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.