Hamiltonian S1-manifolds are uniruled

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 π1 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.

Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

“…Meanwhile the invariants hE i ; E j ; E k i 0;3;ˇw ithˇ0 D 0, ie the invariants hE i ; E j ; E k i 0;3;rE 0 are, according to [43,Lemma 2.3], equal to 0 unless r D 0 (in which case they come from the classical cap product E i \ E j D E iCj ) or r D 1, in which case they are 1 if i C j C k D 2n 1 and zero otherwise. In view of this, we have…”

“…At the other extreme, if M is not uniruled, then the undeformed quantum homology of the blowup has a field direct summand; this fact is proven based on results of McDuff [43] in Entov and Polterovich [14,Section 3], where its discovery is attributed to McDuff.…”

“…Moreover, this B -algebra N is generically field-split. Proof of Lemma 7.17 As in the corresponding result in Bayer [2], we will use properties of the Gromov-Witten invariants of blowups that were discovered by Gathmann [24] in the context of convex algebraic varieties, and which were extended to the symplectic case by Hu [32] and McDuff [43]. (To be specific, we will require extensions to the context of genus zero symplectic Gromov-Witten invariants of Lemmas 2.2 and 2.4 and Proposition 3.1 of [24].…”

Loops in the Hamiltonian group: a survey

“…This was the guiding idea in my recent proof [38] that every closed symplectic manifold that supports a Hamiltonian S 1 action is uniruled. The argument in [38] applies more generally to manifolds with Hamiltonian loops that are nondegenerate Hofer geodesics. This opens up many interesting questions of a more dynamical flavor.…”

Gromov–Witten invariants of blow-ups along submanifolds with convex normal bundles

“…GW-invariants are also used to classify symplectic manifolds in a symplectic birational geometric program in the work of Hu-Li-Ruan [14] and McDuff [27]. Recently, there has also been substantial progress in the crepant resolution conjecture, which roughly says the quantum cohomology is preserved by the crepant resolution after analytic continuation and some changes of parameters.…”