Vector bundles on curves coming from Variation of Hodge Structures

TL;DR

This paper constructs counterexamples to Fujita's question on the semi-ampleness of certain vector bundles arising from variations of Hodge structures, using hypergeometric integrals and explicit algebraic surfaces.

Contribution

It provides an infinite series of counterexamples to Fujita's semi-ampleness question, using hypergeometric integrals and explicit algebraic surfaces with specific fibrations.

Findings

Counterexamples with infinite monodromy representation

Surfaces of general type with positive index as explicit examples

Fibrations with only three singular fibers

Abstract

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.