2025/11/14 by Smilga, A. V.
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2511.11786
openalex publication_date 2025/11/14 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28
In this note, we make two methodical observations. \bullet We prove in a simple explicit way that a necessary and sufficient condition for a Kähler manifold to be hyperkähler is hi k hj l Ω k l = C Ωij, where hi k is a complex metric, Ω is a symplectic matrix and C is a positive constant. \bullet The procedure of Kähler reduction includes two stages. On the first stage, a Kähler manifold of dimension 2n is reduced to a (2n-1) - dimensional manifold, while on the second stage, one arrives at a Kähler manifold of dimension 2(n-1). We note that this second stage has the meaning of Hamiltonian reduction. We illustrate the procedure by discussing a simple toy model when ℝ3 × S1 is reduced down to S2. We elucidate also hyperkähler reduction of ℝ7 × S1 down to the Taub-NUT metric.