Spin(7) metrics from Kähler Geometry

Abstract

We investigate the T 2 -quotient of a torsion free Spin(7)-structure on an 8-manifold under the assumption that the quotient 6-manifold is Kähler. We show that there exists either a Hamiltonian S 1 or T 2 action on the quotient preserving the complex structure. Performing a Kähler reduction in each case reduces the problem of finding Spin(7) metrics to studying a system of PDEs on either a 4-or 2-manifold with trivial canonical bundle, which in the compact case corresponds to either T 4 , a K3 surface or an elliptic curve. By reversing this construction we give infinitely many new explicit examples of Spin(7) holonomy metrics. In the simplest case, our result can be viewed as an extension of the Gibbons-Hawking ansatz. Contents 22 12. Examples of non-constant solutions 23 References 24

Explicit abelian instantons on $S^1$-invariant Kähler Einstein $6$-manifolds

“…One can verify directly by computing the rank of the curvature operator, say using Maple, that indeed the resulting metrics have holonomy equal to SU (3). Similar deformations of G 2 and Spin(7) holonomy metrics were found in [2,16].…”

“…In this section we want to show the simplest instance how one can construct deformations of some of the constant solutions Calabi-Yau metrics by allowing u to vary on M 4 . We follow the approach described in [16,Sect. 9].…”

Einstein metrics on bundles over hyperKähler manifolds

“…Next we consider metrics on L 7 and N 8 . When n = 1 so that M is a hyperKähler 4-manifold then Apostolov-Salamon found the solution (7.2) g = t 2 (t + b) 2 dt 2 + t −2 α 2 + (t + b) −2 ξ 2 + t(t + b)g M on L 7 with holonomy group G 2 ⊂ SO(7) [2] and in [18] we extend their construction and found the solution…”