Families over special base manifolds and a conjecture of Campana

TL;DR

This paper proves Campana's conjecture that families of canonically polarized varieties over special base manifolds are isotrivial, specifically for surfaces and threefolds, using advanced sheaf and vanishing theorems.

Contribution

It establishes the conjecture for surfaces and threefolds by employing sheaves of symmetric differentials and a modified Bogomolov-Sommese Vanishing Theorem within Campana's orbifold theory.

Findings

Proved the conjecture for surfaces and threefolds.

Developed a version of the Bogomolov-Sommese Vanishing Theorem for fractional positivity.

Utilized sheaves of symmetric differentials on log canonical spaces.

Abstract

Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y, all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana. We prove the conjecture when Y is a surface or threefold. The proof uses sheaves of symmetric differentials associated to fractional boundary divisors on log canonical spaces, as introduced by Campana in his theory of Orbifoldes Geometriques. We discuss a weak variant of the Harder-Narasimhan Filtration and prove a version of the Bogomolov-Sommese Vanishing Theorem that take the additional fractional positivity along the boundary into account. A brief, but self-contained introduction to Campana's theory is included for the reader's convenience.