To my limited knowledge, many compact Kähler manifolds have trivial odd Betti numbers. For instance, flag manifolds $G/P$，where $G$ is a semisimple complex Lie group and $P$ a parabolic subgroup, and Fano contact manifolds. But all these manifolds are Fano, i.e., with positive first Chern class.
My question is, does there exist a compact Kähler manifold with trivial odd Betti numbers so that its first Chern class is also trivial? Thanks in advance!