Monodromy in Hamiltonian Floer theory

TL;DR

This paper explores conditions under which spectral invariants descend to the Hamiltonian group in symplectic geometry, demonstrating implications for the structure and size of Hamiltonian groups, including infinite diameter results.

Contribution

It identifies new less restrictive conditions for spectral invariants to descend, extending previous results and applying genus zero Gromov-Witten invariants calculations.

Findings

Spectral invariants descend under specific quantum homology conditions.

Hamiltonian group of the one point blow-up of T^4 admits a Calabi quasimorphism.

Hamiltonian groups have infinite diameter in many cases.

Abstract

Schwarz showed that when a closed symplectic manifold (M,\om) is symplectically aspherical (i.e. the symplectic form and the first Chern class vanish on \pi_2(M)) then the spectral invariants, which are initially defined on the universal cover of the Hamiltonian group, descend to the Hamiltonian group Ham (M,\om). In this note we describe less stringent conditions on the Chern class and quantum homology of M under which the (asymptotic) spectral invariants descend to Ham (M,\om). For example, they descend if the quantum multiplication of M is undeformed and H_2(M) has rank >1, or if the minimal Chern number is at least n+1 (where \dim M=2n) and the even cohomology of M is generated by divisors. The proofs are based on certain calculations of genus zero Gromov--Witten invariants. As an application, we show that the Hamiltonian group of the one point blow up of T^4 admits a Calabi quasimorphism. Moreover, whenever the (asymptotic) spectral invariants descend it is easy to see that Ham (M,\om) has infinite diameter in the Hofer norm. Hence our results establish the infinite diameter of Ham in many new cases. We also show that the area pseudonorm -- a geometric version of the Hofer norm -- is nontrivial on the (compactly supported) Hamiltonian group for all noncompact manifolds as well as for a large class of closed manifolds.