2020/01/30 by Isabelle Charton, Charton, Isabelle
Mathematics · #37J10 #57R91 #57S25 #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #math.AT #msc:37J10 #msc:57R91 #msc:57S25
paper · pdf · doi:10.48550/arxiv.2001.11386
21 pages
arxiv created 2020/01/30 · openalex publication_date 2020/01/30 · arxiv updated 2020/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A complexity-one space is a compact symplectic manifold (M, ω) endowed with an effective Hamiltonian action of a torus T of dimension (1)/(2)dim(M)-1. In this note we prove that for a certain class of complexity-one spaces the Poincaré dual of the Chern class cn-1 can be represented by a collection of (n)/(2)χ(M) symplectic embedded 2-spheres, where χ(M) is the Euler characteristic of M and dim(M)=2n. We call such a collection a toric one-skeleton. The classification of complexity-one spaces is an important subject in symplectic geometry. A nice subcategory of those spaces are the ones which are monotone. The existence of a toric one-skeleton is a useful tool to understand six-dimensional monotone complexity-one spaces. In particular, we will show that the existence of a toric one-skeleton for such a space implies that the second Betti number of M is at most seven. This is a simple application of results by Sabatini-Sepe and Lindsay-Panov.