Vector bundles on curves coming from variation of Hodge structures

Abstract

Fujita’s second theorem for Kähler fibre spaces over a curve asserts, that the direct image [Formula: see text] of the relative dualizing sheaf splits as the direct sum [Formula: see text], where [Formula: see text] is ample and [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] of a Del Pezzo surface [Formula: see text] of degree 5 (branched on the union of the lines of [Formula: see text], 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.