Hamiltonian S1-manifolds are uniruled

Abstract

The main result of this article is that every closed Hamiltonian $S^1$-manifold is uniruled, (i.e., it has a nonzero Gromov-Witten invariant, one of whose constraints is a point). The proof uses the Seidel representation of ${\pi }_1$ of the Hamiltonian group in the small quantum homology of $M$ as well as the blow-up technique recently introduced by Hu, Li, and Ruan [15, Th. 5.15]. It applies more generally to manifolds that have a loop of Hamiltonian symplectomorphisms with a nondegenerate fixed maximum. Some consequences for Hofer geometry are explored. An appendix discusses the structure of the quantum homology ring of uniruled manifolds