Families over special base manifolds and a conjecture of Campana

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.

Final version, to appear in Math. Zeitschrift

References in corpus (4)

• Differential Forms on Log Canonical Spaces
• Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties
• Orbifoldes speciales et classification bimeromorphe des varietes kaehleriennes compactes
• Positive sheaves of differentials coming from coarse moduli spaces

Cited by in corpus (11)

• Differential Forms on Log Canonical Spaces
• Orbifold Stability and Miyaoka-Yau Inequality for minimal pairs
• Generic positivity and applications to hyperbolicity of moduli spaces
• Differential forms on log canonical spaces in positive characteristic
• The isotriviality of families of canonically-polarized manifolds over a special quasi-projective base
• Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks
• Equality in the Miyaoka-Yau inequality and uniformization of non-positively curved klt pairs
• Bogomolov--Sommese type vanishing for globally $F$-regular threefolds
• Bogomolov-Sommese vanishing and liftability for surface pairs in positive characteristic
• Birational stability of the cotangent bundle
• Arakelov-Parshin rigidity of towers of curve fibrations