The direct image of the relative dualizing sheaf needs not be semiample

We provide details for the proof of Fujita's second theorem and prove that for a Kähler fibre space f:X→B$f:X\to B$ over a smooth projective curve B  , the direct image of the relative dualizing sheaf V:=f_?ω_X/B$V:=f_{?}{\omega }_{X/B}$ is the direct sum of an ample and a unitary flat bundle. We also show that V needs not be semiample, which is our main result.