  For a generic conservative diffeomorphism of a 3-manifold M, the Oseledets
splitting is a globally dominated splitting. Moreover, either all Lyapunov
exponents vanish almost everywhere, or else the system is non-uniformly
hyperbolic and ergodic. This is the 3-dimensional version of a well-known
result by Ma\~n\'e-Bochi, stating that a generic conservative surface
diffeomorphism is either Anosov or all Lyapunov exponents vanish almost
everywhere. This result inspired and answers in the positive for dimension 3 a
conjecture by Avila and Bochi. We also prove that all partially hyperbolic sets
with positive measure and center dimension one have a strong homoclinic
intersection. This implies that Cr generically for any r, a diffeomorphism
contains no proper partially hyperbolic sets with positive measure and center
dimension one.
