On holomorphic partially hyperbolic systems

We prove that the center distribution of any fibered holomorphic partially hyperbolic diffeomorphism on a complex 3-fold is holomorphic. In particular, any such a system is a holomorphic skew product over a linear automorphism on a complex 2-torus. In higher dimension, we demonstrate a contrast: for any n 5, we construct a holomorphic fibered partially hyperbolic system on a complex n -fold, where the center distribution is not holomorphic in any open subset.