Local Systems, Algebraic Foliations and Fibrations

TL;DR

This paper establishes a correspondence between foliations on algebraic varieties and local systems on their base, providing new criteria for fibrations and extending classical theorems like Castelnuovo-de Franchis.

Contribution

It introduces a novel link between foliations and local systems in the context of semistable fibrations, with applications to rationally connected fibrations and Iitaka fibrations.

Findings

Established a correspondence between foliations and local systems.

Derived conditions for maximal rationally connected fibrations.

Proved a version of the Castelnuovo-de Franchis theorem for p-forms.

Abstract

Given a semistable fibration $f\colon X\to B$ we introduce a correspondence between foliations $\mathcal{F}$ on $X$ and local systems $\mathbb{L}$ on $B$. Building up on this correspondence we find conditions that give maximal rationally connected fibrations in terms of data on the foliation. We prove the Castelnuovo-de Franchis theorem in the case of $p$-forms and we apply it to show when, under some natural conditions, a line subbundle of the sheaf of $p$-forms induces the Iitaka fibration.