We study different hyperbolicity related problems for complete intersection varieties. Given a smooth complete intersection variety X ⊂ M in a smooth complex projective variety M, we prove that if k is bigger than dimX/codimM X and if the multidegree of X is sufficiently big then there exists invariant jet differential equations on X of order k and degree m for m big enough. Then we study a conjecture of O. Debarre: if X ⊂ P^N is the intersection of at least N/2 generic hypersurfaces of sufficiently high degree, then the cotangent bundle of X is ample. We give different results towards this conjecture. We prove that if X satisfies the hypothesis of the conjecture, then X is hyperbolic and the cotangent bundle of X is numerically positive, big, and ample modulo a subset of codimension at least 2. We also give a strategy to compute explicitly symmetric differential forms on particular complete intersection varieties. Then, we prove a vanishing theorem for the cohomology of the Green-Griffiths jet differential bundles, generalizing a result of Schneider and a result of Diverio. Finally, we study the cohomology of line bundles on the universal hypersurface of divisors in P^1.
Projective varieties with ample cotangent bundle