2014/09/14 by Jongsu Kim, Kim, Jongsu, Chanyoung Sung +1
Mathematics · Physics and Astronomy · #53C15 #53C21 #53D05 #53D35 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53C15 #msc:53C21 #msc:53D05 #msc:53D35
paper · pdf · doi:10.48550/arxiv.1409.4004
19 pages
openalex publication_date 2014/09/14 · arxiv created 2014/09/18 · arxiv updated 2014/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a closed smooth manifold M admitting a symplectic structure, we define a smooth topological invariant Z(M) using almost-Kähler metrics, i.e. Riemannian metrics compatible with symplectic structures. We also introduce Z(M, [[ω]]) depending on symplectic deformation equivalence class [[ω]]. We first prove that there exists a 6-dimensional smooth manifold M with more than one deformation equivalence classes with different signs of Z(M, [[ω]] ). Using Z invariants, we set up a Kazdan-Warner type problem of classifying symplectic manifolds into three categories. We finally prove that on every closed symplectic manifold (M, ω) of dimension ≥ 4, any smooth function which is somewhere negative and somewhere zero can be the scalar curvature of an almost-Kähler metric compatible with a symplectic form which is deformation equivalent to ω.