2012/07/05 by Yi Lin, Lin, Yi, Álvaro Pelayo +1
Mathematics · #Advanced Algebra and Geometry #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1207.1335
openalex publication_date 2012/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a compact, connected symplectic 2n-dimensional manifold on which an(n-2)-dimensional torus T acts effectively and Hamiltonianly. Under the assumption that there is an effective complementary 2-torus acting on M with symplectic orbits, we show that the Duistermaat-Heckman measure of the T-action is log-concave. This verifies the logarithmic concavity conjecture for a class of inequivalent T-actions. Then we use this conjecture to prove the following: if there is an effective symplectic action of an (n-2)-dimensional torus T on a compact, connected symplectic 2n-dimensional manifold that admits an effective complementary symplectic action of a 2-torus with symplectic orbits, then the existence of T-fixed points implies that the T-action is Hamiltonian. As a consequence of this, we give new proofs of a classical theorem by McDuff about S1-actions, and some of its recent extensions.