Hyperbolicity of Generic Surfaces of High Degree in Projective 3-Space
  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 the surface is constant. In 1970, Kobayashi conjectured more generally that a (very) generic hypersurface of sufficiently high degree in projective space is hyperbolic (here, the terminology "very generic" refers to complements of countable unions of proper algebraic subsets). Our technique follows the stream of ideas initiated by Green and Griffiths in 1979, which consists in considering jet differentials and their associated base loci. However, a key ingredient is the use of a different kind of jet bundles, namely the "Semple jet bundles" previously studied by the first named author (Santa Cruz Summer School, July 1995, Proc. Symposia Pure Math., Vol. 62.2, 1997). The base locus calculation is achieved through a sequence of Riemann-Roch formulas combined with a suitable generic vanishing theorem for order 2-jets. Our method covers the case of surfaces of general type with Picard group ${\Bbb Z}$ and $(13+12\theta_2) c_1^2 - 9 c_2 > 0$, where $\theta_2$ is what we call the "2-jet threshold" (the 2-jet threshold turns out to be bounded below by -1/6 for surfaces in ${\Bbb P}^3$). The final conclusion is obtained by using very recent results of McQuillan on holomorphic foliations. 
Jean-Pierre Demailly (Universit\'e de Grenoble I, Institut Fourier),   Jawher El Goul (Universit\'e de Toulouse III)