  Fujita's second theorem for K\"ahler fibre spaces over a curve asserts 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. We focus on our
negative answer (\cite{cd}) to a question by Fujita: is $V$ semiample?
  We give here an infinite series of counterexamples using hypergeometric
integrals and we give a simple argument to show that the monodromy
representation is infinite. Our counterexamples are surfaces of general type
with positive index, explicitly given as abelian coverings with group $(\mathbb
Z/n)^2$ of a Del Pezzo surface of degree 5 (branched on a union of lines
forming a bianticanonical divisor), and endowed with a semistable fibration
with only $3$ singular fibres.
  The simplest such surfaces are the three ball quotients, already considered
in joint work of I. Bauer and the first author, fibred over a curve of genus
$2$, and with fibres of genus $4$.
  These examples are a larger class than the ones corresponding to Shimura
curves in the moduli space of Abelian varieties.
