The classification of compact Kähler and projective manifolds with positive tangent or anticanonical bundles has been the focus of extensive research. Notably, the classification of projective manifolds with nef anticanonical line bundles has been successfully achieved by Cao and H\"oring, resolving a long-standing problem. However, the analogous problem for compact Kähler manifolds remains a formidable challenge.In this presentation, I will discuss recent collaborative work with Matsumura that tackles the three-dimensional case. We employ the Compact Kähler Minimal Model Program in dimension three (as developed by Höring and Peternell) and the positivity properties of direct images following Cao-H\"oring which will be subject of my talk after a general introduction. Additionally, I will introduce the Serge current construction, a method I have developed, which plays a crucial role in our approach.