2026/07/30 by Yichen Liu
Mathematics · #math.SG #msc:53D20
16 pages, 1 figure, comments are welcome
arxiv created 2026/07/30 · arxiv updated 2026/08/03
Complexity one T-spaces are Hamiltonian T-spaces (M,ω,Φ) such that (1)/(2)dim M -dim T=1. The skeleton of a complexity one T-space is an important invariant in the classification and encodes the information about non-generic orbits. In this paper, we prove that the moment image of the skeleton of a compact, connected maximally short complexity one T-space, which is in fact a GKM space, is connected. The proof relies on the well-known fact that each connected component of regular values of a proper moment map is a convex locally polyhedral set. We also gave an elementary proof of that fact along the way. Then we use the connectedness result to estimate the number of symplectic toric (T × S1)-manifolds whose underlying complexity one T-space is the same as the given maximally short complexity one T-space.