Abundance after nonvanishing for compact Kähler fourfolds

Abstract — v1

We prove semiampleness of the actual ℚ-Cartier adjoint $K_X+\Delta$ of a normal connected compact Kähler klt fourfold whenever it is analytically nef and some positive Cartier multiple has a nonzero section. The boundary is effective and rational; the fourfold need not be projective or ℚ-factorial. In Iitaka dimension zero, a positive Cartier multiple is the trivial holomorphic line bundle.