\(\mathrm{Spin}(7)\) metrics from Kähler geometry (Q6598526)

This paper explores the Kähler reduction of torsion-free \(\mathrm{Spin}(7)\)-structures on an \(8\)-manifold with a torus action. When the quotient \(6\)-manifold is Kähler, a symplectic and holomorphic reduction leads to a complex surface or a curve. The main result shows that this process is reversible, allowing the construction of \(\mathbb{T}^2\)-invariant \(\mathrm{Spin}(7)\)-metrics. The paper also introduces new examples of incomplete \(\mathrm{Spin}(7)\)-holonomy metrics.\N\NFor finding \(\mathrm{Spin}(7)\)-structures, \textit{V. Apostolov} and \textit{S. Salamon} [Commun. Math. Phys. 246, No. 1, 43--61 (2004; Zbl 1067.53039)] showed that reducing the problem to Kähler geometry could significantly simplify the involved equations. The author claims to be inspired by their work.\N\NA torsion-free \(\mathrm{Spin}(7)\)-structure on an \(8\)-manifold is defined by a closed \(4\)-form, which in turn defines a Ricci-flat Riemannian metric.\N\NIn general, a closed form can encode information about the manifold's symmetries and induce special geometric structures, such as Kähler structures or \(\mathrm{Spin}(7)\)-structures. This relationship helps to determine how the manifold behaves under various geometric transformations and constraints, and also provides insights on the curvature of an induced metric.\N\NThe main results of this paper are Corollary 4.4 and Theorem 10.1. In Corollary 4.4, the author presents an explicit 4-form that defines a torsion-free \(\mathrm{Spin}(7)\)-structure on an eight-manifold, derived from a \(4\)-manifold satisfying certain initial data.\N\NTheorem 10.1 produces a torsion-free \(\mathrm{Spin}(7)\)-structure from a complex curve, with extra conditions, and satisfying some equations. The author also explicitly provides the corresponding \(\mathrm{Spin}(7)\)-metric.\N\NSection 10.2 contains a procedure to find solutions to the equations from Theorem 10.1 and explains how to find explicit examples.