Campana rational connectedness and weak approximation

Abstract

Campana introduced a notion of Campana rational connectedness for Campana orbifolds. Given a Campana fibration over a complex curve, we prove that a version of weak approximation for Campana sections holds at places of good reduction when the general fiber satisfies a slightly stronger version of Campana rational connectedness. Campana also conjectured that any Fano orbifold is Campana rationally connected; we verify a stronger statement for toric Campana orbifolds. A key tool in our study is log geometry and moduli stacks of stable log maps.