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

2011/05/29 by Egor Shelukhin, Shelukhin, Egor · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Group Theory (math.GR) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1105.5814

openalex publication_date 2011/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We extend the definition of Weinstein's Action homomorphism to Hamiltonian\nactions with equivariant moment maps of (possibly infinite-dimensional) Lie\ngroups on symplectic manifolds, and show that under conditions including a\nuniform bound on the symplectic areas of geodesic triangles the resulting\nhomomorphism extends to a quasimorphism on the universal cover of the group. We\napply these principles to finite dimensional Hermitian Lie groups like\nSp(2n,R), reinterpreting the Guichardet-Wigner quasimorphisms, and to the\ninfinite dimensional groups of Hamiltonian diffeomorphisms Ham(M, om) of closed\nsymplectic manifolds (M, om), that act on the space of compatible almost\ncomplex structures with an equivariant moment map given by the theory of\nDonaldson and Fujiki. We show that the quasimorphism on widetildeHam(M, om)\nobtained in the second case is Symp(M, om)-congjugation-invariant and compute\nits restrictions to \π₁(Ham(M, om)) via a homomorphism introduced by\nLalonde-McDuff-Polterovich, answering a question of Polterovich; to the\nsubgroup Hamiltonian biholomorphisms via the Futaki invariant; and to subgroups\nof diffeomorphisms supported in an embedded ball via the Barge-Ghys average\nMaslov quasimorphism, the Calabi homomorphism and the average Hermitian scalar\ncurvature. We show that when c₁(TM)=0 this quasimorphism is proportional to a\nquasimorphism of Entov and when [ om] is a non-zero multiple of c₁(TM), it is\nproportional to a quasimorphism due to Py. As an application we show that the\nL²₂-distance on widetildeHam(M, om) is unbounded, similarly to the results\nof Eliashberg-Ratiu for the L²₁-distance.\n

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