The Action homomorphism, quasimorphisms and moment maps on the space of compatible almost complex structures

摘要

We extend the definition of Weinstein's Action homomorphism to Hamiltonianactions with equivariant moment maps of (possibly infinite-dimensional) Liegroups on symplectic manifolds, and show that under conditions including auniform bound on the symplectic areas of geodesic triangles the resultinghomomorphism extends to a quasimorphism on the universal cover of the group. Weapply these principles to finite dimensional Hermitian Lie groups likeSp(2n,R), reinterpreting the Guichardet-Wigner quasimorphisms, and to theinfinite dimensional groups of Hamiltonian diffeomorphisms Ham(M,\om) of closedsymplectic manifolds (M,\om), that act on the space of compatible almostcomplex structures with an equivariant moment map given by the theory ofDonaldson and Fujiki. We show that the quasimorphism on \widetilde{Ham}(M,\om)obtained in the second case is Symp(M,\om)-congjugation-invariant and computeits restrictions to \pi_1(Ham(M,\om)) via a homomorphism introduced byLalonde-McDuff-Polterovich, answering a question of Polterovich; to thesubgroup Hamiltonian biholomorphisms via the Futaki invariant; and to subgroupsof diffeomorphisms supported in an embedded ball via the Barge-Ghys averageMaslov quasimorphism, the Calabi homomorphism and the average Hermitian scalarcurvature. We show that when c_1(TM)=0 this quasimorphism is proportional to aquasimorphism of Entov and when [\om] is a non-zero multiple of c_1(TM), it isproportional to a quasimorphism due to Py. As an application we show that theL^2_2-distance on \widetilde{Ham}(M,\om) is unbounded, similarly to the resultsof Eliashberg-Ratiu for the L^2_1-distance.