Vector Bundles On Curves Coming From Variation Of Hodge Structures

Fujita's second theorem for Kahler fibre spaces over a curve asserts, that the direct image V of the relative dualizing sheaf splits as the direct sum V = A circle plus Q, where A is ample and Q is unitary flat. We focus on our negative answer [F. Catanese and M. Dettweiler, Answer to a question by Fujita on variation of Hodge structures, to appear in Adv. Stud. Pure Math.] to a question by Fujita: is V semiample? We give here an infinite series of counter examples 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 (Z/n)(2) of a Del Pezzo surface Z of degree 5 (branched on the union of the lines of Z, which form a bianticanonical divisor), and endowed with a semistable fibration with only three singular fibres. The simplest such surfaces are the three ball quotients considered in [I. C. Bauer and F. Catanese, A volume maximizing canonical surface in 3-space, Comment. Math. Helv. 83(1) (2008) 387-406.], 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.