  We first provide details for the proof of Fujita's second theorem for
K\"ahler fibre spaces over a curve, asserting that the direct image $V$ of the
relative dualizing sheaf splits as the direct sum $ V = A \oplus Q$, where $A$
is ample and $Q$ is unitary flat. Our main result then answers in the negative
the question posed by Fujita whether $V$ is semiample. In fact, $V$ is
semiample if and only if $Q$ is associated to a representation of the
fundamental group of $B$ having finite image. Our examples are based on
hypergeometric integrals.
