Campana rational connectedness and weak approximation

TL;DR

This paper explores Campana rational connectedness in orbifolds, proving weak approximation results for certain fibrations and verifying conjectures for toric cases using log geometry and moduli stacks.

Contribution

It establishes weak approximation for Campana sections over complex curves and confirms Campana's conjecture for toric Fano orbifolds.

Findings

Weak approximation holds at places of good reduction.

Campana conjecture verified for toric Fano orbifolds.

Log geometry and moduli stacks are key tools in the proofs.

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.