Vector bundles on curves coming from Variation of Hodge Structures

摘要

Fujita's second theorem for K\"ahler fibre spaces over a curve asserts thatthe 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 ournegative answer (\cite{cd}) to a question by Fujita: is $V$ semiample? We give here an infinite series of counterexamples using hypergeometricintegrals and we give a simple argument to show that the monodromyrepresentation is infinite. Our counterexamples are surfaces of general typewith positive index, explicitly given as abelian coverings with group $(\mathbbZ/n)^2$ of a Del Pezzo surface of degree 5 (branched on a union of linesforming a bianticanonical divisor), and endowed with a semistable fibrationwith only $3$ singular fibres. The simplest such surfaces are the three ball quotients, already consideredin 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 Shimuracurves in the moduli space of Abelian varieties.