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arXiv · 0706.0675

Hamiltonian S^1 manifolds are uniruled

Dusa McDuff
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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.

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Dusa McDuff. 2009-07-17. Hamiltonian S^1 manifolds are uniruled. https://arxiv.org/abs/0706.0675

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