Kähler reduction of metrics with holonomy \(G_2\) (Q1424141)

The authors study the quotient \(N=Y/S^1\) of a manifold with holonomy \(G_2\) by a free \(S^1\) action: A torsion-free \(G_2\) structure admitting an infinitesimal isometry such that the quotient is a Kähler manifold is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by section order equations. Reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to \(G_2\).