Campana rational-connectedness conjecture for Fano orbifolds
Let be an algebraically closed field of characteristic . Every projective klt Fano orbifold , meaning a klt orbifold pair with ample, is Campana rationally connected.
References
Primary source
Additional references
- Fano orbifolds admit free Campana curves — arXiv — Brian Lehmann, Sho Tanimoto
Progress summary
The general conjecture remains open, but new work proves free curves and settles substantial surface cases without proving rational connectedness in full.
Campana’s conjecture says that every klt Fano orbifold in characteristic with ample anticanonical class is Campana rationally connected. The general statement remains unproved.
Known results
- Chen, Lehmann, and Tanimoto (2025) prove a stronger result for toric Campana orbifolds and establish conditional consequences for weak approximation.
- Lehmann and Tanimoto (2026) prove strong Campana uniruledness, hence rational connectedness, for broad families of non-toric del Pezzo orbifolds.
- Related 2016 work establishes slope rational connectedness for smooth klt Fano orbifold pairs, but not the conjecture’s full conclusion.
2026 free-curve advance
Lehmann and Tanimoto report an unconditional existence theorem for free Campana curves on Fano orbifolds. The rational-connectedness conclusion remains conditional on an auxiliary conjecture, so this is progress rather than a solution.
Current status (as of September 2026): Free-curve existence and several toric and surface cases are established, but Campana rational connectedness for all Fano orbifolds remains open.
Solutions 0
No solutions have been posted yet.