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 degre e at least 21 in complex projective 3-dimensional space is hyperbolic in th e sense of Kobayashi. This means that every entire holomorphic map f: C --> X to the surface is constant. In 1970, Kobayashi conjectured more generall y that a (very) generic hypersurface of sufficiently high degree in project ive space is hyperbolic. Our technique follows the stream of ideas initiate d by Green and Griffiths in 1979, which consists of considering jet differe ntials and their associated base loci. However, a key ingredient is the use of a different kind of jet bundle, namely the "Semple jet bundles" previou sly studied by the first named author. The base locus calculation is achiev ed through a sequence of Riemann-Roch formulas combined with a suitable gen eric vanishing theorem for order 2-jets. Our method covers the case of surf aces of general type with Picard group Z and (13 + 12 theta(2))c(1)(2) - 9( c2) > 0, where theta(2) is the "2-jet threshold" (bounded below by -1/6 for surfaces in P-3). The final conclusion is obtained by using recent results of McQuillan on holomorphic foliations.