Erwan Rousseau

Effective algebraic degeneracy

AbstractWe show that for every smooth projective hypersurface X⊂ℙn+1 of degree d and of arbitrary dimension n≥2, if X is generic, then there exists a proper algebraic subvariety Y⊊X such that every nonconstant entire holomorphic curve f:ℂ→X has image f(ℂ) which lies in Y, as soon as its degree satisfies the effective lower bound

On the logarithmic Kobayashi conjecture

d\geqslant 2^{n^{5}}

The exceptional set and the Green-Griffiths locus do not always coincide

.

On the hyperbolicity of surfaces of general type with small c12

We study the hyperbolicity of the log variety (ℙ n ,X), where X is a very general hypersurface of degree d ≧ 2n + 1 (which is the bound predicted by the Kobayashi conjecture). Using a positivity result for the sheaf of (twisted) logarithmic vector fields, which may be of independent interest, we show that any log-subvariety of (ℙ n , X) is of log-general type, give a new proof of the algebraic hyperbolicity of (ℙ n , X), and exclude the existence of maximal rank families of entire curves in the complement of the universal degree d hypersurface. Moreover, we prove that, as in the compact case, the algebraic hyperbolicity of a log-variety is a necessary condition for the metric one.

Canonical surfaces with big cotangent bundle

We give a very simple criterion in order to ensure that the Green-Griffiths locus of a projective manifold is the whole manifold. Next, we use it to show that the Green-Griffiths locus of any projective manifold uniformized by a bounded symmetric domain of rank greater than one is the whole manifold. In particular, this clarifies an old example given by M. Green to S. Lang.

Algebraic degeneracy for generic projective hypersurfaces

Surfaces of general type with positive second Segre number

Hyperbolicity of generic surfaces in projective 3-space

s_2:=c_1^2-c_2>0

Kobayashi hyperbolicity: basic theory

are known by results of Bogomolov to be quasi-hyperbolic i.e. with finitely many rational and elliptic curves. These results were extended by McQuillan in his proof of the Green-Griffiths conjecture for entire curves on such surfaces. In this work, we study hyperbolic properties of minimal surfaces of general type with minimal

Hyperbolicity and negativity of the curvature

c_1^2

A Survey on Hyperbolicity of Projective Hypersurfaces

, known as Horikawa surfaces. In principle these surfaces should be the most difficult case for the above conjecture as illustrate the quintic surfaces in