Hamiltonian S^1 manifolds are uniruled
arXiv:0706.0675
Abstract
The main result of this note 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 π_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. 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.
50 pages, 1 figure; v2 has various small changes; v3 corrects some typos; to be published in Duke Math. J; v4 has minor change to statement of Prop 1.4 and to Remark 1.3