The $Spin(7)$-structures on complex line bundles and explicit Riemannian metrics with SU(4)-holonomy

TL;DR

This paper investigates the differential equations governing $Spin(7)$-structures on cones over 7-dimensional 3-Sasakian manifolds, constructing explicit solutions that lead to metrics with SU(4) holonomy, generalizing Calabi metrics.

Contribution

It provides a complete analysis of the ODE system for $Spin(7)$-structures and constructs a family of explicit SU(4)-holonomy metrics extending known Calabi metrics.

Findings

Constructed a family of solutions to the $Spin(7)$ ODE system.

Derived explicit metrics with SU(4) holonomy.

Generalized Calabi metrics through new solutions.

Abstract

We completely explore the system of ODE's which is equivalent to the existence of a parallel $Spin(7)$-structure on the cone over a 7-dimensional 3-Sasakian manifold. The one-dimensional family of solutions of this system is constructed. The solutions of this family correspond to metrics with holonomy SU(4) which generalize the Calabi metrics.