Locally Monge-Ampere Foliations

Abstract: It is shown that codimension one parabolic foliations of complex manifolds are holomorphic. This is proved using the fact that codimension one foliations of complex manifolds are necessarily locally Monge-Amp`ere foliations and that parabolic leaves cannot have hyperbolic behavior. The result holds true also for locally Monge-Amp`ere foliations with parabolic leaves of arbitrary codimension.