Fujita decomposition and Massey product for fibered varieties

TL;DR

This paper provides a detailed structural analysis of semistable fibered varieties, interpreting Fujita decomposition via local systems, and explores implications for irrational pencils, monodromy, and Albanese properties.

Contribution

It offers a new interpretation of Fujita decomposition for fibered varieties and applies it to study irrational pencils, monodromy finiteness, and Albanese properties.

Findings

Structural theorem for local systems of relative forms

Existence of higher irrational pencils up to base change

Finiteness of monodromy representations under certain conditions

Abstract

Let $f\colon X\to B$ be a semistable fibration where $X$ is a smooth variety of dimension $n\geq 2$ and $B$ is a smooth curve. We give the structure theorem for the local system of the relative $1$-forms and of the relative top forms. This gives a neat interpretation of the second Fujita decomposition of $f_*\omega_{X/B}$. We apply our interpretation to show the existence, up to base change, of higher irrational pencils and on the finiteness of the associated monodromy representations under natural Castelnuovo-type hypothesis on local subsystems. Finally we give a criterion to have that $X$ is not of Albanese general type if $B=\mathbb{P}^1$.