Classification of Smooth Minimal Kähler Fourfolds Without Effective Divisors and Surfaces

We prove that if $X$ is a compact Kähler fourfold with pseudo–effective canonical bundle and no subvarieties of codimension one or two, then $K_X$ is a torsion line bundle. By the Beauville–Bogomolov decomposition theorem, it follows that $X$ is either a quotient of a complex torus or an irreducible holomorphic symplectic manifold.