Exponential mixing of all orders and CLT for automorphisms of compact K{ä}hler manifolds

We consider the unique measure of maximal entropy of an automorphism of a compact K{ä}hler manifold with simple action on cohomology. We show that it is exponentially mixing of all orders with respect to H{ö}lder observables. It follows that the Central Limit Theorem (CLT) holds for these observables. In particular, our result applies to all automorphisms of compact K{ä}hler surfaces with positive entropy.