve of degree $d$ at least equal to 15 in the projective plane is hyperbolic in the sens of Kobayashi (here, the terminology ``very generic'' refers to complements of countable unions of…
A major unsolved problem (according to Demailly 1997) towards the Kobayashi hyperbolicity conjecture in optimal degree is to understand jet differentials of germs of holomorphic discs that are invariant under any reparametrization of the…
In view of Kobayashi's hyperbolicity conjecture, Demailly-Semple jets of orders 4 and 5 in dimension 2 are studied, some expectations about their algebraic tameness being (dis)proved, after systematic, substantial, formal, manual…
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…
In this article we prove that the complement of a very generic curve of degree at least equal to 14 in the complex projective plane is hyperbolic in the sense of Kobayashi. Thus, using a new method, we improve the former known bound…
For a generic hypersurface $\mathbb{X}^{n-1} \subset \mathbb{P}^n(\mathbb{C})$ of degree \[ d \,\geqslant\, n^{2n} \] (1) $\mathbb{P}^n \big\backslash \mathbb{X}^{n-1}$ is Kobayashi-hyperbolically imbedded in $\mathbb{P}^n$; (2)…
We construct two classes of singular Kobayashi hyperbolic surfaces in $P^3$. The first consists of generic projections of the cartesian square $V = C \times C$ of a generic genus $g \ge 2$ curve $C$ smoothly embedded in $P^5$. These…
The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a…
We describe a new method of constructing Kobayashi-hyperbolic surfaces in complex projective 3-space based on deforming surfaces with a "hyperbolic non-percolation" property. We use this method to show that general small deformations of…
Using degeneration to scrolls, we give an easy proof of non-existence of curves of low genera on general surfaces in P3 of degree d >=5. We show, along the same lines, boundedness of families of curves of small enough genera on general…
Of the two techniques introduced by Bloch, Green-Griffiths and developed by Siu, Demailly to establish Kobayashi hyperbolicity of generic high degree complex algebraic hypersurfaces X in P^(n+1), the second one, initiated by Clemens, Ein,…
We study the hyperbolicity of the log variety $(\mathbb{P}^n, X)$, where $X$ is a very general hypersurface of degree $d\geq 2n+1$ (which is the bound predicted by the Kobayashi conjecture). Using a positivity result for the sheaf of…
We modify the deformation method explored previously in a joint work of B. Shiffman and the author, in order to construct further examples of Kobayashi hyperbolic surfaces in the projective 3-space of any even degree starting with degree 8.
We give a new version of a recent result of B{\'e}rczi-Kirwan, proving the Kobayashi and Green-Griffiths-Lang conjectures for generic hypersurfaces in the projective space , with a polynomial lower bound on the degree. Our strategy again…
In 1981 J.Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate. In the present paper we are interested in the case of…
The paper is a contribution of the conjecture of Kobayashi that the complement of a generic plain curve of degree at least five is hyperbolic. The main result is that the complement of a generic configuration of three quadrics is hyperbolic…
Applying the Second Main Theorem we deal with the algebraic degeneracy of entire holomorphic curves from the complex plane into a complex algebraic normal variety of positive log Kodaira dimension that admits a finite proper morphism to a…
We give two new elementary proofs of the complete Kobayashi hyperbolicity of the twice-punctured complex plane. We also present an extremely short proof that bounded domains are complete Kobayashi hyperbolic. Our proofs rely neither on the…
We are interested in the study of parabolic equations on a multi-dimensional junction, i.e. the union of a finite number of copies of a half-hyperplane of dimension d + 1 whose boundaries are identified. The common boundary is referred to…