The Hamiltonian no-torsion uniruledness conjecture

A Hamiltonian torsion element is a non-trivial finite-order element of the Hamiltonian diffeomorphism group of a closed symplectic manifold. A closed symplectic manifold is uniruled if it admits a non-constant genus-zero holomorphic curve through a generic point.

Hamiltonian no-torsion uniruledness conjecture. Each closed symplectic manifold with non-trivial Hamiltonian torsion must be uniruled.

The conjecture proposes a geometric restriction on Hamiltonian torsion, extending known results that symplectically aspherical, symplectically Calabi–Yau, and negative monotone manifolds do not admit Hamiltonian torsion. In the positive monotone case, the paper proves that non-trivial Hamiltonian torsion implies Steenrod-uniruledness for certain primes and hence geometric uniruledness; the general statement remains open.