Massey products and Fujita decomposition over higher dimensional base

TL;DR

This paper investigates the structure of local systems in semistable fibrations over higher-dimensional bases, linking Massey products to Fujita decompositions and monodromy finiteness, with implications for Castelnuovo-de Franchis theorems.

Contribution

It extends the analysis of Massey products and Fujita decompositions to higher dimensions, establishing their relation to monodromy and Castelnuovo-de Franchis theorems.

Findings

Finiteness conditions for monodromy representations are derived.

The relation between Massey products and Castelnuovo-de Franchis theorems is clarified.

The second Fujita decomposition is explicitly connected to local systems of top forms.

Abstract

Let $f\colon X\to Y$ be a semistable fibration between smooth complex varieties of dimension $n$ and $m$. This paper contains an analysis of the local systems of de Rham closed relative one forms and top forms on the fibers. In particular the latter recovers the local system of the second Fujita decomposition of $f_*\omega_{X/Y}$ over higher dimensional base. The so called theory of Massey products allows, under natural Castelnuovo-type hypothesis, to study the finiteness of the associated monodromy representations. Motivated by this result, we also make precise the close relation between Massey products and Castelnuovo-de Franchis type theorems.